rprev then\n10:\nE ←ContextUpdate(E, esi, rsi)\n// Update context if confidence improves\n11:\nend if\n12:\nend for\n13: end for\n14: return f, E\n4\nExperiment\n4.1\nSetup\nWe evaluated our approach using two schemas: the\nsource schema was the Microsoft Defender for End-\npoint schema containing 195 fields, and the target\nwas our Common Schema with 137 fields. The\nground truth consisted of 66 manually verified\nfield mappings, curated collaboratively by domain,\nthreat intelligence, and product experts to cover a\ndiverse range of field types and complexity levels.\nWe employed GPT-4o as the underlying model and\nassessed performance using two key metrics: (1)\nConfidence score—measuring the consistency of\npredictions across multiple attempts within a single\niteration, with special handling for empty outputs;\nand (2) Accuracy—defined as the percentage of\ncorrectly predicted mappings among the 66 verified\npairs.\nFor each field mapping iteration, the model was\nexecuted three times (n=3), and both the confidence\nscore and accuracy were computed. While accu-\nracy depends on the ground truth and is used solely\nfor evaluation, confidence scores—which can be\ncalculated without ground truth—are leveraged to\nguide the reinforcement learning process and are\ntherefore suitable for production deployment.\nConfidence Score Calculation:\nWe calcu-\nlate confidence as the consistency of predic-\ntions across multiple attempts using a modified\nfrequency-based approach. For a given field, if\nwe collect predictions P\n= [p1, p2, p3] across\nthree iterations, the confidence score is: C =\ncount(most_frequent_prediction)\nadjusted_total\nwhere the adjusted to-\ntal accounts for empty predictions with reduced\nweight (0.5 instead of 1.0). The design aligns with\nthe findings of Kalai et al. (2025), who argue that\ncurrent training and evaluation paradigms implic-\nitly reward guessing behavior in language models,\nleading to hallucinations. By contrast, our scoring\nmechanism provides a quantitative incentive for\nuncertainty acknowledgment, steering the model\ntoward more trustworthy behavior.\nWe prepare two baselines for comparison. Base-\nline 1: We used GPT-4o with a single-shot prompt\ncontaining only the field name and value, resulting\nin an accuracy of 56.36%. Baseline 2: We used\nGPT-4o with a single-shot prompt enhanced with\nadditional field descriptions, data types, and sam-\nple data context from our internal knowledge base,\nresulting in an accuracy of 72.73%.\n4.2\nPerformance Improvements\nStarting from a Baseline 2: accuracy of 72.73%\nwith GPT-4o in the first iteration, our method\nachieved significant improvements throughout the\nexperiment. By the end of the 100-iteration exper-\niment, the model demonstrated consistent perfor-\nmance with accuracy reaching 93.94%.\nOur analysis of the 100-iteration experiment re-\nveals several key performance characteristics. GPT-\n4o collects 81 evidence tuples across 100 iterations,\nwith the most significant accuracy gains occurring\nin the early stages—for example, iteration 1 shows\n\nFigure 2: Comprehensive performance analysis of the reinforcement learning agent over 100 iterations. Top:\nAccuracy improves from 72.73% to 93.94%, while confidence trends upward and approaches 1.0, highlighting\na persistent overconfidence gap. Middle: Accuracy variability is high in early iterations but stabilizes over time.\nBottom: Conflict field count decreases from 26 to 4(85% reduction), demonstrating reduced ambiguity as evidence\naccumulates.\na +21.21% gain, iteration 5 achieves +9.60%, and\niteration 6 yields +10.71%. The system makes de-\ncisions solely based on confidence scores, without\nreferencing accuracy at any point during execution.\nThis reflects real-world conditions in which\nground truth labels are unavailable. In these sce-\nnarios, accuracy can only be measured retrospec-\ntively to confirm whether the solution performed\ncorrectly. Notably, in 19 of the 100 iterations, the\nsystem rejected newly proposed evidence, demon-\nstrating that it learned to recognize when gather-\ning additional information was unlikely to improve\nresults. This selective behavior preserves a high-\nquality evidence set, which is critical not only for\nvalidating outcomes but also for providing trans-\nparency into how the agent arrived at its decisions.\nBy the conclusion of the 100 iterations, the agent\nresolved 22 conflicts, only 4 fields were flagged as\nlow-confidence and requiring expert review—down\nfrom 26 initially—representing an 85% reduction\nin the verification burden on security analysts and\nallowing them to focus on the most ambiguous or\ncritical cases.\nTo evaluate the robustness of our approach, we\nran the full 100-iteration process multiple times.\nThe final accuracy consistently reached 93-94%,\nwith minimal variance (standard deviation less than\n0.01) in the later iterations. This demonstrates that\nthe improvement is reliable and not due to chance.\n4.3\nEvidence Collection Analysis\nFor this experiment, we equipped the AI agent with\na tool that enables internet searches for additional\nfacts and evidence. Specifically, we used a generic\nsearch utility powered by Bing. In each iteration,\nthe agent analyzes conflicts identified across three\nprompt variants, formulates a targeted search query,\nand retrieves relevant results from the tool. This\nsetup demonstrates one possible method of evi-\ndence collection; other approaches could include\n\nFigure 3: The relationship between confidence and accu-\nracy. Path 1 shows the starting point. Path 2 illustrates\nhow confidence as a proxy reward improves accuracy.\nPath 3 highlights the challenge of overconfidence, where\nconfidence saturates while accuracy lags behind. There-\nfore, more engineering and research efforts are needed\nto bring the confidence curve down. The diagonal line\nindicates perfect calibration, and the curves above it\nshow overconfidence.\nconsulting human experts, performing advanced\nreasoning, or validating findings against produc-\ntion data.\nWe organized the collected evidence into three-\nelement tuples, each comprising the detected con-\nflict, the resolution plan, and the retrieved evidence.\nThis structured representation ensures systematic\nevidence collection and evaluation. Evidence is re-\ntained only if it demonstrably increases the LLM’s\nconfidence in its mapping decisions. Helpful ev-\nidence contributes in 4 ways, (1) by enhancing\nawareness through the identification of ambigu-\nous fields and conflicting mappings; (2) by en-\nabling self-correction through improved reasoning\nabout relationships among fields; (3) by improving\nclarity via authoritative information on how fields\nare defined and used in both the common schema\nand vendor-specific schemas (e.g., Microsoft De-\nfender); and (4) by revealing context-dependent\nmappings, in which the correct correspondence\nvaries by scenario and requires additional contex-\ntual understanding.\n5\nConclusion\nThis paper presents a test-time learning framework\nin which a reinforcement learning agent improves\nschema mapping accuracy through iterative evi-\ndence collection and context refinement—without\nrequiring model weight updates or pre-collected\ntraining data.\nUnlike conventional approaches\nthat rely on all labeled data before inference, our\nmethod actively gathers new evidence during test-\ntime to refine its decisions. This design eliminates\nthe high computational cost, GPU dependency, and\npotential instability associated with model retrain-\ning, making it more practical for real-world de-\nployment. A confidence-based score serves as a\nproxy reward for accuracy, providing a novel mech-\nanism that evaluates model consistency across mul-\ntiple mapping attempts within a single iteration.\nThrough systematic evidence collection and evalu-\nation, the agent resolves mapping ambiguities with\ntransparent reasoning that facilitates expert review\nand validation. Applied to Microsoft Defender\nfor Endpoint logs mapped to a common schema,\nour approach improves accuracy from 72.73% to\n93.94%, while reducing the number of fields requir-\ning expert review by 85%—from 26 to 4.\nA key insight from our work is the use of confi-\ndence scores as a proxy for ground-truth labels, as\nconceptually illustrated in Figure 3. While confi-\ndence can effectively guide accuracy improvement,\nour results reveal persistent overconfidence, where\nconfidence values exceed actual accuracy. This\nobservation underscores the need for better confi-\ndence definition and calibration. In our current ap-\nproach, confidence is measured as the consistency\nof predictions across three trials. Future research\ncould extend this by (1) employing multiple mod-\nels to generate diverse predictions and compute\nensemble-based confidence, or (2) directly prompt-\ning LLMs to provide explicit self-assessed confi-\ndence scores alongside their outputs. Additional\nexperiments extending the number of inference at-\ntempts from three to ten show encouraging signs of\nreducing the overconfidence gap (see Appendix B),\nalbeit with increased computational cost. These\nenhancements could yield better-calibrated confi-\ndence measures that more accurately reflect true\nprediction quality.\nLimitations\nWhile our approach demonstrates significant im-\nprovements, several limitations should be acknowl-\nedged:\n1. Evidence Quality Dependency: The sys-\ntem’s performance is dependent on the quality\nand availability of external evidence sources.\nIn domains where documentation is sparse or\ninconsistent, the improvement may be limited.\n2. Computational Overhead: The iterative ev-\nidence collection process requires multiple\n\nLLM calls and external searches, which may\nincrease computational costs compared to\nsingle-shot approaches.\n3. Domain Specificity: Our evaluation focuses\non cybersecurity schemas.\nWhile the ap-\nproach is general, validation in other domains\n(healthcare, finance, etc.) would strengthen\nthe generalizability claims.\n4. Scope Limitations: We focus on 1-to-1 map-\npings to address the core challenge of knowl-\nedge scarcity. Extension to more complex\nmapping cardinalities (1-to-N, N-to-M) re-\nmains future work.\nEthical Considerations\nThis study does not involve human subjects or per-\nsonal data. However, the method collects web\nevidence at test time, which could occasionally\ninclude malicious or adversarial content. Practi-\ntioners reproducing this system should apply input\nsanitization, source filtering, and sandboxing to\nprevent prompt-injection or security risks.\nReferences\nThomas Hazel. 2021. Harnessing the value of log data\nanalytics. Techstrong Research. Accessed: 2025-06-\n23.\nBowen Jin, Hansi Zeng, Zhenrui Yue, Jinsung Yoon,\nSercan Arik, Dong Wang, Hamed Zamani, and Jiawei\nHan. 2025. Search-r1: Training llms to reason and\nleverage search engines with reinforcement learning.\narXiv preprint arXiv:2503.09516.\nAdam Tauman Kalai, Ofir Nachum, Santosh S Vem-\npala, and Edwin Zhang. 2025. Why language models\nhallucinate. arXiv preprint arXiv:2509.04664.\nG. Krishna. 2023. Security logging and monitoring\nfailures: A comprehensive guide. Accessed: 2025-\n06-23.\nMicrosoft. 2024. Microsoft sentinel pricing calculator.\nAccessed: 2025-06-23.\nMarcel Parciak, Brecht Vandevoort, Frank Neven,\nLiesbet M Peeters, and Stijn Vansummeren. 2024.\nSchema matching with large language models: an ex-\nperimental study. arXiv preprint arXiv:2407.11852.\nColin Reid. 2023. Searching for a siem solution? here\nare 7 things it likely needs.\nGartner.\nAccessed:\n2025-06-23.\nNabeel Seedat and Mihaela van der Schaar. 2024.\nMatchmaker: Self-improving large language model\nprograms for schema matching.\narXiv preprint\narXiv:2410.24105.\nEitam Sheetrit, Menachem Brief, Moshik Mishaeli,\nand Oren Elisha. 2024.\nRematch: Retrieval en-\nhanced schema matching with llms. arXiv preprint\narXiv:2403.01567.\nNoah Shinn, Federico Cassano, Beck Labash, Ash-\nwin Gopinath, Karthik Narasimhan, and Shunyu\nYao. 2023. Reflexion: Language agents with ver-\nbal reinforcement learning, 2023. URL https://arxiv.\norg/abs/2303.11366, 1.\nXuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le,\nEd Chi, Sharan Narang, Aakanksha Chowdhery, and\nDenny Zhou. 2022. Self-consistency improves chain\nof thought reasoning in language models.\narXiv\npreprint arXiv:2203.11171.\nAoxiao Zhong, Dengyao Mo, Guiyang Liu, Jinbu Liu,\nQingda Lu, Qi Zhou, Jiesheng Wu, Quanzheng Li,\nand Qingsong Wen. 2024. Logparser-llm: Advancing\nefficient log parsing with large language models. In\nProceedings of the 30th ACM SIGKDD Conference\non Knowledge Discovery and Data Mining, pages\n4559–4570.\nYuxin Zuo, Kaiyan Zhang, Li Sheng, Shang Qu, Ganqu\nCui, Xuekai Zhu, Haozhan Li, Yuchen Zhang, Xin-\nwei Long, Ermo Hua, and 1 others. 2025.\nTtrl:\nTest-time reinforcement learning.\narXiv preprint\narXiv:2504.16084.\n\nA\nCase Study: Direction-Sensitive Port\nMapping (Iteration 49)\nTo illustrate the practical difficulty of schema map-\nping, we examine a representative example from\niteration 49 involving the common schema field\ndpt (destination port). The field dpt is defined in\nthe Trend Micro Common Schema as “the service\ndestination port of the private application server\n(dstport).” In Microsoft Defender for Endpoint,\nhowever, two candidate fields exist: LocalPort\n(TCP port on the local device used during commu-\nnication) and RemotePort (TCP port on the remote\ndevice being connected to).\nMapping ambiguity.\nBoth candidates are seman-\ntically plausible depending on the traffic direction.\nFor outbound connections, the local device is the\nsource, so the destination port resides on the remote\nendpoint—corresponding to RemotePort.\nCon-\nversely, for inbound connections (e.g., an RDP\nsession initiated by a remote host), the local device\nbecomes the destination, and thus the correct map-\nping is LocalPort. Without an explicit indicator of\nconnection direction, an LLM can easily misassign\nthe field.\nWhy fine-tuning does not help.\nSimply fine-\ntuning model weights cannot reliably resolve this\nambiguity. The correct mapping is not a mem-\norized association (dpt →RemotePort) but a\nconditional rule that depends on runtime con-\ntext—information that is absent from the training\ncorpus. Updating parameters may reinforce spu-\nrious correlations rather than teach the model to\nreason over directionality, resulting in brittle behav-\nior across different network scenarios.\nEvidence-based reasoning.\nDuring iteration 49,\nthe agent retrieved definitional evidence clarifying\nthat:\n• LocalPort refers to the port on the local de-\nvice.\n• RemotePort refers to the port on the remote\ndevice being connected to.\n• dpt represents the service destination port.\nAlthough this evidence did not directly yield the\nfinal mapping, it revealed that the correct cor-\nrespondence varies by communication context\nand motivated the agent to infer direction from\nauxiliary fields such as RemoteIP, LocalIP, and\nActionType. The agent’s confidence increased\nfrom 0.67 to 1.0, reflecting a clearer and more con-\nsistent conceptual understanding.\nCategorization of helpful evidence.\nAccord-\ning to the taxonomy defined in the main paper,\nthis instance exemplifies Category (4): revealing\ncontext-dependent mappings. The evidence was\nhelpful because it exposed the conditional nature\nof the mapping—showing that correctness depends\non dynamic context rather than static field align-\nment—and thereby guided subsequent reasoning\ntoward more generalizable, context-aware mapping\nrules.\nDerived practical rule.\ndpt =\nRemotePort,\nif Direction = Outbound;\nLocalPort,\nif Direction = Inbound;\n(defer/flag),\nif Direction is unknown.\nThis case highlights that progress in schema align-\nment arises not from parameter updates but from\nthe integration of structured evidence and contex-\ntual inference.\nB\nOverconfidence Mitigation\nExperiments\nTo address the overconfidence phenomenon ob-\nserved in our main experiments (Figure 3), we\nconducted an additional study to examine how in-\ncreasing the number of inference attempts affects\nconfidence calibration. Our hypothesis was that\na larger number of inference samples would im-\nprove statistical robustness and reduce systematic\noverconfidence.\nB.1\nExperimental Setup\nWe extended the original setup by increasing the\nnumber of inference attempts per iteration from 3\nto 10. This change allows for more reliable con-\nfidence estimation through better sampling of the\nmodel’s output distribution. All other experimental\nparameters remained identical to the main study:\nthe same source and target schemas (Microsoft De-\nfender for Endpoint with 195 fields and the Com-\nmon Schema with 137 fields), the same 66 manu-\nally verified ground-truth mappings, and the same\nGPT-4o model configuration.\nThe confidence score computation followed the\nsame principle of prediction consistency but with\n\na larger sample size (10 vs. 3). This design pro-\nvides a more statistically grounded measure of un-\ncertainty and is expected to yield better-calibrated\nconfidence estimates.\nB.2\nResults and Analysis\nTable 1 presents a comparison between the 3- and\n10-inference settings.\nIncreasing the number of inference attempts\nfrom 3 to 10 effectively narrows the gap between\nconfidence and accuracy. With 10 inferences, the\nmean confidence (89.3%) aligns much more closely\nwith the observed accuracy (92.1%), whereas with\n3 inferences, confidence (95.2%) noticeably ex-\nceeds accuracy (93.94%), indicating mild overcon-\nfidence.\nThis calibration improvement comes with a mod-\nest cost: while the confidence estimates become\nmore realistic, overall accuracy decreases slightly\n(93.94% →92.1%). Nevertheless, the more conser-\nvative confidence levels offer better guidance for\nexpert review prioritization. The reduction in low-\nconfidence fields remains substantial under both\nsettings (85% and 80% reductions), demonstrat-\ning that the key benefit—reducing expert review\neffort—is preserved.\nB.3\nImplications and Trade-offs\nThese findings highlight a clear balance between\ncalibration quality and computational cost. Increas-\ning the number of inference attempts leads to better-\ncalibrated confidence scores but requires more com-\nputation. In practical terms:\n• For high-precision scenarios, increasing at-\ntempts (e.g., to 10) provides more reliable\nuncertainty estimates and tighter alignment\nbetween confidence and accuracy.\n• For cost-sensitive deployments, even three\nattempts already achieve strong accuracy and\nsubstantial reduction in expert workload, mak-\ning it a highly efficient operational setting.\nThese results demonstrate improved calibra-\ntion—confidence more accurately tracks empir-\nical accuracy as the number of inferences in-\ncreases—without the need for additional metric\nreporting.\nC\nPrompt Architecture\nTable 2 summarizes the three prompts used in our\nschema mapping system. Each serves a distinct\nfunction in guiding the agent’s reasoning process\nacross sessions and mapping iterations.\nSystem Prompt Example. The system prompt de-\nfines the AI agent’s expertise and reasoning frame-\nwork. It is shown below for reference.\nYou are a Trend Micro cybersecurity data expert spe-\ncializing in Trend Micro’s Common Schema across\nmultiple layers, including endpoint, network, mes-\nsaging, and cloud. You have extensive expertise in\nprocessing and mapping third-party product and log\nschemas to Trend Micro’s Common Data Schema,\nenabling cross-log correlation, advanced threat detec-\ntion, and compliance reporting.\nYou routinely perform professional schema mapping\nfor new third-party log sources with a focus on accu-\nracy. Your approach follows a layered reasoning pro-\ncess: (1) identify core entities such as IP addresses,\nfilenames, and hashes (e.g., SHA1); (2) narrow down\ncandidate fields based on data flow direction and con-\ntext; (3) make precise mapping decisions supported\nby semantic consistency.\nFor\nexample:\n–\nsrc_ip\nvs.\ndst_ip\nde-\npends\non\nwhether\ntraffic\nis\ninbound\nor\noutbound.\n–\nSHA1\nhashes\nmay\nrepresent\nparent_process_sha1, launched_process_sha1,\nor dropped_object_sha1.\nIf no suitable mapping exists, respond professionally\nwith NOT_COVERED.\n\nSetting\nFinal Accuracy\nMean Confidence\nLow-Confidence Fields\n3 Inferences\n93.94%\n95.2%\n26 →4 (-85%)\n10 Inferences\n92.1%\n89.3%\n31 →6 (-80%)\nTable 1: Comparison of overconfidence mitigation between 3 and 10 inferences per iteration. The 10-inference\nsetting improves calibration by bringing confidence values more closely aligned with actual accuracy.\n#\nPrompt Name\nPurpose\nUsage\n1\nSystem Prompt\nDefines the AI agent’s role as a cy-\nbersecurity data expert specializing in\nschema mapping. Establishes reasoning\nrules and guidelines.\nLoaded once per ses-\nsion\n2\nUser Prompt\nCarries the per-request payload: (a)\ncurated facts from prior conflicts (if\nany), (b) RAG-assembled source/target\nschema context (descriptions, sample\nvalues, types), and (c) the mapping\ntask/question. Enforces a strict XML re-\nsponse (CSV decision, 1–5 confidence,\nreasoning) for deterministic parsing.\nFor every mapping\nrequest\n3\nSearch Prompt\nGenerates\ntargeted\nInternet\nsearch\nqueries string to resolve ambiguous\nfield mappings using prior conflict in-\nformation.\nInvoked only during\nconflict resolution\nTable 2: Overview of the three prompts used in the schema mapping system, detailing their roles and usage.\n"},"cleaned":{"kind":"string","value":"Mapping Smarter, Not Harder: A Test-Time Reinforcement Learning Agent That Improves Without Labels or Model Updates Wen-Kwang Tsao*, Yao-Ching Yu*, Chien-Ming Huang AI Lab, TrendMicro { Abstract The Enterprise Intelligence Platform must integrate logs from numerous third-party vendors in order to perform various downstream tasks. However, vendor documentation is often unavailable at test time. It is either misplaced, mismatched, poorly formatted, or incomplete, which makes schema mapping challenging. We introduce a reinforcement learning agent that can self-improve without labeled examples or model weight updates. During inference, the agent first identifies ambiguous field-mapping attempts, then generates targeted web-search queries to gather external evidence, and finally applies a confidence-based reward to iteratively refine its mappings. To demonstrate this concept, we converted Microsoft Defender for Endpoint logs into a common schema. Our method increased mapping accuracy from 56.4% (LLM-only) to 72.73% (RAG) to 93.94% over 100 iterations using GPT-4o. At the same time, it reduced the number of low-confidence mappings requiring expert review by 85%. This new approach provides an evidence-driven, transparent method for solving future industry problems, paving the way for more robust, accountable, scalable, efficient, flexible, adaptable, and collaborative solutions. 1 Introduction Enterprise IT environments are rapidly evolving toward proactive, agent-driven workflows. At the core of these systems lies a foundational requirement: the ability to efficiently process and semantically interpret logs from a wide range of thirdparty sources. Such capability enables agents to remain context-aware, access real-time data, and make well-informed decisions. For cybersecurity use cases, enterprises must ingest massive volumes of logs-often terabytes per day-from heterogeneous sources such as firewalls, servers, endpoints, cloud applications, net- *Corresponding author. work flows, policy events, file operations, and API calls, in order to enable effective security operations and real-time threat detection (Reid, 2023). The cost of Security Information and Event Management (SIEM) ingestion is substantial, exceeding 1 billion (Krishna, 2023). The emergence of large language models (LLMs) with robust natural language processing capabilities has the potential to transform third-party log integration. These models could dramatically reduce the need for labor-intensive processes that require experts. The full log integration pipeline comprises four stages: processing raw logs into structured data, mapping source schemas to target schemas, generating data transformation code, and deploying to production with ongoing monitoring. Schema mapping is the critical decision point that underpins the success of the entire integration process. This paper focuses on the practical challenge of schema mapping, a topic that is often overlooked in current literature. Our focus is on a scenario in which enterprises ingest new data into their platforms. Therefore, the target schema is usually well-documented and stable. By contrast, incoming data source schemas are often poorly documented, typically due to their origin in legacy systems or outdated software versions. Unlike conventional machine learning tasks-where the challenge is extracting key features from abundant data-the difficulty here is the opposite: the source provides too little context. Because these vendor schemas lack labeled training 16 Oct 2025 data, fine-tuning or other supervised methods are impractical. Although expert review remains necessary, it is resource-intensive and must be carefully prioritized. To address these challenges, we propose a testtime reinforcement learning (RL) agent that can self-improve without updating model weights or relying on pre-defined labeled data. Our approach is inspired by the TTRL framework (Zuo et al., 2025), which improves model performance on unlabeled data through test-stage rewards. In the absence of ground-truth labels, we introduce a confidencebased score as a proxy reward signal to guide the agent toward more accurate mappings. Our design explicitly targets industrially practical constraints: model updates are costly, GPU-intensive, and operationally complex. Instead, our method adapts the system prompt at inference time by iteratively enriching the context in a verbal RL-driven manner, thereby enabling continual self-improvement under real-world deployment conditions. The confidence-based proxy reward is both intuitive and interpretable. The agent identifies conflicts and ambiguities in its prior mapping attempts and formulates targeted search queries to gather external evidence. Confidence scores then serve as reward signals to determine whether the collected evidence should be retained or discarded, guiding iterative refinement. Furthermore, the agent produces transparent reasoning traces, enabling human experts to focus their review on low-confidence mappings, thereby reducing manual verification costs while preserving overall reliability. This method has several advantages over existing approaches. First, it achieves significantly higher mapping accuracy through iterative refinement, reducing reliance on static, one-time attempts. Second, it uses confidence-based rewards instead of ground truth labels, enabling effective learning in settings where labeled data is unavailable. Finally, it promotes transparency by revealing its reasoning and evidence collection processes. This empowers security experts to understand decisions and prioritize low-confidence mappings. Overall, our key contributions are: 1) identifying the challenge of schema mapping where prompting, fine-tuning, and retrieval-augmented generation all face limitations and no ground truth labels are available, and 2) proposing a test-time reinforcement learning framework that enables an agent to improve accuracy over time. This approach opens a new research direction by using confidence as a proxy signal to guide the agent's learning process. Log Raw CEF:0|h1|h2|001|5| LocalPort=443 Log Parsed Vendor=h1 Product=h2 DeviceVersion=001 Severity=5 LocalPort=443 Target schema Common Schema fields • vendor • pname • severity • src • spt • dst • dpt KB AI Response: the answer is ... src more context - layer:network - type:array[string] - desc: the source IP address RQ1: Can we detect the ambiguity in the LocalPort mapping? RQ2: Can we improve without label data at test time? RQ3: What if we don't have high-quality documents in our internal KB? Prompt: Which field in target schema should LocalPort map to? Context: A: Log parsing B: Field mapping C: Field retrieve for augment generation D: E2e test-time reinforcement learning agent Figure 1: Position of our test-time RL agent relative to prior work in the schema matching pipeline. In Section A, Logparser-LLM Zhong et al. (2024) handles raw-to-structured parsing. In section B, SchemaMatching-LLM Parciak et al. (2024) shows baseline one-shot mapping capability. In section C, ReMatch Sheetrit et al. (2024) adds retrieval when full documentation exists. MatchMaker Seedat and van der Schaar (2024) pre-embeds target schemas for reuse. In section D, Self-consistency Wang et al. (2022) improves chain-of-thought reasoning. Search-R1 Jin et al. (2025) enhances LLM reasoning with search-augmented RL. This example shows that the log LocalPort is ambiguous for decision making; we need more context to determine whether this field should map to src or dst. Our research uniquely extends rightward beyond traditional enterprise knowledge bases to handle newly seen logs with ill-formatted or incomplete documentation. Unlike fine-tuning approaches that require labeled data and risk overfitting, our agent operates without test-time labels, conducting internet searches to gather evidence outside the enterprise KB scope. This addresses the critical gap where traditional methods fail on unseen vendor schemas with insufficient documentation. 2 Related Work The foundation of schema mapping begins with converting raw logs into structured data. Zhong et al. (2024) introduced LogParser-LLM, which addresses the initial step of our pipeline by transforming unstructured log messages into key-value pairs. Their hybrid approach combines an LLMbased template extractor with a prefix-tree clustering method, achieving efficient parsing; the authors report ~272.5 LLM calls amortized across large log sets. This work eliminates the need for hyperparameter tuning or labeled data, enabling rapid adaptability to new log formats. LogParser-LLM's contribution is complementary to our work: while they focus on the raw-tostructured parsing phase. In contrast, our approach begins where their process ends: once logs have already been parsed into structured records, we take these structured logs and further map them into standardized, common schemas that enable consistent downstream analysis, correlation, and integration across diverse sources. Parciak et al. (2024) conducted a comprehensive experimental study of LLM capabilities in schema matching, focusing on off-the-shelf performance using only element names and descriptions. Their work provides crucial insights into the baseline capabilities of LLMs for schema matching tasks, demonstrating that context scope significantly affects performance-neither too little nor too much context leads to optimal results. Their findings directly inform our approach in several ways: (1) they validate that LLMs can perform meaningful schema matching without requiring data instances, which aligns with our privacysensitive enterprise scenarios; (2) their context optimization insights guide our prompt engineering; and (3) their baseline performance metrics provide a foundation for measuring the improvements achieved by our reinforcement learning approach. However, their work focuses on one-shot matching capabilities, while our approach addresses the iterative improvement challenge through test-time learning. ReMatch, introduced by Sheetrit et al. (2024), leverages retrieval-augmented prompting to reduce the target schema search space before matching. Their approach assumes the availability of comprehensive documentation and uses embedding-based retrieval to balance recall and precision in schema matching tasks. ReMatch demonstrates strong performance in healthcare schema matching scenarios where complete documentation is available. Our work differs from ReMatch in a fundamental assumption: while ReMatch operates in environments with well-documented schemas and comprehensive knowledge bases, our approach is designed for practical enterprise scenarios where such documentation is often incomplete or unavailable. ReMatch's retrieval mechanism works within a closed set of known mappings, whereas our agent dynamically discovers and accumulates evidence from external sources to handle previously unseen vendor schemas. Seedat and van der Schaar (2024) introduced MatchMaker, which decomposes schema matching into candidate generation, refinement, and confidence scoring steps using LLMs and vector retrieval. Their approach focuses on embedding target schemas for better retrieval and future reuse, generating synthetic in-context examples on-the-fly for zero-shot self-improvement. MatchMaker's compositional approach and confidence scoring align with our methodology, but their self-improvement mechanism differs significantly from ours. While MatchMaker generates synthetic examples for in-context learning, our approach accumulates real-world evidence through external search and interaction. MatchMaker's focus on target schema embedding optimization complements our source schema adaptation strategy, suggesting potential for hybrid approaches in future work. Two key techniques underpin our reinforcement learning framework: search-enhanced reasoning and self-consistency validation. Jin et al. (2025) demonstrated how external search can enhance LLM reasoning capabilities, providing a foundation for our evidence collection mechanism. Their Search-R1 approach shows that targeted web search can significantly improve reasoning accuracy in complex tasks. Wang et al. (2022) established that consistency across multiple reasoning paths serves as a reliable indicator of correctness, forming the basis of our confidence-based reward system. Their selfconsistency approach demonstrates that taking the majority answer from multiple sampled reasoning paths significantly improves accuracy, which directly informs our confidence score calculation. A closely related line of research is the Reflexion framework (Shinn et al., 2023), which introduces verbal reinforcement learning as a way for language agents to improve through self-generated feedback rather than model weight updates. Reflexion agents reflect on task feedback, store these reflections in episodic memory, and leverage them to guide future decisions. This approach demonstrates that linguistic feedback-whether scalar or free-form-can significantly enhance agent performance across sequential decision-making, coding, and reasoning tasks, achieving state-of-the-art results on benchmarks such as HumanEval. Our work shares a similar philosophy in emphasizing memory- and feedback-driven improvement at test time; however, we focus specifically on schema mapping under industrial constraints where ground truth labels and comprehensive documentation are unavailable. In this setting, we extend the notion of verbal reinforcement learning by integrating external evidence collection with confidence-based rewards to refine mappings dynamically. Our work addresses the critical gap where existing methods fail on newly encountered vendor schemas with ill-formatted or incomplete documentation. Unlike prior approaches that operate within controlled environments with welldocumented schemas, our test-time reinforcement learning agent adapts dynamically by accumulating external evidence without requiring model updates or labeled training data. 3 Methodology 3.1 Problem Formulation We formally define the schema mapping problem as follows. Given a source schema S = {s1, s2, . . . , sn} and a target schema T = {t1, t2, . . . , tm}, we seek to find a mapping function f : S →T ∪{∅} such that f(si) = tj represents a semantic correspondence between source field si and target field tj, or f(si) = ∅indicates no corresponding field exists in the target schema. Our objective is to maximize the correctness of this mapping function: maxf Pn i=1 I[f(si) = f∗(si)] where f∗(si) represents the ground truth mapping and I[·] is the indicator function. However, in practical deployment scenarios-particularly when processing third-party vendor logs ground truth mappings f∗are unavailable. To address this challenge, we use confidence scores as a proxy for correctness. We define confidence C(f(si)) as the consistency of mapping predictions across multiple inferences, serving as a surrogate objective: maxf Pn i=1 C(f(si)). 3.2 Test-Time Reinforcement Learning Framework Our test-time reinforcement learning agent improves schema mapping policy's accuracy by iteratively collecting evidence and reinforcing useful context. The agent starts from an empty evidence set, identifies inconsistencies through multiple mapping attempts, and formulates targeted search queries for ambiguous fields. Collected evidence is added to the context, and confidence-based reward signals guide which evidence to retain or discard. The complete algorithm is detailed in Algorithm 1. 3.3 Reinforcement Learning Formulation To formalize our approach, we define the reinforcement learning components as follows: State: The current state st consists of the schema mapping hypothesis from source to target fields and the collected evidence at time t. Specifically: st = {Mt, Et} where Mt is the current mapping hypothesis and Et is the set of evidence collected up to time t. Action: The action at involves selecting a field or conflict to investigate and executing a targeted search query. The action space includes all possible fields that could be investigated and the possible search queries that could be formulated. Reward: The reward rt is derived from the change in confidence score after adding new evidence: rt = Ct+1 -Ct where Ct is the confidence score at time t. The confidence score serves as a proxy for accuracy when ground truth labels are unavailable. Policy: The agent's policy π(a|s) specifies the suggested mapping from source schema fields to target schema fields. While the policy can be implemented in various ways, one practical approach is to leverage an LLM's reasoning process. In this setup, the LLM performs the mapping by combining its prior knowledge with contextual evidence, enabling the system to detect conflicts and generate targeted search queries. We define a conflict as a disagreement among the n prompt-variant predictions for the same source field within a single iteration (as opposed to cross-run drift). Learning: The goal of learning is to adapt a policy's behavior in a beneficial direction by leveraging rewards that reflect the situation. Instead of updating the LLM's model weights, learning takes place through the accumulation of useful evidence in the agent's context. Evidence that increases confidence is preserved, while unhelpful evidence is discarded. This process reflects a form of verbal, memory-based learning in which the agent's knowledge base is continuously refined. We use verbal RL in the Reflexion sense-policy adaptation via context/memory rather than parameter updates; rewards are intrinsic confidence deltas, not ground-truth returns. Algorithm 1 Iterative RL Schema Mapping with Confidence Improvement Inputs: Source schema S = {s1, . . . , sn}; target schema T = {t1, . . . , tm}; iteration limit α (default: 100); conflict detection attempts n (default: 3); initial evidence context E = ∅; generative LLM F. Outputs: Mapping function f; refined evidence context E. 1: for i ←1 to α do 2: f ←F(S, E) // Generate initial mappings using current evidence 3: C ←ConflictDetection(f, n) // Detect inconsistent mappings 4: for each source field si ∈C do 5: Qsi ←QueryFormulation(si) // Formulate search queries 6: esi ←EvidenceCollection(Qsi) // Collect external evidence 7: rsi ←ConfidenceEvaluation(f(si), esi) // Evaluate new confidence 8: rprev ←Confidence(f(si)) // Retrieve previous confidence 9: if rsi > rprev then 10: E ←ContextUpdate(E, esi, rsi) // Update context if confidence improves 11: end if 12: end for 13: end for 14: return f, E 4 Experiment 4.1 Setup We evaluated our approach using two schemas: the source schema was the Microsoft Defender for Endpoint schema containing 195 fields, and the target was our Common Schema with 137 fields. The ground truth consisted of 66 manually verified field mappings, curated collaboratively by domain, threat intelligence, and product experts to cover a diverse range of field types and complexity levels. We employed GPT-4o as the underlying model and assessed performance using two key metrics: (1) Confidence score-measuring the consistency of predictions across multiple attempts within a single iteration, with special handling for empty outputs; and (2) Accuracy-defined as the percentage of correctly predicted mappings among the 66 verified pairs. For each field mapping iteration, the model was executed three times (n=3), and both the confidence score and accuracy were computed. While accuracy depends on the ground truth and is used solely for evaluation, confidence scores-which can be calculated without ground truth-are leveraged to guide the reinforcement learning process and are therefore suitable for production deployment. Confidence Score Calculation: We calculate confidence as the consistency of predictions across multiple attempts using a modified frequency-based approach. For a given field, if we collect predictions P = [p1, p2, p3] across three iterations, the confidence score is: C = count(most_frequent_prediction) adjusted_total where the adjusted total accounts for empty predictions with reduced weight (0.5 instead of 1.0). The design aligns with the findings of Kalai et al. (2025), who argue that current training and evaluation paradigms implicitly reward guessing behavior in language models, leading to hallucinations. By contrast, our scoring mechanism provides a quantitative incentive for uncertainty acknowledgment, steering the model toward more trustworthy behavior. We prepare two baselines for comparison. Baseline 1: We used GPT-4o with a single-shot prompt containing only the field name and value, resulting in an accuracy of 56.36%. Baseline 2: We used GPT-4o with a single-shot prompt enhanced with additional field descriptions, data types, and sample data context from our internal knowledge base, resulting in an accuracy of 72.73%. 4.2 Performance Improvements Starting from a Baseline 2: accuracy of 72.73% with GPT-4o in the first iteration, our method achieved significant improvements throughout the experiment. By the end of the 100-iteration experiment, the model demonstrated consistent performance with accuracy reaching 93.94%. Our analysis of the 100-iteration experiment reveals several key performance characteristics. GPT4o collects 81 evidence tuples across 100 iterations, with the most significant accuracy gains occurring in the early stages-for example, iteration 1 shows Figure 2: Comprehensive performance analysis of the reinforcement learning agent over 100 iterations. Top: Accuracy improves from 72.73% to 93.94%, while confidence trends upward and approaches 1.0, highlighting a persistent overconfidence gap. Middle: Accuracy variability is high in early iterations but stabilizes over time. Bottom: Conflict field count decreases from 26 to 4(85% reduction), demonstrating reduced ambiguity as evidence accumulates. a +21.21% gain, iteration 5 achieves +9.60%, and iteration 6 yields +10.71%. The system makes decisions solely based on confidence scores, without referencing accuracy at any point during execution. This reflects real-world conditions in which ground truth labels are unavailable. In these scenarios, accuracy can only be measured retrospectively to confirm whether the solution performed correctly. Notably, in 19 of the 100 iterations, the system rejected newly proposed evidence, demonstrating that it learned to recognize when gathering additional information was unlikely to improve results. This selective behavior preserves a highquality evidence set, which is critical not only for validating outcomes but also for providing transparency into how the agent arrived at its decisions. By the conclusion of the 100 iterations, the agent resolved 22 conflicts, only 4 fields were flagged as low-confidence and requiring expert review-down from 26 initially-representing an 85% reduction in the verification burden on security analysts and allowing them to focus on the most ambiguous or critical cases. To evaluate the robustness of our approach, we ran the full 100-iteration process multiple times. The final accuracy consistently reached 93-94%, with minimal variance (standard deviation less than 0.01) in the later iterations. This demonstrates that the improvement is reliable and not due to chance. 4.3 Evidence Collection Analysis For this experiment, we equipped the AI agent with a tool that enables internet searches for additional facts and evidence. Specifically, we used a generic search utility powered by Bing. In each iteration, the agent analyzes conflicts identified across three prompt variants, formulates a targeted search query, and retrieves relevant results from the tool. This setup demonstrates one possible method of evidence collection; other approaches could include Figure 3: The relationship between confidence and accuracy. Path 1 shows the starting point. Path 2 illustrates how confidence as a proxy reward improves accuracy. Path 3 highlights the challenge of overconfidence, where confidence saturates while accuracy lags behind. Therefore, more engineering and research efforts are needed to bring the confidence curve down. The diagonal line indicates perfect calibration, and the curves above it show overconfidence. consulting human experts, performing advanced reasoning, or validating findings against production data. We organized the collected evidence into threeelement tuples, each comprising the detected conflict, the resolution plan, and the retrieved evidence. This structured representation ensures systematic evidence collection and evaluation. Evidence is retained only if it demonstrably increases the LLM's confidence in its mapping decisions. Helpful evidence contributes in 4 ways, (1) by enhancing awareness through the identification of ambiguous fields and conflicting mappings; (2) by enabling self-correction through improved reasoning about relationships among fields; (3) by improving clarity via authoritative information on how fields are defined and used in both the common schema and vendor-specific schemas (e.g., Microsoft Defender); and (4) by revealing context-dependent mappings, in which the correct correspondence varies by scenario and requires additional contextual understanding. 5 Conclusion This paper presents a test-time learning framework in which a reinforcement learning agent improves schema mapping accuracy through iterative evidence collection and context refinement-without requiring model weight updates or pre-collected training data. Unlike conventional approaches that rely on all labeled data before inference, our method actively gathers new evidence during testtime to refine its decisions. This design eliminates the high computational cost, GPU dependency, and potential instability associated with model retraining, making it more practical for real-world deployment. A confidence-based score serves as a proxy reward for accuracy, providing a novel mechanism that evaluates model consistency across multiple mapping attempts within a single iteration. Through systematic evidence collection and evaluation, the agent resolves mapping ambiguities with transparent reasoning that facilitates expert review and validation. Applied to Microsoft Defender for Endpoint logs mapped to a common schema, our approach improves accuracy from 72.73% to 93.94%, while reducing the number of fields requiring expert review by 85%-from 26 to 4. A key insight from our work is the use of confidence scores as a proxy for ground-truth labels, as conceptually illustrated in Figure 3. While confidence can effectively guide accuracy improvement, our results reveal persistent overconfidence, where confidence values exceed actual accuracy. This observation underscores the need for better confidence definition and calibration. In our current approach, confidence is measured as the consistency of predictions across three trials. Future research could extend this by (1) employing multiple models to generate diverse predictions and compute ensemble-based confidence, or (2) directly prompting LLMs to provide explicit self-assessed confidence scores alongside their outputs. Additional experiments extending the number of inference attempts from three to ten show encouraging signs of reducing the overconfidence gap (see Appendix B), albeit with increased computational cost. These enhancements could yield better-calibrated confidence measures that more accurately reflect true prediction quality. Limitations While our approach demonstrates significant improvements, several limitations should be acknowledged: 1. Evidence Quality Dependency: The system's performance is dependent on the quality and availability of external evidence sources. In domains where documentation is sparse or inconsistent, the improvement may be limited. 2. Computational Overhead: The iterative evidence collection process requires multiple LLM calls and external searches, which may increase computational costs compared to single-shot approaches. 3. Domain Specificity: Our evaluation focuses on cybersecurity schemas. While the approach is general, validation in other domains (healthcare, finance, etc.) would strengthen the generalizability claims. 4. Scope Limitations: We focus on 1-to-1 mappings to address the core challenge of knowledge scarcity. Extension to more complex mapping cardinalities (1-to-N, N-to-M) remains future work. Ethical Considerations This study does not involve human subjects or personal data. However, the method collects web evidence at test time, which could occasionally include malicious or adversarial content. Practitioners reproducing this system should apply input sanitization, source filtering, and sandboxing to prevent prompt-injection or security risks. References Thomas Hazel. 2021. Harnessing the value of log data analytics. Techstrong Research. Accessed: 2025-0623. Bowen Jin, Hansi Zeng, Zhenrui Yue, Jinsung Yoon, Sercan Arik, Dong Wang, Hamed Zamani, and Jiawei Han. 2025. Search-r1: Training llms to reason and leverage search engines with reinforcement learning. arXiv preprint . Adam Tauman Kalai, Ofir Nachum, Santosh S Vempala, and Edwin Zhang. 2025. Why language models hallucinate. arXiv preprint . G. Krishna. 2023. Security logging and monitoring failures: A comprehensive guide. Accessed: 202506-23. Microsoft. 2024. Microsoft sentinel pricing calculator. Accessed: 2025-06-23. Marcel Parciak, Brecht Vandevoort, Frank Neven, Liesbet M Peeters, and Stijn Vansummeren. 2024. Schema matching with large language models: an experimental study. arXiv preprint . Colin Reid. 2023. Searching for a siem solution? here are 7 things it likely needs. Gartner. Accessed: 2025-06-23. Nabeel Seedat and Mihaela van der Schaar. 2024. Matchmaker: Self-improving large language model programs for schema matching. arXiv preprint . Eitam Sheetrit, Menachem Brief, Moshik Mishaeli, and Oren Elisha. 2024. Rematch: Retrieval enhanced schema matching with llms. arXiv preprint . Noah Shinn, Federico Cassano, Beck Labash, Ashwin Gopinath, Karthik Narasimhan, and Shunyu Yao. 2023. Reflexion: Language agents with verbal reinforcement learning, 2023. URL https://arxiv. org/abs/2303.11366, 1. Xuezhi Wang, Jason Wei, Dale Schuurmans, Quoc Le, Ed Chi, Sharan Narang, Aakanksha Chowdhery, and Denny Zhou. 2022. Self-consistency improves chain of thought reasoning in language models. arXiv preprint . Aoxiao Zhong, Dengyao Mo, Guiyang Liu, Jinbu Liu, Qingda Lu, Qi Zhou, Jiesheng Wu, Quanzheng Li, and Qingsong Wen. 2024. Logparser-llm: Advancing efficient log parsing with large language models. In Proceedings of the 30th ACM SIGKDD Conference on Knowledge Discovery and Data Mining, pages 4559-4570. Yuxin Zuo, Kaiyan Zhang, Li Sheng, Shang Qu, Ganqu Cui, Xuekai Zhu, Haozhan Li, Yuchen Zhang, Xinwei Long, Ermo Hua, and 1 others. 2025. Ttrl: Test-time reinforcement learning. arXiv preprint . A Case Study: Direction-Sensitive Port Mapping (Iteration 49) To illustrate the practical difficulty of schema mapping, we examine a representative example from iteration 49 involving the common schema field dpt (destination port). The field dpt is defined in the Trend Micro Common Schema as \"the service destination port of the private application server (dstport).\" In Microsoft Defender for Endpoint, however, two candidate fields exist: LocalPort (TCP port on the local device used during communication) and RemotePort (TCP port on the remote device being connected to). Mapping ambiguity. Both candidates are semantically plausible depending on the traffic direction. For outbound connections, the local device is the source, so the destination port resides on the remote endpoint-corresponding to RemotePort. Conversely, for inbound connections (e.g., an RDP session initiated by a remote host), the local device becomes the destination, and thus the correct mapping is LocalPort. Without an explicit indicator of connection direction, an LLM can easily misassign the field. Why fine-tuning does not help. Simply finetuning model weights cannot reliably resolve this ambiguity. The correct mapping is not a memorized association (dpt →RemotePort) but a conditional rule that depends on runtime context-information that is absent from the training corpus. Updating parameters may reinforce spurious correlations rather than teach the model to reason over directionality, resulting in brittle behavior across different network scenarios. Evidence-based reasoning. During iteration 49, the agent retrieved definitional evidence clarifying that: • LocalPort refers to the port on the local device. • RemotePort refers to the port on the remote device being connected to. • dpt represents the service destination port. Although this evidence did not directly yield the final mapping, it revealed that the correct correspondence varies by communication context and motivated the agent to infer direction from auxiliary fields such as RemoteIP, LocalIP, and ActionType. The agent's confidence increased from 0.67 to 1.0, reflecting a clearer and more consistent conceptual understanding. Categorization of helpful evidence. According to the taxonomy defined in the main paper, this instance exemplifies Category (4): revealing context-dependent mappings. The evidence was helpful because it exposed the conditional nature of the mapping-showing that correctness depends on dynamic context rather than static field alignment-and thereby guided subsequent reasoning toward more generalizable, context-aware mapping rules. Derived practical rule. dpt = RemotePort, if Direction = Outbound; LocalPort, if Direction = Inbound; (defer/flag), if Direction is unknown. This case highlights that progress in schema alignment arises not from parameter updates but from the integration of structured evidence and contextual inference. B Overconfidence Mitigation Experiments To address the overconfidence phenomenon observed in our main experiments (Figure 3), we conducted an additional study to examine how increasing the number of inference attempts affects confidence calibration. Our hypothesis was that a larger number of inference samples would improve statistical robustness and reduce systematic overconfidence. B.1 Experimental Setup We extended the original setup by increasing the number of inference attempts per iteration from 3 to 10. This change allows for more reliable confidence estimation through better sampling of the model's output distribution. All other experimental parameters remained identical to the main study: the same source and target schemas (Microsoft Defender for Endpoint with 195 fields and the Common Schema with 137 fields), the same 66 manually verified ground-truth mappings, and the same GPT-4o model configuration. The confidence score computation followed the same principle of prediction consistency but with a larger sample size (10 vs. 3). This design provides a more statistically grounded measure of uncertainty and is expected to yield better-calibrated confidence estimates. B.2 Results and Analysis Table 1 presents a comparison between the 3- and 10-inference settings. Increasing the number of inference attempts from 3 to 10 effectively narrows the gap between confidence and accuracy. With 10 inferences, the mean confidence (89.3%) aligns much more closely with the observed accuracy (92.1%), whereas with 3 inferences, confidence (95.2%) noticeably exceeds accuracy (93.94%), indicating mild overconfidence. This calibration improvement comes with a modest cost: while the confidence estimates become more realistic, overall accuracy decreases slightly (93.94% →92.1%). Nevertheless, the more conservative confidence levels offer better guidance for expert review prioritization. The reduction in lowconfidence fields remains substantial under both settings (85% and 80% reductions), demonstrating that the key benefit-reducing expert review effort-is preserved. B.3 Implications and Trade-offs These findings highlight a clear balance between calibration quality and computational cost. Increasing the number of inference attempts leads to bettercalibrated confidence scores but requires more computation. In practical terms: • For high-precision scenarios, increasing attempts (e.g., to 10) provides more reliable uncertainty estimates and tighter alignment between confidence and accuracy. • For cost-sensitive deployments, even three attempts already achieve strong accuracy and substantial reduction in expert workload, making it a highly efficient operational setting. These results demonstrate improved calibration-confidence more accurately tracks empirical accuracy as the number of inferences increases-without the need for additional metric reporting. C Prompt Architecture Table 2 summarizes the three prompts used in our schema mapping system. Each serves a distinct function in guiding the agent's reasoning process across sessions and mapping iterations. System Prompt Example. The system prompt defines the AI agent's expertise and reasoning framework. It is shown below for reference. You are a Trend Micro cybersecurity data expert specializing in Trend Micro's Common Schema across multiple layers, including endpoint, network, messaging, and cloud. You have extensive expertise in processing and mapping third-party product and log schemas to Trend Micro's Common Data Schema, enabling cross-log correlation, advanced threat detection, and compliance reporting. You routinely perform professional schema mapping for new third-party log sources with a focus on accuracy. Your approach follows a layered reasoning process: (1) identify core entities such as IP addresses, filenames, and hashes (e.g., SHA1); (2) narrow down candidate fields based on data flow direction and context; (3) make precise mapping decisions supported by semantic consistency. For example: - src_ip vs. dst_ip depends on whether traffic is inbound or outbound. - SHA1 hashes may represent parent_process_sha1, launched_process_sha1, or dropped_object_sha1. If no suitable mapping exists, respond professionally with NOT_COVERED. Setting Final Accuracy Mean Confidence Low-Confidence Fields 3 Inferences 93.94% 95.2% 26 →4 (-85%) 10 Inferences 92.1% 89.3% 31 →6 (-80%) Table 1: Comparison of overconfidence mitigation between 3 and 10 inferences per iteration. The 10-inference setting improves calibration by bringing confidence values more closely aligned with actual accuracy. # Prompt Name Purpose Usage 1 System Prompt Defines the AI agent's role as a cybersecurity data expert specializing in schema mapping. Establishes reasoning rules and guidelines. Loaded once per session 2 User Prompt Carries the per-request payload: (a) curated facts from prior conflicts (if any), (b) RAG-assembled source/target schema context (descriptions, sample values, types), and (c) the mapping task/question. Enforces a strict XML response (CSV decision, 1-5 confidence, reasoning) for deterministic parsing. For every mapping request 3 Search Prompt Generates targeted Internet search queries string to resolve ambiguous field mappings using prior conflict information. Invoked only during conflict resolution Table 2: Overview of the three prompts used in the schema mapping system, detailing their roles and usage."}}},{"rowIdx":83,"cells":{"id":{"kind":"string","value":"2510.14897"},"raw":{"kind":"string","value":"Forecasting Quantum Observables:\nA Compressed Sensing Approach with Performance Guarantees\nVíctor Valls,1 Albert Akhriev,1 Olatz Sanz Larrarte,2 Javier Oliva del Moral,2, 3\nŠtěpán Šmíd,4, 5 Josu Etxezarreta Martinez,2 Sergiy Zhuk,1 and Dmytro Mishagli1, 5\n1IBM Quantum, IBM Research Europe – Dublin\n2Department of Basic Sciences, Tecnun - University of Navarra, 20018 San Sebastian, Spain\n3Donostia International Physics Center, 20018 San Sebastián, Spain\n4Department of Computing, Imperial College London, UK\n5IBM Quantum, IBM Research Europe – UK\n(Dated: October 17, 2025)\nNoise in near-term quantum devices limits the timescales over which measurements can be reliably\nobtained. Existing data-driven extrapolation methods extend the dynamics of quantum observables\nfrom measurements, but they cannot guarantee that the recovered model reflects the true dynamics.\nIn this paper, we introduce atomic norm minimization as a framework to certify that a model\nlearned by any algorithm accurately captures the underlying dynamics of quantum observables.\nThis certification is valid when the system is governed by a small number of well-separated Bohr\nfrequencies. We validate the framework across multiple algorithms on numerical experiments with\nspin-chain Hamiltonians involving 8–20 qubits. Comparing with exact diagonalization, certification\nyields an average forecasting error below 0.1 (within the observable’s range of [−1, 1]) in 98% of cases\nand below 0.05 in 89–97% of cases, depending on the forecasting algorithm. Even in the presence\nof realistic shot noise, certified models remain robust, with success rates decreasing only to 88–95%\nfor the 0.1 error threshold.\nI.\nINTRODUCTION\nQuantum spin Hamiltonians are natural testbeds for\nnear-term quantum devices because their local interac-\ntion structure maps directly onto existing qubit connec-\ntivity [1]. Recent experiments have already probed dy-\nnamics with more than one hundred qubits and circuit\ndepths of several hundred two-qubit gates [2, 3]. How-\never, noise grows rapidly with circuit depth [4], limiting\nthe timescales over which reliable quantum time evolu-\ntion (TE) can be observed.\nOne way to extend this limit is to infer the dy-\nnamics from early, low-noise measurement data [5–8].\nThese data-driven forecasting methods are appealing\nfor their simplicity and scalability, in contrast to com-\nputationally heavier model-based approaches that de-\nmand explicit Hamiltonian knowledge (e.g., Krylov sub-\nspace methods [9, 10]). The effectiveness of this data-\ndriven paradigm—which traces its origins to classical\nsignal recovery methods like Prony’s method [11, 12]—\nhas been empirically demonstrated, for example, in pre-\ndicting the dynamics of the transverse-field Ising model\n(TFIM) using a truncated Dynamic Mode Decomposition\n(DMD) [7] or a spin-boson system using ESPRIT [5].\nDespite their empirical success, a critical drawback\nof these data-driven forecasting methods is their funda-\nmental lack of reliable guarantees on the long-term fore-\ncast values. To address this limitation, we introduce a\nnovel framework that uses Atomic Norm Minimization\n(ANM) [13, 14] to certify the correctness of candidate TE\nmodels. While ANM was originally devised as a robust\ntechnique for signal recovery in off-the-grid compressed\nsensing [13, 15], it has a high computation cost, making\nit impractical for a large number of measurements. Here,\nwe repurpose the ANM theoretical guarantees to create\na rigorous validation framework that ensures that a fore-\ncast, obtained by any algorithm, accurately captures the\nunderlying quantum dynamics.\nThe paper makes the following contributions:\n• We introduce a framework that leverages ANM\nto certify the correctness of candidate quantum\nTE models obtained by any forecasting algorithm\n(Sec. III B). The framework requires that the\nunderlying model has a small number of well-\nseparated Bohr frequencies relative to the number\nof measurements.\n• We validate the certification framework through ex-\ntensive experiments on spin-chain time evolutions\n(TEs) with 8–20 qubits (Sec. IV). Our results show\nthat, for exact TEs, certifications yields an average\nforecasting error below 0.1 (within the observable’s\nrange of [−1, 1]) in 98% of cases and below 0.05\nin 89% of cases. For TEs with shot noise, these\nrates decrease to 88–95% for a 0.1 error threshold\nand 53–74% for a 0.05 threshold, depending on the\nalgorithm.\nII.\nPROBLEM STATEMENT\nConsider a quantum system with n qubits that evolves\naccording to a time-independent Hamiltonian H\n∈\nC2n×2n.\nLet |ψ(0)⟩∈C2n be an initial state vec-\ntor.\nThe quantum TE of the state vector is given by\n|ψ(τk)⟩= e−iHτk|ψ(0)⟩where τk = ∆(k −1) is a discrete\ntime, ∆> 0, and k ∈[m] := {1, . . . , m} with m the to-\ntal number of time steps. The k-th measurement of an\narXiv:2510.14897v1 [quant-ph] 16 Oct 2025\n\n2\n-0.1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n-0.2\n-0.1\n0\n0.1\n0.2\n(a)\n|ψ(0)⟩= |0⟩⊗n\n-0.2\n-0.1\n0\n0.1\n0.2\n(b)\n|ψ(0)⟩= |1⟩⊗|0⟩⊗n−1\nO = X1\nR(c(f))\nBohr frequency\nO = X1\nBohr frequency\nFIG.\n1.\nAmplitudes\nR(c(f))\nfor\nTE\nunder\nH\n=\nPn\ni=1 σz\ni σz\ni+1 + Pn\ni=1 σx\ni with n = 8 qubits, two initial states,\nand periodic boundary conditions (site n+1 = 1). The imag-\ninary part of c(f) is zero for both cases.\nobservable O ∈C2n×2n is given by\nyk := ⟨ψ(τk)|O|ψ(τk)⟩=\nX\nf∈Ω\nc(f)ei2πfτk,\n(1)\nwhere c(f) ∈C, and f ∈Ω⊂[−1\n2, 1\n2] is a Bohr frequency,\ni.e., normalized difference of the eigenvalues of H.\nThe set Ωcontains only the frequencies with non-zero\namplitude, which depend on the spectral information of\nH, the initial state |ψ(0)⟩, and observable O. Note that\nΩis generally unknown for large systems since we cannot\ncompute the spectral information of H. Fig. 1 shows the\namplitudes R(c(f)) for all f ∈Ωfor a TFIM Hamiltonian\nwith two initial states and the same observable. Note\nthat the number of non-zero amplitudes varies. Specif-\nically, subfigure (a) has fewer and more well-separated\nfrequencies than (b).\nAlso, note that the spectrum is\nsymmetric.\nThe TE forecasting problem is the following. For m\ngiven measurements y := (y1, . . . , ym) ∈[−1, 1]m and\ntolerance ǫ ≥0, find a forecast sequence xk with k ∈[m′]\nsuch that\n1\nm′\nm′\nX\nk=1\n|xk −⟨ψ(τm+k)|O|ψ(τm+k)⟩| ≤ǫ.\n(2)\nThat is, the goal is to predict m′ future values of the time\nseries based on the m available measurements.\nIII.\nCERTIFICATION VIA ANM\nIn this section, we briefly review atomic norm mini-\nmization (ANM, Sec. III A) and describe how it can be\nused to certify the frequencies and amplitudes recovered\nby any spectral estimation algorithm (Sec. III B).\nA.\nAtomic norm minimization\nANM aims to minimize the atomic norm: ∥y∥A :=\ninf{t ≥0 : y ∈t · conv(A)}, where A is a collection\nof “atoms” (vectors) and conv(A) its convex hull. The\n−1\n−0.5\n0\n0.5\n1\n−0.4\n−0.2\n0\n0.2\n0.4\nR(c(f))\nQ(f)\nBohr frequency\nFIG. 2. Illustration of the dual polynomial Q(f) for the set-\nting in Fig. 1 (a). The quantum TE is generated over 10s with\na sampling rate of 5 steps per second (i.e., m = 50, ∆= 1/5).\nOnly frequencies with amplitude magnitude larger than 0.005\nare shown. The markers (small crosses) indicate where the\nQ(f) matches a frequency.\natomic norm can be seen as the continuous analogue of\nthe ℓ1 norm, since A may contain an uncountable set of\natoms corresponding to all possible continuous-frequency\nsinusoidal components.\nFor our TE forecasting problem, A contains vectors of\nthe form\na(f) := (ei2πfτ1, . . . , ei2πfτm),\n(3)\nwhere f ∈[−1\n2, 1\n2], and the atomic norm minimization\nfinds the sparsest combination of these continuous atoms\nthat match the measurements vector y. In some cases,\nthis representation is unique, enabling exact recovery of\nthe true frequencies and amplitudes underlying y. For\nthe representation to be unique, two sufficient condi-\ntions must be met: the number of measurements is large\nenough, and the true frequencies are sufficiently sepa-\nrated (i.e., |fi −fj| ≥\n4\nm for all fi, fj ∈Ωwith i ̸= j);\nsee details in [15, Theorem 1.2] and [16, Theorem 1.1].\nIn practice, we can solve the ANM with atoms of the\nform in Eq. (3) via a semidefinite program (SDP):\nminimize\nT,t>0\n1\n2mTr(T ) + 1\n2t\ns.t.\n\u0014\nT\ny\ny∗t\n\u0015\n⪰0,\n(4)\nwhere T ∈Cm×m is a Hermitian Toeplitz matrix that\nadmits a decomposition T = P\nf∈F w(f)a(f)a(f)∗(see\n[16, Lemma 2.2]), where w(f) > 0 and F ⊂[−1\n2, 1\n2] de-\nnote the support set of active frequencies. In brief, the\nlinear matrix inequality in Eq. (4) ensures that y lies in\nthe subspace spanned by vectors a(f), while the objec-\ntive promotes low-rank T , thus enforcing sparsity in the\nfrequency domain. Once the optimal T is found, the fre-\nquencies can be recovered from T using Prony’s method.\nCrucially, the correctness of these recovered frequen-\ncies can be certified by solving the associated dual prob-\nlem:\nmaximize\nq∈Cm\n⟨y, q⟩\ns.t.\nsup\nf∈[−1\n2 , 1\n2 ]\n|Q(f)| ≤1,\n(5)\nwhere Q(f) := ⟨a(f), q⟩is the dual polynomial. The so-\nlution is certified as unique when the dual polynomial\n\n3\nreaches its maximum value |Q(f)| = 1 precisely for all\nf ∈Ωand |Q(f)| < 1 otherwise (see [16, Proposition\n2.4]).\nAn example of this dual certificate is shown in\nFig. 2 for the setting used in Fig. 1 (a), where Q(f) is\nreal-valued and touches ±1 precisely at the true frequen-\ncies (indicated with markers).\nFinally, we note that solving the ANM with the SDP\nformulation in Eq. (4) is computationally impractical for\na large number of measurements (see Appendix D). In the\nfollowing section, we show how ANM provides a princi-\npled way to certify whether, under sparsity and frequency\nseparation conditions, a candidate model accurately re-\nproduces the true dynamics in Eq. (1).\nB.\nCertification pipeline\nThe proposed certification pipeline is illustrated in\nFig. 3.\nAn algorithm (e.g., ESPRIT) takes as input\na vector y with m measurements and returns candi-\ndate frequencies Ωand amplitudes c(f) such that yk ≈\nP\nf∈Ωc(f) e−i2πfτk, k ∈{1, . . ., m}.\nThese candidates\nare then certified by solving the ANM dual problem in\nEq. (5).\nCertificates. To certify optimality, we evaluate three\nkey criteria: the duality gap, frequency separation, and\nthe dual polynomial.\nThe duality gap measures the difference between the\nprimal and dual values (Eqs. (4) and (5)). The gap is\nalways non-negative and, due to strong duality, is zero\nwhen both problems are solved optimally. We can ob-\ntain an upper bound on the optimal primal objective by\nconstructing T = P\nf∈Ω|c(f)|a(f)a(f)∗, and then set-\nting t = y∗T †y. This construction ensures that the con-\nstraints in Eq. (4) are satisfied by the Schur complement\ncondition. A lower bound on the dual value can be ob-\ntained by solving the dual problem approximately (as\ndiscussed later). The duality gap is considered admissi-\nble if this is below a prescribed tolerance.\nFor frequency separation, we need to check that the\nminimum distance between the candidate frequencies in\nΩexceeds 4/m, which is required for unique recovery in\nANM.\nFinally, the dual polynomial has to satisfy |Q(f)| ≈1\nfor all f ∈Ωand |Q(f)| < 1, otherwise, within numerical\ntolerances. Computing the dual polynomial is straight-\nforward once we have a solution to the dual problem.\nSolving the dual problem. The dual problem in\nEq. (5) is infinite-dimensional and must be discretized\nto make it tractable. We therefore sample the frequency\ndomain finely, transforming the problem into a second-\norder cone program. In addition, since the measurement\nvector y is real-valued (which is always the case in quan-\ntum TEs), the dual problem can be further simplified\nto\nmax\nq∈Rm ⟨y, q⟩\ns.t.\n−1 ≤⟨a(f), q⟩≤1, ∀f ∈F,\n(6)\nFind frequencies and\namplitudes such that \nCertify solution\nGap\noptimality certificate\nFIG. 3. Schematic illustration of the proposed certification\npipeline. The process verifies frequency and amplitude esti-\nmates obtained from any algorithm using the dual formulation\nof ANM.\nwhere F is a finely discretized set of frequencies in [0, 1\n2],\nsince the frequency spectrum is symmetric. Finally, al-\nthough discretization introduces a small approximation\nerror, it remains negligible when the frequency grid is\nsufficiently fine.\nIV.\nEXPERIMENTS\nThis section evaluates the proposed ANM certification\nmethodology using several existing algorithms. The goal\nis to demonstrate that, when a certificate of optimality\nis obtained, the correct model can be recovered, enabling\nreliable forecasting of the TE. Note that successful cer-\ntification also depends on the true model being sparse\nand having well-separated frequencies—conditions that\nare typically unknown in practice.\nA.\nSetup\nDataset.\nWe consider quantum TEs generated by\n15 spin-chain Hamiltonians (4 TFIM and 11 Heisenberg)\nwith 8–20 qubits and periodic boundary conditions [17–\n26] (see details in Appendix B). For all TEs, we consider\nfive initial states—Néel, dimerized, paramagnetic, ferro-\nmagnetic, and the zero state—and measure six observ-\nables (X, Y , Z, XX, Y Y , and ZZ); see Appendix B.\nThis results in a total of 5850 TEs. The TEs are gen-\nerated using Qutip [27] for 40 time units with dt = 0.2,\nyielding m = 200 measurements per TE. A time unit is\nt·J, where J is the coupling strength (interaction energy)\nin a Hamiltonian.\nSpectral estimation algorithms. We use four al-\ngorithms to compute the frequencies and amplitudes:\nProny, OMP, FISTA, and ESPRIT. Prony [11] is a\nmethod that can provably recover the true frequencies\nand amplitudes under ideal conditions but is highly sen-\nsitive to noise. OMP [28, 29] is a widely used heuristic in\nsignal processing for sparse signal recovery. FISTA [30]\nis a fast gradient method for solving Lasso problems, and\nESPRIT [31, 32] is a well-known algorithm in spectrum\nestimation, also applied in quantum TE forecasting [5].\nThe algorithms are implemented with default or minimal\n\n4\nExact TE\nTE with shot noise\nǫ\nPRO\nOMP\nFIS\nESP\nPRO\nOMP\nFIS\nESP\n0.01\n83.4\n75.1\n68.7\n70.8\n50.2\n20.2\n10.3\n21.3\n0.05\n97.4\n96.4\n95.8\n89.5\n58.3\n69.9\n74.6\n53.2\n0.10\n98.7\n99.2\n99.7\n98.8\n82.2\n95.1\n95.4\n83.1\nTABLE I. SCs rate for different algorithms and forecasting\ntolerances (ǫ). Noisy TEs use 1000 shots.\nExact TE\nTE with shot noise\nǫ\nPRO\nOMP\nFIS\nESP\nPRO\nOMP\nFIS\nESP\n0.01\n15.8\n11.2\n2.9\n13.8\n6.7\n7.1\n1.3\n6.6\n0.05\n18.4\n14.3\n4.1\n17.5\n7.7\n24.5\n9.0\n16.4\n0.10\n18.6\n14.7\n4.3\n19.3\n10.9\n33.4\n11.6\n25.6\nTABLE II. Absolute SCs rate for different algorithms and\nforecasting tolerances (ǫ). The Absolute SC rate is the per-\ncentage of SCs over the total number of runs. Noisy TEs use\n1000 shots.\ntuning settings, as our focus is on evaluating the certifica-\ntion methodology. A brief description of each algorithm\nand its implementation is provided in Appendix A.\nForecasting procedure. Each algorithm is applied\nto the first k measurements, where k ∈{10, . . ., 160},\nand forecasting is evaluated on the remaining 200 −k\nmeasurements, as defined in Eq. (2).\nCertification. For each algorithm run, ANM certifi-\ncation is performed (150 times per algorithm and TE).\nThis is done by solving Eq. (6) on a frequency grid of 1000\nuniformly spaced points in [0, 1\n2] [33], augmented with\nthe candidate frequencies estimated by the algorithm—\nto ensure that the dual polynomial is evaluated at the\nexact/candidate frequencies. To account for numerical\ninaccuracies, empirically chosen tolerances are applied.\nSpecifically, for exact TEs, the dual polynomial must sat-\nisfy |Q(f)| ≥0.98 at all candidate frequencies, and the\nduality gap must be smaller than 0.05. For TEs with\nshot noise, the thresholds are |Q(f)| ≥0.95 and a dual-\nity gap smaller than 0.5. In addition, we only consider\namplitudes with magnitude larger than 0.025 for exact\nTEs and 0.05 for TEs with shot noise.\nB.\nResults\nWe evaluate forecasting performance using tolerances\nǫ ∈{0.01, 0.05, 0.1} (see Sec. II), for both exact TEs and\nTEs with shot noise. We consider the output of a spectral\nalgorithm (i.e., frequencies and amplitudes) certified if it\nmeets the criteria in Sec. III B. A successful certification\n(SC) occurs when a certified run also satisfies the fore-\ncasting error in Eq. (2) for the remaining measurements,\nwhich range from 190 (k = 10) down to 40 (k = 160).\nWe define the SC rate as the percentage of SCs among\n0\n0.2\n0.4\n0.6\n0.8\n1\n0\n0.05\n0.1\n0.15\n0.2\n0.25\n0\n0.05\n0.1\n0.15\n0.2\n0.25\n0.3\nFraction of certified TEs\nError\nExact TE\nProny\nOMP\nFISTA\nESPRIT\nError\n1000-shot noise\nFIG. 4. Distribution of the forecasting error upon the first\ncertification, with exact and noisy TEs.\n0\n0.2\n0.4\n0.6\n0.8\n1\n0\n0.05\n0.1\n0.15\n0.2\n0.25\n0\n0.05\n0.1\n0.15\n0.2\n0.25\n0.3\nFraction of certified TEs\nError\nExact TE\nProny\nOMP\nFISTA\nESPRIT\nError\n1000-shot noise\nFIG. 5. Distribution of the forecasting error upon the first\ncertification for certificates issued for 30 or more measure-\nments, with exact and noisy TEs.\nall certified runs (successful or not).\nAverage performance. Table I reports the SC rate\nfor each algorithm and tolerance ǫ. For exact TEs, all\nalgorithms achieve high SC rates, exceeding 98% for ǫ ≤\n0.1. When shot noise is introduced (1000 shots), SC rates\ndecrease, particularly for ǫ ≤0.01.\nOMP and FISTA generally outperform Prony and ES-\nPRIT on noisy TEs; however, FISTA certifies a smaller\nnumber of runs compared to the other algorithms. Ta-\nble II shows the percentage of SCs over all algorithm runs\n(877,500 runs per algorithm: 150 runs per TE across 5850\nTEs). Only a fraction of the runs yield a certificate, with\nProny and ESPRIT achieving SC around 18% of all the\nruns for ǫ ≤0.05.\nInterestingly, the absolute SC rate for OMP and ES-\nPRIT (Table II) increases with noisy TEs. This is likely\ndue to the looser certification tolerances applied under\nnoise, which increase the total number of certified runs.\nPerformance on first certification. Fig. 4 shows\nthe distribution of forecasting errors at the first certifi-\ncation (the first k ∈{10, . . ., 160} that meets the crite-\nria in Sec. III B). Overall, the algorithms exhibit similar\nbehavior. For exact TEs, Prony and OMP achieve er-\nrors below 0.05 and 0.1 in approximately 50% and 75%\nof cases, respectively. For noisy TEs, the proportion of\nforecasts with errors below 0.05 drops sharply, and both\nProny and OMP achieve errors below 0.1 in only about\n50% of cases.\nThese results contrast with the high SC rates in Ta-\n\n5\nble I, possibly due to insufficient measurements used dur-\ning spectral estimation or certification.\nTo investigate\nthis, Fig. 5 shows forecasting errors for first certifications\nissued only after 30 or more measurements. Performance\nimproves noticeably: for exact TEs, Prony achieves er-\nrors below 0.05 and 0.1 in 83.4% and 93.4% of cases,\nrespectively, while OMP achieves the same thresholds in\n76% and 100% of cases. For noisy TEs, all algorithms re-\nduce the proportion of forecasts with errors exceeding 0.1\n(cf. Fig. 4 and Fig. 5), with OMP achieving errors below\n0.1 and 0.15 in 75% and 97% of cases, respectively.\nThese results indicate that a sufficiently large number\nof measurements is crucial for reliable certification; while\nthe minimum is unknown, monitoring stability, as sug-\ngested in [5, Sec. III-C], or setting a minimum measure-\nment threshold (e.g., k ≥30) can significantly enhance\nperformance.\nV.\nCONCLUSIONS\nWe introduced ANM as a framework to certify the\nspectral models produced by data-driven forecasting al-\ngorithms. Our numerical experiments show that, with\ncarefully chosen tolerance thresholds, ANM can success-\nfully certify both exact and noisy quantum TEs.\nA key limitation is that certification is guaranteed only\nwhen the underlying dynamics are sparse and the Bohr\nfrequencies are well-separated. Nevertheless, our results\nindicate that ANM can still provide reliable certificates\nin realistic scenarios. A promising direction for future\nwork is to characterize which time evolutions—defined\nby specific Hamiltonians, initial states, and observables—\nsatisfy the framework conditions, thereby pinpointing the\nregimes where reliable forecasting is theoretically guaran-\nteed.\nVI.\nACKNOWLEDGMENTS\nSZ and DM are grateful to Sergey Bravyi, Ewout van\nden Berg, and Kristan Temme for stimulating discus-\nsions. 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Kailath, Esprit-estimation of signal\nparameters via rotational invariance techniques, IEEE\nTransactions on acoustics, speech, and signal processing\n37, 984 (2002).\n[33] It is sufficient to consider [0, 1\n2] instead of [−1\n2, 1\n2] since\nthe frequency spectrum is symmetric.\n[34] D. P. Kingma and J. Ba, Adam: A method for stochastic\noptimization, arXiv preprint arXiv:1412.6980 (2014).\n[35] M. ApS, MOSEK Optimization Software (2025), version\n11.0.\n\n7\nAppendix A: Algorithms\n1.\nAlgorithm for finding frequencies and amplitudes\na.\nProny’s method\nProny’s method is implemented as described in [11]. The method requires that the number of measurements m\nsatisfies ≥2|Ω|, where |Ω| denotes the number of frequencies. However, |Ω| is typically unknown. To address this, we\nexecute Prony’s method for a range of candidate values of |Ω|. Specifically, given m measurements, we apply Prony’s\nmethod using the first 2k measurements for k = 1, . . . , ⌊m/2⌋. For each k, we compute the error ∥y −x∥2, where\nx, y ∈Rm denote the reconstructed and measured signals, respectively. The algorithm selects the model (i.e., the set\nof frequencies and amplitudes defining x) that yields the smallest error.\nb.\nOMP\nOrthogonal Matching Pursuit (OMP) is implemented as described in [29, Algorithm 1], using atoms defined by\na(f, φ) = (cos(2πfτ1 + φ), . . . , cos(2πfτm + φ)),\n(A1)\nwhere f ∈F := {0, 1\n2L, . . . , L\n2L} and Φ := {0, 2π\nP , . . . , 2π(P −1)\nP\n}, with L = 10000 and P = 10. We use the atoms in\nEq. (A1) instead of those in Eq. (3) because y is real-valued. Consequently, its frequency spectrum is symmetric, and\nthe corresponding amplitudes occur in complex-conjugate pairs. We make the algorithm terminate after 250 iterations\nor when the recovered signal, x, satisfies ∥y −x∥2 ≤ǫ for ǫ = 10−2. The algorithm returns the frequencies, phases,\nand amplitude magnitudes of the selected atoms.\nc.\nFISTA\nThe Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is implemented as described in [30] to solve the\noptimization problem\nminimize\nx∈RLP\n1\n2∥Ax −y∥2\n2 + λ∥x∥1,\n(A2)\nwhere y ∈Rm denotes the vector of measurements and A ∈Rm×LP is a matrix whose columns correspond to the\natoms defined in Eq. (A1), with L = 1000 and P = 4. The algorithm outputs the frequencies, phases, and amplitude\nmagnitudes associated with the selected atoms (i.e., the active columns of A). To account for numerical issues, we\nonly consider an atom is selected if the associated amplitude magnitude is larger than 10−4.\nd.\nESPRIT\nWe have adjusted the method [31, 32] for a signal (1), generated by a Hamiltonian evolution, as follows. For N\nsamples of a signal y = Pr\nk=1 ak z n\nk and for a window length d = m + 1 with 2 ≤d ≤N −1, we form delay Hankel\nmatrices Y0, Y1 ∈Cd×L (L = N −d) that are shifted by one sample. Here m is a parameter that controls the size of\na subspace, chosen such that N ≥m + 2. We then compute a rank-r singular value decomposition of Y0 = U Σ V ∗\nand project the one-step shift matrix Y1 onto the r-dimensional signal subspace spanned by columns of U by forming\nthe reduced operator\nΦ = U ∗Y1 V Σ−1 ∈Cr×r.\n(A3)\nUsing Y0 = UΣV ∗and Y1 = V ΛW, we have U ∗Y1V Σ−1 ≈U ∗(V ΛW)V Σ−1, and since U spans the columns of V , Φ\nis similar to Λ on the signal subspace. Thus\neig(Φ) = {z1, . . . , zr}.\n(A4)\nHaving the poles {zk}, we form the Vandermonde matrix on the full sample range,\n\u0002\nV\n\u0003\nnk\n=\nz n\nk where n =\n0, . . . , N −1, and find the amplitudes from least squares: mina∈Cr \r\ry −V a\n\r\r2\n2. The frequencies follow from ωk\n=\n−arg(zk).\n\n8\nInitial State\nDefinition\nZero\n|0⟩⊗N\nNéel\n|01⟩⊗N/2\nDimerized\n(|01⟩−|10⟩)⊗N/2\nParamagnetic\n(|0⟩+ |1⟩)⊗N\nFerromagnetic\n|1⟩⊗N\nTABLE III. Initial states and their definitions (normalization constants are omitted for clarity of presentation).\nAdditionally, we assumed unitary dynamics and, thus, project zk to a unit circle, zk 7→ei arg zk. The subspace\nparameter m was chosen as max(2, ⌊N\n3 ⌋) providing the requirement N ≥m + 2. The number of samples N varied\nin simulations as N ≡k ∈{1−, . . ., 160}. When the SVD procedure failed, we considered that as a negative TE\ninstance.\n2.\nAlgorithm to solve the dual problem in Eq. (5)\nWe solve the problem in Eq. (5) with Adam [34]. The objective function is\nf(q) = ⟨y, q⟩+ λ\nL\nX\nj=1\nmax{0, |(Aq)j| −1}2,\n(A5)\nwhere y ∈Rm is the vector of measurements, and A ∈RL×m a matrix with rows equal to the atoms in Eq. (3). We\nuse L = 1000 and λ = 15. We set the learning rate equal to 0.1 and run the algorithm for 500 iterations.\nAppendix B: Hamiltonians and Initial States\nWe consider one–dimensional spin- 1\n2 chains. For convenience, we write all Hamiltonians in the Pauli convention,\ni.e. with Pauli matrices X, Y , and Z acting on site k:\nHXYZ = Jx\nX\nk\nXkXk+1 + Jy\nX\nk\nYkYk+1 + Jz\nX\nk\nZkZk+1 + hx\nX\nk\nXk + hy\nX\nk\nYk + hz\nX\nk\nZk.\nWhen restricted to Ising–type couplings and a transverse field, we use:\nHTFIM = Jz\nX\nk\nZkZk+1 + hx\nX\nk\nXk.\nHere the index k runs over lattice sites (boundary conditions are specified elsewhere, mostly as “periodic”), and the\ncouplings {Jα} and fields {hα} are given in units where energy scales are O(1).\nSometimes, spin Hamiltonians are written with spin operators Sα\nk rather than Pauli matrices σα\nk . Since Sα\nk = 1\n2 σα\nk ,\nour parameters map to the S-operator convention as:\nJ(S)\nα\n= 4 Jα,\nh(S)\nα\n= 2 hα,\nand respectively,\nJα = J(S)\nα /4,\nhα = h(S)\nα /2.\nSigns are unaffected by this rescaling.\nOur study focuses on spin- 1\n2 models in the above Pauli normalization for implementation simplicity. The parameter\nchoices we explore were inspired by Hamiltonians used in diverse physical simulations (see Table IV). We note,\nhowever, that several of those source models were originally formulated for spin-1 or higher; when importing their\ncoefficients into our convention one should apply the normalization map above. Spin- 1\n2 and spin-1 chains can exhibit\nqualitatively different physics. Because our primary objective here is predictive performance rather than a faithful\ncomparison of phase structures, we do not distinguish between spin- 1\n2 and spin-1 models in the experiments. Results\nshould therefore be interpreted with this caveat in mind.\nThe initial states used in all simulations (Table III) are standard in the literature and admit shallow state-preparation\ncircuits, making them both representative and practical for our benchmarks.\n\n9\nParameters\nDescription\nJx = 0, Jy = 0, Jz = −2,\nhx = 1, hy = 0, hz = 0\nIsing ferromagnet.\nJx = 0, Jy = 0, Jz = 2,\nhx = 1, hy = 0, hz = 0\nIsing antiferromagnet.\nJx = 0, Jy = 0, Jz = 1,\nhx = 2, hy = 0, hz = 0\nIsing paramagnet.\nJx = 0, Jy = 0, Jz = 1,\nhx = 1, hy = 0, hz = 0\nIsing critical model.\nJx = Jy = Jz = 1,\nhx = hy = 0, hz = 0.2\nIsotropic Haldane, tiny longitudinal field, [17].\nJx = Jy = Jz = 1,\nhx = hy = 0, hz = 0.4105\nAntiferromagnetic quantum chain at the lower critical field, [18].\nJx = Jy = Jz = 1,\nhx = hy = 0, hz = 0.6\nField-induced Tomonaga–Luttinger liquid, [19].\nJx = 0.7, Jy = 0.7, Jz = 1.3,\nhx = 0, hy = 0, hz = 0.1\nEasy-axis XXZ with weak field, [20].\nJx = 1.0, Jy = 1.0, Jz = 0.6,\nhx = 0.2, hy = 0, hz = 0\nEasy-plane XXZ, transverse probe, [21],\nJx = Jy = Jz = 1,\nhx = 0.3, hy = 0, hz = 0.3\nTilted field, symmetry breaking, [22].\nJx = 1.2, Jy = 0.8, Jz = 1,\nhx = 0.9, hy = 0, hz = 0\nAnisotropic XYZ + transverse field, [23].\nJx = Jy = Jz = 1,\nhx = 0, hy = 0, hz = 1\nIsotropic AFM, intermediate longitudinal field, [24].\nJx = Jy = Jz = −1,\nhx = 0, hy = 0, hz = 0.1\nIsotropic ferromagnetic Heisenberg chain, tiny field, [25].\nJx = 1, Jy = 1, Jz = −0.7,\nhx = 0.15, hy = 0, hz = 0.15\nMixed-sign anisotropy, easy-plane exchange but ferromagmetic Jz,\n[19].\nJx = Jy = Jz = 1,\nhx = 1.5, hy = 0, hz = 0\nIsotropic AFM with strong transverse field (near product paramagnet)\nJz, [26].\nTABLE IV. Hamiltonians selected for TE.\n\n10\nAppendix C: Additional experiments\n1.\nError growth after first certification\nThe following figures illustrate the evolution of the forecasting error after the first certification, measured on the\nsamples predicted beyond that point. For example, if the first certification occurs at sample 50 in one TE and at\nsample 100 in another, the first forecast samples correspond to 51 and 101, respectively. The average error is indicated\nby a thick line, and the standard deviation is shown as a shaded area. The figures display the average results for all\nalgorithms and TEs, with the experimental setup described in Sec. IV.\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0\n20\n40\n60\n80 100 120 140 0\n20\n40\n60\n80 100 120 140\nNo shot noise\nError\n250 shot noise\n500 shot noise\nError\nForecast measurements\n1000 shot noise\nForecast measurements\nFIG. 6. Average per-sample forecasting error after the first certification for exact TEs and three shot noise levels (250, 500,\nand 1000 shots). The error is averaged over all algorithms.\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0\n0.1\n0.2\n0.3\n0.4\n0.5\n0.6\n0\n20\n40\n60\n80 100 120 140 0\n20\n40\n60\n80 100 120 140\nNo shot noise\nError\n250 shot noise\n500 shot noise\nError\nForecast measurements\n1000 shot noise\nForecast measurements\nFIG. 7. Average per-sample forecasting error after the first certification after 30 or more measurements, for exact TEs and\nthree shot noise levels (250, 500, and 1000 shots). The error is averaged over all algorithms.\n\n11\n2.\nIllustrative examples\n−1\n−0.5\n0\n0.5\n1\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n0\n0.2\n0.4\n0.6\n0.8\n1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\nMeasurements\nForecasting\nMeasurement\n|Q(f)|\nf ∈Ω\nBohr frequency\nFIG. 8. Forecasting with OMP: 18 qubits Anisotropic XYZ + transverse field Hamiltonian [23], dimerized initial state, and\nO = X9X10. The red vertical line marks the training interval [0 . . . 100), where we fit the sparse model; the remaining data\n[100 . . . 200) are reserved for out-of-sample verification.\n−1\n−0.5\n0\n0.5\n1\n0\n20\n40\n60\n80\n100\n120\n140\n160\n180\n200\n0\n0.2\n0.4\n0.6\n0.8\n1\n0\n0.1\n0.2\n0.3\n0.4\n0.5\nMeasurements\nForecasting\nMeasurement\n|Q(f)|\nf ∈Ω\nBohr frequency\nFIG. 9. Forecasting with OMP: 20 qubit easy-plane XXZ transverse probe Hamiltonian [21], paramagnetic initial state, and\nO = Y10Y11. The red vertical line marks the training interval [0 . . . 100), where we fit the sparse model; the remaining data\n[100 . . . 200) are reserved for out-of-sample verification.\n\n12\nAppendix D: SDP Running Times with MOSEK\nFig. 10 shows the time it takes MOSEK [35] (a commercial SDP solver) to solve the problem in Eq. (4). The SDP\nis run on an Intel i7-14700 CPU with 32 GB of memory.\n0\n50\n100\n150\n200\n250\n0\n50\n100\n150\n200\n250\n300\n350\n400\nTime (s)\nNumber of measurements\nFIG. 10. Time to solve the SDP in Eq. (4) with MOSEK [35] for different number of measurements.\n"},"cleaned":{"kind":"string","value":"Forecasting Quantum Observables: A Compressed Sensing Approach with Performance Guarantees Víctor Valls,1 Albert Akhriev,1 Olatz Sanz Larrarte,2 Javier Oliva del Moral,2, 3 Štěpán Šmíd,4, 5 Josu Etxezarreta Martinez,2 Sergiy Zhuk,1 and Dmytro Mishagli1, 5 1IBM Quantum, IBM Research Europe - Dublin 2 - 20018 San Sebastian, Spain 3Donostia International Physics Center, 20018 San Sebastián, Spain 4 5IBM Quantum, IBM Research Europe - UK (Dated: October 17, 2025) Noise in near-term quantum devices limits the timescales over which measurements can be reliably obtained. Existing data-driven extrapolation methods extend the dynamics of quantum observables from measurements, but they cannot guarantee that the recovered model reflects the true dynamics. In this paper, we introduce atomic norm minimization as a framework to certify that a model learned by any algorithm accurately captures the underlying dynamics of quantum observables. This certification is valid when the system is governed by a small number of well-separated Bohr frequencies. We validate the framework across multiple algorithms on numerical experiments with spin-chain Hamiltonians involving 8-20 qubits. Comparing with exact diagonalization, certification yields an average forecasting error below 0.1 (within the observable's range of [-1, 1]) in 98% of cases and below 0.05 in 89-97% of cases, depending on the forecasting algorithm. Even in the presence of realistic shot noise, certified models remain robust, with success rates decreasing only to 88-95% for the 0.1 error threshold. I. INTRODUCTION Quantum spin Hamiltonians are natural testbeds for near-term quantum devices because their local interaction structure maps directly onto existing qubit connectivity [1]. Recent experiments have already probed dynamics with more than one hundred qubits and circuit depths of several hundred two-qubit gates [2, 3]. However, noise grows rapidly with circuit depth [4], limiting the timescales over which reliable quantum time evolution (TE) can be observed. One way to extend this limit is to infer the dynamics from early, low-noise measurement data [5-8]. These data-driven forecasting methods are appealing for their simplicity and scalability, in contrast to computationally heavier model-based approaches that demand explicit Hamiltonian knowledge (e.g., Krylov subspace methods [9, 10]). The effectiveness of this datadriven paradigm-which traces its origins to classical signal recovery methods like Prony's method [11, 12]- has been empirically demonstrated, for example, in predicting the dynamics of the transverse-field Ising model (TFIM) using a truncated Dynamic Mode Decomposition (DMD) [7] or a spin-boson system using ESPRIT [5]. Despite their empirical success, a critical drawback of these data-driven forecasting methods is their fundamental lack of reliable guarantees on the long-term forecast values. To address this limitation, we introduce a novel framework that uses Atomic Norm Minimization (ANM) [13, 14] to certify the correctness of candidate TE models. While ANM was originally devised as a robust technique for signal recovery in off-the-grid compressed sensing [13, 15], it has a high computation cost, making it impractical for a large number of measurements. Here, we repurpose the ANM theoretical guarantees to create a rigorous validation framework that ensures that a forecast, obtained by any algorithm, accurately captures the underlying quantum dynamics. The paper makes the following contributions: • We introduce a framework that leverages ANM to certify the correctness of candidate quantum TE models obtained by any forecasting algorithm (Sec. III B). The framework requires that the underlying model has a small number of wellseparated Bohr frequencies relative to the number of measurements. • We validate the certification framework through extensive experiments on spin-chain time evolutions (TEs) with 8-20 qubits (Sec. IV). Our results show that, for exact TEs, certifications yields an average forecasting error below 0.1 (within the observable's range of [-1, 1]) in 98% of cases and below 0.05 in 89% of cases. For TEs with shot noise, these rates decrease to 88-95% for a 0.1 error threshold and 53-74% for a 0.05 threshold, depending on the algorithm. II. PROBLEM STATEMENT Consider a quantum system with n qubits that evolves according to a time-independent Hamiltonian H ∈ C2n×2n. Let |ψ(0)⟩∈C2n be an initial state vector. The quantum TE of the state vector is given by |ψ(τk)⟩= e-iHτk|ψ(0)⟩where τk = ∆(k -1) is a discrete time, ∆> 0, and k ∈[m] := {1, . . . , m} with m the total number of time steps. The k-th measurement of an 16 Oct 2025 2 -0.1 0 0.1 0.2 0.3 0.4 0.5 -0.2 -0.1 0 0.1 0.2 (a) |ψ(0)⟩= |0⟩⊗n -0.2 -0.1 0 0.1 0.2 (b) |ψ(0)⟩= |1⟩⊗|0⟩⊗n-1 O = X1 R(c(f)) Bohr frequency O = X1 Bohr frequency FIG. 1. Amplitudes R(c(f)) for TE under H = Pn i=1 σz i σz i+1 + Pn i=1 σx i with n = 8 qubits, two initial states, and periodic boundary conditions (site n+1 = 1). The imaginary part of c(f) is zero for both cases. observable O ∈C2n×2n is given by yk := ⟨ψ(τk)|O|ψ(τk)⟩= X f∈Ω c(f)ei2πfτk, (1) where c(f) ∈C, and f ∈Ω⊂[-1 2, 1 2] is a Bohr frequency, i.e., normalized difference of the eigenvalues of H. The set Ωcontains only the frequencies with non-zero amplitude, which depend on the spectral information of H, the initial state |ψ(0)⟩, and observable O. Note that Ωis generally unknown for large systems since we cannot compute the spectral information of H. Fig. 1 shows the amplitudes R(c(f)) for all f ∈Ωfor a TFIM Hamiltonian with two initial states and the same observable. Note that the number of non-zero amplitudes varies. Specifically, subfigure (a) has fewer and more well-separated frequencies than (b). Also, note that the spectrum is symmetric. The TE forecasting problem is the following. For m given measurements y := (y1, . . . , ym) ∈[-1, 1]m and tolerance ǫ ≥0, find a forecast sequence xk with k ∈[m′] such that 1 m′ m′ X k=1 |xk -⟨ψ(τm+k)|O|ψ(τm+k)⟩| ≤ǫ. (2) That is, the goal is to predict m′ future values of the time series based on the m available measurements. III. CERTIFICATION VIA ANM In this section, we briefly review atomic norm minimization (ANM, Sec. III A) and describe how it can be used to certify the frequencies and amplitudes recovered by any spectral estimation algorithm (Sec. III B). A. Atomic norm minimization ANM aims to minimize the atomic norm: ∥y∥A := inf{t ≥0 : y ∈t · conv(A)}, where A is a collection of \"atoms\" (vectors) and conv(A) its convex hull. The -1 -0.5 0 0.5 1 -0.4 -0.2 0 0.2 0.4 R(c(f)) Q(f) Bohr frequency FIG. 2. Illustration of the dual polynomial Q(f) for the setting in Fig. 1 (a). The quantum TE is generated over 10s with a sampling rate of 5 steps per second (i.e., m = 50, ∆= 1/5). Only frequencies with amplitude magnitude larger than 0.005 are shown. The markers (small crosses) indicate where the Q(f) matches a frequency. atomic norm can be seen as the continuous analogue of the l1 norm, since A may contain an uncountable set of atoms corresponding to all possible continuous-frequency sinusoidal components. For our TE forecasting problem, A contains vectors of the form a(f) := (ei2πfτ1, . . . , ei2πfτm), (3) where f ∈[-1 2, 1 2], and the atomic norm minimization finds the sparsest combination of these continuous atoms that match the measurements vector y. In some cases, this representation is unique, enabling exact recovery of the true frequencies and amplitudes underlying y. For the representation to be unique, two sufficient conditions must be met: the number of measurements is large enough, and the true frequencies are sufficiently separated (i.e., |fi -fj| ≥ 4 m for all fi, fj ∈Ωwith i ̸= j); see details in [15, Theorem 1.2] and [16, Theorem 1.1]. In practice, we can solve the ANM with atoms of the form in Eq. (3) via a semidefinite program (SDP): minimize T,t>0 1 2mTr(T ) + 1 2t s.t. T y y∗t ⪰0, (4) where T ∈Cm×m is a Hermitian Toeplitz matrix that admits a decomposition T = P f∈F w(f)a(f)a(f)∗(see [16, Lemma 2.2]), where w(f) > 0 and F ⊂[-1 2, 1 2] denote the support set of active frequencies. In brief, the linear matrix inequality in Eq. (4) ensures that y lies in the subspace spanned by vectors a(f), while the objective promotes low-rank T , thus enforcing sparsity in the frequency domain. Once the optimal T is found, the frequencies can be recovered from T using Prony's method. Crucially, the correctness of these recovered frequencies can be certified by solving the associated dual problem: maximize q∈Cm ⟨y, q⟩ s.t. sup f∈[-1 2 , 1 2 ] |Q(f)| ≤1, (5) where Q(f) := ⟨a(f), q⟩is the dual polynomial. The solution is certified as unique when the dual polynomial 3 reaches its maximum value |Q(f)| = 1 precisely for all f ∈Ωand |Q(f)| < 1 otherwise (see [16, Proposition 2.4]). An example of this dual certificate is shown in Fig. 2 for the setting used in Fig. 1 (a), where Q(f) is real-valued and touches ±1 precisely at the true frequencies (indicated with markers). Finally, we note that solving the ANM with the SDP formulation in Eq. (4) is computationally impractical for a large number of measurements (see Appendix D). In the following section, we show how ANM provides a principled way to certify whether, under sparsity and frequency separation conditions, a candidate model accurately reproduces the true dynamics in Eq. (1). B. Certification pipeline The proposed certification pipeline is illustrated in Fig. 3. An algorithm (e.g., ESPRIT) takes as input a vector y with m measurements and returns candidate frequencies Ωand amplitudes c(f) such that yk ≈ P f∈Ωc(f) e-i2πfτk, k ∈{1, . . ., m}. These candidates are then certified by solving the ANM dual problem in Eq. (5). Certificates. To certify optimality, we evaluate three key criteria: the duality gap, frequency separation, and the dual polynomial. The duality gap measures the difference between the primal and dual values (Eqs. (4) and (5)). The gap is always non-negative and, due to strong duality, is zero when both problems are solved optimally. We can obtain an upper bound on the optimal primal objective by constructing T = P f∈Ω|c(f)|a(f)a(f)∗, and then setting t = y∗T †y. This construction ensures that the constraints in Eq. (4) are satisfied by the Schur complement condition. A lower bound on the dual value can be obtained by solving the dual problem approximately (as discussed later). The duality gap is considered admissible if this is below a prescribed tolerance. For frequency separation, we need to check that the minimum distance between the candidate frequencies in Ωexceeds 4/m, which is required for unique recovery in ANM. Finally, the dual polynomial has to satisfy |Q(f)| ≈1 for all f ∈Ωand |Q(f)| < 1, otherwise, within numerical tolerances. Computing the dual polynomial is straightforward once we have a solution to the dual problem. Solving the dual problem. The dual problem in Eq. (5) is infinite-dimensional and must be discretized to make it tractable. We therefore sample the frequency domain finely, transforming the problem into a secondorder cone program. In addition, since the measurement vector y is real-valued (which is always the case in quantum TEs), the dual problem can be further simplified to max q∈Rm ⟨y, q⟩ s.t. -1 ≤⟨a(f), q⟩≤1, ∀f ∈F, (6) Find frequencies and amplitudes such that Certify solution Gap optimality certificate FIG. 3. Schematic illustration of the proposed certification pipeline. The process verifies frequency and amplitude estimates obtained from any algorithm using the dual formulation of ANM. where F is a finely discretized set of frequencies in [0, 1 2], since the frequency spectrum is symmetric. Finally, although discretization introduces a small approximation error, it remains negligible when the frequency grid is sufficiently fine. IV. EXPERIMENTS This section evaluates the proposed ANM certification methodology using several existing algorithms. The goal is to demonstrate that, when a certificate of optimality is obtained, the correct model can be recovered, enabling reliable forecasting of the TE. Note that successful certification also depends on the true model being sparse and having well-separated frequencies-conditions that are typically unknown in practice. A. Setup Dataset. We consider quantum TEs generated by 15 spin-chain Hamiltonians (4 TFIM and 11 Heisenberg) with 8-20 qubits and periodic boundary conditions [1726] (see details in Appendix B). For all TEs, we consider five initial states-Néel, dimerized, paramagnetic, ferromagnetic, and the zero state-and measure six observables (X, Y , Z, XX, Y Y , and ZZ); see Appendix B. This results in a total of 5850 TEs. The TEs are generated using Qutip [27] for 40 time units with dt = 0.2, yielding m = 200 measurements per TE. A time unit is t·J, where J is the coupling strength (interaction energy) in a Hamiltonian. Spectral estimation algorithms. We use four algorithms to compute the frequencies and amplitudes: Prony, OMP, FISTA, and ESPRIT. Prony [11] is a method that can provably recover the true frequencies and amplitudes under ideal conditions but is highly sensitive to noise. OMP [28, 29] is a widely used heuristic in signal processing for sparse signal recovery. FISTA [30] is a fast gradient method for solving Lasso problems, and ESPRIT [31, 32] is a well-known algorithm in spectrum estimation, also applied in quantum TE forecasting [5]. The algorithms are implemented with default or minimal 4 Exact TE TE with shot noise ǫ PRO OMP FIS ESP PRO OMP FIS ESP 0.01 83.4 75.1 68.7 70.8 50.2 20.2 10.3 21.3 0.05 97.4 96.4 95.8 89.5 58.3 69.9 74.6 53.2 0.10 98.7 99.2 99.7 98.8 82.2 95.1 95.4 83.1 TABLE I. SCs rate for different algorithms and forecasting tolerances (ǫ). Noisy TEs use 1000 shots. Exact TE TE with shot noise ǫ PRO OMP FIS ESP PRO OMP FIS ESP 0.01 15.8 11.2 2.9 13.8 6.7 7.1 1.3 6.6 0.05 18.4 14.3 4.1 17.5 7.7 24.5 9.0 16.4 0.10 18.6 14.7 4.3 19.3 10.9 33.4 11.6 25.6 TABLE II. Absolute SCs rate for different algorithms and forecasting tolerances (ǫ). The Absolute SC rate is the percentage of SCs over the total number of runs. Noisy TEs use 1000 shots. tuning settings, as our focus is on evaluating the certification methodology. A brief description of each algorithm and its implementation is provided in Appendix A. Forecasting procedure. Each algorithm is applied to the first k measurements, where k ∈{10, . . ., 160}, and forecasting is evaluated on the remaining 200 -k measurements, as defined in Eq. (2). Certification. For each algorithm run, ANM certification is performed (150 times per algorithm and TE). This is done by solving Eq. (6) on a frequency grid of 1000 uniformly spaced points in [0, 1 2] [33], augmented with the candidate frequencies estimated by the algorithmto ensure that the dual polynomial is evaluated at the exact/candidate frequencies. To account for numerical inaccuracies, empirically chosen tolerances are applied. Specifically, for exact TEs, the dual polynomial must satisfy |Q(f)| ≥0.98 at all candidate frequencies, and the duality gap must be smaller than 0.05. For TEs with shot noise, the thresholds are |Q(f)| ≥0.95 and a duality gap smaller than 0.5. In addition, we only consider amplitudes with magnitude larger than 0.025 for exact TEs and 0.05 for TEs with shot noise. B. Results We evaluate forecasting performance using tolerances ǫ ∈{0.01, 0.05, 0.1} (see Sec. II), for both exact TEs and TEs with shot noise. We consider the output of a spectral algorithm (i.e., frequencies and amplitudes) certified if it meets the criteria in Sec. III B. A successful certification (SC) occurs when a certified run also satisfies the forecasting error in Eq. (2) for the remaining measurements, which range from 190 (k = 10) down to 40 (k = 160). We define the SC rate as the percentage of SCs among 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0.3 Fraction of certified TEs Error Exact TE Prony OMP FISTA ESPRIT Error 1000-shot noise FIG. 4. Distribution of the forecasting error upon the first certification, with exact and noisy TEs. 0 0.2 0.4 0.6 0.8 1 0 0.05 0.1 0.15 0.2 0.25 0 0.05 0.1 0.15 0.2 0.25 0.3 Fraction of certified TEs Error Exact TE Prony OMP FISTA ESPRIT Error 1000-shot noise FIG. 5. Distribution of the forecasting error upon the first certification for certificates issued for 30 or more measurements, with exact and noisy TEs. all certified runs (successful or not). Average performance. Table I reports the SC rate for each algorithm and tolerance ǫ. For exact TEs, all algorithms achieve high SC rates, exceeding 98% for ǫ ≤ 0.1. When shot noise is introduced (1000 shots), SC rates decrease, particularly for ǫ ≤0.01. OMP and FISTA generally outperform Prony and ESPRIT on noisy TEs; however, FISTA certifies a smaller number of runs compared to the other algorithms. Table II shows the percentage of SCs over all algorithm runs (877,500 runs per algorithm: 150 runs per TE across 5850 TEs). Only a fraction of the runs yield a certificate, with Prony and ESPRIT achieving SC around 18% of all the runs for ǫ ≤0.05. Interestingly, the absolute SC rate for OMP and ESPRIT (Table II) increases with noisy TEs. This is likely due to the looser certification tolerances applied under noise, which increase the total number of certified runs. Performance on first certification. Fig. 4 shows the distribution of forecasting errors at the first certification (the first k ∈{10, . . ., 160} that meets the criteria in Sec. III B). Overall, the algorithms exhibit similar behavior. For exact TEs, Prony and OMP achieve errors below 0.05 and 0.1 in approximately 50% and 75% of cases, respectively. For noisy TEs, the proportion of forecasts with errors below 0.05 drops sharply, and both Prony and OMP achieve errors below 0.1 in only about 50% of cases. These results contrast with the high SC rates in Ta5 ble I, possibly due to insufficient measurements used during spectral estimation or certification. To investigate this, Fig. 5 shows forecasting errors for first certifications issued only after 30 or more measurements. Performance improves noticeably: for exact TEs, Prony achieves errors below 0.05 and 0.1 in 83.4% and 93.4% of cases, respectively, while OMP achieves the same thresholds in 76% and 100% of cases. For noisy TEs, all algorithms reduce the proportion of forecasts with errors exceeding 0.1 (cf. Fig. 4 and Fig. 5), with OMP achieving errors below 0.1 and 0.15 in 75% and 97% of cases, respectively. These results indicate that a sufficiently large number of measurements is crucial for reliable certification; while the minimum is unknown, monitoring stability, as suggested in [5, Sec. III-C], or setting a minimum measurement threshold (e.g., k ≥30) can significantly enhance performance. V. CONCLUSIONS We introduced ANM as a framework to certify the spectral models produced by data-driven forecasting algorithms. Our numerical experiments show that, with carefully chosen tolerance thresholds, ANM can successfully certify both exact and noisy quantum TEs. A key limitation is that certification is guaranteed only when the underlying dynamics are sparse and the Bohr frequencies are well-separated. Nevertheless, our results indicate that ANM can still provide reliable certificates in realistic scenarios. A promising direction for future work is to characterize which time evolutions-defined by specific Hamiltonians, initial states, and observablessatisfy the framework conditions, thereby pinpointing the regimes where reliable forecasting is theoretically guaranteed. VI. ACKNOWLEDGMENTS SZ and DM are grateful to Sergey Bravyi, Ewout van den Berg, and Kristan Temme for stimulating discussions. We all wish to thank the members of the Quantum Information Lab at Tecnun. 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[35] M. ApS, MOSEK Optimization Software (2025), version 11.0. 7 Appendix A: Algorithms 1. Algorithm for finding frequencies and amplitudes a. Prony's method Prony's method is implemented as described in [11]. The method requires that the number of measurements m satisfies ≥2|Ω|, where |Ω| denotes the number of frequencies. However, |Ω| is typically unknown. To address this, we execute Prony's method for a range of candidate values of |Ω|. Specifically, given m measurements, we apply Prony's method using the first 2k measurements for k = 1, . . . , ⌊m/2⌋. For each k, we compute the error ∥y -x∥2, where x, y ∈Rm denote the reconstructed and measured signals, respectively. The algorithm selects the model (i.e., the set of frequencies and amplitudes defining x) that yields the smallest error. b. OMP Orthogonal Matching Pursuit (OMP) is implemented as described in [29, Algorithm 1], using atoms defined by a(f, φ) = (cos(2πfτ1 + φ), . . . , cos(2πfτm + φ)), (A1) where f ∈F := {0, 1 2L, . . . , L 2L} and Φ := {0, 2π P , . . . , 2π(P -1) P }, with L = 10000 and P = 10. We use the atoms in Eq. (A1) instead of those in Eq. (3) because y is real-valued. Consequently, its frequency spectrum is symmetric, and the corresponding amplitudes occur in complex-conjugate pairs. We make the algorithm terminate after 250 iterations or when the recovered signal, x, satisfies ∥y -x∥2 ≤ǫ for ǫ = 10-2. The algorithm returns the frequencies, phases, and amplitude magnitudes of the selected atoms. c. FISTA The Fast Iterative Shrinkage-Thresholding Algorithm (FISTA) is implemented as described in [30] to solve the optimization problem minimize x∈RLP 1 2∥Ax -y∥2 2 + λ∥x∥1, (A2) where y ∈Rm denotes the vector of measurements and A ∈Rm×LP is a matrix whose columns correspond to the atoms defined in Eq. (A1), with L = 1000 and P = 4. The algorithm outputs the frequencies, phases, and amplitude magnitudes associated with the selected atoms (i.e., the active columns of A). To account for numerical issues, we only consider an atom is selected if the associated amplitude magnitude is larger than 10-4. d. ESPRIT We have adjusted the method [31, 32] for a signal (1), generated by a Hamiltonian evolution, as follows. For N samples of a signal y = Pr k=1 ak z n k and for a window length d = m + 1 with 2 ≤d ≤N -1, we form delay Hankel matrices Y0, Y1 ∈Cd×L (L = N -d) that are shifted by one sample. Here m is a parameter that controls the size of a subspace, chosen such that N ≥m + 2. We then compute a rank-r singular value decomposition of Y0 = U Σ V ∗ and project the one-step shift matrix Y1 onto the r-dimensional signal subspace spanned by columns of U by forming the reduced operator Φ = U ∗Y1 V Σ-1 ∈Cr×r. (A3) Using Y0 = UΣV ∗and Y1 = V ΛW, we have U ∗Y1V Σ-1 ≈U ∗(V ΛW)V Σ-1, and since U spans the columns of V , Φ is similar to Λ on the signal subspace. Thus eig(Φ) = {z1, . . . , zr}. (A4) Having the poles {zk}, we form the Vandermonde matrix on the full sample range, V nk = z n k where n = 0, . . . , N -1, and find the amplitudes from least squares: mina∈Cr y -V a 2 2. The frequencies follow from ωk = -arg(zk). 8 Initial State Definition Zero |0⟩⊗N Néel |01⟩⊗N/2 Dimerized (|01⟩-|10⟩)⊗N/2 Paramagnetic (|0⟩+ |1⟩)⊗N Ferromagnetic |1⟩⊗N TABLE III. Initial states and their definitions (normalization constants are omitted for clarity of presentation). Additionally, we assumed unitary dynamics and, thus, project zk to a unit circle, zk 7→ei arg zk. The subspace parameter m was chosen as max(2, ⌊N 3 ⌋) providing the requirement N ≥m + 2. The number of samples N varied in simulations as N ≡k ∈{1-, . . ., 160}. When the SVD procedure failed, we considered that as a negative TE instance. 2. Algorithm to solve the dual problem in Eq. (5) We solve the problem in Eq. (5) with Adam [34]. The objective function is f(q) = ⟨y, q⟩+ λ L X j=1 max{0, |(Aq)j| -1}2, (A5) where y ∈Rm is the vector of measurements, and A ∈RL×m a matrix with rows equal to the atoms in Eq. (3). We use L = 1000 and λ = 15. We set the learning rate equal to 0.1 and run the algorithm for 500 iterations. Appendix B: Hamiltonians and Initial States We consider one-dimensional spin- 1 2 chains. For convenience, we write all Hamiltonians in the Pauli convention, i.e. with Pauli matrices X, Y , and Z acting on site k: HXYZ = Jx X k XkXk+1 + Jy X k YkYk+1 + Jz X k ZkZk+1 + hx X k Xk + hy X k Yk + hz X k Zk. When restricted to Ising-type couplings and a transverse field, we use: HTFIM = Jz X k ZkZk+1 + hx X k Xk. Here the index k runs over lattice sites (boundary conditions are specified elsewhere, mostly as \"periodic\"), and the couplings {Jα} and fields {hα} are given in units where energy scales are O(1). Sometimes, spin Hamiltonians are written with spin operators Sα k rather than Pauli matrices σα k . Since Sα k = 1 2 σα k , our parameters map to the S-operator convention as: J(S) α = 4 Jα, h(S) α = 2 hα, and respectively, Jα = J(S) α /4, hα = h(S) α /2. Signs are unaffected by this rescaling. Our study focuses on spin- 1 2 models in the above Pauli normalization for implementation simplicity. The parameter choices we explore were inspired by Hamiltonians used in diverse physical simulations (see Table IV). We note, however, that several of those source models were originally formulated for spin-1 or higher; when importing their coefficients into our convention one should apply the normalization map above. Spin- 1 2 and spin-1 chains can exhibit qualitatively different physics. Because our primary objective here is predictive performance rather than a faithful comparison of phase structures, we do not distinguish between spin- 1 2 and spin-1 models in the experiments. Results should therefore be interpreted with this caveat in mind. The initial states used in all simulations (Table III) are standard in the literature and admit shallow state-preparation circuits, making them both representative and practical for our benchmarks. 9 Parameters Description Jx = 0, Jy = 0, Jz = -2, hx = 1, hy = 0, hz = 0 Ising ferromagnet. Jx = 0, Jy = 0, Jz = 2, hx = 1, hy = 0, hz = 0 Ising antiferromagnet. Jx = 0, Jy = 0, Jz = 1, hx = 2, hy = 0, hz = 0 Ising paramagnet. Jx = 0, Jy = 0, Jz = 1, hx = 1, hy = 0, hz = 0 Ising critical model. Jx = Jy = Jz = 1, hx = hy = 0, hz = 0.2 Isotropic Haldane, tiny longitudinal field, [17]. Jx = Jy = Jz = 1, hx = hy = 0, hz = 0.4105 Antiferromagnetic quantum chain at the lower critical field, [18]. Jx = Jy = Jz = 1, hx = hy = 0, hz = 0.6 Field-induced Tomonaga-Luttinger liquid, [19]. Jx = 0.7, Jy = 0.7, Jz = 1.3, hx = 0, hy = 0, hz = 0.1 Easy-axis XXZ with weak field, [20]. Jx = 1.0, Jy = 1.0, Jz = 0.6, hx = 0.2, hy = 0, hz = 0 Easy-plane XXZ, transverse probe, [21], Jx = Jy = Jz = 1, hx = 0.3, hy = 0, hz = 0.3 Tilted field, symmetry breaking, [22]. Jx = 1.2, Jy = 0.8, Jz = 1, hx = 0.9, hy = 0, hz = 0 Anisotropic XYZ + transverse field, [23]. Jx = Jy = Jz = 1, hx = 0, hy = 0, hz = 1 Isotropic AFM, intermediate longitudinal field, [24]. Jx = Jy = Jz = -1, hx = 0, hy = 0, hz = 0.1 Isotropic ferromagnetic Heisenberg chain, tiny field, [25]. Jx = 1, Jy = 1, Jz = -0.7, hx = 0.15, hy = 0, hz = 0.15 Mixed-sign anisotropy, easy-plane exchange but ferromagmetic Jz, [19]. Jx = Jy = Jz = 1, hx = 1.5, hy = 0, hz = 0 Isotropic AFM with strong transverse field (near product paramagnet) Jz, [26]. TABLE IV. Hamiltonians selected for TE. 10 Appendix C: Additional experiments 1. Error growth after first certification The following figures illustrate the evolution of the forecasting error after the first certification, measured on the samples predicted beyond that point. For example, if the first certification occurs at sample 50 in one TE and at sample 100 in another, the first forecast samples correspond to 51 and 101, respectively. The average error is indicated by a thick line, and the standard deviation is shown as a shaded area. The figures display the average results for all algorithms and TEs, with the experimental setup described in Sec. IV. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 No shot noise Error 250 shot noise 500 shot noise Error Forecast measurements 1000 shot noise Forecast measurements FIG. 6. Average per-sample forecasting error after the first certification for exact TEs and three shot noise levels (250, 500, and 1000 shots). The error is averaged over all algorithms. 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 20 40 60 80 100 120 140 0 20 40 60 80 100 120 140 No shot noise Error 250 shot noise 500 shot noise Error Forecast measurements 1000 shot noise Forecast measurements FIG. 7. Average per-sample forecasting error after the first certification after 30 or more measurements, for exact TEs and three shot noise levels (250, 500, and 1000 shots). The error is averaged over all algorithms. 11 2. Illustrative examples -1 -0.5 0 0.5 1 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 Measurements Forecasting Measurement |Q(f)| f ∈Ω Bohr frequency FIG. 8. Forecasting with OMP: 18 qubits Anisotropic XYZ + transverse field Hamiltonian [23], dimerized initial state, and O = X9X10. The red vertical line marks the training interval [0 . . . 100), where we fit the sparse model; the remaining data [100 . . . 200) are reserved for out-of-sample verification. -1 -0.5 0 0.5 1 0 20 40 60 80 100 120 140 160 180 200 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 Measurements Forecasting Measurement |Q(f)| f ∈Ω Bohr frequency FIG. 9. Forecasting with OMP: 20 qubit easy-plane XXZ transverse probe Hamiltonian [21], paramagnetic initial state, and O = Y10Y11. The red vertical line marks the training interval [0 . . . 100), where we fit the sparse model; the remaining data [100 . . . 200) are reserved for out-of-sample verification. 12 Appendix D: SDP Running Times with MOSEK Fig. 10 shows the time it takes MOSEK [35] (a commercial SDP solver) to solve the problem in Eq. (4). The SDP is run on an Intel i7-14700 CPU with 32 GB of memory. 0 50 100 150 200 250 0 50 100 150 200 250 300 350 400 Time (s) Number of measurements FIG. 10. Time to solve the SDP in Eq. (4) with MOSEK [35] for different number of measurements."}}},{"rowIdx":84,"cells":{"id":{"kind":"string","value":"2510.14898"},"raw":{"kind":"string","value":"CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN\nGENERAL ACTION SPACES\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nAbstract. We prove the stability and global convergence of a coupled actor-critic gradient flow for\ninfinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous state and\naction space with linear function approximation under Q-function realisability. We consider a version\nof the actor critic gradient flow where the critic is updated using temporal difference (TD) learning while\nthe policy is updated using a policy mirror descent method on a separate timescale. We demonstrate\nstability and exponential convergence of the actor critic flow to the optimal policy. Finally, we address\nthe interplay of the timescale separation and entropy regularisation and its effect on stability and\nconvergence.\n1. Introduction\nIn reinforcement learning (RL) an agent aims to learn an optimal policy that maximizes the expected\ncumulative reward through repeated interactions with its environment. Such methods typically involve\ntwo key components: policy evaluation and policy improvement. During policy evaluation, the advantage\nfunction corresponding to a policy, or its function approximation, is updated using state, action and\nreward data generated under this policy. Policy improvement then uses this approximate advantage\nfunction to update the policy, most commonly through some policy gradient method. Algorithms that\nexplicitly combine these two components are known as actor–critic (AC) methods [13], where the actor\ncorresponds to policy improvement and the critic to policy evaluation.\nThere are many policy gradient methods to choose from.\nIn the last decade trust region policy\noptimization (TRPO) methods [20] and methods inspired by these like PPO [21] have become increasingly\nwell-established due to their impressive empirical performance. Largely, this is because they alleviate the\ndifficulty in choosing appropriate step sizes for the policy gradient updates: for vanilla policy gradient\neven a small change in the parameter may result in large change in the policy, leading to instability,\nbut TRPO prevents this by explicitly ensuring the KL divergence between successive updates is smaller\nthan some tolerance. Mirror descent replaces the TRPO’s hard constraint with a penalty leading to a\nfirst order method which is also ameanable to analysis. Indeed, at least for direct parametrisation, it is\nknown to converge with sub-linear and even linear rate for entropy regularized problems (depending on\nexact assumptions) [9, 14, 11].\nDue to the favourable analytical properties of mirror descent, in this paper we consider a version of\nthe actor critic gradient flow where the policy is updated using a policy mirror descent method while the\ncritic is updated using temporal difference (TD) on a separate timescale.\nEntropy-regularised MDPs are widely used in practice since the entropic regularizer leads to a number\nof desirable properties: it has a natural interpretation as something that drives exploration, it ensures\nthat there is a unique optimal policy and it can accelerate convergence of mirror descent [11], as well as\nclassical policy gradient [18]. However, analysing the stability and convergence of actor–critic methods\nin this entropy-regularized setting with general state and action spaces remains highly non-trivial due to\nlack of a priori bounds on the value functions.\nTo address the actor critic methods for entropy regularised MDPs in general action spaces, a careful\ntreatment of tools from two timescale analysis, convex analysis over both Euclidean spaces and measure\nspaces must be deployed.\nIn this paper, we address precisely this challenge. We study the stability and convergence of a widely\nused actor–critic algorithm in which the critic is updated using Temporal Difference (TD) learning [22],\nand the policy is updated through Policy Mirror Descent [9]. Our analysis employs a two-timescale\nupdate scheme [3], where both the actor and critic are updated at each iteration with the critic updated\non a faster timescale.\nKeywords: Reinforcement learning, Actor-Critic method, Entropy regularisation, Approximate gradient flow, Non-\nconvex optimization, Global convergence, Function approximation.\n1\narXiv:2510.14898v1 [math.OC] 16 Oct 2025\n\n2\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\n1.1. Related works. We focus on the subset of RL literature that address the convergence of coupled\nactor-critic algorithms. In the unregularised setting, actor–critic methods have been studied extensively.\nThe first convergence results in the two-timescale regime established asymptotic convergence in the\ncontinuous-time limit of coupled updates ([3, 13]). Most modern research employs linear function ap-\nproximation for the critic, where linear convergence rates have been obtained under various assumptions\non the step-sizes of the actor and critic ([1, 24, 8]).\nMost closely related to our work is [25], which considers the same two-timescale actor–critic scheme\nin the continuous-time limit for unregularised MDPs, with an overparameterized neural network used\nfor the critic. However, convergence to the optimal policy was only achieved up to a neighbourhood of\na scaling factor, and a restarting mechanism was required to ensure the stability of the dynamics.\nIn the entropy-regularised setting, [5, 6] address the convergence of a natural actor critic algorithm.\nHowever, the convergence and stability of these results rely on the finite cardinality of the action space\nin presence of entropy regularisation.\n1.2. Our Contribution. Under linear Q-realisability assumption, we address the following question:\n“Are actor–critic methods for entropy-regularized MDPs in general action spaces stable and convergent,\nand if so, at what rate?”\nOur main contributions are as follows:\n• We study a common variant of actor–critic where the critic is updated using temporal difference\n(TD) learning and the policy is updated using mirror descent. Similarly to [13, 25], we analyse\nthe coupled updates in the continuous-time limit, resulting in a dynamical system where the\ncritic flow is captured by a semi-gradient flow and the actor flow corresponds to an approximate\nFisher–Rao gradient flow over the space of probability kernels.\n• By combining convex analysis over the space of probability measures with classical Euclidean\nconvex analysis, we develop a Lyapunov-based stability framework that captures the interplay\nbetween entropy regularisation and timescale separation, and establish stability of the resulting\ndynamics.\n• We prove convergence of the actor–critic dynamics for entropy-regularized MDPs with infinite\naction spaces.\n1.3. Notation. Let (E, d) denote a Polish space (i.e., a complete separable metric space). We always\nequip a Polish space with its Borel sigma-field B(E). Denote by Bb(E) the space of bounded measurable\nfunctions f : E →R endowed with the supremum norm |f|Bb(E) = supx∈E |f(x)|. Denote by M(E) the\nBanach space of finite signed measures µ on E endowed with the total variation norm |µ|M(E) = |µ|(E),\nwhere |µ| is the total variation measure. Recall that if µ = f dρ, where ρ ∈M+(E) is a nonnegative\nmeasure and f ∈L1(E, ρ), then |µ|M(E) = |f|L1(E,ρ). Denote by P(E) ⊂M(E) the set of probability\nmeasures on E. Moreover, we denote the Euclidean norm on RN by | · | with inner product ⟨·, ·⟩. Given\nsome A, B ∈RN×N, we denote by λmin(A) the minimum eigenvalue of A and denote A ⪰B if and only\nif A −B is positive semidefinite. Moreover, we denote by |A|op the operator norm of A induced by the\nEuclidean norm, |A|op := sup|x|̸=0\n|Ax|\n|x| .\n1.4. Entropy Regularised Markov Decision Processes. Consider an infinite horizon Markov De-\ncision Process (S, A, P, c, γ), where the state space S and action space A are Polish, P ∈P(S|S × A) is\nthe state transition probability kernel, c is a bounded cost function and γ ∈(0, 1) is a discount factor.\nLet µ ∈P(A) denote a reference probability measure and τ > 0 denote a regularisation parameter.\nTo ease notation, for each π ∈P(A|S) we define the kernels Pπ(ds′|s) :=\nR\nA P(ds′|s, a)π(da|s) and\nP π(ds′, da′|s, a) := P(ds′|s, a)π(da′|s′). Denoting Eπ\ns = Eπ\nδs where δs ∈P(S) denotes the Dirac measure\nat s ∈S, for each stochastic policy π ∈P(A|S) and s ∈S, define the regularised value function by\n(1)\nV π\nτ (s) = Eπ\ns\n\" ∞\nX\nn=0\nγn\u0010\nc(sn, an) + τ KL(π(·|sn)|µ)\n\u0011#\n∈R ∪{∞} ,\nwhere KL(π(·|s)|µ) is the Kullback-Leibler (KL) divergence of π(·|s) with respect to µ, define as KL(π(·|s)|µ) :=\nR\nA ln dπ\ndµ(a|s)π(da|s) if π(·|s) is absolutely continuous with respect to µ, and infinity otherwise.\nFor each π ∈P(A|S), we define the state-action value function Qπ\nτ ∈Bb(S × A) by\n(2)\nQπ\nτ (s, a) = c(s, a) + γ\nZ\nS\nV π\nτ (s′)P(ds′|s, a) .\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n3\nBy the Dynamic Programming Principle, Qπ\nτ : S × A →R is the unique fixed point of the Bellman\noperator Tπ : Bb(S × A) →Bb(S × A), which for any f ∈Bb(S × A) is defined as\n(3)\nTπf(s, a) = c(s, a) + γ\nZ\nS×A\nf(s′, a′)P π(ds′, da′|s, a) + τγ\nZ\nS\nKL(π(·|s′)|µ)P(ds′|s, a).\nThe state-occupancy kernel dπ ∈P(S|S) is defined by\n(4)\ndπ(ds′|s) = (1 −γ)\n∞\nX\nn=0\nγnP n\nπ (ds′|s) ,\nwhere P n\nπ is the n-times product of the kernel Pπ with P 0\nπ(ds′|s) := δs(ds′). Moreover, for each π ∈\nP(A|S) and (s, a) ∈S × A, we define the state-action occupancy kernel as\n(5)\ndπ(ds, da|s, a) = (1 −γ)\n∞\nX\nn=0\nγn(P π)n(ds, da|s, a)\nwhere (P π)n is the n-times product of the kernel P π with (P π)0(ds′, da′|s, a) := δ(s,a)(ds′, da′). Given\nsome initial state-action distribution β ∈P(S × A) with initial state distribution given by ρ(ds) =\nR\nA β(da, ds), we define the state-occupancy and state-action occupancy measures as\n(6)\ndπ\nρ(ds) =\nZ\nS\ndπ(ds|s′)ρ(ds′),\ndπ\nβ(ds, da) =\nZ\nS×A\ndπ(ds, da|s′, a′)β(da′, ds′).\nNote that for all E ∈B(S × A), by defining the linear operator Jπ : P(S × A) →P(S × A) as\n(7)\nJπβ(E) =\nZ\nS×A\nP π(E|s′, a′)β(ds′, da′),\nit directly holds that\n(8)\ndπ\nβ(da, ds) = (1 −γ)\n∞\nX\nn=0\nγnJn\nπ β(da, ds),\nwith Jn\nπ the n-fold product of the operator Jπ with J0\nπ = I, the identity operator on P(S × A). By\nchoosing β = ρ ⊗π, we retrieve the classical state-action occupancy measure dπ\nβ = dπ\nρπ.\nFor a given initial distribution ρ ∈P(S), the optimal value function is defined as\n(9)\nV ∗\nτ (ρ) =\nmin\nπ∈P(A|S) V π\nτ (ρ),\nwith V π\nτ (ρ) :=\nZ\nS\nV π\nτ (s)ρ(ds)\nand we refer to π∗∈P(A|S) as the optimal policy if V ∗\nτ (ρ) = V π∗\nτ\n(ρ). Due to [11, Theorem B.1, Lemma\nB.2] we have the following dynamical programming principle for entropy regularised MDPs.\nTheorem 1.1 (Dynamical Programming Principle). Let τ > 0. The optimal value function V ∗\nτ is the\nunique bounded solution of the following Bellman equation:\nV ∗\nτ (s) = −τ ln\nZ\nA\nexp\n\u0012\n−1\nτ Q∗\nτ(s, a)\n\u0013\nµ(da),\nwhere Q∗\nτ ∈Bb(S × A) is defined by\nQ∗\nτ(s, a) = c(s, a) + γ\nZ\nS\nV ∗\nτ (s′)P(ds′|s, a) ,\n∀(s, a) ∈S × A .\nMoreover, there is an optimal policy π∗\nτ ∈P(A|S) given by\nπ∗\nτ(da|s) = exp\n\u0012\n−1\nτ (Q∗\nτ(s, a) −V ∗\nτ (s))\n\u0013\nµ(da) ,\n∀s ∈S.\nFinally, the value function V π\nτ is the unique bounded solution of the following Bellman equation for all\ns ∈S\nV π\nτ (s) =\nZ\nA\n\u0012\nQπ\nτ (s, a) + τ ln dπ\ndµ(a, s)\n\u0013\nπ(da|s) .\nTheorem 1.1 suggests that, without loss of generality, it suffices to minimise (9) over the class of\npolicies that are equivalent to the reference measure µ.\nDefinition 1.1 (Admissible Policies). Let Πµ denote the class of policies for which there exists f ∈\nBb(S × A) with\nπ(da|s) =\nexp(f(s, a))\nR\nA exp(f(s, a))µ(da)µ(da).\n\n4\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nThe performance difference lemma, first introduced for tabular unregularised MDPs, has become\nfundamental in the analysis of MDPs as it acts a substitute for the strong convexity of the π 7→V π\nτ if\nthe state-occupancy measure dπ\nρ is ignored (e.g [10], [25], [9]). By virtue of [11], we have the following\nperformance difference for entropy regularised MDPs in Polish state and action spaces.\nLemma 1.1 (Performance difference). For all ρ ∈P(S) and π, π′ ∈Πµ,\nV π\nτ (ρ) −V π′\nτ (ρ)\n=\n1\n1 −γ\nZ\nS\n\u0014 Z\nA\n\u0012\nQπ′\nτ (s, a) + τ ln dπ′\ndµ (a, s)\n\u0013\n(π −π′)(da|s) + τ KL(π(·|s)|π′(·|s))\n\u0015\ndπ\nρ(ds) .\n2. Mirror-Descent and the Fisher-Rao Gradient flow\nDefining the soft advantage function as\nAπ\nτ (s, a) := Qπ\nτ (s, a) + τ ln dπ\ndµ(s, a) −V π\nτ (s),\nthen for some λ > 0 and π0 ∈Πµ, the Policy Mirror Descent update rule reads as\nπn+1(·|s) = arg min\nm∈P(A)\n\u0014Z\nA\nAπn\nτ (s, a)(m(da) −πn(da|s)) + 1\nλ KL(m|πn(·|s))\n\u0015\n(10)\n= arg min\nm∈P(A)\n\u0014Z\nA\n\u0012\nQπn\nτ (s, a) + τ ln dπn\ndµ (s, a)\n\u0013\n(m(da) −πn(da|s)) + 1\nλ KL(m|πn(·|s))\n\u0015\n.\n(11)\n[7] shows that the pointwise optimisation is achieved by\n(12)\ndπn+1\ndπn (a, s) =\nexp\n\u0000−λAπn\nτ (s, a)\n\u0001\nR\nA exp (−λAπn\nτ (s, a)) πn(da|s).\nObserve that for any π ∈P(A|S), it holds that\nR\nA Aπ\nτ (s, a)π(da|s) = 0. Hence taking the logarithm of\n(12) we have\nln dπn+1\ndµ\n(s, a) −ln dπn\ndµ (s, a) = −λAπn\nτ (s, a) −ln\nZ\nA\ne−λAπn\nτ\n(s,a)πn(da|s).\nInterpolating in the time variable and letting λ →0 we retrieve the Fisher-Rao gradient flow for the\npolicies\n(13)\n∂t ln dπt\ndµ (s, a) = −\n\u0012\nAπt\nτ (s, a) −\nZ\nA\nAπt\nτ (s, a)πt(da|s)\n\u0013\n= −Aπt\nτ (s, a).\nNote that the soft advantage formally corresponds to the functional derivative of the value function with\nrespect to the policy πn and thus (13) can be seen as a gradient flow of the value function over the space\nof kernels P(A|S) (see [11] for a detailed description of the functional derivative).\nIn the case where the advantage function is fully accessible for all t ≥0, [11][Theorem 2.8] shows that\nthe entropy regularisation in the value function induces an exponential convergence to the optimal policy.\nIn the following section we define the approximate Fisher Rao dynamics arising from approximating the\nadvantage for all t ≥0.\n3. Actor Critic Methods\nGiven some feature mapping ϕ : S × A →RN, we parametrise the state-action value function as\nQ(s, a; θ) := ⟨θ, ϕ(s, a)⟩. Moreover, we denote the approximate soft Advantage function as\n(14)\nA(s, a; θ) = Q(s, a; θ) + τ ln dπ\ndµ(s, a) −\nZ\nA\n\u0012\nQ(s, a; θ) + τ ln dπ\ndµ(s, a)\n\u0013\nπ(da|s).\nThe Mean Squared Bellman Error (MSBE) is defined as\n(15)\nMSBE(θ, π) = 1\n2\nZ\nS×A\n(Q(s, a; θ) −TπQ(s, a; θ))2dπ\nβ(da, ds)\nwhere dπ\nβ ∈P(S × A) is the state-action occupancy measure defined in (6). Given that β ∈P(S × A)\nhas full support, by (3) it holds that MSBE(θ, π) = 0 if and only if Q(s, a; θ) = Qπ\nτ (s, a) for all s ∈S\nand a ∈A. Hence one approach to implementing the Policy Mirror Descent updates is to calculate the\noptimal parameters for Q(s, a; θ) by minimising the MSBE at each policy mirror descent iteration\n(16)\nθn+1 = arg min\nθ∈RN\nMSBE(θ, πn),\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n5\n(17)\ndπn+1\ndπn (a, s) =\nexp\n\u0000−λA(s, a; θn+1)\n\u0001\nR\nA exp (−λA(s, a; θn+1)) πn(da|s).\nTo avoid fully solving the optimisation in (16) for each policy update, one can update the critic using\na semi-gradient descent on a different timescale to the policy update. Let hn, λn > 0 be the step-sizes of\nthe critic and actor respectively at iteration n ≥0. Let the semi-gradient g : RN × P(A|S) →RN of the\nMSBE with respect to θ be\n(18)\ng(θ, π) :=\nZ\nS×A\n(Q(s, a; θ) −TπQ(s, a; θ))ϕ(s, a)dπ\nβ(da, ds).\nThe two-timescale actor-critic Mirror Descent scheme reads as\n(19)\nθn+1 = θn −hng(θn, πn),\ndπn+1\ndπn (a, s) =\nexp\n\u0000−λnA(s, a; θn+1)\n\u0001\nR\nA exp (−λnA(s, a; θn+1)) πn(da|s).\n(20)\nwhere timescale separation ηn := hn\nλn > 1 ensures that the critic is updated on a much faster timescale\nthan the policy to improve the local estimation of the policy updates. As pointed out in [25], even with\nthe KL penalty in (10) the critic may still be far away from the true state-action value function, resulting\nin unstable updates.\n4. Dynamics\nIn this paper, we study the stability and convergence of the two-timescale actor-critic Mirror Descent\nscheme in the continuous-time limit.\nLet Qt(s, a) := Q(s, a; θt) and At(s, a) := A(s, a; θt).\nLet η :\n[0, ∞) →[1, ∞) be a non-decreasing function representing the timescale separation, then for some θ0 ∈\nRN and π0 ∈Πµ we have the following coupled dynamics\n(21)\ndθt\ndt = −ηtg(θt, πt)\n∂tπt(da|s) = −At(s, a)πt(da|s)\n(22)\nwhere g : RN × P(A|S) is the semi-gradient of the MSBE defined in (18).\nWe refer to (22) as the\nApproximate Fisher Rao Gradient flow.\nWe perform our analysis under the following assumptions.\nAssumption 4.1 (Qπ\nτ -realisability). For all π ∈Πµ and (s, a) ∈S × A, there exists θπ ∈RN such that\nQπ(s, a) = ⟨θπ, ϕ(s, a)⟩.\nA simple example of when this holds is in the tabular case, where one can choose ϕ to be a one-hot\nencoding of the state-action space. Moreover, all linear MDPs are Qπ-realisable. In a linear MDP there\nexists exists ϕ : S × A →RN, w ∈RN and a sequence {ψi}N\ni=1 with ψi ∈M(S) such that for all\n(s, a) ∈S × A,\nc(s, a) = ⟨w, ϕ(s, a)⟩,\nP(ds′ | s, a) =\nN\nX\ni=1\nϕi(s, a)ψi(ds′).\nIn this case it holds that (θπ)i = wi +\nR\nS V π(s′)ψi(ds′). Assumption 4.1 can be seen as a convention to\nomit function approximation errors in the final convergence results. This assumption, or the presence\nof approximation errors in convergence results, are widely present in the actor-critic literature ([5], [24],\n[23], [6], [8], [19]).\nMore recently, [16] derives some weaker ordering conditions in the bandit case (empty state space)\nwhich guarantee the convergence of soft-max policy gradient in the tabular setting beyond realisability.\nHowever as of now it is unclear how this applies to MDPs and also fundamentally depends on the finite\ncardinality of the action space.\nBy [4], Assumption 4.1 holds in the limit N →∞when ϕi are the basis functions of L2(ρ ⊗µ) for\nsome ρ ⊗µ ∈P(S × A). However, analysis in such a Hilbert space becomes more involved and intricate\nand is the result of ongoing work.\nAssumption 4.2. For all (s, a) ∈S × A it holds that |ϕ(s, a)| ≤1.\nAssumption 4.2 is purely for convention and is without loss of generality in the finite-dimensional case.\n\n6\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nAssumption 4.3. Let β ∈P(S × A) be fixed. Then\nλβ := λmin\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤β(ds da)\n\u0013\n> 0.\nNote that unlike the analogous assumptions imposed in [8], Assumption 4.3 is independent of the\npolicy. This property allows us to remove any dependence on the continuity of eigenvalues.\nDefinition 4.1. For all π ∈Πµ and ζ ∈P(S × A), the squared loss with respect to ζ is defined as\n(23)\nL(θ, π; ζ) = 1\n2\nZ\nS×A\n(⟨θ, ϕ(s, a)⟩−Qπ(s, a))2ζ(da, ds)\nwhere Qπ\nτ is defined in (2).\nA straightforward calculation given in Lemma A.4 shows that due to Lemma 5.1 and Assumption 4.3,\nfor any π ∈Πµ it holds that L(·, π; dπ\nβ) is (1 −γ)λβ-strongly convex.\nThe following result then connects the geometry of the semi-gradient of the MSBE and the gradient\nof L(θ, π; β), which can be seen as an extension of Lemma 3 of [2] to the current entropy regularised\nsetting and where the measure of integration in the MSBE is not necessarily stationary.\nLemma 4.1. Let Assumption 4.1 hold. Then for all θ ∈RN and π ∈Πµ it holds that\n(24)\n−⟨g(θ, π), θ −θπ⟩≤−(1 −√γ)(1 −γ) ⟨∇θL(θ, π; β), θ −θπ⟩\nwith\n∇θL(θ, π; β) =\nZ\nS×A\n(⟨θ, ϕ(s, a)⟩−Qπ\nτ (s, a))ϕ(s, a)β(da, ds).\nSee Appendix A.1 for a proof.\n5. Stability\nIn this section we analyse the stability of the coupled Actor Critic flow. Throughout this section, to\nease notation we let\nΓ := λβ(1 −γ)(1 −√γ),\nKt := sup\ns∈S\nKL(πt(·|s)|µ),\nwith λβ > 0 the constant from Assumption 4.3.\nThe following lemma establishes properties of the state-action occupancy measure defined in (6) and\nwhich are useful in the proofs.\nLemma 5.1. For all π ∈P(A|S), β ∈P(S × A) and E ∈B(S × A) it holds that\n(25)\ndπ\nJπβ(E) = Jπdπ\nβ(E).\nMoreover, for all γ ∈(0, 1) we have\n(26)\ndπ\nβ(E) −γdπ\nJπβ(E) = (1 −γ)β(E).\nSee Appendix A.2 for a proof. Lemma 5.2 then establishes the effect of the coupling and timescale\nseparation in the actor-critic flow and its effect on the stability of the critic parameters.\nLemma 5.2. Let Assumptions 4.2 and 4.3 hold. Then for all t ≥0 it holds that\n(27)\n1\n2ηt\nd\ndt|θt|2 ≤−Γ\n2 |θt|2 + τ 2γ2K2\nt\nΓ\n+\n|c|2\nBb(S×A)\nΓ\nSee Appendix B.1 for a proof. By connecting the result from Lemma 5.2 with the approximate Fisher\nRao gradient flow, we are able to establish a Gronwall-type inequality for the KL divergence of the\npolicies with respect to the reference measure.\nTheorem 5.1. Let Assumptions 4.2 and 4.3 hold. Let η0 > τ\nΓ. Then there exists constants\na1 = a1\n \nτ, η0, γ, λβ, |c|Bb(S×A),\n\f\f\f\f\ndπ0\ndµ\n\f\f\f\f\nBb(S×A)\n!\n> 0\nand a2 = a2(τ, η0, γ, λβ) > 0 such that for all γ ∈(0, 1) and t ≥0 it holds that\n(28)\nK2\nt ≤a1 + a2\nZ t\n0\ne−τ(t−r)K2\nr dr.\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n7\nSee Appendix B.2 for a proof. Through applications of Gronwall’s Lemma (Lemma A.1), two direct\ncorollaries of Theorem 5.1 show that the KL divergence of the policies with respect to the reference\nmeasure and the critic parameters do not blow up in finite time.\nCorollary 5.1 (Stability). Under the same assumptions as Theorem 5.1, for all γ ∈(0, 1), s ∈S and\nt ≥0 it holds that\n(29)\nKL(πt(·|s)|µ)2 ≤a1ea2t.\nCorollary 5.2. Under the same assumptions as Theorem 5.1, suppose that there exists α > 0 such that\nd\ndtηt ≤αηt. Then for all γ ∈(0, 1) there exists r1, r2 > 0 such that for all t ≥0 it holds that\n(30)\n|θt| ≤r1er2t.\nSee Appendix B.3 and B.4 for the proofs.\nCorollaries 5.3 and 5.4 then show that if the MDP is\nsufficiently regularised through a sufficiently small discounting factor, the KL divergence of the policies\nwith respect to the reference measure remains uniformly bounded along the flow.\nCorollary 5.3 (Uniform boundedness). Under the same assumptions as Theorem 5.1, for γ ∈(0, 1)\nsuch that\n64γ2\nΓ2−Γτ\nη0\n< 1 it holds that a2 < τ and for all t ≥0 it holds that\n(31)\nKL(πt(·|s)|µ)2 ≤\na1τ\nτ −a2\nCorollary 5.4. Under the conditions of Corollary 5.3, there exists R > 0 such that for all t ≥0 it holds\nthat\n(32)\n|θt| ≤R\nSee Appendix B.5 and B.6 for the proofs.\n6. Convergence\nIn this section we will present three convergence results of the coupled actor-critic flow. Firstly, we\ncharacterise the time derivative of the state-action value function along the approximate gradient flow\nfor the policies.\nLemma 6.1. For all t ≥0 and (s, a) ∈S × A, it holds that\n(33)\nd\ndtQπt\nτ (s, a) =\nγ\n1 −γ\nZ\nS\n\u0012Z\nS×A\nAπt\nτ (s′′, a′′)∂tπt(da′′|s′′)dπt(ds′′|s′)\n\u0013\nP(ds′|s, a)\nSee Appendix C.1 for a proof. Observe that in the exact setting, (13), we obtain the dissipative\nproperty of {Qπt\nτ }t≥0 along the flow\nd\ndtQπt\nτ (s, a) =\n−γ\n1 −γ\nZ\nS\n\u0012Z\nS×A\nAπt\nτ (s′′, a′′)2dπt(ds′′|s′)\n\u0013\nP(ds′|s, a) ≤0.\nFurthermore, Theorem 6.1 shows that the actor-critic flow maintains the exponential convergence to\nthe optimal policy induced by the τ-regularisation up to a error term arising from not solving the critic\nto full accuracy.\nTheorem 6.1. Let {πt, θt}t≥0 be the trajectories of the actor critic flow. Let Assumptions 4.1 and 4.2\nhold. Then for all t > 0 it holds that\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ≤\nτ\n2(1 −γ)(1 −e−τ\n2 t)\n \ne−τ\n2 t\nZ\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(34)\n+ 1\n2τ\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr\n!\n(35)\nSee Appendix C.2 for a proof. Note that by Theorem 1.1, it holds that KL(π∗(·|s)|π0(·|s)) < ∞. For\nPolish state and action spaces, in the unregularised case (τ = 0), KL(π∗(·|s)|π0(·|s)) will typically be\ninfinite (see [15][Theorem 2.1]).\nTheorem 6.1 shows that the exponentially weighted error term determines the rate of convergence of\nthe actor-critic dynamics. On this note, Theorem 6.2 shows that this error term decays exponentially\nup to an integral which now depends on the rate of change of the true state-action value function and\nthe timescale separation.\n\n8\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nTheorem 6.2. Let Assumptions 4.1, 4.2 and 4.3 hold. Let η0 > 1\nΓ and 0 < τ < 1. Then for all t ≥0\nthere exists constants b1, b2 > 0 such that\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤b1e−τ\n2 t + b2\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndt\n\f\f\f\f\n2\ndr.\n(36)\nSee Appendix C.3 for a proof. The following two results then connects Corollaries 5.1 and 5.2, Lemma\n6.1 and Theorem 6.1 to demonstrate an exponential convergence to the optimal policy for all γ ∈(0, 1)\nwhen the critic is updated sufficiently fast.\nTheorem 6.3. Under the same assumptions as Theorem 6.2, there exists k1 > 0 with ηt = η0ek1t and\nk2 > 0 such that for all γ ∈(0, 1) and t > 0 it holds that\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ≤\nτe−τ\n2 t\n2(1 −γ)(1 −e−τ\n2 t)\n Z\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds) + k2\n2τ\n!\n(37)\nSee Appendix C.4 for a proof. A direct consequence arising from the proof of Theorem 6.3 shows that\nif the MDP is sufficiently regularised through the same small discounting factor condition as in Corollary\n5.3, one can arrive at convergence for a much more general class of functions ηt.\nCorollary 6.1. Under the same assumptions as Theorem 6.2, for γ ∈(0, 1) such that\n2\n√\n2γ\nq\nΓ2−Γτ\nη0\n< 1 there\nexists d1 > 0 such that for all t ≥0,\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ≤\nτ\n2(1 −γ)(1 −e−τ\n2 t)\n \ne−τ\n2 t\nZ\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(38)\n+ d1\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\ndr\n!\n.\n(39)\nSee Appendix C.3 for a proof. For example, suppose the small discounting factor condition is satisfied,\nchoosing ηt = t\n1\n2 + η0 with η0 > 1\nΓ and τ = 0.5, it can be shown that asymptotically\n(40)\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ∼1\n√\nt.\n7. Limitations\nIn this work, we only study the continuous-time dynamics of the actor-critic algorithm. Although this\nformulation gives insights into the discrete counterpart, a rigorous treatment of the discrete-time setting\nis more realistic for practical purposes and is left for future research.\nMoreover, for the purposes of analysis our critic approximation is linear while in practice non-linear\nneural networks are used to approximate the critic.\nFinally, our work assumes all integrals are evaluated exactly, in particular the semi-gradient (18). In\npractice these would need to be estimated from samples leading to additional Monte-Carlo errors. To\nfully analyse this is left for future work.\n8. Acknowledgements\nDZ was supported by the EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis\nand Computation (MAC-MIGS) funded by the UK Engineering and Physical Sciences Research Coun-\ncil (grant EP/S023291/1), Heriot-Watt University and the University of Edinburgh. We acknowledge\nfunding from the UKRI Prosperity Partnerships grant APP43592: AI2 – Assurance and Insurance for\nArtificial Intelligence, which supported this work.\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n9\nReferences\n[1] Anas Barakat, Pascal Bianchi, and Julien Lehmann, Analysis of a target-based actor-critic algorithm with linear\nfunction approximation, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics\n(Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera, eds.), Proceedings of Machine Learning Research, vol.\n151, PMLR, 28–30 Mar 2022, pp. 991–1040.\n[2] Jalaj Bhandari, Daniel Russo, and Raghav Singal, A finite time analysis of temporal difference learning with linear\nfunction approximation, Operations Research 69 (2021), no. 3, 950–973.\n[3] V. S. Borkar and V. R. 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Wortman Vaughan, eds.), 2021.\n\n10\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nAppendix A. Technical details\nIn this section, we present some calculations that will be used in the proofs of the main results.\nLemma A.1 (Gronwall). Let λ(s) ≥0, a = a(s), b = b(s) and y = y(s) be locally integrable, real-valued\nfunctions defined on [0, T] such that y is also locally integrable and for almost all s ∈[0, T],\ny(s) + a(s) ≤b(s) +\nZ s\n0\nλ(t)y(t)dt.\nThen\ny(s) + a(s) ≤b(s) +\nZ s\n0\nλ(t)\n\u0014Z t\n0\nλ(r)(b(r) −a(r))dr\n\u0015\ndt,\n∀s ∈[0, T].\nFurthermore, if b is monotone increasing and a is non-negative, then\ny(s) + a(s) ≤b(s)e\nR s\n0 λ(r)dr,\n∀s ∈[0, T].\nLemma A.2. For some β ∈P(S × A), let dπ\nβ ∈P(S × A) be the state-action occupancy measure.\nMoreover let κ(ds, da, ds′, da′) := P π(ds′, da′|s, a)dπ\nβ(ds, da). Then for any π ∈Πµ and any integrable\nf : S × A →R, it holds that\n(41)\nZ\nS×A×S×A\nf(s, a)f(s′, a′)κ(ds, da, ds′, da′) ≤\n1\n√γ\nZ\nS×A\nf(s, a)2dπ\nβ(ds, da)\nProof. By H¨older’s inequality, it holds that\nZ\nS×A×S×A\nf(s, a)f(s′, a′)κ(ds, da, ds′, da′)\n(42)\n≤\n\u0012Z\nS×A×S×A\nf(s, a)2κ(ds, da, ds′, da′)\n\u0013 1\n2 \u0012Z\nS×A×S×A\nf(s′, a′)2κ(ds, da, ds′, da′)\n\u0013 1\n2\n.\n(43)\nMoreover, observe that\nZ\nS×A×S×A\nf(s, a)2κ(ds, da, ds′, da′) =\nZ\nS×A\n\u0012Z\nS×A\nP π(ds′, da′|s, a)\n\u0013\nf(s, a)2dπ\nβ(ds, da)\n(44)\n=\nZ\nS×A\nf(s, a)2dπ\nβ(ds, da),\n(45)\nhence (42) becomes\n\u0012Z\nS×A×S×A\nf(s, a)2κ(ds, da, ds′, da′)\n\u0013 1\n2 \u0012Z\nS×A×S×A\nf(s′, a′)2κ(ds, da, ds′, da′)\n\u0013 1\n2\n(46)\n≤\n\u0012Z\nS×A\nf(s, a)2dπ\nβ(ds, da)\n\u0013 1\n2 \u0012Z\nS×A×S×A\nf(s′, a′)2κ(ds, da, ds′, da′)\n\u0013 1\n2\n.\n(47)\nNow by the first part of Lemma 5.1, it holds that\nZ\nS×A×S×A\nf(s′, a′)2κ(ds, da, ds′, da′) =\nZ\nS×A×S×A\nf(s′, a′)2P π(ds′, da′|s, a)dπ\nβ(ds, da)\n(48)\n=\nZ\nS×A\nf(s, a)2dπ\nJπβ(ds, da),\n(49)\nwhere Jπ : P(S × A) →P(S × A) is defined in (7). Then by the second part of Lemma 5.1 we have\n\u0012Z\nS×A\nf(s, a)2dπ\nβ(ds, da)\n\u0013 1\n2 \u0012Z\nS×A×S×A\nf(s′, a′)2κ(ds, da, ds′, da′)\n\u0013 1\n2\n(50)\n≤\n\u0012Z\nS×A\nf(s, a)2dπ\nβ(ds, da)\n\u0013 1\n2 \u0012Z\nS×A\nf(s, a)2dπ\nJπβ(ds, da)\n\u0013 1\n2\n(51)\n≤\n1\n√γ\nZ\nS×A\nf(s, a)2dπ\nβ(ds, da),\n(52)\nwhich concludes the proof.\n□\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n11\nLemma A.3. For some θ0 ∈RN and π0 ∈Πµ, let {πt, θt}t≥0 be the trajectory of coupled actor-critic\nflow. Moreover let Kt = sups∈S KL(πt(·|s)|µ). There exists C1 > 0 such that for all t ≥0 it holds that\nsup\ns∈S\n|∂tπt(·|s)|M(A) ≤|At|Bb(S×A) ,\n(53)\n|At|Bb(S×A) ≤2 |Qt|Bb(S×A) + 2τ\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n,\n(54)\n|Qπt\nτ |Bb(S×A) ≤\n1\n1 −γ\n\u0010\n|c|Bb(S×A) + τγKt\n\u0011\n,\n(55)\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n≤C1 + 2\nτ\nsup\nr∈[0,t]\n|θr| + sup\nr∈[0,t]\nKr.\n(56)\nProof. The first claim sups∈S |∂tπt(·|s)|M(A) ≤|Aπt\nτ |Bb(S×A) follows trivially from the definition of the\napproximate Fisher Rao gradient flow defined in (22). Moreover, it holds that\n|At|Bb(S×A) =\n\f\f\f\fQt + τ ln dπt\ndµ −\nZ\nA\n\u0012\nQt(·, a) + τ ln dπt\ndµ (·, a)\n\u0013\nπt(da|·)\n\f\f\f\f\nBb(S×A)\n(57)\n≤2\n\f\f\f\fQt + τ ln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n(58)\n≤2 |Qt|Bb(S×A) + 2τ\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n(59)\nwhere we used the triangle inequality in the final inequality. Moreover, the state-action value function\nQπt\nτ\nis a fixed point of the Bellman operator defined in (3). Hence, for all (s, a) ∈S × A, we have\nQπt\nτ (s, a) = c(s, a) + γ\nZ\nS×A\nQπt\nτ (s′, a′) P πt(ds′, da′|s, a) + τγ\nZ\nS\nKL(πt(·|s′)∥µ) P(ds′|s, a).\n(60)\nTaking absolute values and using the triangle inequality we have\n|Qπt\nτ (s, a)| ≤|c|Bb(S×A) + γ |Qπt\nτ |Bb(S×A) + τγ sup\ns′∈S\nKL(πt(·|s′)∥µ)\n(61)\n= |c|Bb(S×A) + γ |Qπt\nτ |Bb(S×A) + τγKt.\n(62)\nTaking the supremum over (s, a) ∈S × A on the left-hand side yields\n(63)\n|Qπt\nτ |Bb(S×A) ≤|c|Bb(S×A) + γ |Qπt\nτ |Bb(S×A) + τγKt.\nRearranging gives\n(64)\n(1 −γ) |Qπt\nτ |Bb(S×A) ≤|c|Bb(S×A) + τγKt,\nwhich is the desired bound. Recall the approximate Fisher-Rao gradient flow for the policies {πt}t≥0,\nwhich for all t ≥0 and for all (s, a) ∈S × A is given by\n(65)\n∂t ln dπt\ndµ (s, a) = −\n\u0012\nQt(s, a) + τ ln dπt\ndµ (a, s) −\nZ\nA\n\u0012\nQt(s, a′) + τ ln dπt\ndµ (a′, s)\n\u0013\nπt(da′|s)\n\u0013\n.\nDuhamel’s principle yields for all t ≥0 that\nln dπt\ndµ (s, a) = e−τt ln dπ0\ndµ (a, s) +\nZ t\n0\ne−τ(t−r)\n\u0012Z\nA\nQr(s, a′)πr(da′|s) −Qr(s, a)\n\u0013\ndr\n(66)\n+ τ\nZ t\n0\ne−τ(t−r) KL(πr(·|s)|µ)dr.\n(67)\n\n12\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nSince π0 ∈Πµ, there exists C1 ≥1 such that\n\f\f\fln dπ0\ndµ\n\f\f\f\nBb(S×A) ≤C1. Then by Assumption 4.2 we have\nthat for all t ≥0,\n\f\f\f\fln dπt\ndµ (s, a)\n\f\f\f\f ≤C1 +\nZ t\n0\ne−τ(t−r)\n\f\f\f\f\nZ\nA\nQr(s, a′)πr(da′|s) −Qr(s, a)\n\f\f\f\f dr\n(68)\n+ τ\nZ t\n0\ne−τ(t−r) KL(πr(·|s)∥µ) dr\n(69)\n≤C1 + 2\nZ t\n0\ne−τ(t−r) |θr| dr + τ\nZ t\n0\ne−τ(t−r)Kr dr\n(70)\n≤C1 + 2\nτ\nsup\nr∈[0,t]\n|θr| + sup\nr∈[0,t]\nKr,\n(71)\nwhere in the last inequality we used\nR t\n0 e−τ(t−r)dr ≤1\nτ . Taking the supremum over (s, a) ∈S × A yields\n(72)\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n≤C1 + 2\nτ\nsup\nr∈[0,t]\n|θr| + sup\nr∈[0,t]\nKr,\nwhich is the desired bound.\n□\nLemma A.4. Let Assumption 4.3 hold. Then for all π ∈Πµ, it holds that L(·, π; dπ\nβ) is λβ(1−γ)-strongly\nconvex.\nProof. For any ξ ∈P(S × A), let Σξ :=\nR\nS×A ϕ(s, a)ϕ(s, a)⊤ξ(ds, da) ∈RN×N. Then by Lemma 5.1 and\nAssumption 4.3 it holds that Σdπ\nβ ⪰(1 −γ)Σβ ⪰(1 −γ)λβI and thus L(·, π; dπ\nβ) is λβ(1 −γ)-strongly\nconvex.\n□\nA.1. Proof of Lemma 4.1.\nProof. Recall that Q(s, a) = ⟨θ, ϕ(s, a)⟩for some θ ∈RN and that for all π ∈Πµ, there exists θπ ∈RN\nsuch that Qπ(s, a) = ⟨θπ, ϕ(s, a)⟩by Assumption 4.1. Then by definition of the semi-gradient of the\nMSBE g : RN × P(A|S) →RN in (18), it holds that\n⟨g(θ, π), θ −θπ⟩=\n\u001cZ\nS×A\n(Q(s, a) −TπQ(s, a)) ϕ(s, a)dπ\nβ(da, ds), θ −θπ\n\u001d\n(73)\n=\n\u001cZ\nS×A\n(Q(s, a) −Qπ\nτ (s, a))ϕ(s, a)dπ\nβ(da, ds), θ −θπ\n\u001d\n(74)\n+\n\u001cZ\nS×A\n(Qπ\nτ (s, a) −TπQ(s, a)ϕ(s, a)dπ\nβ(da, ds), θ −θπ\n\u001d\n(75)\n=\n\u001cZ\nS×A\n(Q(s, a) −Qπ\nτ (s, a))ϕ(s, a)dπ\nβ(da, ds), θ −θπ\n\u001d\n(76)\n−γ\n\u001cZ\nS×A×S×A\n(Q(s′, a′) −Qπ\nτ (s′, a′))ϕ(s, a)P π(ds′, da′|s, a)dπ\nβ(ds, da), θ −θπ\n\u001d\n,\n(77)\nwhere we added and subtracted the true state-action value function Qπ\nτ ∈Bb(S × A) in the second\nequality and used the fact that it is a fixed point of the Bellman operator defined in (3).\nTo ease\nnotation, let ε(s, a) := Q(s, a) −Qπ\nτ (s, a). Multiplying both sides by −1 and using the associativity of\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n13\nthe inner product, we have\n−⟨g(θ, π), θ −θπ⟩\n(78)\n= −\n\u001cZ\nS×A\nε(s, a)ϕ(s, a)dπ\nβ(da, ds), θ −θπ\n\u001d\n(79)\n+ γ\n\u001cZ\nS×A\nε(s′, a′)ϕ(s, a)P π(ds′, da′|s, a)dπ\nβ(ds, da), θ −θπ\n\u001d\n(80)\n= −\nZ\nS×A\nε(s, a) ⟨ϕ(s, a), θ −θπ⟩dπ\nβ(da, ds)\n(81)\n+ γ\nZ\nS×A\nε(s′, a′) ⟨ϕ(s, a), θ −θπ⟩P π(ds′, da′|s, a)dπ\nβ(ds, da)\n(82)\n= −\nZ\nS×A\nε(s, a)2dπ\nβ(da, ds)\n(83)\n+ γ\nZ\nS×A×S×A\nε(s, a)ε(s′, a′)P π(ds′, da′|s, a)dπ\nβ(ds, da)\n(84)\n= I(1) + γI(2).\n(85)\nNow applying Lemma A.2 to I(2) we have\nI(2) :=\nZ\nS×A×S×A\nε(s, a)ε(s′, a′)P π(ds′, da′|s, a)dπ\nβ(ds, da)\n(86)\n≤\n1\n√γ\nZ\nS×A\nε(s, a)2dπ\nβ(ds, da).\n(87)\nThus it holds that\n−⟨g(θ, π), θ −θπ⟩≤I(1) + γI(2)\n(88)\n≤−(1 −√γ)\nZ\nS×A\nϵ(s, a)2dπ\nβ(da, ds)\n(89)\n= −(1 −√γ)\nZ\nS×A\n(Q(s, a) −Qπ\nτ (s, a))2dπ\nβ(da, ds)\n(90)\n= −(1 −√γ)\n\n∇θL(θ, π; dπ\nβ), θ −θπ\n\u000b\n,\n(91)\nwhere the last inequality follows from the Assumption 4.1 and the definition of Q(s, a) = ⟨θ, ϕ(s, a)⟩.\n□\nA.2. Proof of Lemma 5.1.\nProof. For any β ∈P(S × A), π ∈P(A|S) and E ∈B(S × A), it holds that\ndπ\nJπβ(E) = (1 −γ)\n∞\nX\nn=0\nγn(Jn\nπ Jπβ)(E)\n(92)\n= Jπdπ\nβ(E)\n(93)\nwhere we just used the associativity of the operator Jπ. Furthermore by letting m = n + 1 it holds that\ndπ\nJπβ(E) = (1 −γ)\n∞\nX\nn=0\nγnJn+1\nπ\nβ(E)\n(94)\n= (1 −γ)\n∞\nX\nm=1\nγm−1Jm\nπ β(E)\n(95)\n= 1 −γ\nγ\n∞\nX\nm=1\nγmJm\nπ β(E)\n(96)\n= 1\nγ (dπ\nβ(E) −(1 −γ)β(E)).\n(97)\nRearranging concludes the proof.\n□\n\n14\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nAppendix B. Proof of Regularity Results\nB.1. Proof of Lemma 5.2.\nProof. Consider the following equation\n1\n2ηt\nd\ndt |θt|2 = 1\nηt\n\u001c d\ndtθt, θt\n\u001d\n(98)\n= −⟨g(θt, πt), θt⟩\n(99)\n= −\n\u001cZ\nS×A\n(Qt(s, a) −T πtQt(s, a)) ϕ(s, a) dπt\nβ (da, ds), θt\n\u001d\n(100)\n= −\n\u001cZ\nS×A\nQt(s, a)ϕ(s, a) dπt\nβ (da, ds), θt\n\u001d\n(101)\n+\n\u001cZ\nS×A\nT πtQt(s, a)ϕ(s, a) dπt\nβ (da, ds), θt\n\u001d\n(102)\n:= −J(1)\nt\n+ J(2)\nt\n(103)\nwhere we used the θt dynamics from (21) in the second equality and the definition of the semi-gradient\nin the third equality. For any π ∈Πµ, let Σπ ∈RN×N be\n(104)\nΣπ =\nZ\nS×A\nϕ(s, a)ϕ(s, a)⊤dπ\nβ(da, ds).\nThen by definition we have that Qt(s, a) = ⟨θt, ϕ(s, a)⟩, hence for J(1)\nt\nwe have\nJ(1)\nt\n=\n\u001cZ\nS×A\nQt(s, a)ϕ(s, a) dπt\nβ (da, ds), θt\n\u001d\n(105)\n=\n\u001cZ\nS×A\n⟨θt, ϕ(s, a)⟩ϕ(s, a)dπt\nβ (da, ds), θt\n\u001d\n(106)\n=\n\u001c\nθt,\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤dπt\nβ (da, ds)\n\u0013\nθt\n\u001d\n(107)\n= ⟨θt, Σπtθt⟩\n(108)\nNow dealing with J(1)\nt\n, expanding the Bellman operator defined in (3) we have\nJ(2)\nt\n=\n\u001cZ\nS×A\nTπtQt(s, a)ϕ(s, a) dπt\nβ (da, ds), θt\n\u001d\n(109)\n=\n\u001cZ\nS×A\nc(s, a)ϕ(s, a)dπt\nβ (da, ds), θt\n\u001d\n(110)\n+ γ\n\u001cZ\nS×A\n⟨θt, ϕ(s′, a′)⟩ϕ(s, a)P πt(ds′, da′|s, a)dπt\nβ (da, ds), θt\n\u001d\n(111)\n+ τγ\n\u001cZ\nS×A\n\u0012Z\nS\nKL(πt(·|s′), µ)P(ds′|s, a)ϕ(s, a)dπt\nβ (da, ds)\n\u0013\n, θt\n\u001d\n(112)\n≤|c|Bb(S×A)|θt| + γI(1)\nt\n+ τγI(2)\nt\n(113)\nwhere we defined\nI(1)\nt\n=\n\u001cZ\nS×A\n⟨θt, ϕ(s′, a′)⟩ϕ(s, a)P πt(ds′, da′|s, a)dπt\nβ (da, ds), θt\n\u001d\n,\nI(2)\nt\n=\n\u001cZ\nS×A\n\u0012Z\nS\nKL(πt(·|s′), µ)P(ds′|s, a)ϕ(s, a)dπ\nβ(da, ds)\n\u0013\n, θt\n\u001d\n.\nMoreover, to ease notation let\nKt := sup\ns∈S\nKL(πt(·|s)|µ)\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n15\nand temporarily let κt(ds, da, ds′, da′) := P πt(ds′, da′|s, a)dπt\nβ (da, ds). Now focusing on I(1)\nt\n, it holds that\nI(1)\nt\n=\n\u001cZ\nS×A×S×A\n⟨θt, ϕ(s′, a′)⟩ϕ(s, a)κt(da′, ds′, da, ds), θt\n\u001d\n(114)\n=\nZ\nS×A×S×A\n⟨θt, ϕ(s, a)⟩⟨θt, ϕ(s′, a′)⟩κt(ds′, da′, ds, da).\n(115)\nNow using Lemma A.2 with f = ⟨θ, ϕ(·, ·)⟩we have\nI(1)\nt\n≤\n1\n√γ\n\u0012Z\nS×A\n⟨θt, ϕ(s, a)⟩2 dπt\nβ (ds, da)\n\u0013 1\n2 \u0012Z\nS×A\n⟨θt, ϕ(s, a)⟩2 dπt\nβ (ds, da)\n\u0013 1\n2\n(116)\n=\n1\n√γ\nZ\nS×A\n⟨θt, ϕ(s, a)⟩2 dπt\nβ (ds, da)\n(117)\n=\n1\n√γ ⟨θt, Σπtθt⟩.\n(118)\nThus all together it holds that\n(119)\nγI(1)\nt\n≤√γ ⟨θt, Σπtθt⟩.\nNow focusing on I(2)\nt\n, we have\nI(2)\nt\n=\n\u001cZ\nS×A\n\u0012Z\nS\nKL(πt(·|s′), µ)P(ds′|s, a)\n\u0013\nϕ(s, a)dπt\nβ (da, ds), θt\n\u001d\n(120)\n≤Kt\n\f\f\f\f\nZ\nS×A\nϕ(s, a)dπt\nβ (ds, da)\n\f\f\f\f |θt|\n(121)\n≤Kt|θt|\n(122)\nwhere we used Assumption 4.2 in the final inequality. Hence along with (108), (98) becomes\n1\n2ηt\nd\ndt|θt|2 ≤−J(1)\nt\n+ J(2)\nt\n(123)\n≤−⟨θt, Σπtθt⟩+ |c|Bb(S×A)|θt| + γI(1)\nt\n+ τγI(2)\nt\n(124)\n≤−⟨θt, Σπtθt⟩+ √γ ⟨θt, Σπtθt⟩+ |c|Bb(S×A)|θt| + τγKt|θt|\n(125)\n= −(1 −√γ) ⟨θt, Σπtθt⟩+\n\u0000|c|Bb(S×A) + τγKt\n\u0001\n|θt|.\n(126)\nObserve that by (26) and Assumption 4.2, Σπ ∈RN×N is positive definite for all π ∈P(A|S), hence it\nholds that\n(127)\n⟨θt, Σπtθt⟩≥(1 −γ)λβ |θt|2 .\nTherefore (123) becomes\n(128)\n1\n2ηt\nd\ndt|θt|2 ≤−(1 −√γ)(1 −γ)λβ |θt|2 + (|c|Bb(S×A) + τγKt)|θt|\nLet Γ := λβ(1 −γ)(1 −√γ). By Young’s inequality, there exists ϵ > 0 such that\n1\n2ηt\nd\ndt|θt|2 ≤−Γ|θt|2 + ϵ\n2|θt|2 + (|c|Bb(S×A) + τγKt)2\n2ϵ\n(129)\n≤−Γ|θt|2 + ϵ\n2|θt|2 +\n|c|2\nBb(S×A) + τ 2γ2K2\nt\nϵ\n,\n(130)\nwhere we used the identity (a + b)2 ≤2a2 + 2b2. Choosing ϵ = Γ we arrive at\n(131)\n1\n2ηt\nd\ndt|θt|2 ≤−Γ\n2 |θt|2 + τ 2γ2K2\nt\nΓ\n+\n|c|2\nBb(S×A)\nΓ\nwhich concludes the proof.\n□\n\n16\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nB.2. Proof of Theorem 5.1.\nProof. By Lemma 5.2, we have that for all r ≥0\n(132)\n1\n2ηr\nd\ndr|θr|2 ≤−Γ\n2 |θr|2 + τ 2γ2K2\nr\nΓ\n+\n|c|2\nBb(S×A)\nΓ\n.\nRearranging, it holds that for all t ≥0\n|θr|2 ≤−1\nΓηr\nd\ndr|θr|2 +\n2|c|2\nBb(S×A) + 2τ 2γ2K2\nr\nΓ2\n.\n(133)\nMultiplying both sides by e−τ(t−r) and integrating over r from 0 to t we have that for all t ≥0\nZ t\n0\ne−τ(t−r)|θr|2dr ≤−1\nΓ\nZ t\n0\ne−τ(t−r) 1\nηr\nd\ndr|θr|2dr +\n2|c|2\nBb(S×A)\nΓ2\nZ t\n0\ne−τ(t−r)dr\n(134)\n+ 2τ 2γ2\nΓ2\nZ t\n0\ne−τ(t−r)K2\nrdr\n(135)\n≤−1\nΓ\nZ t\n0\ne−τ(t−r) 1\nηr\nd\ndr|θr|2dr +\n2|c|2\nBb(S×A)\nΓ2τ\n+ 2τ 2γ2\nΓ2\nZ t\n0\ne−τ(t−r)K2\nrdr,\n(136)\nwhere we used that\nR t\n0 e−τ(t−r)dr ≤1\nτ . Integrating the first term by parts, we have\n−\nZ t\n0\ne−τ(t−r) 1\nηr\nd\ndr|θr|2dr = −|θt|2\nηt\n+ e−τt |θ0|2\nη0\n+ τ\nZ t\n0\n|θr|2 e−τ(t−r)\nηr\ndr\n(137)\n−\nZ t\n0\n|θr|2 e−τ(t−r) d\ndrηr\nη2r\ndr.\n(138)\nSince by definition we have that for all t ≥0, ηt ≥1 and\nd\ndtηt ≥0 it holds that\n(139)\nZ t\n0\n|θr|2 e−τ(t−r) d\ndrηr\nη2r\ndr ≥0.\nHence dropping the negative terms on the right hand side of (137) and using that ηt ≥η0 for all t ≥0,\nwe have\n−1\nΓ\nZ t\n0\ne−τ(t−r) 1\nηr\nd\ndr|θr|2dr ≤e−τt |θ0|2\nΓη0\n+\nτ\nΓη0\nZ t\n0\ne−τ(t−r)|θr|2dr.\n(140)\nSubstituting this back into (134), for all t ≥0 we have that\nZ t\n0\ne−τ(t−r)|θr|2dr ≤e−τt |θ0|2\nΓη0\n+\nτ\nΓη0\nZ t\n0\ne−τ(t−r)|θr|2dr\n(141)\n+\n2|c|2\nBb(S×A)\nΓ2τ\n+ 2τ 2γ2\nΓ2\nZ t\n0\ne−τ(t−r)K2\nrdr.\n(142)\nGrouping like terms we have\n\u0012\n1 −\nτ\nΓη0\n\u0013 Z t\n0\ne−τ(t−r)|θr|2dr ≤e−τt |θ0|2\nΓη0\n+\n2|c|2\nBb(S×A)\nΓ2τ\n+ 2τ 2γ2\nΓ2\nZ t\n0\ne−τ(t−r)K2\nrdr.\n(143)\nRecall that we have η0 > τ\nΓ to ensure that 1 −\nτ\nΓη0 > 0. Dividing through by 1 −\nτ\nΓη0 gives for all t ≥0\nthat\nZ t\n0\ne−τ(t−r)|θr|2dr ≤σ1 + σ2\nZ t\n0\ne−τ(t−r)K2\nrdr\n(144)\nwhere we’ve set\nσ1 :=\n|θ0|2\nΓη0\n\u0010\n1 −\nτ\nΓη0\n\u0011 +\n2|c|2\nBb(S×A)\nΓ2τ\n\u0010\n1 −\nτ\nΓη0\n\u0011,\nσ2 :=\n2τ 2γ2\nΓ2\n\u0010\n1 −\nτ\nΓη0\n\u0011.\nRecall the approximate Fisher Rao gradient flow for the policies {πt}t≥0, which for all t ≥0 and for all\ns ∈S, a ∈A is\n(145)\n∂t ln dπt\ndµ (s, a) = −\n\u0012\nQt(s, a) + τ ln dπt\ndµ (a, s) −\nZ\nA\n\u0012\nQt(s, a) + τ ln dπt\ndµ (a, s)\n\u0013\nπt(da|s)\n\u0013\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n17\nDuhamel’s principle yields for all t ≥0 that\nln dπt\ndµ (s, a) = e−τt ln dπ0\ndµ (a, s) +\nZ t\n0\ne−τ(t−r)\n\u0012Z\nA\nQr(s, a)πr(da|s) −Qr(s, a)\n\u0013\ndr\n(146)\n+ τ\nZ t\n0\ne−τ(t−r) KL(πr(·|s)|µ)dr\n(147)\nObserve that since π0 ∈Πµ, there exists C1 ≥1 such that ln\n\f\f\f dπt\ndµ\n\f\f\f\nBb(S×A) ≤C1. Assumption 4.2 gives\nthat for all t ≥0\nln dπt\ndµ (s, a) ≤C1 + 2\nZ t\n0\ne−τ(t−r)|θr|dr + τ\nZ t\n0\ne−τ(t−r) KL(πr(·|s)|µ)dr\n(148)\n≤C1 + 2\nZ t\n0\ne−τ(t−r)|θr|dr + τ\nZ t\n0\ne−τ(t−r)Krdr\n(149)\nIntegrating over the actions with respect to πt(·|s) ∈P(A) gives for all t ≥0 that\n(150)\nKL(πt(·|s)|µ) ≤C1 + 2\nZ t\n0\ne−τ(t−r)|θr|dr + τ\nZ t\n0\ne−τ(t−r)Krdr\nwhere we again use that Kr = sups∈S KL(πr(·|s)|µ). Following from the techniques in [17], observe that\nfrom (66) and Assumption 4.2 we similarly get for all t ≥0 that\n(151)\nln dµ\ndπt\n(a, s) = −ln dπt\ndµ (s, a) ≤C1 + 2\nZ t\n0\ne−τ(t−r)|θr|dr −τ\nZ t\n0\ne−τ(t−r)Krdr.\nNow integrating over the actions with respect to the reference measure µ ∈P(A) we have\n(152)\nKL(µ|πt(·|s)) ≤C1 + 2\nZ t\n0\ne−τ(t−r)|θr|dr −τ\nZ t\n0\ne−τ(t−r)Krdr\nMoreover, using the non-negativity of the KL divergence, it holds for all t ≥0 that\n(153)\nKL(πt(·|s)|µ) ≤KL(πt(·|s)|µ) + KL(µ|πt(·|s)) ≤2C1 + 4\nZ t\n0\ne−τ(t−r)|θr|dr\nSince this holds for any s ∈S, it holds for all t ≥0 that\n(154)\nKt ≤2C1 + 4\nZ t\n0\ne−τ(t−r)|θr|dr\nNow squaring both sides and using the H¨older’s inequality, we have\nK2\nt ≤\n\u0012\n2C1 + 4\nZ t\n0\ne−τ(t−r)|θr|dr\n\u00132\n(155)\n≤8(C1)2 + 32\n\u0012Z t\n0\ne−τ(t−r)|θr|dr\n\u00132\n(156)\n= 8(C1)2 + 32\n\u0012Z t\n0\ne−τ\n2 (t−r)e−τ\n2 (t−r)|θr|dr\n\u00132\n(157)\n≤8(C1)2 + 32\n\u0012Z t\n0\ne−τ(t−r)dr\n\u0013 \u0012Z t\n0\ne−τ(t−r)|θr|2dr\n\u0013\n(158)\n≤8(C1)2 + 32\nτ\nZ t\n0\ne−τ(t−r)|θr|2dr,\n(159)\nwhere we again used\nR t\n0 e−τ(t−r)dr ≤1\nτ . We can now substitute (144) into (159) to arrive at\nK2\nt ≤8(C1)2 + 32\nτ σ1 + 32\nτ σ2\nZ t\n0\ne−τ(t−r)K2\nrdr\n(160)\n:= a1 + a2\nZ t\n0\ne−τ(t−r)K2\nrdr\n(161)\nwith a1 = 8(C1)2 + 32\nτ σ1 and a2 = 32σ2\nτ\n.\n□\n\n18\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nB.3. Proof of Corollary 5.1.\nProof. By Theorem 5.1 it holds that\n(162)\nK2\nt ≤a1 + a2\nZ t\n0\ne−τ(t−r)K2\nrdr.\nObserve that by multiplying through by eτt, we can rewrite this as\n(163)\neτtK2\nt ≤eτta1 + a2\nZ t\n0\neτrK2\nrdr.\nHence after defining g(t) = eτtK2\nt and applying Gronwall’s inequality (Lemma A.1), for all γ ∈(0, 1) it\nholds for all t ≥0 that\n(164)\nK2\nt ≤a1ea2t.\n□\nB.4. Proof of Corollary 5.2.\nProof. By Corollary 5.1 and Lemma 5.2, for all γ ∈(0, 1) it holds that\n1\n2\nd\ndt |θt|2 ≤−Γ\n2 ηt|θt|2 + btηt\n(165)\nsuch that\n(166)\nbt =\n \n2|c|2\nBb(S×A) + 2τ 2γ2a1ea2t\nΓ2\n!\n.\nRecall that there exists α > 0 such that\nd\ndtηt ≤αηt, another application of Gronwall’s Lemma then\nconcludes the proof.\n□\nB.5. Proof of Corollary 5.3.\nProof. By Theorem 5.1 we have that\n(167)\nK2\nt ≤a1 + a2\nZ t\n0\ne−τ(t−r)K2\nrdr.\nTaking the supremum over [0, t] on the right hand side, we have\n(168)\nK2\nt ≤a1 + a2\nτ\nsup\nr∈[0,t]\nK2\nr.\nSince this holds for all t ≥0, we have\n(169)\nsup\nr∈[0,t]\nK2\nr ≤a1 + a2\nτ\nsup\nr∈[0,t]\nK2\nr.\nNow forcing 1 −a2\nτ > 0, which is equivalent to the condition\n64γ2\nΓ2 −Γτ\nη0\n< 1.\nHence after rearranging we have\n(170)\nK2\nt ≤sup\nr∈[0,t]\nK2\nr ≤\na1τ\nτ −a2\n□\nB.6. Proof of Corollary 5.4.\nProof. By Corollary 5.3, for sufficiently small γ > 0 it holds that for all t ≥0,\nK2\nt ≤\na1τ\nτ −a2\n.\nHence by Lemma 5.2 we have\n1\n2\nd\ndt |θt|2 ≤−ηt\nΓ\n2 |θt|2 + ηt\n\n\n2|c|2\nBb(S×A) + 2τ 2γ2 \u0010\na1τ\nτ−a2\n\u0011\nΓ2\n\n.\n(171)\nThe uniform boundedness in time of |θt| then follows by Gronwall’s Lemma (Lemma A.1).\n□\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n19\nAppendix C. Proof of Convergence Results\nC.1. Proof of Lemma 6.1.\nProof. By the definition of the state-action value function (2) it holds that\nd\ndtQπt\nτ (s, a) = lim\nh→0\nQπt+h\nτ\n(s, a) −Qπt\nτ (s, a)\nh\n(172)\n= γ\nZ\nS\nd\ndtV πt\nτ (s′)P(ds′|s, a).\n(173)\nNow observe that by [11][Proof of Proposition 2.6], we have\n(174)\nd\ndtV πt\nτ (s) =\n1\n1 −γ\nZ\nS×A\nAπt\nτ (s, a)∂tπt(da|s′)dπt(ds′|s).\nThus we have\nd\ndtQπt\nτ (s, a) =\nγ\n1 −γ\nZ\nS\n\u0012Z\nS×A\nAπt\nτ (s′′, a′′)∂tπt(da′′|s′′)dπt(ds′′|s′)\n\u0013\nP(ds′|s, a).\n(175)\n□\nC.2. Proof of Theorem 6.1.\nProof. Recall the performance difference Lemma (Lemma 1.1): for all ρ ∈P(S) and π, π′ ∈Πµ,\nV π\nτ (ρ) −V π′\nτ (ρ)\n(176)\n=\n1\n1 −γ\nZ\nS\n\u0014 Z\nA\n\u0012\nQπ′\nτ (s, a) + τ ln dπ′\ndµ (a, s)\n\u0013\n(π −π′)(da|s) + τ KL(π(·|s)|π′(·|s))\n\u0015\ndπ\nρ(ds) .\n(177)\nNow let π = π∗and π′ = πt and multiply both sides by −1 we have\nV πt\nτ (ρ) −V π∗\nτ\n(ρ) =\n−1\n1 −γ\nZ\nS\n Z\nA\n\u0012\nQπt(s, a) + τ ln dπt\ndµ (a, s)\n\u0013\n(π∗−πt)(da|s)\n(178)\n+ τ KL(π∗(·|s)|πt(·|s))\n!\ndπ∗\nρ (ds).\n(179)\nRecall the approximate Fisher Rao dynamics, which we write as\n(180)\n∂t ln dπt\ndµ (s, a) +\n\u0012\nQt(s, a) + τ ln dπt\ndµ (a, s) −\nZ\nA\n\u0012\nQt(s, a′) + τ ln dπt\ndµ (a′, s)\n\u0013\nπt(da′|s)\n\u0013\n= 0.\nObserve that since the normalisation constant (enforcing the conservation of mass along the flow)\nR\nA\n\u0010\nQt(s, a) + τ ln dπt\ndµ (a, s)\n\u0011\nπt(da|s) is independent of a ∈A, it holds that\nZ\nA\n\u0012Z\nA\n\u0012\nQt(s, a′) + τ ln dπt\ndµ (a′, s)\n\u0013\nπt(da′|s)\n\u0013\n(π∗−πt)(da|s) = 0.\nHence adding 0 in the form of (180) into (178) it holds that for all t ≥0\nV πt\nτ (ρ) −V π∗\nτ\n(ρ) =\n1\n1 −γ\n Z\nS×A\n∂t ln dπt\ndµ (a, s)(π∗−πt)(da|s)dπ∗\nρ (ds)\n(181)\n+\nZ\nS×A\n(Qt(s, a) −Qπt\nτ (s, a))(π∗−πt)(da|s)dπ∗\nρ (ds) −τ\nZ\nS\nKL(π∗(·|s)|πt(·|s)dπ∗\nρ (ds)\n!\n.\n(182)\nBy [12, Lemma 3.8] and Corollary 5.1, for any fixed ν ∈Πµ, the map t →KL(ν|πt) is differentiable.\nHence we have\nZ\nA\n∂t ln dπt\ndµ (s, a)(π∗−πt)(da|s) =\nZ\nA\n∂t ln dπt\ndµ (s, a)π∗(da|s) −\nZ\nA\n∂t ln dπt\ndµ (s, a)πt(da|s)\n(183)\n=\nZ\nA\n∂t ln dπt\ndµ (s, a)π∗(da|s)\n(184)\n= −d\ndt KL(π∗(·|s)|πt(·|s)),\n(185)\n\n20\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nwhere we used the conservation of mass of the policy dynamics in the second equality. Substituting this\ninto (181) we have\nV πt\nτ (ρ) −V π∗\nτ\n(ρ) =\n1\n1 −γ\n \n−d\ndt\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds)\n(186)\n+\nZ\nS×A\n(Qt(s, a) −Qπt\nτ (s, a))(π∗−πt)(da|s)dπ∗\nρ (ds) −τ\nZ\nS\nKL(π∗(·|s)|πt(·|s)dπ∗\nρ (ds)\n!\n.\n(187)\nFocusing on the second term, we have\nZ\nS×A\n(Qt(s, a) −Qπt\nτ (s, a))(π∗−πt)(da|s)dπ∗\nρ (ds)\n(188)\n≤|Qt(s, a) −Qπt\nτ (s, a)|Bb(S×A)\nZ\nS\nTV(π∗(·|s), πt(·|s))dπ∗\nρ (ds)\n(189)\n≤\n1\n√\n2|θt −θπt|\nZ\nS\nKL(π∗(·|s)|πt(·|s))\n1\n2 dπ∗\nρ (ds)\n(190)\n≤\n1\n√\n2|θt −θπt|\n\u0012Z\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds)\n\u0013 1\n2\n,\n(191)\nwhere we used Pinsker’s Inequality in the second inequality and H¨older’s inequality in the final inequality.\nNow applying Young’s inequality, there exists ϵ > 0 such that\n(192)\n|θt −θπt|\n\u0012Z\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds)\n\u0013 1\n2\n≤1\n2ϵ|θt −θπt|2 + ϵ\n2\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds).\nSubstituting this back into (186) and choosing ϵ =\n√\n2τ we have\nV πt\nτ (ρ) −V π∗\nτ\n(ρ) =\n1\n1 −γ\n \n−d\ndt\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds)\n(193)\n−τ\n2\nZ\nS\nKL(π∗(·|s)|πt(·|s)dπ∗\nρ (ds) + 1\n4τ |θt −θπt|2\n!\n.\n(194)\nRearranging, we arrive at\nd\ndt\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds) ≤−τ\n2\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds)\n(195)\n−(1 −γ)\n\u0010\nV πt\nτ (ρ) −V π∗\nτ\n(ρ)\n\u0011\n+ 1\n4τ |θt −θπt|2.\n(196)\nApplying Duhamel’s principle yields\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds) ≤e−τ\n2 t\nZ\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(197)\n−(1 −γ)\nZ t\n0\ne−τ\n2 (t−r)(V πr\nτ (ρ) −V π∗\nτ\n(ρ))dr + 1\n2τ\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr.\n(198)\nNow using that\nR t\n0 e−τ\n2 (t−r)dr = 2(1−e−τ\n2 )\nτ\n, we have\nZ\nS\nKL(π∗(·|s)|πt(·|s))dπ∗\nρ (ds) ≤e−τ\n2 t\nZ\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(199)\n−2(1 −γ)(1 −e−τ\n2 )\nτ\nmin\nr∈[0,t]\n\u0010\nV πr\nτ (ρ) −V π∗\nτ\n(ρ)\n\u0011\n+ 1\n2τ\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr.\n(200)\nRearranging, we have\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ≤\nτ\n2(1 −γ)(1 −e−τ\n2 )\n \ne−τ\n2 t\nZ\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(201)\n+ 1\n2τ\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr\n!\n.\n(202)\nwhich concludes the proof.\n□\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n21\nC.3. Proof of Theorem 6.2.\nProof. Using the chain rule and the critic dynamics in (21), we have that for all r ≥0\n1\n2ηr\nd\ndr|θr −θπr|2 = 1\nηr\n\u0012\u001cdθr\ndr , θr −θπr\n\u001d\n−\n\u001cdθπr\ndr , θr −θπr\n\u001d\u0013\n(203)\n= −⟨g(θr, πr), θr −θπr⟩−1\nηr\n\u001cdθπr\ndr , θr −θπr\n\u001d\n(204)\nLet Γ = λβ(1 −γ)(1 −√γ). Using Lemma 4.1 and the λβ-strong convexity of L(·, π; β) and recalling\nthat L(θπr, πr) = 0 for all r ≥0, it holds for all r ≥0 that\n1\n2ηt\nd\ndt|θt −θπt|2 = −⟨g(θt, πt), θt −θπt⟩−1\nηt\n\u001cdθπt\ndt , θt −θπt\n\u001d\n(205)\n≤−(1 −γ)(1 −√γ) ⟨∇θL(θt, πt; β), θt −θπt⟩−1\nηt\n\u001cdθπt\ndt , θt −θπt\n\u001d\n(206)\n≤−(1 −γ)(1 −√γ)L(θt, πt; β) −Γ\n2 |θt −θπt|2 −1\nηt\n\u001cdθπt\ndt , θt −θπt\n\u001d\n(207)\n≤−(1 −γ)(1 −√γ)L(θt, πt; β) −Γ\n2 |θt −θπt|2 + 1\n2ηt\n \f\f\f\f\ndθπt\ndt\n\f\f\f\f\n2\n+ |θt −θπt|2\n!\n(208)\n= −(1 −γ)(1 −√γ)L(θt, πt; β) −\n\u0012Γ\n2 −1\n2ηt\n\u0013\n|θt −θπt|2 + 1\n2ηt\n\f\f\f\f\ndθπt\ndt\n\f\f\f\f\n2\n,\n(209)\nwhere we used H¨older’s and Young’s inequalities in (208). Since η0 >\n1\nΓ and ηt is a non-decreasing\nfunction, it holds that ηt > 1\nΓ for all t ≥0. Hence Γ\n2 −\n1\n2ηt > 0 and thus we can drop the second term.\nMoreover the λβ-strong convexity of L(·, π; β) along with L(θπ, π; β) = 0 and ∇θL(θπ, π) = 0 for all\nπ ∈Πµ gives that\n|θt −θπt|2 ≤2\nλβ\nL(θt, πt; β).\nHence for all r ≥0 we arrive at\n1\n2ηr\nd\ndr|θr −θπr|2 ≤−Γ\n2 |θr −θπr|2 +\n1\n2ηr\n\f\f\f\f\ndθπr\ndr\n\f\f\f\f\n2\n.\n(210)\nRearranging, multiplying by e−τ(t−r) and integrating over r from 0 to t, it holds for all t ≥0 that\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤−1\nΓ\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\nd\ndr|θr −θπr|2dr + 1\nΓ\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndt\n\f\f\f\f\n2\ndr.\n(211)\nIntegrating the first term by parts (identically to (137) from the proof of Theorem 5.1), we have\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤1\nΓ\n \n−|θt −θπt|2\nηt\n+ e−τ\n2 t |θ0 −θπ0|2\nη0\n(212)\n+ τ\n2\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n|θr −θπr|2dr −\nZ t\n0\n|θr −θπr|2 e−τ\n2 (t−r) d\ndrηr\nη2r\ndr\n(213)\n+\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndr\n\f\f\f\f\n2\ndr\n!\n.\n(214)\nSince for all t ≥0 it holds that ηt ≥1 and\nd\ndtηt ≥0, we have that\nZ t\n0\n|θr −θπr|2 e−τ\n2 (t−r) d\ndrηr\nη2r\ndr ≥0.\nThus after dropping all negative terms and using that ηt ≥η0 for all t ≥0, we have\n(215)\n\u0012\n1 −\nτ\n2Γη0\n\u0013 Z t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤e−τ\n2 |θ0 −θπ0|2\nΓη0\n+\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndr\n\f\f\f\f\n2\ndr.\nSince η0 >\n1\n2Γ and τ < 1, it holds that 1 −\nτ\n2Γη0 > 0 and hence it holds that\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤e−τ\n2\n|θ0 −θπ0|2\nΓη0\n\u0010\n1 −\nτ\n2Γη0\n\u0011 +\n1\n\u0010\n1 −\nτ\n2Γη0\n\u0011\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndr\n\f\f\f\f\n2\ndr,\n(216)\n\n22\nDENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH\nwhich concludes the proof.\n□\nC.4. Proof of Theorem 6.3.\nProof. By Theorem 6.2, we have\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤e−τ\n2\n|θ0 −θπ0|2\nΓη0\n\u0010\n1 −\nτ\n2Γη0\n\u0011 +\n1\n\u0010\n1 −\nτ\n2Γη0\n\u0011\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\n\f\f\f\f\ndθπr\ndr\n\f\f\f\f\n2\ndr.\n(217)\nHence it remains to characterise the growth of the final integral. Observe that for all π ∈P(A|S),\nθπ ∈RN satisfies the least-squares optimality condition given by\n(218) θπ = arg min\nθ\nL(θ, π; β) =\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤β(da, ds)\n\u0013−1 \u0012Z\nS×A\nϕ(s, a)Qπ\nτ (s, a) β(ds, da)\n\u0013\n.\nSetting π = πt and differentiating time we arrive at\n(219)\ndθπt\ndt\n=\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤β(da, ds)\n\u0013−1 \u0012Z\nS×A\nϕ(s, a) d\ndtQπt(s, a) β(ds, da)\n\u0013\n.\nHence by Lemma 6.1, Assumption 4.2 and Assumption 4.3, for all t ≥0 it holds that\n\f\f\f\f\ndθπt\ndt\n\f\f\f\f =\n\f\f\f\f\f\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤β(da, ds)\n\u0013−1 \u0012Z\nS×A\nϕ(s, a) d\ndtQπt\nτ (s, a) β(ds, da)\n\u0013\f\f\f\f\f\n(220)\n≤\n\f\f\f\f\f\n\u0012Z\nS×A\nϕ(s, a)ϕ(s, a)⊤β(da, ds)\n\u0013−1\f\f\f\f\f\nop\n\f\f\f\f\nd\ndtQπt\nτ\n\f\f\f\f\nBb(S×A)\n(221)\n= 1\nλβ\n\f\f\f\f\nd\ndtQπt\n\f\f\f\f\nBb(S×A)\n(222)\n=\nγ\nλβ(1 −γ)\n\f\f\f\f\nZ\nS\n\u0012Z\nS×A\nAπt\nτ (s′′, a′′)∂tπt(da′′|s′′) dπt(ds′′|s′)\n\u0013\nP(ds′|·, ·)\n\f\f\f\f\nBb(S×A)\n(223)\n≤\nγ\nλβ(1 −γ) |Aπt\nτ |Bb(S×A) sup\ns∈S\n|∂tπt(·|s)|M(A) .\n(224)\nNow using Lemma A.3, it holds that\n|Aπt\nτ |Bb(S×A) sup\ns∈S\n|∂tπt(·|s)|M(A) ≤|Aπt\nτ |Bb(S×A) |At|Bb(S×A)\n(225)\n≤\n \n2 |Qπt\nτ |Bb(S×A) + 2τ\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n! \n2 |Qt|Bb(S×A) + 2τ\n\f\f\f\fln dπt\ndµ\n\f\f\f\f\nBb(S×A)\n!\n.\n(226)\nHence by Corollaries 5.1 and 5.2 and Lemma A.3, there exists α1, α2 > 0 such that\n\f\f\f\f\ndθπt\ndt\n\f\f\f\f\n2\n≤α1eα2t.\nThus Theorem 6.2 becomes\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤e−τ\n2\n|θ0 −θπ0|2\nΓη0\n\u0010\n1 −\nτ\n2Γη0\n\u0011 +\nα1\n\u0010\n1 −\nτ\n2Γη0\n\u0011\nZ t\n0\ne−τ\n2 (t−r) eα2r\nηr\ndr.\n(227)\nLet ηt = η0ek1t for any k1 > τ\n2 + α2. Then observe that\nZ t\n0\ne−τ\n2 (t−r) eα2r\nηr\ndr = 1\nη0\ne−τ\n2 t\nZ t\n0\ne( τ\n2 +α2−k1)rdr\n(228)\n≤1\nη0\ne−τ\n2 t\n \ne( τ\n2 +α2−k1)t −1\nτ\n2 + α2 −k1\n!\n(229)\n≤\ne−τ\n2 t\nη0\n\u0000 τ\n2 + α2 −k1\n\u0001,\n(230)\nhence all together it holds that\nZ t\n0\ne−τ\n2 (t−r)|θr −θπr|2dr ≤e−τ\n2\n|θ0 −θπ0|2\nΓη0\n\u0010\n1 −\nτ\n2Γη0\n\u0011 + e−τ\n2 t\nα1\n\u0000η0 −\nτ\n2Γ\n\u0001 \u0000 τ\n2 + α2 −k1\n\u0001.\n(231)\n\nCONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES\n23\nSubstituting this into the result from Theorem 6.2 concludes the proof.\n□\nC.5. Proof of Theorem 6.3. Following completely identically to the proof of Theorem 6.3, we have\n\f\f\f\f\ndθπt\ndt\n\f\f\f\f ≤\nγ\nλβ(1 −γ) |Aπt\nτ |Bb(S×A) sup\ns∈S\n|∂tπt(·|s)|M(A)\n(232)\n≤\n4\n(1 −γ)2\n\u0010\n|c|Bb(S×A) + Kt\n\u00112\n+ 4τ\n \nC1 + 2\nτ\nsup\nr∈[0,t]\n|θr| + sup\nr∈[0,t]\nKr\n!2\n.\n(233)\nThen by Corollaries 5.3 and 5.4, there exists b2 > 0 such that\n\f\f\f\ndθπt\ndt\n\f\f\f\n2\n≤d1. Hence by Theorem 6.2 we\nhave\nmin\nr∈[0,t] V πr\nτ (ρ) −V π∗\nτ\n(ρ) ≤\nτ\n2(1 −γ)(1 −e−τ\n2 )\n \ne−τ\n2 t\n Z\nS\nKL(π∗(·|s)|π0(·|s))dπ∗\nρ (ds)\n(234)\n+ d1\nZ t\n0\ne−τ\n2 (t−r) 1\nηr\ndr\n!\n.\n(235)\nSchool of Mathematics, University of Edinburgh, UK\nEmail address: ezorba@ed.ac.uk\nSchool of Mathematics, University of Edinburgh, UK\nEmail address: d.siska@ed.ac.uk\nSchool of Mathematics, University of Edinburgh, UK, The Alan Turing Institute, UK, and Simtopia, UK\nEmail address: l.szpruch@ed.ac.uk\n"},"cleaned":{"kind":"string","value":"CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Abstract. We prove the stability and global convergence of a coupled actor-critic gradient flow for infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous state and action space with linear function approximation under Q-function realisability. We consider a version of the actor critic gradient flow where the critic is updated using temporal difference (TD) learning while the policy is updated using a policy mirror descent method on a separate timescale. We demonstrate stability and exponential convergence of the actor critic flow to the optimal policy. Finally, we address the interplay of the timescale separation and entropy regularisation and its effect on stability and convergence. 1. Introduction In reinforcement learning (RL) an agent aims to learn an optimal policy that maximizes the expected cumulative reward through repeated interactions with its environment. Such methods typically involve two key components: policy evaluation and policy improvement. During policy evaluation, the advantage function corresponding to a policy, or its function approximation, is updated using state, action and reward data generated under this policy. Policy improvement then uses this approximate advantage function to update the policy, most commonly through some policy gradient method. Algorithms that explicitly combine these two components are known as actor-critic (AC) methods [13], where the actor corresponds to policy improvement and the critic to policy evaluation. There are many policy gradient methods to choose from. In the last decade trust region policy optimization (TRPO) methods [20] and methods inspired by these like PPO [21] have become increasingly well-established due to their impressive empirical performance. Largely, this is because they alleviate the difficulty in choosing appropriate step sizes for the policy gradient updates: for vanilla policy gradient even a small change in the parameter may result in large change in the policy, leading to instability, but TRPO prevents this by explicitly ensuring the KL divergence between successive updates is smaller than some tolerance. Mirror descent replaces the TRPO's hard constraint with a penalty leading to a first order method which is also ameanable to analysis. Indeed, at least for direct parametrisation, it is known to converge with sub-linear and even linear rate for entropy regularized problems (depending on exact assumptions) [9, 14, 11]. Due to the favourable analytical properties of mirror descent, in this paper we consider a version of the actor critic gradient flow where the policy is updated using a policy mirror descent method while the critic is updated using temporal difference (TD) on a separate timescale. Entropy-regularised MDPs are widely used in practice since the entropic regularizer leads to a number of desirable properties: it has a natural interpretation as something that drives exploration, it ensures that there is a unique optimal policy and it can accelerate convergence of mirror descent [11], as well as classical policy gradient [18]. However, analysing the stability and convergence of actor-critic methods in this entropy-regularized setting with general state and action spaces remains highly non-trivial due to lack of a priori bounds on the value functions. To address the actor critic methods for entropy regularised MDPs in general action spaces, a careful treatment of tools from two timescale analysis, convex analysis over both Euclidean spaces and measure spaces must be deployed. In this paper, we address precisely this challenge. We study the stability and convergence of a widely used actor-critic algorithm in which the critic is updated using Temporal Difference (TD) learning [22], and the policy is updated through Policy Mirror Descent [9]. Our analysis employs a two-timescale update scheme [3], where both the actor and critic are updated at each iteration with the critic updated on a faster timescale. Keywords: Reinforcement learning, Actor-Critic method, Entropy regularisation, Approximate gradient flow, Nonconvex optimization, Global convergence, Function approximation. 1 16 Oct 2025 2 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH 1.1. Related works. We focus on the subset of RL literature that address the convergence of coupled actor-critic algorithms. In the unregularised setting, actor-critic methods have been studied extensively. The first convergence results in the two-timescale regime established asymptotic convergence in the continuous-time limit of coupled updates ([3, 13]). Most modern research employs linear function approximation for the critic, where linear convergence rates have been obtained under various assumptions on the step-sizes of the actor and critic ([1, 24, 8]). Most closely related to our work is [25], which considers the same two-timescale actor-critic scheme in the continuous-time limit for unregularised MDPs, with an overparameterized neural network used for the critic. However, convergence to the optimal policy was only achieved up to a neighbourhood of a scaling factor, and a restarting mechanism was required to ensure the stability of the dynamics. In the entropy-regularised setting, [5, 6] address the convergence of a natural actor critic algorithm. However, the convergence and stability of these results rely on the finite cardinality of the action space in presence of entropy regularisation. 1.2. Our Contribution. Under linear Q-realisability assumption, we address the following question: \"Are actor-critic methods for entropy-regularized MDPs in general action spaces stable and convergent, and if so, at what rate?\" Our main contributions are as follows: • We study a common variant of actor-critic where the critic is updated using temporal difference (TD) learning and the policy is updated using mirror descent. Similarly to [13, 25], we analyse the coupled updates in the continuous-time limit, resulting in a dynamical system where the critic flow is captured by a semi-gradient flow and the actor flow corresponds to an approximate Fisher-Rao gradient flow over the space of probability kernels. • By combining convex analysis over the space of probability measures with classical Euclidean convex analysis, we develop a Lyapunov-based stability framework that captures the interplay between entropy regularisation and timescale separation, and establish stability of the resulting dynamics. • We prove convergence of the actor-critic dynamics for entropy-regularized MDPs with infinite action spaces. 1.3. Notation. Let (E, d) denote a Polish space (i.e., a complete separable metric space). We always equip a Polish space with its Borel sigma-field B(E). Denote by Bb(E) the space of bounded measurable functions f : E →R endowed with the supremum norm |f|Bb(E) = supx∈E |f(x)|. Denote by M(E) the Banach space of finite signed measures μ on E endowed with the total variation norm |μ|M(E) = |μ|(E), where |μ| is the total variation measure. Recall that if μ = f dρ, where ρ ∈M+(E) is a nonnegative measure and f ∈L1(E, ρ), then |μ|M(E) = |f|L1(E,ρ). Denote by P(E) ⊂M(E) the set of probability measures on E. Moreover, we denote the Euclidean norm on RN by | · | with inner product ⟨·, ·⟩. Given some A, B ∈RN×N, we denote by λmin(A) the minimum eigenvalue of A and denote A ⪰B if and only if A -B is positive semidefinite. Moreover, we denote by |A|op the operator norm of A induced by the Euclidean norm, |A|op := sup|x|̸=0 |Ax| |x| . 1.4. Entropy Regularised Markov Decision Processes. Consider an infinite horizon Markov Decision Process (S, A, P, c, γ), where the state space S and action space A are Polish, P ∈P(S|S × A) is the state transition probability kernel, c is a bounded cost function and γ ∈(0, 1) is a discount factor. Let μ ∈P(A) denote a reference probability measure and τ > 0 denote a regularisation parameter. To ease notation, for each π ∈P(A|S) we define the kernels Pπ(ds′|s) := R A P(ds′|s, a)π(da|s) and P π(ds′, da′|s, a) := P(ds′|s, a)π(da′|s′). Denoting Eπ s = Eπ δs where δs ∈P(S) denotes the Dirac measure at s ∈S, for each stochastic policy π ∈P(A|S) and s ∈S, define the regularised value function by (1) V π τ (s) = Eπ s \" ∞ X n=0 γn c(sn, an) + τ KL(π(·|sn)|μ) # ∈R ∪{∞} , where KL(π(·|s)|μ) is the Kullback-Leibler (KL) divergence of π(·|s) with respect to μ, define as KL(π(·|s)|μ) := R A ln dπ dμ(a|s)π(da|s) if π(·|s) is absolutely continuous with respect to μ, and infinity otherwise. For each π ∈P(A|S), we define the state-action value function Qπ τ ∈Bb(S × A) by (2) Qπ τ (s, a) = c(s, a) + γ Z S V π τ (s′)P(ds′|s, a) . CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 3 By the Dynamic Programming Principle, Qπ τ : S × A →R is the unique fixed point of the Bellman operator Tπ : Bb(S × A) →Bb(S × A), which for any f ∈Bb(S × A) is defined as (3) Tπf(s, a) = c(s, a) + γ Z S×A f(s′, a′)P π(ds′, da′|s, a) + τγ Z S KL(π(·|s′)|μ)P(ds′|s, a). The state-occupancy kernel dπ ∈P(S|S) is defined by (4) dπ(ds′|s) = (1 -γ) ∞ X n=0 γnP n π (ds′|s) , where P n π is the n-times product of the kernel Pπ with P 0 π(ds′|s) := δs(ds′). Moreover, for each π ∈ P(A|S) and (s, a) ∈S × A, we define the state-action occupancy kernel as (5) dπ(ds, da|s, a) = (1 -γ) ∞ X n=0 γn(P π)n(ds, da|s, a) where (P π)n is the n-times product of the kernel P π with (P π)0(ds′, da′|s, a) := δ(s,a)(ds′, da′). Given some initial state-action distribution β ∈P(S × A) with initial state distribution given by ρ(ds) = R A β(da, ds), we define the state-occupancy and state-action occupancy measures as (6) dπ ρ(ds) = Z S dπ(ds|s′)ρ(ds′), dπ β(ds, da) = Z S×A dπ(ds, da|s′, a′)β(da′, ds′). Note that for all E ∈B(S × A), by defining the linear operator Jπ : P(S × A) →P(S × A) as (7) Jπβ(E) = Z S×A P π(E|s′, a′)β(ds′, da′), it directly holds that (8) dπ β(da, ds) = (1 -γ) ∞ X n=0 γnJn π β(da, ds), with Jn π the n-fold product of the operator Jπ with J0 π = I, the identity operator on P(S × A). By choosing β = ρ ⊗π, we retrieve the classical state-action occupancy measure dπ β = dπ ρπ. For a given initial distribution ρ ∈P(S), the optimal value function is defined as (9) V ∗ τ (ρ) = min π∈P(A|S) V π τ (ρ), with V π τ (ρ) := Z S V π τ (s)ρ(ds) and we refer to π∗∈P(A|S) as the optimal policy if V ∗ τ (ρ) = V π∗ τ (ρ). Due to [11, Theorem B.1, Lemma B.2] we have the following dynamical programming principle for entropy regularised MDPs. Theorem 1.1 (Dynamical Programming Principle). Let τ > 0. The optimal value function V ∗ τ is the unique bounded solution of the following Bellman equation: V ∗ τ (s) = -τ ln Z A exp -1 τ Q∗ τ(s, a) μ(da), where Q∗ τ ∈Bb(S × A) is defined by Q∗ τ(s, a) = c(s, a) + γ Z S V ∗ τ (s′)P(ds′|s, a) , ∀(s, a) ∈S × A . Moreover, there is an optimal policy π∗ τ ∈P(A|S) given by π∗ τ(da|s) = exp -1 τ (Q∗ τ(s, a) -V ∗ τ (s)) μ(da) , ∀s ∈S. Finally, the value function V π τ is the unique bounded solution of the following Bellman equation for all s ∈S V π τ (s) = Z A Qπ τ (s, a) + τ ln dπ dμ(a, s) π(da|s) . Theorem 1.1 suggests that, without loss of generality, it suffices to minimise (9) over the class of policies that are equivalent to the reference measure μ. Definition 1.1 (Admissible Policies). Let Πμ denote the class of policies for which there exists f ∈ Bb(S × A) with π(da|s) = exp(f(s, a)) R A exp(f(s, a))μ(da)μ(da). 4 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH The performance difference lemma, first introduced for tabular unregularised MDPs, has become fundamental in the analysis of MDPs as it acts a substitute for the strong convexity of the π 7→V π τ if the state-occupancy measure dπ ρ is ignored (e.g [10], [25], [9]). By virtue of [11], we have the following performance difference for entropy regularised MDPs in Polish state and action spaces. Lemma 1.1 (Performance difference). For all ρ ∈P(S) and π, π′ ∈Πμ, V π τ (ρ) -V π′ τ (ρ) = 1 1 -γ Z S Z A Qπ′ τ (s, a) + τ ln dπ′ dμ (a, s) (π -π′)(da|s) + τ KL(π(·|s)|π′(·|s)) dπ ρ(ds) . 2. Mirror-Descent and the Fisher-Rao Gradient flow Defining the soft advantage function as Aπ τ (s, a) := Qπ τ (s, a) + τ ln dπ dμ(s, a) -V π τ (s), then for some λ > 0 and π0 ∈Πμ, the Policy Mirror Descent update rule reads as πn+1(·|s) = arg min m∈P(A) Z A Aπn τ (s, a)(m(da) -πn(da|s)) + 1 λ KL(m|πn(·|s)) (10) = arg min m∈P(A) Z A Qπn τ (s, a) + τ ln dπn dμ (s, a) (m(da) -πn(da|s)) + 1 λ KL(m|πn(·|s)) . (11) [7] shows that the pointwise optimisation is achieved by (12) dπn+1 dπn (a, s) = exp -λAπn τ (s, a) R A exp (-λAπn τ (s, a)) πn(da|s). Observe that for any π ∈P(A|S), it holds that R A Aπ τ (s, a)π(da|s) = 0. Hence taking the logarithm of (12) we have ln dπn+1 dμ (s, a) -ln dπn dμ (s, a) = -λAπn τ (s, a) -ln Z A e-λAπn τ (s,a)πn(da|s). Interpolating in the time variable and letting λ →0 we retrieve the Fisher-Rao gradient flow for the policies (13) ∂t ln dπt dμ (s, a) = - Aπt τ (s, a) - Z A Aπt τ (s, a)πt(da|s) = -Aπt τ (s, a). Note that the soft advantage formally corresponds to the functional derivative of the value function with respect to the policy πn and thus (13) can be seen as a gradient flow of the value function over the space of kernels P(A|S) (see [11] for a detailed description of the functional derivative). In the case where the advantage function is fully accessible for all t ≥0, [11][Theorem 2.8] shows that the entropy regularisation in the value function induces an exponential convergence to the optimal policy. In the following section we define the approximate Fisher Rao dynamics arising from approximating the advantage for all t ≥0. 3. Actor Critic Methods Given some feature mapping φ : S × A →RN, we parametrise the state-action value function as Q(s, a; θ) := ⟨θ, φ(s, a)⟩. Moreover, we denote the approximate soft Advantage function as (14) A(s, a; θ) = Q(s, a; θ) + τ ln dπ dμ(s, a) - Z A Q(s, a; θ) + τ ln dπ dμ(s, a) π(da|s). The Mean Squared Bellman Error (MSBE) is defined as (15) MSBE(θ, π) = 1 2 Z S×A (Q(s, a; θ) -TπQ(s, a; θ))2dπ β(da, ds) where dπ β ∈P(S × A) is the state-action occupancy measure defined in (6). Given that β ∈P(S × A) has full support, by (3) it holds that MSBE(θ, π) = 0 if and only if Q(s, a; θ) = Qπ τ (s, a) for all s ∈S and a ∈A. Hence one approach to implementing the Policy Mirror Descent updates is to calculate the optimal parameters for Q(s, a; θ) by minimising the MSBE at each policy mirror descent iteration (16) θn+1 = arg min θ∈RN MSBE(θ, πn), CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 5 (17) dπn+1 dπn (a, s) = exp -λA(s, a; θn+1) R A exp (-λA(s, a; θn+1)) πn(da|s). To avoid fully solving the optimisation in (16) for each policy update, one can update the critic using a semi-gradient descent on a different timescale to the policy update. Let hn, λn > 0 be the step-sizes of the critic and actor respectively at iteration n ≥0. Let the semi-gradient g : RN × P(A|S) →RN of the MSBE with respect to θ be (18) g(θ, π) := Z S×A (Q(s, a; θ) -TπQ(s, a; θ))φ(s, a)dπ β(da, ds). The two-timescale actor-critic Mirror Descent scheme reads as (19) θn+1 = θn -hng(θn, πn), dπn+1 dπn (a, s) = exp -λnA(s, a; θn+1) R A exp (-λnA(s, a; θn+1)) πn(da|s). (20) where timescale separation ηn := hn λn > 1 ensures that the critic is updated on a much faster timescale than the policy to improve the local estimation of the policy updates. As pointed out in [25], even with the KL penalty in (10) the critic may still be far away from the true state-action value function, resulting in unstable updates. 4. Dynamics In this paper, we study the stability and convergence of the two-timescale actor-critic Mirror Descent scheme in the continuous-time limit. Let Qt(s, a) := Q(s, a; θt) and At(s, a) := A(s, a; θt). Let η : [0, ∞) →[1, ∞) be a non-decreasing function representing the timescale separation, then for some θ0 ∈ RN and π0 ∈Πμ we have the following coupled dynamics (21) dθt dt = -ηtg(θt, πt) ∂tπt(da|s) = -At(s, a)πt(da|s) (22) where g : RN × P(A|S) is the semi-gradient of the MSBE defined in (18). We refer to (22) as the Approximate Fisher Rao Gradient flow. We perform our analysis under the following assumptions. Assumption 4.1 (Qπ τ -realisability). For all π ∈Πμ and (s, a) ∈S × A, there exists θπ ∈RN such that Qπ(s, a) = ⟨θπ, φ(s, a)⟩. A simple example of when this holds is in the tabular case, where one can choose φ to be a one-hot encoding of the state-action space. Moreover, all linear MDPs are Qπ-realisable. In a linear MDP there exists exists φ : S × A →RN, w ∈RN and a sequence {ψi}N i=1 with ψi ∈M(S) such that for all (s, a) ∈S × A, c(s, a) = ⟨w, φ(s, a)⟩, P(ds′ | s, a) = N X i=1 φi(s, a)ψi(ds′). In this case it holds that (θπ)i = wi + R S V π(s′)ψi(ds′). Assumption 4.1 can be seen as a convention to omit function approximation errors in the final convergence results. This assumption, or the presence of approximation errors in convergence results, are widely present in the actor-critic literature ([5], [24], [23], [6], [8], [19]). More recently, [16] derives some weaker ordering conditions in the bandit case (empty state space) which guarantee the convergence of soft-max policy gradient in the tabular setting beyond realisability. However as of now it is unclear how this applies to MDPs and also fundamentally depends on the finite cardinality of the action space. By [4], Assumption 4.1 holds in the limit N →∞when φi are the basis functions of L2(ρ ⊗μ) for some ρ ⊗μ ∈P(S × A). However, analysis in such a Hilbert space becomes more involved and intricate and is the result of ongoing work. Assumption 4.2. For all (s, a) ∈S × A it holds that |φ(s, a)| ≤1. Assumption 4.2 is purely for convention and is without loss of generality in the finite-dimensional case. 6 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Assumption 4.3. Let β ∈P(S × A) be fixed. Then λβ := λmin Z S×A φ(s, a)φ(s, a)⊤β(ds da) > 0. Note that unlike the analogous assumptions imposed in [8], Assumption 4.3 is independent of the policy. This property allows us to remove any dependence on the continuity of eigenvalues. Definition 4.1. For all π ∈Πμ and ζ ∈P(S × A), the squared loss with respect to ζ is defined as (23) L(θ, π; ζ) = 1 2 Z S×A (⟨θ, φ(s, a)⟩-Qπ(s, a))2ζ(da, ds) where Qπ τ is defined in (2). A straightforward calculation given in Lemma A.4 shows that due to Lemma 5.1 and Assumption 4.3, for any π ∈Πμ it holds that L(·, π; dπ β) is (1 -γ)λβ-strongly convex. The following result then connects the geometry of the semi-gradient of the MSBE and the gradient of L(θ, π; β), which can be seen as an extension of Lemma 3 of [2] to the current entropy regularised setting and where the measure of integration in the MSBE is not necessarily stationary. Lemma 4.1. Let Assumption 4.1 hold. Then for all θ ∈RN and π ∈Πμ it holds that (24) -⟨g(θ, π), θ -θπ⟩≤-(1 -√γ)(1 -γ) ⟨∇θL(θ, π; β), θ -θπ⟩ with ∇θL(θ, π; β) = Z S×A (⟨θ, φ(s, a)⟩-Qπ τ (s, a))φ(s, a)β(da, ds). See Appendix A.1 for a proof. 5. Stability In this section we analyse the stability of the coupled Actor Critic flow. Throughout this section, to ease notation we let Γ := λβ(1 -γ)(1 -√γ), Kt := sup s∈S KL(πt(·|s)|μ), with λβ > 0 the constant from Assumption 4.3. The following lemma establishes properties of the state-action occupancy measure defined in (6) and which are useful in the proofs. Lemma 5.1. For all π ∈P(A|S), β ∈P(S × A) and E ∈B(S × A) it holds that (25) dπ Jπβ(E) = Jπdπ β(E). Moreover, for all γ ∈(0, 1) we have (26) dπ β(E) -γdπ Jπβ(E) = (1 -γ)β(E). See Appendix A.2 for a proof. Lemma 5.2 then establishes the effect of the coupling and timescale separation in the actor-critic flow and its effect on the stability of the critic parameters. Lemma 5.2. Let Assumptions 4.2 and 4.3 hold. Then for all t ≥0 it holds that (27) 1 2ηt d dt|θt|2 ≤-Γ 2 |θt|2 + τ 2γ2K2 t Γ + |c|2 Bb(S×A) Γ See Appendix B.1 for a proof. By connecting the result from Lemma 5.2 with the approximate Fisher Rao gradient flow, we are able to establish a Gronwall-type inequality for the KL divergence of the policies with respect to the reference measure. Theorem 5.1. Let Assumptions 4.2 and 4.3 hold. Let η0 > τ Γ. Then there exists constants a1 = a1 τ, η0, γ, λβ, |c|Bb(S×A), dπ0 dμ Bb(S×A) ! > 0 and a2 = a2(τ, η0, γ, λβ) > 0 such that for all γ ∈(0, 1) and t ≥0 it holds that (28) K2 t ≤a1 + a2 Z t 0 e-τ(t-r)K2 r dr. CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 7 See Appendix B.2 for a proof. Through applications of Gronwall's Lemma (Lemma A.1), two direct corollaries of Theorem 5.1 show that the KL divergence of the policies with respect to the reference measure and the critic parameters do not blow up in finite time. Corollary 5.1 (Stability). Under the same assumptions as Theorem 5.1, for all γ ∈(0, 1), s ∈S and t ≥0 it holds that (29) KL(πt(·|s)|μ)2 ≤a1ea2t. Corollary 5.2. Under the same assumptions as Theorem 5.1, suppose that there exists α > 0 such that d dtηt ≤αηt. Then for all γ ∈(0, 1) there exists r1, r2 > 0 such that for all t ≥0 it holds that (30) |θt| ≤r1er2t. See Appendix B.3 and B.4 for the proofs. Corollaries 5.3 and 5.4 then show that if the MDP is sufficiently regularised through a sufficiently small discounting factor, the KL divergence of the policies with respect to the reference measure remains uniformly bounded along the flow. Corollary 5.3 (Uniform boundedness). Under the same assumptions as Theorem 5.1, for γ ∈(0, 1) such that 64γ2 Γ2-Γτ η0 0 such that for all t ≥0 it holds that (32) |θt| ≤R See Appendix B.5 and B.6 for the proofs. 6. Convergence In this section we will present three convergence results of the coupled actor-critic flow. Firstly, we characterise the time derivative of the state-action value function along the approximate gradient flow for the policies. Lemma 6.1. For all t ≥0 and (s, a) ∈S × A, it holds that (33) d dtQπt τ (s, a) = γ 1 -γ Z S Z S×A Aπt τ (s′′, a′′)∂tπt(da′′|s′′)dπt(ds′′|s′) P(ds′|s, a) See Appendix C.1 for a proof. Observe that in the exact setting, (13), we obtain the dissipative property of {Qπt τ }t≥0 along the flow d dtQπt τ (s, a) = -γ 1 -γ Z S Z S×A Aπt τ (s′′, a′′)2dπt(ds′′|s′) P(ds′|s, a) ≤0. Furthermore, Theorem 6.1 shows that the actor-critic flow maintains the exponential convergence to the optimal policy induced by the τ-regularisation up to a error term arising from not solving the critic to full accuracy. Theorem 6.1. Let {πt, θt}t≥0 be the trajectories of the actor critic flow. Let Assumptions 4.1 and 4.2 hold. Then for all t > 0 it holds that min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ≤ τ 2(1 -γ)(1 -e-τ 2 t) e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (34) + 1 2τ Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ! (35) See Appendix C.2 for a proof. Note that by Theorem 1.1, it holds that KL(π∗(·|s)|π0(·|s)) 1 Γ and 0 0 such that Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤b1e-τ 2 t + b2 Z t 0 e-τ 2 (t-r) 1 ηr dθπr dt 2 dr. (36) See Appendix C.3 for a proof. The following two results then connects Corollaries 5.1 and 5.2, Lemma 6.1 and Theorem 6.1 to demonstrate an exponential convergence to the optimal policy for all γ ∈(0, 1) when the critic is updated sufficiently fast. Theorem 6.3. Under the same assumptions as Theorem 6.2, there exists k1 > 0 with ηt = η0ek1t and k2 > 0 such that for all γ ∈(0, 1) and t > 0 it holds that min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ≤ τe-τ 2 t 2(1 -γ)(1 -e-τ 2 t) Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) + k2 2τ ! (37) See Appendix C.4 for a proof. A direct consequence arising from the proof of Theorem 6.3 shows that if the MDP is sufficiently regularised through the same small discounting factor condition as in Corollary 5.3, one can arrive at convergence for a much more general class of functions ηt. Corollary 6.1. Under the same assumptions as Theorem 6.2, for γ ∈(0, 1) such that 2 √ 2γ q Γ2-Γτ η0 0 such that for all t ≥0, min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ≤ τ 2(1 -γ)(1 -e-τ 2 t) e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (38) + d1 Z t 0 e-τ 2 (t-r) 1 ηr dr ! . (39) See Appendix C.3 for a proof. For example, suppose the small discounting factor condition is satisfied, choosing ηt = t 1 2 + η0 with η0 > 1 Γ and τ = 0.5, it can be shown that asymptotically (40) min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ∼1 √ t. 7. Limitations In this work, we only study the continuous-time dynamics of the actor-critic algorithm. Although this formulation gives insights into the discrete counterpart, a rigorous treatment of the discrete-time setting is more realistic for practical purposes and is left for future research. Moreover, for the purposes of analysis our critic approximation is linear while in practice non-linear neural networks are used to approximate the critic. Finally, our work assumes all integrals are evaluated exactly, in particular the semi-gradient (18). In practice these would need to be estimated from samples leading to additional Monte-Carlo errors. To fully analyse this is left for future work. 8. Acknowledgements DZ was supported by the EPSRC Centre for Doctoral Training in Mathematical Modelling, Analysis and Computation (MAC-MIGS) funded by the UK Engineering and Physical Sciences Research Council (grant EP/S023291/1), Heriot-Watt University and the . We acknowledge funding from the UKRI Prosperity Partnerships grant APP43592: AI2 - Assurance and Insurance for Artificial Intelligence, which supported this work. CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 9 References [1] Anas Barakat, Pascal Bianchi, and Julien Lehmann, Analysis of a target-based actor-critic algorithm with linear function approximation, Proceedings of The 25th International Conference on Artificial Intelligence and Statistics (Gustau Camps-Valls, Francisco J. R. Ruiz, and Isabel Valera, eds.), Proceedings of Machine Learning Research, vol. 151, PMLR, 28-30 Mar 2022, pp. 991-1040. 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[24] Shangtong Zhang, Bo Liu, Hengshuai Yao, and Shimon Whiteson, Provably convergent two-timescale off-policy actorcritic with function approximation, Proceedings of the 37th International Conference on Machine Learning (Hal Daum ́e III and Aarti Singh, eds.), Proceedings of Machine Learning Research, vol. 119, PMLR, 13-18 Jul 2020, pp. 1120411213. [25] Yufeng Zhang, Siyu Chen, Zhuoran Yang, Michael Jordan, and Zhaoran Wang, Wasserstein flow meets replicator dynamics: A mean-field analysis of representation learning in actor-critic, Advances in Neural Information Processing Systems (A. Beygelzimer, Y. Dauphin, P. Liang, and J. Wortman Vaughan, eds.), 2021. 10 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Appendix A. Technical details In this section, we present some calculations that will be used in the proofs of the main results. Lemma A.1 (Gronwall). Let λ(s) ≥0, a = a(s), b = b(s) and y = y(s) be locally integrable, real-valued functions defined on [0, T] such that y is also locally integrable and for almost all s ∈[0, T], y(s) + a(s) ≤b(s) + Z s 0 λ(t)y(t)dt. Then y(s) + a(s) ≤b(s) + Z s 0 λ(t) Z t 0 λ(r)(b(r) -a(r))dr dt, ∀s ∈[0, T]. Furthermore, if b is monotone increasing and a is non-negative, then y(s) + a(s) ≤b(s)e R s 0 λ(r)dr, ∀s ∈[0, T]. Lemma A.2. For some β ∈P(S × A), let dπ β ∈P(S × A) be the state-action occupancy measure. Moreover let κ(ds, da, ds′, da′) := P π(ds′, da′|s, a)dπ β(ds, da). Then for any π ∈Πμ and any integrable f : S × A →R, it holds that (41) Z S×A×S×A f(s, a)f(s′, a′)κ(ds, da, ds′, da′) ≤ 1 √γ Z S×A f(s, a)2dπ β(ds, da) Proof. By H ̈older's inequality, it holds that Z S×A×S×A f(s, a)f(s′, a′)κ(ds, da, ds′, da′) (42) ≤ Z S×A×S×A f(s, a)2κ(ds, da, ds′, da′) 1 2 Z S×A×S×A f(s′, a′)2κ(ds, da, ds′, da′) 1 2 . (43) Moreover, observe that Z S×A×S×A f(s, a)2κ(ds, da, ds′, da′) = Z S×A Z S×A P π(ds′, da′|s, a) f(s, a)2dπ β(ds, da) (44) = Z S×A f(s, a)2dπ β(ds, da), (45) hence (42) becomes Z S×A×S×A f(s, a)2κ(ds, da, ds′, da′) 1 2 Z S×A×S×A f(s′, a′)2κ(ds, da, ds′, da′) 1 2 (46) ≤ Z S×A f(s, a)2dπ β(ds, da) 1 2 Z S×A×S×A f(s′, a′)2κ(ds, da, ds′, da′) 1 2 . (47) Now by the first part of Lemma 5.1, it holds that Z S×A×S×A f(s′, a′)2κ(ds, da, ds′, da′) = Z S×A×S×A f(s′, a′)2P π(ds′, da′|s, a)dπ β(ds, da) (48) = Z S×A f(s, a)2dπ Jπβ(ds, da), (49) where Jπ : P(S × A) →P(S × A) is defined in (7). Then by the second part of Lemma 5.1 we have Z S×A f(s, a)2dπ β(ds, da) 1 2 Z S×A×S×A f(s′, a′)2κ(ds, da, ds′, da′) 1 2 (50) ≤ Z S×A f(s, a)2dπ β(ds, da) 1 2 Z S×A f(s, a)2dπ Jπβ(ds, da) 1 2 (51) ≤ 1 √γ Z S×A f(s, a)2dπ β(ds, da), (52) which concludes the proof. □ CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 11 Lemma A.3. For some θ0 ∈RN and π0 ∈Πμ, let {πt, θt}t≥0 be the trajectory of coupled actor-critic flow. Moreover let Kt = sups∈S KL(πt(·|s)|μ). There exists C1 > 0 such that for all t ≥0 it holds that sup s∈S |∂tπt(·|s)|M(A) ≤|At|Bb(S×A) , (53) |At|Bb(S×A) ≤2 |Qt|Bb(S×A) + 2τ ln dπt dμ Bb(S×A) , (54) |Qπt τ |Bb(S×A) ≤ 1 1 -γ |c|Bb(S×A) + τγKt , (55) ln dπt dμ Bb(S×A) ≤C1 + 2 τ sup r∈[0,t] |θr| + sup r∈[0,t] Kr. (56) Proof. The first claim sups∈S |∂tπt(·|s)|M(A) ≤|Aπt τ |Bb(S×A) follows trivially from the definition of the approximate Fisher Rao gradient flow defined in (22). Moreover, it holds that |At|Bb(S×A) = Qt + τ ln dπt dμ - Z A Qt(·, a) + τ ln dπt dμ (·, a) πt(da|·) Bb(S×A) (57) ≤2 Qt + τ ln dπt dμ Bb(S×A) (58) ≤2 |Qt|Bb(S×A) + 2τ ln dπt dμ Bb(S×A) (59) where we used the triangle inequality in the final inequality. Moreover, the state-action value function Qπt τ is a fixed point of the Bellman operator defined in (3). Hence, for all (s, a) ∈S × A, we have Qπt τ (s, a) = c(s, a) + γ Z S×A Qπt τ (s′, a′) P πt(ds′, da′|s, a) + τγ Z S KL(πt(·|s′)∥μ) P(ds′|s, a). (60) Taking absolute values and using the triangle inequality we have |Qπt τ (s, a)| ≤|c|Bb(S×A) + γ |Qπt τ |Bb(S×A) + τγ sup s′∈S KL(πt(·|s′)∥μ) (61) = |c|Bb(S×A) + γ |Qπt τ |Bb(S×A) + τγKt. (62) Taking the supremum over (s, a) ∈S × A on the left-hand side yields (63) |Qπt τ |Bb(S×A) ≤|c|Bb(S×A) + γ |Qπt τ |Bb(S×A) + τγKt. Rearranging gives (64) (1 -γ) |Qπt τ |Bb(S×A) ≤|c|Bb(S×A) + τγKt, which is the desired bound. Recall the approximate Fisher-Rao gradient flow for the policies {πt}t≥0, which for all t ≥0 and for all (s, a) ∈S × A is given by (65) ∂t ln dπt dμ (s, a) = - Qt(s, a) + τ ln dπt dμ (a, s) - Z A Qt(s, a′) + τ ln dπt dμ (a′, s) πt(da′|s) . Duhamel's principle yields for all t ≥0 that ln dπt dμ (s, a) = e-τt ln dπ0 dμ (a, s) + Z t 0 e-τ(t-r) Z A Qr(s, a′)πr(da′|s) -Qr(s, a) dr (66) + τ Z t 0 e-τ(t-r) KL(πr(·|s)|μ)dr. (67) 12 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Since π0 ∈Πμ, there exists C1 ≥1 such that ln dπ0 dμ Bb(S×A) ≤C1. Then by Assumption 4.2 we have that for all t ≥0, ln dπt dμ (s, a) ≤C1 + Z t 0 e-τ(t-r) Z A Qr(s, a′)πr(da′|s) -Qr(s, a) dr (68) + τ Z t 0 e-τ(t-r) KL(πr(·|s)∥μ) dr (69) ≤C1 + 2 Z t 0 e-τ(t-r) |θr| dr + τ Z t 0 e-τ(t-r)Kr dr (70) ≤C1 + 2 τ sup r∈[0,t] |θr| + sup r∈[0,t] Kr, (71) where in the last inequality we used R t 0 e-τ(t-r)dr ≤1 τ . Taking the supremum over (s, a) ∈S × A yields (72) ln dπt dμ Bb(S×A) ≤C1 + 2 τ sup r∈[0,t] |θr| + sup r∈[0,t] Kr, which is the desired bound. □ Lemma A.4. Let Assumption 4.3 hold. Then for all π ∈Πμ, it holds that L(·, π; dπ β) is λβ(1-γ)-strongly convex. Proof. For any ξ ∈P(S × A), let Σξ := R S×A φ(s, a)φ(s, a)⊤ξ(ds, da) ∈RN×N. Then by Lemma 5.1 and Assumption 4.3 it holds that Σdπ β ⪰(1 -γ)Σβ ⪰(1 -γ)λβI and thus L(·, π; dπ β) is λβ(1 -γ)-strongly convex. □ A.1. Proof of Lemma 4.1. Proof. Recall that Q(s, a) = ⟨θ, φ(s, a)⟩for some θ ∈RN and that for all π ∈Πμ, there exists θπ ∈RN such that Qπ(s, a) = ⟨θπ, φ(s, a)⟩by Assumption 4.1. Then by definition of the semi-gradient of the MSBE g : RN × P(A|S) →RN in (18), it holds that ⟨g(θ, π), θ -θπ⟩= Z S×A (Q(s, a) -TπQ(s, a)) φ(s, a)dπ β(da, ds), θ -θπ (73) = Z S×A (Q(s, a) -Qπ τ (s, a))φ(s, a)dπ β(da, ds), θ -θπ (74) + Z S×A (Qπ τ (s, a) -TπQ(s, a)φ(s, a)dπ β(da, ds), θ -θπ (75) = Z S×A (Q(s, a) -Qπ τ (s, a))φ(s, a)dπ β(da, ds), θ -θπ (76) -γ Z S×A×S×A (Q(s′, a′) -Qπ τ (s′, a′))φ(s, a)P π(ds′, da′|s, a)dπ β(ds, da), θ -θπ , (77) where we added and subtracted the true state-action value function Qπ τ ∈Bb(S × A) in the second equality and used the fact that it is a fixed point of the Bellman operator defined in (3). To ease notation, let ε(s, a) := Q(s, a) -Qπ τ (s, a). Multiplying both sides by -1 and using the associativity of CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 13 the inner product, we have -⟨g(θ, π), θ -θπ⟩ (78) = - Z S×A ε(s, a)φ(s, a)dπ β(da, ds), θ -θπ (79) + γ Z S×A ε(s′, a′)φ(s, a)P π(ds′, da′|s, a)dπ β(ds, da), θ -θπ (80) = - Z S×A ε(s, a) ⟨φ(s, a), θ -θπ⟩dπ β(da, ds) (81) + γ Z S×A ε(s′, a′) ⟨φ(s, a), θ -θπ⟩P π(ds′, da′|s, a)dπ β(ds, da) (82) = - Z S×A ε(s, a)2dπ β(da, ds) (83) + γ Z S×A×S×A ε(s, a)ε(s′, a′)P π(ds′, da′|s, a)dπ β(ds, da) (84) = I(1) + γI(2). (85) Now applying Lemma A.2 to I(2) we have I(2) := Z S×A×S×A ε(s, a)ε(s′, a′)P π(ds′, da′|s, a)dπ β(ds, da) (86) ≤ 1 √γ Z S×A ε(s, a)2dπ β(ds, da). (87) Thus it holds that -⟨g(θ, π), θ -θπ⟩≤I(1) + γI(2) (88) ≤-(1 -√γ) Z S×A ε(s, a)2dπ β(da, ds) (89) = -(1 -√γ) Z S×A (Q(s, a) -Qπ τ (s, a))2dπ β(da, ds) (90) = -(1 -√γ) ∇θL(θ, π; dπ β), θ -θπ , (91) where the last inequality follows from the Assumption 4.1 and the definition of Q(s, a) = ⟨θ, φ(s, a)⟩. □ A.2. Proof of Lemma 5.1. Proof. For any β ∈P(S × A), π ∈P(A|S) and E ∈B(S × A), it holds that dπ Jπβ(E) = (1 -γ) ∞ X n=0 γn(Jn π Jπβ)(E) (92) = Jπdπ β(E) (93) where we just used the associativity of the operator Jπ. Furthermore by letting m = n + 1 it holds that dπ Jπβ(E) = (1 -γ) ∞ X n=0 γnJn+1 π β(E) (94) = (1 -γ) ∞ X m=1 γm-1Jm π β(E) (95) = 1 -γ γ ∞ X m=1 γmJm π β(E) (96) = 1 γ (dπ β(E) -(1 -γ)β(E)). (97) Rearranging concludes the proof. □ 14 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Appendix B. Proof of Regularity Results B.1. Proof of Lemma 5.2. Proof. Consider the following equation 1 2ηt d dt |θt|2 = 1 ηt d dtθt, θt (98) = -⟨g(θt, πt), θt⟩ (99) = - Z S×A (Qt(s, a) -T πtQt(s, a)) φ(s, a) dπt β (da, ds), θt (100) = - Z S×A Qt(s, a)φ(s, a) dπt β (da, ds), θt (101) + Z S×A T πtQt(s, a)φ(s, a) dπt β (da, ds), θt (102) := -J(1) t + J(2) t (103) where we used the θt dynamics from (21) in the second equality and the definition of the semi-gradient in the third equality. For any π ∈Πμ, let Σπ ∈RN×N be (104) Σπ = Z S×A φ(s, a)φ(s, a)⊤dπ β(da, ds). Then by definition we have that Qt(s, a) = ⟨θt, φ(s, a)⟩, hence for J(1) t we have J(1) t = Z S×A Qt(s, a)φ(s, a) dπt β (da, ds), θt (105) = Z S×A ⟨θt, φ(s, a)⟩φ(s, a)dπt β (da, ds), θt (106) = θt, Z S×A φ(s, a)φ(s, a)⊤dπt β (da, ds) θt (107) = ⟨θt, Σπtθt⟩ (108) Now dealing with J(1) t , expanding the Bellman operator defined in (3) we have J(2) t = Z S×A TπtQt(s, a)φ(s, a) dπt β (da, ds), θt (109) = Z S×A c(s, a)φ(s, a)dπt β (da, ds), θt (110) + γ Z S×A ⟨θt, φ(s′, a′)⟩φ(s, a)P πt(ds′, da′|s, a)dπt β (da, ds), θt (111) + τγ Z S×A Z S KL(πt(·|s′), μ)P(ds′|s, a)φ(s, a)dπt β (da, ds) , θt (112) ≤|c|Bb(S×A)|θt| + γI(1) t + τγI(2) t (113) where we defined I(1) t = Z S×A ⟨θt, φ(s′, a′)⟩φ(s, a)P πt(ds′, da′|s, a)dπt β (da, ds), θt , I(2) t = Z S×A Z S KL(πt(·|s′), μ)P(ds′|s, a)φ(s, a)dπ β(da, ds) , θt . Moreover, to ease notation let Kt := sup s∈S KL(πt(·|s)|μ) CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 15 and temporarily let κt(ds, da, ds′, da′) := P πt(ds′, da′|s, a)dπt β (da, ds). Now focusing on I(1) t , it holds that I(1) t = Z S×A×S×A ⟨θt, φ(s′, a′)⟩φ(s, a)κt(da′, ds′, da, ds), θt (114) = Z S×A×S×A ⟨θt, φ(s, a)⟩⟨θt, φ(s′, a′)⟩κt(ds′, da′, ds, da). (115) Now using Lemma A.2 with f = ⟨θ, φ(·, ·)⟩we have I(1) t ≤ 1 √γ Z S×A ⟨θt, φ(s, a)⟩2 dπt β (ds, da) 1 2 Z S×A ⟨θt, φ(s, a)⟩2 dπt β (ds, da) 1 2 (116) = 1 √γ Z S×A ⟨θt, φ(s, a)⟩2 dπt β (ds, da) (117) = 1 √γ ⟨θt, Σπtθt⟩. (118) Thus all together it holds that (119) γI(1) t ≤√γ ⟨θt, Σπtθt⟩. Now focusing on I(2) t , we have I(2) t = Z S×A Z S KL(πt(·|s′), μ)P(ds′|s, a) φ(s, a)dπt β (da, ds), θt (120) ≤Kt Z S×A φ(s, a)dπt β (ds, da) |θt| (121) ≤Kt|θt| (122) where we used Assumption 4.2 in the final inequality. Hence along with (108), (98) becomes 1 2ηt d dt|θt|2 ≤-J(1) t + J(2) t (123) ≤-⟨θt, Σπtθt⟩+ |c|Bb(S×A)|θt| + γI(1) t + τγI(2) t (124) ≤-⟨θt, Σπtθt⟩+ √γ ⟨θt, Σπtθt⟩+ |c|Bb(S×A)|θt| + τγKt|θt| (125) = -(1 -√γ) ⟨θt, Σπtθt⟩+ |c|Bb(S×A) + τγKt |θt|. (126) Observe that by (26) and Assumption 4.2, Σπ ∈RN×N is positive definite for all π ∈P(A|S), hence it holds that (127) ⟨θt, Σπtθt⟩≥(1 -γ)λβ |θt|2 . Therefore (123) becomes (128) 1 2ηt d dt|θt|2 ≤-(1 -√γ)(1 -γ)λβ |θt|2 + (|c|Bb(S×A) + τγKt)|θt| Let Γ := λβ(1 -γ)(1 -√γ). By Young's inequality, there exists ε > 0 such that 1 2ηt d dt|θt|2 ≤-Γ|θt|2 + ε 2|θt|2 + (|c|Bb(S×A) + τγKt)2 2ε (129) ≤-Γ|θt|2 + ε 2|θt|2 + |c|2 Bb(S×A) + τ 2γ2K2 t ε , (130) where we used the identity (a + b)2 ≤2a2 + 2b2. Choosing ε = Γ we arrive at (131) 1 2ηt d dt|θt|2 ≤-Γ 2 |θt|2 + τ 2γ2K2 t Γ + |c|2 Bb(S×A) Γ which concludes the proof. □ 16 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH B.2. Proof of Theorem 5.1. Proof. By Lemma 5.2, we have that for all r ≥0 (132) 1 2ηr d dr|θr|2 ≤-Γ 2 |θr|2 + τ 2γ2K2 r Γ + |c|2 Bb(S×A) Γ . Rearranging, it holds that for all t ≥0 |θr|2 ≤-1 Γηr d dr|θr|2 + 2|c|2 Bb(S×A) + 2τ 2γ2K2 r Γ2 . (133) Multiplying both sides by e-τ(t-r) and integrating over r from 0 to t we have that for all t ≥0 Z t 0 e-τ(t-r)|θr|2dr ≤-1 Γ Z t 0 e-τ(t-r) 1 ηr d dr|θr|2dr + 2|c|2 Bb(S×A) Γ2 Z t 0 e-τ(t-r)dr (134) + 2τ 2γ2 Γ2 Z t 0 e-τ(t-r)K2 rdr (135) ≤-1 Γ Z t 0 e-τ(t-r) 1 ηr d dr|θr|2dr + 2|c|2 Bb(S×A) Γ2τ + 2τ 2γ2 Γ2 Z t 0 e-τ(t-r)K2 rdr, (136) where we used that R t 0 e-τ(t-r)dr ≤1 τ . Integrating the first term by parts, we have - Z t 0 e-τ(t-r) 1 ηr d dr|θr|2dr = -|θt|2 ηt + e-τt |θ0|2 η0 + τ Z t 0 |θr|2 e-τ(t-r) ηr dr (137) - Z t 0 |θr|2 e-τ(t-r) d drηr η2r dr. (138) Since by definition we have that for all t ≥0, ηt ≥1 and d dtηt ≥0 it holds that (139) Z t 0 |θr|2 e-τ(t-r) d drηr η2r dr ≥0. Hence dropping the negative terms on the right hand side of (137) and using that ηt ≥η0 for all t ≥0, we have -1 Γ Z t 0 e-τ(t-r) 1 ηr d dr|θr|2dr ≤e-τt |θ0|2 Γη0 + τ Γη0 Z t 0 e-τ(t-r)|θr|2dr. (140) Substituting this back into (134), for all t ≥0 we have that Z t 0 e-τ(t-r)|θr|2dr ≤e-τt |θ0|2 Γη0 + τ Γη0 Z t 0 e-τ(t-r)|θr|2dr (141) + 2|c|2 Bb(S×A) Γ2τ + 2τ 2γ2 Γ2 Z t 0 e-τ(t-r)K2 rdr. (142) Grouping like terms we have 1 - τ Γη0 Z t 0 e-τ(t-r)|θr|2dr ≤e-τt |θ0|2 Γη0 + 2|c|2 Bb(S×A) Γ2τ + 2τ 2γ2 Γ2 Z t 0 e-τ(t-r)K2 rdr. (143) Recall that we have η0 > τ Γ to ensure that 1 - τ Γη0 > 0. Dividing through by 1 - τ Γη0 gives for all t ≥0 that Z t 0 e-τ(t-r)|θr|2dr ≤σ1 + σ2 Z t 0 e-τ(t-r)K2 rdr (144) where we've set σ1 := |θ0|2 Γη0 1 - τ Γη0 + 2|c|2 Bb(S×A) Γ2τ 1 - τ Γη0 , σ2 := 2τ 2γ2 Γ2 1 - τ Γη0 . Recall the approximate Fisher Rao gradient flow for the policies {πt}t≥0, which for all t ≥0 and for all s ∈S, a ∈A is (145) ∂t ln dπt dμ (s, a) = - Qt(s, a) + τ ln dπt dμ (a, s) - Z A Qt(s, a) + τ ln dπt dμ (a, s) πt(da|s) CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 17 Duhamel's principle yields for all t ≥0 that ln dπt dμ (s, a) = e-τt ln dπ0 dμ (a, s) + Z t 0 e-τ(t-r) Z A Qr(s, a)πr(da|s) -Qr(s, a) dr (146) + τ Z t 0 e-τ(t-r) KL(πr(·|s)|μ)dr (147) Observe that since π0 ∈Πμ, there exists C1 ≥1 such that ln dπt dμ Bb(S×A) ≤C1. Assumption 4.2 gives that for all t ≥0 ln dπt dμ (s, a) ≤C1 + 2 Z t 0 e-τ(t-r)|θr|dr + τ Z t 0 e-τ(t-r) KL(πr(·|s)|μ)dr (148) ≤C1 + 2 Z t 0 e-τ(t-r)|θr|dr + τ Z t 0 e-τ(t-r)Krdr (149) Integrating over the actions with respect to πt(·|s) ∈P(A) gives for all t ≥0 that (150) KL(πt(·|s)|μ) ≤C1 + 2 Z t 0 e-τ(t-r)|θr|dr + τ Z t 0 e-τ(t-r)Krdr where we again use that Kr = sups∈S KL(πr(·|s)|μ). Following from the techniques in [17], observe that from (66) and Assumption 4.2 we similarly get for all t ≥0 that (151) ln dμ dπt (a, s) = -ln dπt dμ (s, a) ≤C1 + 2 Z t 0 e-τ(t-r)|θr|dr -τ Z t 0 e-τ(t-r)Krdr. Now integrating over the actions with respect to the reference measure μ ∈P(A) we have (152) KL(μ|πt(·|s)) ≤C1 + 2 Z t 0 e-τ(t-r)|θr|dr -τ Z t 0 e-τ(t-r)Krdr Moreover, using the non-negativity of the KL divergence, it holds for all t ≥0 that (153) KL(πt(·|s)|μ) ≤KL(πt(·|s)|μ) + KL(μ|πt(·|s)) ≤2C1 + 4 Z t 0 e-τ(t-r)|θr|dr Since this holds for any s ∈S, it holds for all t ≥0 that (154) Kt ≤2C1 + 4 Z t 0 e-τ(t-r)|θr|dr Now squaring both sides and using the H ̈older's inequality, we have K2 t ≤ 2C1 + 4 Z t 0 e-τ(t-r)|θr|dr 2 (155) ≤8(C1)2 + 32 Z t 0 e-τ(t-r)|θr|dr 2 (156) = 8(C1)2 + 32 Z t 0 e-τ 2 (t-r)e-τ 2 (t-r)|θr|dr 2 (157) ≤8(C1)2 + 32 Z t 0 e-τ(t-r)dr Z t 0 e-τ(t-r)|θr|2dr (158) ≤8(C1)2 + 32 τ Z t 0 e-τ(t-r)|θr|2dr, (159) where we again used R t 0 e-τ(t-r)dr ≤1 τ . We can now substitute (144) into (159) to arrive at K2 t ≤8(C1)2 + 32 τ σ1 + 32 τ σ2 Z t 0 e-τ(t-r)K2 rdr (160) := a1 + a2 Z t 0 e-τ(t-r)K2 rdr (161) with a1 = 8(C1)2 + 32 τ σ1 and a2 = 32σ2 τ . □ 18 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH B.3. Proof of Corollary 5.1. Proof. By Theorem 5.1 it holds that (162) K2 t ≤a1 + a2 Z t 0 e-τ(t-r)K2 rdr. Observe that by multiplying through by eτt, we can rewrite this as (163) eτtK2 t ≤eτta1 + a2 Z t 0 eτrK2 rdr. Hence after defining g(t) = eτtK2 t and applying Gronwall's inequality (Lemma A.1), for all γ ∈(0, 1) it holds for all t ≥0 that (164) K2 t ≤a1ea2t. □ B.4. Proof of Corollary 5.2. Proof. By Corollary 5.1 and Lemma 5.2, for all γ ∈(0, 1) it holds that 1 2 d dt |θt|2 ≤-Γ 2 ηt|θt|2 + btηt (165) such that (166) bt = 2|c|2 Bb(S×A) + 2τ 2γ2a1ea2t Γ2 ! . Recall that there exists α > 0 such that d dtηt ≤αηt, another application of Gronwall's Lemma then concludes the proof. □ B.5. Proof of Corollary 5.3. Proof. By Theorem 5.1 we have that (167) K2 t ≤a1 + a2 Z t 0 e-τ(t-r)K2 rdr. Taking the supremum over [0, t] on the right hand side, we have (168) K2 t ≤a1 + a2 τ sup r∈[0,t] K2 r. Since this holds for all t ≥0, we have (169) sup r∈[0,t] K2 r ≤a1 + a2 τ sup r∈[0,t] K2 r. Now forcing 1 -a2 τ > 0, which is equivalent to the condition 64γ2 Γ2 -Γτ η0 0 it holds that for all t ≥0, K2 t ≤ a1τ τ -a2 . Hence by Lemma 5.2 we have 1 2 d dt |θt|2 ≤-ηt Γ 2 |θt|2 + ηt 2|c|2 Bb(S×A) + 2τ 2γ2 a1τ τ-a2 Γ2 . (171) The uniform boundedness in time of |θt| then follows by Gronwall's Lemma (Lemma A.1). □ CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 19 Appendix C. Proof of Convergence Results C.1. Proof of Lemma 6.1. Proof. By the definition of the state-action value function (2) it holds that d dtQπt τ (s, a) = lim h→0 Qπt+h τ (s, a) -Qπt τ (s, a) h (172) = γ Z S d dtV πt τ (s′)P(ds′|s, a). (173) Now observe that by [11][Proof of Proposition 2.6], we have (174) d dtV πt τ (s) = 1 1 -γ Z S×A Aπt τ (s, a)∂tπt(da|s′)dπt(ds′|s). Thus we have d dtQπt τ (s, a) = γ 1 -γ Z S Z S×A Aπt τ (s′′, a′′)∂tπt(da′′|s′′)dπt(ds′′|s′) P(ds′|s, a). (175) □ C.2. Proof of Theorem 6.1. Proof. Recall the performance difference Lemma (Lemma 1.1): for all ρ ∈P(S) and π, π′ ∈Πμ, V π τ (ρ) -V π′ τ (ρ) (176) = 1 1 -γ Z S Z A Qπ′ τ (s, a) + τ ln dπ′ dμ (a, s) (π -π′)(da|s) + τ KL(π(·|s)|π′(·|s)) dπ ρ(ds) . (177) Now let π = π∗and π′ = πt and multiply both sides by -1 we have V πt τ (ρ) -V π∗ τ (ρ) = -1 1 -γ Z S Z A Qπt(s, a) + τ ln dπt dμ (a, s) (π∗-πt)(da|s) (178) + τ KL(π∗(·|s)|πt(·|s)) ! dπ∗ ρ (ds). (179) Recall the approximate Fisher Rao dynamics, which we write as (180) ∂t ln dπt dμ (s, a) + Qt(s, a) + τ ln dπt dμ (a, s) - Z A Qt(s, a′) + τ ln dπt dμ (a′, s) πt(da′|s) = 0. Observe that since the normalisation constant (enforcing the conservation of mass along the flow) R A Qt(s, a) + τ ln dπt dμ (a, s) πt(da|s) is independent of a ∈A, it holds that Z A Z A Qt(s, a′) + τ ln dπt dμ (a′, s) πt(da′|s) (π∗-πt)(da|s) = 0. Hence adding 0 in the form of (180) into (178) it holds that for all t ≥0 V πt τ (ρ) -V π∗ τ (ρ) = 1 1 -γ Z S×A ∂t ln dπt dμ (a, s)(π∗-πt)(da|s)dπ∗ ρ (ds) (181) + Z S×A (Qt(s, a) -Qπt τ (s, a))(π∗-πt)(da|s)dπ∗ ρ (ds) -τ Z S KL(π∗(·|s)|πt(·|s)dπ∗ ρ (ds) ! . (182) By [12, Lemma 3.8] and Corollary 5.1, for any fixed ν ∈Πμ, the map t →KL(ν|πt) is differentiable. Hence we have Z A ∂t ln dπt dμ (s, a)(π∗-πt)(da|s) = Z A ∂t ln dπt dμ (s, a)π∗(da|s) - Z A ∂t ln dπt dμ (s, a)πt(da|s) (183) = Z A ∂t ln dπt dμ (s, a)π∗(da|s) (184) = -d dt KL(π∗(·|s)|πt(·|s)), (185) 20 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH where we used the conservation of mass of the policy dynamics in the second equality. Substituting this into (181) we have V πt τ (ρ) -V π∗ τ (ρ) = 1 1 -γ -d dt Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) (186) + Z S×A (Qt(s, a) -Qπt τ (s, a))(π∗-πt)(da|s)dπ∗ ρ (ds) -τ Z S KL(π∗(·|s)|πt(·|s)dπ∗ ρ (ds) ! . (187) Focusing on the second term, we have Z S×A (Qt(s, a) -Qπt τ (s, a))(π∗-πt)(da|s)dπ∗ ρ (ds) (188) ≤|Qt(s, a) -Qπt τ (s, a)|Bb(S×A) Z S TV(π∗(·|s), πt(·|s))dπ∗ ρ (ds) (189) ≤ 1 √ 2|θt -θπt| Z S KL(π∗(·|s)|πt(·|s)) 1 2 dπ∗ ρ (ds) (190) ≤ 1 √ 2|θt -θπt| Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) 1 2 , (191) where we used Pinsker's Inequality in the second inequality and H ̈older's inequality in the final inequality. Now applying Young's inequality, there exists ε > 0 such that (192) |θt -θπt| Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) 1 2 ≤1 2ε|θt -θπt|2 + ε 2 Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds). Substituting this back into (186) and choosing ε = √ 2τ we have V πt τ (ρ) -V π∗ τ (ρ) = 1 1 -γ -d dt Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) (193) -τ 2 Z S KL(π∗(·|s)|πt(·|s)dπ∗ ρ (ds) + 1 4τ |θt -θπt|2 ! . (194) Rearranging, we arrive at d dt Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) ≤-τ 2 Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) (195) -(1 -γ) V πt τ (ρ) -V π∗ τ (ρ) + 1 4τ |θt -θπt|2. (196) Applying Duhamel's principle yields Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) ≤e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (197) -(1 -γ) Z t 0 e-τ 2 (t-r)(V πr τ (ρ) -V π∗ τ (ρ))dr + 1 2τ Z t 0 e-τ 2 (t-r)|θr -θπr|2dr. (198) Now using that R t 0 e-τ 2 (t-r)dr = 2(1-e-τ 2 ) τ , we have Z S KL(π∗(·|s)|πt(·|s))dπ∗ ρ (ds) ≤e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (199) -2(1 -γ)(1 -e-τ 2 ) τ min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) + 1 2τ Z t 0 e-τ 2 (t-r)|θr -θπr|2dr. (200) Rearranging, we have min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ≤ τ 2(1 -γ)(1 -e-τ 2 ) e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (201) + 1 2τ Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ! . (202) which concludes the proof. □ CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 21 C.3. Proof of Theorem 6.2. Proof. Using the chain rule and the critic dynamics in (21), we have that for all r ≥0 1 2ηr d dr|θr -θπr|2 = 1 ηr dθr dr , θr -θπr - dθπr dr , θr -θπr (203) = -⟨g(θr, πr), θr -θπr⟩-1 ηr dθπr dr , θr -θπr (204) Let Γ = λβ(1 -γ)(1 -√γ). Using Lemma 4.1 and the λβ-strong convexity of L(·, π; β) and recalling that L(θπr, πr) = 0 for all r ≥0, it holds for all r ≥0 that 1 2ηt d dt|θt -θπt|2 = -⟨g(θt, πt), θt -θπt⟩-1 ηt dθπt dt , θt -θπt (205) ≤-(1 -γ)(1 -√γ) ⟨∇θL(θt, πt; β), θt -θπt⟩-1 ηt dθπt dt , θt -θπt (206) ≤-(1 -γ)(1 -√γ)L(θt, πt; β) -Γ 2 |θt -θπt|2 -1 ηt dθπt dt , θt -θπt (207) ≤-(1 -γ)(1 -√γ)L(θt, πt; β) -Γ 2 |θt -θπt|2 + 1 2ηt dθπt dt 2 + |θt -θπt|2 ! (208) = -(1 -γ)(1 -√γ)L(θt, πt; β) - Γ 2 -1 2ηt |θt -θπt|2 + 1 2ηt dθπt dt 2 , (209) where we used H ̈older's and Young's inequalities in (208). Since η0 > 1 Γ and ηt is a non-decreasing function, it holds that ηt > 1 Γ for all t ≥0. Hence Γ 2 - 1 2ηt > 0 and thus we can drop the second term. Moreover the λβ-strong convexity of L(·, π; β) along with L(θπ, π; β) = 0 and ∇θL(θπ, π) = 0 for all π ∈Πμ gives that |θt -θπt|2 ≤2 λβ L(θt, πt; β). Hence for all r ≥0 we arrive at 1 2ηr d dr|θr -θπr|2 ≤-Γ 2 |θr -θπr|2 + 1 2ηr dθπr dr 2 . (210) Rearranging, multiplying by e-τ(t-r) and integrating over r from 0 to t, it holds for all t ≥0 that Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤-1 Γ Z t 0 e-τ 2 (t-r) 1 ηr d dr|θr -θπr|2dr + 1 Γ Z t 0 e-τ 2 (t-r) 1 ηr dθπr dt 2 dr. (211) Integrating the first term by parts (identically to (137) from the proof of Theorem 5.1), we have Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤1 Γ -|θt -θπt|2 ηt + e-τ 2 t |θ0 -θπ0|2 η0 (212) + τ 2 Z t 0 e-τ 2 (t-r) 1 ηr |θr -θπr|2dr - Z t 0 |θr -θπr|2 e-τ 2 (t-r) d drηr η2r dr (213) + Z t 0 e-τ 2 (t-r) 1 ηr dθπr dr 2 dr ! . (214) Since for all t ≥0 it holds that ηt ≥1 and d dtηt ≥0, we have that Z t 0 |θr -θπr|2 e-τ 2 (t-r) d drηr η2r dr ≥0. Thus after dropping all negative terms and using that ηt ≥η0 for all t ≥0, we have (215) 1 - τ 2Γη0 Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤e-τ 2 |θ0 -θπ0|2 Γη0 + Z t 0 e-τ 2 (t-r) 1 ηr dθπr dr 2 dr. Since η0 > 1 2Γ and τ 0 and hence it holds that Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤e-τ 2 |θ0 -θπ0|2 Γη0 1 - τ 2Γη0 + 1 1 - τ 2Γη0 Z t 0 e-τ 2 (t-r) 1 ηr dθπr dr 2 dr, (216) 22 DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH which concludes the proof. □ C.4. Proof of Theorem 6.3. Proof. By Theorem 6.2, we have Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤e-τ 2 |θ0 -θπ0|2 Γη0 1 - τ 2Γη0 + 1 1 - τ 2Γη0 Z t 0 e-τ 2 (t-r) 1 ηr dθπr dr 2 dr. (217) Hence it remains to characterise the growth of the final integral. Observe that for all π ∈P(A|S), θπ ∈RN satisfies the least-squares optimality condition given by (218) θπ = arg min θ L(θ, π; β) = Z S×A φ(s, a)φ(s, a)⊤β(da, ds) -1 Z S×A φ(s, a)Qπ τ (s, a) β(ds, da) . Setting π = πt and differentiating time we arrive at (219) dθπt dt = Z S×A φ(s, a)φ(s, a)⊤β(da, ds) -1 Z S×A φ(s, a) d dtQπt(s, a) β(ds, da) . Hence by Lemma 6.1, Assumption 4.2 and Assumption 4.3, for all t ≥0 it holds that dθπt dt = Z S×A φ(s, a)φ(s, a)⊤β(da, ds) -1 Z S×A φ(s, a) d dtQπt τ (s, a) β(ds, da) (220) ≤ Z S×A φ(s, a)φ(s, a)⊤β(da, ds) -1 op d dtQπt τ Bb(S×A) (221) = 1 λβ d dtQπt Bb(S×A) (222) = γ λβ(1 -γ) Z S Z S×A Aπt τ (s′′, a′′)∂tπt(da′′|s′′) dπt(ds′′|s′) P(ds′|·, ·) Bb(S×A) (223) ≤ γ λβ(1 -γ) |Aπt τ |Bb(S×A) sup s∈S |∂tπt(·|s)|M(A) . (224) Now using Lemma A.3, it holds that |Aπt τ |Bb(S×A) sup s∈S |∂tπt(·|s)|M(A) ≤|Aπt τ |Bb(S×A) |At|Bb(S×A) (225) ≤ 2 |Qπt τ |Bb(S×A) + 2τ ln dπt dμ Bb(S×A) ! 2 |Qt|Bb(S×A) + 2τ ln dπt dμ Bb(S×A) ! . (226) Hence by Corollaries 5.1 and 5.2 and Lemma A.3, there exists α1, α2 > 0 such that dθπt dt 2 ≤α1eα2t. Thus Theorem 6.2 becomes Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤e-τ 2 |θ0 -θπ0|2 Γη0 1 - τ 2Γη0 + α1 1 - τ 2Γη0 Z t 0 e-τ 2 (t-r) eα2r ηr dr. (227) Let ηt = η0ek1t for any k1 > τ 2 + α2. Then observe that Z t 0 e-τ 2 (t-r) eα2r ηr dr = 1 η0 e-τ 2 t Z t 0 e( τ 2 +α2-k1)rdr (228) ≤1 η0 e-τ 2 t e( τ 2 +α2-k1)t -1 τ 2 + α2 -k1 ! (229) ≤ e-τ 2 t η0 τ 2 + α2 -k1 , (230) hence all together it holds that Z t 0 e-τ 2 (t-r)|θr -θπr|2dr ≤e-τ 2 |θ0 -θπ0|2 Γη0 1 - τ 2Γη0 + e-τ 2 t α1 η0 - τ 2Γ τ 2 + α2 -k1 . (231) CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES 23 Substituting this into the result from Theorem 6.2 concludes the proof. □ C.5. Proof of Theorem 6.3. Following completely identically to the proof of Theorem 6.3, we have dθπt dt ≤ γ λβ(1 -γ) |Aπt τ |Bb(S×A) sup s∈S |∂tπt(·|s)|M(A) (232) ≤ 4 (1 -γ)2 |c|Bb(S×A) + Kt 2 + 4τ C1 + 2 τ sup r∈[0,t] |θr| + sup r∈[0,t] Kr !2 . (233) Then by Corollaries 5.3 and 5.4, there exists b2 > 0 such that dθπt dt 2 ≤d1. Hence by Theorem 6.2 we have min r∈[0,t] V πr τ (ρ) -V π∗ τ (ρ) ≤ τ 2(1 -γ)(1 -e-τ 2 ) e-τ 2 t Z S KL(π∗(·|s)|π0(·|s))dπ∗ ρ (ds) (234) + d1 Z t 0 e-τ 2 (t-r) 1 ηr dr ! . (235) : : :"}}},{"rowIdx":85,"cells":{"id":{"kind":"string","value":"2510.14901"},"raw":{"kind":"string","value":"Reasoning with Sampling:\nYour Base Model is Smarter Than You Think\nAayush Karan1, Yilun Du1\n1Harvard University\n Website\n Code\nAbstract\nFrontier reasoning models have exhibited incredible capabilities across a wide\narray of disciplines, driven by posttraining large language models (LLMs) with\nreinforcement learning (RL). However, despite the widespread success of this\nparadigm, much of the literature has been devoted to disentangling truly novel be-\nhaviors that emerge during RL but are not present in the base models. In our work,\nwe approach this question from a different angle, instead asking whether compa-\nrable reasoning capabilites can be elicited from base models at inference time by\npure sampling, without any additional training. Inspired by Markov chain Monte\nCarlo (MCMC) techniques for sampling from sharpened distributions, we pro-\npose a simple iterative sampling algorithm leveraging the base models’ own like-\nlihoods. Over different base models, we show that our algorithm offers substantial\nboosts in reasoning that nearly match and even outperform those from RL on a\nwide variety of single-shot tasks, including MATH500, HumanEval, and GPQA.\nMoreover, our sampler avoids the collapse in diversity over multiple samples that\nis characteristic of RL-posttraining. Crucially, our method does not require train-\ning, curated datasets, or a verifier, suggesting broad applicability beyond easily\nverifiable domains.\n1\nIntroduction\nReinforcement learning (RL) has become the dominant paradigm for enhancing the reasoning ca-\npabilities of large language models (LLMs) [Guo et al., 2025, Hu et al., 2025]. Equipped with a\nreward signal that is typically automatically verifiable, popular RL techniques have been success-\nfully applied to posttrain frontier models, leading to sizeable performance gains in domains like\nmath, coding, and science [Hendrycks et al., 2021, Li et al., 2022, Rein et al., 2024].\nDespite the widespread empirical success of RL for LLMs, a large body of literature has centered\naround the following question: are the capabilities that emerge during RL-posttraining fundamen-\ntally novel behaviors that are not present in the base models? This is the question of distribution\nsharpening [He et al., 2025, Shao et al., 2025, Yue et al., 2025]: that is, whether the posttrained\ndistribution is simply a “sharper” version of the base model distribution, instead of placing mass on\nreasoning traces the base model is unlikely to generate.\nSeveral works point towards the difficulty in learning new capabilities with RL-posttraining. He\net al. [2025], Song et al. [2025] compare the pass@k (multi-shot) scores of base models with post-\ntrained models, finding that for large k, base models actually outperform while the latter suffer from\ndegraded generation diversity. In such cases, RL appears to redistribute pass@k performance to\nsingle-shot performance at the expense of multi-shot reasoning. Yue et al. [2025] also notes that\nthe reasoning traces post-RL are tightly concentrated at high likelihoods/confidences under the base\nmodel, seemingly drawing from existing high-likelihood capabilities. We illustrate this point in our\nown experiments in Figure 4. Regardless, the advantage of RL-posttraining for single-shot reasoning\nhas remained, as of yet, undeniable.\nPreprint. Under review.\narXiv:2510.14901v1 [cs.LG] 16 Oct 2025\n\nMATH500\nHumanEval\nGPQA\n10\n20\n30\n40\n50\n60\n70\n80\nAccuracy (%)\n49.6\n32.9\n27.8\n78.5\n53.7\n39.9\n74.8\n57.3\n38.9\nReasoning\nAlpacaEval2.0\n0.5\n1.0\n1.5\n2.0\n2.5\n3.0\nScore\n1.61\n2.38\n2.88\nGeneral\nBase (Qwen2.5-Math-7B)\nGRPO (RL)\nOurs (Training-free)\nFigure 1: Our sampling algorithm can match and outperform RL-posttraining. Left: we compare our sam-\npling algorithm (ours) against the base model (base) and RL-posttraining (GRPO) on three verifiable reasoning\ntasks (MATH500, HumanEval, GPQA). Right: we compare them on an unverifiable general task (AlpacaE-\nval2.0). Our algorithm achieves comparable performance to GRPO within the posttraining domain (MATH500)\nbut can outperform on out-of-domain tasks such as HumanEval and AlpacaEval.\nIn this paper, we present a surprising result: sampling directly from the base model can achieve\nsingle-shot reasoning capabilites on par with those from RL.\nWe propose a sampling algorithm for base models that leverages additional compute at inference\ntime, achieving single-shot performance that nearly matches RL-posttraining on in-domain reason-\ning tasks and can even outperform on out-of-domain reasoning tasks. Furthermore, we observe that\ngeneration diversity does not degrade with our sampler; in fact, our pass@k (multi-shot) perfor-\nmance strongly outperforms RL. We benchmark specifically against Group Relative Policy Opti-\nmization (GRPO), which is the standard RL algorithm for enhancing LLM reasoning [Shao et al.,\n2024].\nCrucially, our algorithm is training-free, dataset-free, and verifier-free, avoiding some of the inher-\nent weaknesses of RL methods including extensive hyperparameter sweeps to avoid training insta-\nbilities, the need to curate a diverse and expansive posttraining dataset, and the lack of guaranteed\naccess to a ground truth verifier/reward signal [Prabhudesai et al., 2025].\nOur contributions can be summarized as follows:\ni) We introduce the power distribution as a useful sampling target for reasoning tasks. Since it can\nbe explicitly specified with a base LLM, no additional training is required.\nii) We further introduce an approximate sampling algorithm for the power distribution using a\nMarkov chain Monte Carlo (MCMC) algorithm that iteratively resamples token subsequences\naccording to their base model likelihoods.\niii) We empirically demonstrate the effectiveness of our algorithm over a range of models (Qwen2.5-\nMath-7B, Qwen2.5-7B, Phi-3.5-mini-instruct) and reasoning tasks (MATH500, HumanEval,\nGPQA, AlpacaEval 2.0). Our results show that sampling directly from the base model can achieve\nresults on par with GRPO. In fact, for some out-of-domain tasks, our algorithm consistently out-\nperforms the RL baseline. Moreover, over multiple samples, we avoid the collapse in diversity\nafflicting RL-posttraining, achieving the best of both worlds in terms of single-to-few-shot rea-\nsoning capabilities as well as sample diversity.\nOur results collectively illustrate that existing base models are much more capable at single-shot\nreasoning than current sampling methods reveal.\n2\n\n2\nRelated Works\nReinforcement learning for LLMs. RL has been instrumental in posttraining LLMs. Early on, RL\nwith human feedback (RLHF) [Ouyang et al., 2022] was developed as a technique to align LLMs\nwith human preferences using a trained reward model. Recently, RL with verifiable rewards (RLVR)\nhas emerged as a powerful new posttraining technique, where many works [Guo et al., 2025, Lam-\nbert et al., 2024, Hu et al., 2025, Zeng et al., 2025] discovered that a simple, end-of-generation\nreward given by an automated verifier could substantially enhance performance on difficult reason-\ning tasks in mathematics and coding. The Group Relative Policy Optimization (GRPO) algorithm\nwas at the center of these advances [Shao et al., 2024]. Building off of this success, many subsequent\nworks have examined using reward signals derived from internal signals such as self-entropy [Zhao\net al., 2025], confidence [Prabhudesai et al., 2025], and even random rewards [Shao et al., 2025].\nSimilar to these works, our paper examines base model likelihoods as a mechanism for improving\nreasoning performance, but crucially, our technique is training-free.\nAutoregressive MCMC sampling with LLMs.\nPrior works have explored integrating classic\nMCMC techniques with autoregressive sampling. Many settings including red-teaming, prompt-\nengineering, and personalized generation can be framed as targeting sampling from the base LLM\ndistribution but tilted towards an external reward function. Zhao et al. [2024] proposes learning\nintermediate value functions that are used in a Sequential Monte Carlo (SMC) framework [Chopin,\n2004], where multiple candidate sequences are maintained and updated according to their expected\nfuture reward. Similarly, Faria et al. [2024] proposes a Metropolis-Hastings (MH) algorithm, which\ninstead of maintaining multiple candidates performs iterative resampling, again updating according\nto expected reward. Methodologically, our sampling algorithm is most similar to this latter work,\nbut the crucial difference is that our target sampling distribution is completely specified by the base\nLLM, avoiding the need for an external reward.\nAnnealed sampling for diffusion. In the statistical physics and Monte Carlo literature, sampling\nfrom pα is known as sampling from an annealed, or tempered, distribution [Neal, 1998] and has\ninspired a new wave of interest within the diffusion community. Indeed, in traditional MCMC\nsampling, annealing is used as a way to avoid mode-collapse during sampling and more accurately\nsample from complex multimodal distributions [Łatuszy´nski et al., 2025]. This has re-emerged as\ninference-time sampling methods for diffusion that aim to steer a pretrained model towards “tilted\ndistributions” [Du et al., 2023, Kim et al., 2025, Karan et al., 2025, Wang et al., 2025, Kong et al.,\n2025, Zhang et al., 2025]. Where traditional RL techniques exhibit mode collapse, applications in\nthe physical sciences [Sambridge, 2014] require multimodal sampling. To this end, works such as\nDu et al. [2023], Wang et al. [2025], Kim et al. [2025] construct sequences of annealed distributions\nto ease the transition from base diffusion distribution to tilted distribution. Other works [Skreta et al.,\n2025, Xu et al., 2025] intentionally target sampling from pα for α > 1 as a means of generating\nhigher quality samples from the base diffusion model, which is particularly popular for generating\nmore designable proteins [Geffner et al., 2025].\n3\nPreliminaries\nLet X be a finite vocabulary of tokens, and let X T denote the set of finite sequences of tokens x0:T =\n(x0, x1, . . . , xT ), where xi ∈X for all i and T ∈Z≥0 is some nonnegative integer. For convenience,\nfor a given t, let x<t = (x0, . . . , xt−1) and x>t = (xt+1, . . . , xT ), with similar definitions for x≤t\nand x≥t. In general, x refers to a token sequence x0:T , where T is implicitly given.\nThen an LLM defines a distribution p over token sequences X T by autoregressively learning the\nconditional token distributions p(xt|x<t) for all t, giving the joint distribution via the identity\np(x0:T ) =\nT\nY\nt=0\np(xt|x<t).\n(1)\nTo sample a sequence from p, we simply sample from the LLM token by token using the conditional\ndistributions, which by (1) directly samples from the joint distribution.\n3\n\n4\nMCMC Sampling for Power Distributions\np\npα (α = 4.0)\nFigure 2: A toy example of distribution\nsharpening. Here p is a mixture of Gaus-\nsians, which we plot against pα (α = 4.0).\nIn this section, we introduce our sampling algorithm for\nbase models. Our core intuition is derived from the notion\nof distribution sharpening posed in Section 1. Sharpen-\ning a reference distribution refers to reweighting the dis-\ntribution so that high likelihood regions are further up-\nweighted while low likelihood regions are downweighted,\nbiasing samples heavily towards higher likelihoods under\nthe reference. Then if RL posttrained models really are\njust sharpened versions of the base model, we should be\nable to explicitly specify a target sampling distribution that\nachieves the same effect.\nWe organize this section as follows. Section 4.1 presents\nthis target sharpened distribution and provides some math-\nematical motivation for why its samples are amenable for\nreasoning tasks. Section 4.2 introduces a general class of Markov chain Monte Carlo (MCMC) al-\ngorithms aimed at actually sampling from this target distribution, and finally, Section 4.3 details our\nspecific implementation for LLMs.\n4.1\nReasoning with Power Distributions\nOne natural way to sharpen a distribution p is to sample from the power distribution pα. Since\np(x) > p(x′) =⇒\np(x)α\np(x′)α > p(x)\np(x′)\n(α ∈[1, ∞]),\n(2)\nit follows that exponentiating p increases the relative weight on higher likelihood sequences (x)\nwhile decreasing the relative weight on lower likelihood ones (x′) (see Figure 2 for a visualization).\nA related but well-known sharpening strategy is low-temperature sampling [Wang et al., 2020],\nwhich exponentiates the conditional next-token distributions at each step:\nptemp(xt|x0 . . . xt−1) =\np(xt|xt−1 . . . x0)α\nP\nx′\nt∈X p(x′\nt|xt−1 . . . x0)α ,\n(3)\nwhere the temperature is τ = 1/α. A common misconception is that sampling with (3) over T\ntokens is equivalent to sampling from pα; however, this is false in a subtle yet crucial way, as we\nilluminate in the following.\nProposition 1. Low-temperature sampling does not sample from the power distribution pα.\nProof. We show that the associated conditional next-token distributions are distinct at each timestep\nt. The conditional distribution on xt for pα is given by\nppow(xt|x0 . . . xt−1) =\nP\nx>t p(x0, . . . , xt, . . . , xT )α\nP\nx≥t p(x0, . . . , xt, . . . , xT )α .\n(4)\nUsing Bayes rule\np(xt|xt−1 . . . x0) =\np(x0, . . . , xt)\np(x0, . . . , xt−1) =\nP\nx>t p(x0, . . . , xt, . . . , xT )\nP\nx≥t p(x0, . . . , xt, . . . , xT ),\n(5)\nwe can rewrite the low-temperature marginal (3) as\nptemp(xt|x0 . . . xt−1) =\n\u0010P\nx>t p(x0, . . . , xt, . . . , xT )\n\u0011α\nP\nx′\nt\n\u0010P\nx>t p(x0, . . . , xt, . . . , xT )\n\u0011α .\n(6)\nIgnoring normalizations for clarity, the relative weight on token xt for sampling from pα is given by\na sum of exponents\nppow(xt|x<t) ∝\nX\nx>t\np(x0, . . . , xt, . . . , xT )α.\n(7)\n4\n\nMeanwhile, the relative weight for low-temperature sampling is given by an exponent of sums\nptemp(xt|x<t) ∝\n X\nx>t\np(x0, . . . , xt, . . . , xT )\n!α\n.\n(8)\nSince the relative weights of next-token prediction are distinct for each sampling strategy, it fol-\nlows that the joint distribution over seqeunces must also be distinct for each sampler. Hence, the\ndistribution on sequences given by low-temperature sampling is not the same as the one given by\npα.\nOne intuitive way to understand this difference is that low-temperature sampling does not account\nfor how exponentiation sharpens the likelihoods of “future paths” at time step t, instead “greedily”\naveraging all these future likelihoods (exponent of sums (8)). On the other hand, sampling from pα\ninherently accounts for future completions as it exponentiates all future paths (sum of exponents (7))\nbefore computing the weights for next-token prediction. This has the following consequence:\nObservation 1. The power distribution upweights tokens with few but high likelihood future paths,\nwhile low-temperature sampling upweights tokens with several but low likelihood completions.\nExample 1. We can observe this phenomenon with a simple example. Let us consider the token\nvocabulary X = {a, b} and restrict our attention to two-token sequences (x0, x1): aa, ab, ba, bb. Let\np(aa) = 0.00,\np(ab) = 0.40,\np(ba) = 0.25,\np(bb) = 0.25,\nso that\np(x0 = a) = 0.40,\np(x0 = b) = 0.50.\nLet α = 2.0. Under pα, we have\nppow(x0 = a) ∝0.002 + 0.402 = 0.160,\nppow(x0 = b) ∝0.252 + 0.252 = 0.125,\nso pα prefers sampling a over b. Under low-temperature sampling,\nptemp(x0 = a) ∝(0.00 + 0.40)2 = 0.160,\nptemp(x0 = b) ∝(0.25 + 0.25)2 = 0.250,\npreferring sampling b over a. If pα samples x0 = a, there is only one future path with likelihood\n0.40. If ptemp samples x0 = b, there are two future paths ba, bb, but either choice has likelihood 0.25.\nIn other words, even though a has lower conditional likelihood under both p and ptemp, pα upweights\na and samples the highest likelihood two-token sequence. b has many future paths contributing to\na higher likelihood under p and ptemp, but leads to low likelihood sequences. We provide a stronger\nformalization of this phenomenon in Appendix A.1.\nThus, sampling from pα encourages sampling tokens which have fewer but higher likelihood “future\npaths”, as opposed to tokens with several lower likelihood completions. This type of behavior is\nimmensely valuable for reasoning tasks. For example, choosing “wrong” tokens that have high\naverage likelihoods but trap outputs in low likelihood individual futures are examples of critical\nwindows or pivotal tokens [Li et al., 2025, Abdin et al., 2024], a phenomenon where a few tokens\nare highly influential in the correctness of language model outputs. In fact, sharp critical windows\nhave been shown to correlate strongly with reasoning failures [Li et al., 2025]. Instead, embedded in\nsampling from the power distribution is an implicit bias towards planning for future high likelihood\ntokens.\n4.2\nThe Metropolis-Hastings Algorithm\nNow that we have seen how sampling from pα can in theory assist the underlying LLM’s ability\nto reason, our aim now turns towards proposing an algorithm to accurately sample from it. Given\nan LLM p, we have access to the values pα over any sequence length; however, these values are\nunnormalized. Direct sampling from the true probabilities requires normalizing over all sequences\n(x0, . . . , xT ) ∈X T , which is computationally intractable.\nTo get around this, we invoke a Markov Chain Monte Carlo (MCMC) algorithm known as\nMetropolis-Hastings (MH) [Metropolis et al., 1953], which targets exactly what we want: approx-\nimate sampling from an unnormalized probability distribution. The MH algorithm constructs a\n5\n\nFigure 3: Illustrating Metropolis-Hastings with random resampling. A random index t is selected and a new\ncandidate is generated by resampling. Based on the relative likelihoods, the candidate is accepted or rejected,\nand the process repeats.\nMarkov chain of sample sequences (x0, x1, . . . , xn) using an arbitrary proposal distribution q(x|xi)\nto select the next candidate xi+1. With probability\nA(x, xi) = min\n\u001a\n1, pα(x) · q(xi|x)\npα(xi) · q(x|xi)\n\u001b\n,\n(9)\ncandidate x is accepted as xi+1; otherwise, MH sets xi+1 = xi. This algorithm is especially\nconvenient as it only requires the relative weights given by pα (as the normalization weights in A\ncancel) and works with any generic but tractable sampler q with minimal restrictions. Remarkably,\nfor large enough n, this process converges to sampling from the target distribution pα under the\nfollowing (quite minimal) conditions on the proposal distribution [Neal, 1993]:\nDefinition 1. The proposal distribution q is irreducible if for any set X with nonzero mass under\nthe target distribution pα, q has nonzero probability of eventually sampling from X. The proposal\nis aperiodic if the induced chain of samples does not return to the same sample after a fixed interval\nnumber of steps.\nThus, we must simply ensure that our proposal distribution satisfies irreducibility and aperiodicity,\nand Metropolis-Hastings takes care of the rest. On a practical level, we would also like both q(x|xi)\nand its reverse q(xi|x) to be easily computable.\nConsider the following family of random resampling proposal distributions (see Figure 3). Let pprop\nbe a proposal LLM. With uniform probability 1\nT , select a random t ∈[1, T] and resample the se-\nquence starting at index t using pprop. Then the transition likelihood q(x|xi) is simply the likelihood\nof the resampling. Note that at each candidate selection step, we have a nonzero probability of\ntransitioning between any two sequences x, x′ ∈X, since with some probability we can always re-\nsample as early as the beginning of x. This ensures our proposal distribution is both irreducible and\naperiodic. Moreover, q(xi|x) is easy to calculate by symmetry, since we can treat xi as a resampled\nversion of x.\nWith the flexibility endowed by Metropolis-Hastings, we can choose the proposal LLM pprop to be\nany LLM with any sampling strategy (e.g., low-temperature sampling).\n4.3\nPower Sampling with Autoregressive MCMC\nA direct implementation of Metropolis-Hastings for LLMs would involve initializing with a sampled\ntoken sequence of length T, subsequently generating new candidates of length T with (9) over\nmany, many iterations. This process is computationally expensive, however, due to the repeated, full\nsequence inference calls to the LLM.\nIn fact, the main downside to MCMC algorithms in practice is the potential for an exponential mixing\ntime [Gheissari et al., 2017], where a poor choice of initialization or proposal distribution can result\nin an exponentially large number of samples required before convergence to the target distribution.\nThis problem is exacerbated if the sample space has high dimensionality [Bandeira et al., 2022,\nSchmidler and Woodard, 2013], which is precisely exhibited by the sequence space of tokens X T ,\nespecially for long sequences/large values of T.\nTo remedy this, we propose an algorithm that leverages the sequential structure of autoregressive\nsampling. We define a series of intermediate distributions which we progressively sample from, until\nconverging to the target distribution pα. In particular, samples from one intermediate distribution\ninitiate a Metropolis-Hastings process for the next, helping avoid pathological initializations.\n6\n\nAlgorithm 1: Power Sampling for Autoregressive Models\nInput\n: base p; proposal pprop; power α; length T\nHyperparams: block size B; MCMC steps NMCMC\nOutput\n: (x0, . . . , xT ) ∼pα\n1 Notation: Define the unnormalized intermediate target\nπk(x0:kB) ∝p(x0:kB)α.\n2 for k ←0 to ⌈T\nB ⌉−1 do\n3\nGiven prefix x0:kB, we wish to sample from πk+1. Construct initialization x0 by extending\nautoregressively with pprop:\nx(0)\nt\n∼pprop\n\u0000xt | x<t\n\u0001\n,\nfor kB + 1 ≤t ≤(k + 1)B.\nSet the current state x ←x0.\n4\nfor n ←1 to NMCMC do\n5\nSample an index m ∈{1, . . . , (k + 1)B} uniformly.\n6\nConstruct proposal sequence x′ with prefix x0:m−1 and resampled completion:\nx′\nt ∼pprop\n\u0000xt | x<t\n\u0001\n,\nfor m ≤t ≤(k + 1)B.\n7\nCompute acceptance ratio (9)\nA(x′, x) ←min\n(\n1, πk(x′)\nπk(x) · pprop(x | x′)\npprop(x′ | x)\n)\n.\nDraw u ∼Uniform(0, 1);\n8\nif u ≤A(x′, x) then accept and set x ←x′\n9\nend\n10\nSet x0:(k+1)B ←x to fix the new prefix sequence for the next stage.\n11 end\n12 return x0:T\nFix block size B and proposal LLM pprop, and consider the sequence of (unnormalized) distributions\n∅−→p(x0, . . . , xB)α −→p(x0, . . . , x2B)α −→· · · −→p(x0, . . . , xT )α,\n(10)\nwhere p(x0, . . . , xkB) denotes the joint distribution over token sequences of length kB, for any k.\nFor convenience, let πk denote the distribution given by\nπk(x0:kB) ∝p(x0:kB)α.\n(11)\nSuppose we have a sample from πk. To obtain a sample from πk+1, we initialize a Metropolis-\nHastings process by sampling the next B tokens xkB+1:(k+1)B with pprop. We subsequently run the\nMCMC sampling procedure for NMCMC steps, using the random resampling proposal distribution q\nfrom the previous section. The full details are presented in Algorithm 1.\nNote that Algorithm 1 is single-shot: even though multiple inference calls are made, the decision to\naccept vs. reject new tokens is made purely by base model likelihoods to simulate sampling a single\nsequence from pα. We can interpret this as a new axis for inference-time scaling, as we expend\nadditional compute during sampling to obtain a higher quality/likelihood sample.\nTo quantify the scaling, we can estimate the average number of tokens generated by Algorithm 1.\nNote that each candidate generation step when sampling from πk(x0:kB resamples an average of kB\n2\ntokens, NMCMC times. Summing over all k, the expected number of tokens generated is\nEtokens = NMCMC\n⌈T/B⌉\nX\nk=1\nkB\n2\n≈NMCMCT 2\n4B\n.\n(12)\nThe key tradeoff here is between the block size B and number of MCMC steps NMCMC. A larger B\nrequires larger “jumps” between intermediate distributions, requiring a larger NMCMC to adequately\n7\n\ntransition. In Section 5, we empirically find a value for B that makes Algorithm 1 performant for\nrelatively small values of NMCMC.\n5\nExperiments\n5.1\nExperimental Setup\nEvaluation. We use a standard suite of reasoning benchmarks ranging across mathematics, coding,\nand STEM (MATH500, HumanEval, GPQA), along with a non-verifiable benchmark (AlpacaEval\n2.0) evaluating general helpfulness. We evaluate all of our methods and baselines single-shot; i.e.,\non one final response string.\n• MATH500: The MATH dataset [Lightman et al., 2024] consists of competition math problems\nspanning seven categories including geometry, number theory, and precalculus. There are 12500\nproblems total, with 7500 training problems and 5000 test problems. MATH500 is a specific\nrandomly chosen subset of the test set standardized by OpenAI.\n• HumanEval: HumanEval is a set of 164 handwritten programming problems covering algorihtms,\nreasoning, mathematics, and language comprehension [Chen et al., 2021]. Each problem has an\naverage of 7.7 associated unit tests, where solving the problem corresponds to passing all unit\ntests.\n• GPQA: GPQA [Rein et al., 2024] is a dataset of multiple-choice science questions (physics, chem-\nistry, and biology) which require advanced reasoning skills to solve. We use subset GPQA Dia-\nmond for evaluation, which consists of 198 questions which represent the highest quality subset\nof the GPQA dataset.\n• AlpacaEval 2.0: The AlpacaEval dataset is a collection of 805 prompts [Dubois et al., 2024]\nthat gauge general helpfulness with questions asking e.g., for movie reviews, recommendations,\nand reading emails. The model responses are graded by an automated LLM judge (GPT-4-turbo),\nwhich determines a preference for the model responses over those from a baseline (also GPT-4-\nturbo). The resulting score is a win rate of model responses normalized for the length of the model\nresponse.\nModels. To demonstrate the efficacy of our sampling algorithm, we use the base models Qwen2.5-\nMath-7B, Qwen2.5-7B, and Phi-3.5-mini-instruct. For our RL baselines, we use the implementation\nof GRPO in Shao et al. [2025], which posttrains these models on the MATH training split. For both\nthe Qwen2.5 models, we use the default hyperparameters used to benchmark their performance in\nShao et al. [2025]. For the Phi-3.5 model, we use a set of hyperparameters selected from Abdin et al.\n[2024] that avoids training instabilities and converges to improvement over the base model over a\nlarge number of epochs.\nSampling Algorithm. For our implementation of power sampling (Algorithm 1), we set the max-\nimum T to be Tmax = 3072 (termination can happen earlier with an EOS token) and block size\nB = 3072/16 = 192. Empirically, we find α = 4.0 coupled with a proposal LLM pprop chosen as\nthe base model with sampling temperature 1/α to be most performant for reasoning tasks. For Al-\npacaEval 2.0, we find that having a proposal distribution of higher temperature (τ = 0.5) improves\nperformance.\n5.2\nResults\nMain results. We display our main results in Table 1. Across base models of different families,\nour sampling algorithm achieves massive, near-universal boosts in single-shot accuracies and scores\nover different reasoning and evaluation tasks that reach, e.g., up to +51.9% on HumanEval with\nPhi-3.5-mini and +25.2% on MATH500 with Qwen2.5-Math. In particular, on MATH500, which\nis in-domain for RL-posttraining, power sampling achieves accuracies that are on par with those\nobtained by GRPO. Furthermore, on out-of-domain reasoning, our algorithm again matches GRPO\non GPQA and actually outperforms on HumanEval by up to +59.8%. Similarly, power sampling\nconsistently outperforms on the non-verifiable AlpacaEval 2.0, suggesting a generalizability of our\nboosts to domains beyond verifiability.\nThe surprising success of this fundamentally simple yet training-free sampling algorithm under-\nscores the latent reasoning capabilities of existing base models.\n8\n\nMATH500\nHumanEval\nGPQA\nAlpacaEval2.0\nQwen2.5-Math-7B\nBase\n0.496\n0.329\n0.278\n1.61\nLow-temperature\n0.690\n0.512\n0.353\n2.09\nPower Sampling (ours)\n0.748\n0.573\n0.389\n2.88\nGRPO (MATH)\n0.785\n0.537\n0.399\n2.38\nQwen2.5-7B\nBase\n0.498\n0.329\n0.278\n7.05\nLow-temperature\n0.628\n0.524\n0.303\n5.29\nPower Sampling (ours)\n0.706\n0.622\n0.318\n8.59\nGRPO (MATH)\n0.740\n0.561\n0.354\n7.62\nPhi-3.5-mini-instruct\nBase\n0.400\n0.213\n0.273\n14.82\nLow-temperature\n0.478\n0.585\n0.293\n18.15\nPower Sampling (ours)\n0.508\n0.732\n0.364\n17.65\nGRPO (MATH)\n0.406\n0.134\n0.359\n16.74\nTable 1: Power sampling (ours) matches and even outperforms GRPO across model families and tasks.\nWe benchmark the performance of our sampling algorithm on MATH500, HumanEval, GPQA, and AlpacaEval\n2.0. We bold the scores of both our method and GRPO, and underline whenever our method outperforms GRPO.\nAcross models, we see that power sampling is comparable to GRPO on in-domain reasoning (MATH500), and\ncan outperform GRPO on out-of-domain tasks.\n5.3\nAnalysis\nWe analyze how the reasoning characteristics of power sampling relate to those of GRPO. We\npresent an example in Table 2, with further examples in Appendix A.3.\nReasoning trace likelihoods and confidences. By design, power sampling targets sampling higher\nlikelihood sequences from the base model. In Figure 4, the left graph plots a histogram of the out-\nput sequence log-likelihoods (averaged by length) of the base model, power sampling, and GRPO\nresponses on MATH500, where likelihoods are taken relative to the Qwen2.5-Math-7B base model.\nOur method samples from higher likelihood regions of the base model, as intended, but still main-\ntains noticeable spread. Meanwhile, GRPO samples are heavily concentrated at the highest likeli-\nhood peak.\nFigure 4:\nBase model (Qwen2.5-Math-7B) likelihoods and confidences for MATH500 responses. Left:\nWe plot the log-likelihoods (relative to the base model) of original, power sampling, and GRPO responses\nover MATH500. Right: We do the same but for confidences relative to the base model. We observe that GRPO\nsamples from the highest likelihood and confidence regions with power sampling close behind, which correlates\nwith higher empirical accuracy.\n9\n\nFilter an input list of strings only for ones that start with a given prefix. (Phi-3.5-mini-instruct: HumanEval)\nMethod\nResponse\nPassed\nOurs\nreturn [s for s in\nstrings\nif s.startswith (prefix)]\ntrue\nGRPO\nreturn [string\nfor\nstring\nin\nstrings\nif\nstring. startswith (f’{prefix}’*2)]\nfalse\nTable 2: Sample responses on HumanEval: Phi-3.5-mini-instruct. We present an example where our method\nsolves a simple coding question, but GRPO does not.\nWe also plot the base model confidence of MATH500 responses, defined to be the average negative\nentropy (uncertainty) of the next-token distributions [Prabhudesai et al., 2025]:\nConf(x0:T ) =\n1\nT + 1\nT\nX\nt=0\nX\nx∈X\np(x|x<t) log p(x|x<t).\n(13)\nThe right plot of Figure 4 demonstrates that our method’s and GRPO responses sample from sim-\nilarly high confidence regions from the base model, which again correspond to regions of higher\nlikelihood and correct reasoning.\nReasoning trace lengths. Another defining characteristic of RL-posttraining is long-form reasoning\n[Guo et al., 2025], where samples tend to exhibit longer responses. On MATH500, Qwen2.5-Math-\n7B averages a response length of 600 tokens, while GRPO averages 671 tokens. Surprisingly, power\nsampling achieves a similar average length of 679 tokens, without explicitly being encouraged to\nfavor longer generations. This emerges naturally from the sampling procedure.\nDiversity and pass@k performance. Again, notice the peaked and highly concentrated likelihood-\ns/confidences of GRPO relative to the distributional spread of power sampling in Figure 4. This\nsuggests GRPO exhibits a collapse in diversity while our sampler does not, aligning with the obser-\nvation that RL-posttraining strongly sharpens the base model distribution at the expense of diversity\n[Song et al., 2025]. To quantify the comparative diversity of power sampling relative to GRPO, we\ncan plot the pass@k accuracy rate, where a question is solved if at least one of k samples is accu-\nrate. Figure 5 shows exactly this: unlike GRPO, whose pass@k performance tapers off for large\nk, power sampling strongly outperforms for k > 1. Moreover, our performance curve supersedes\nthat of the base model until finally converging in performance. In particular, we are able to achieve\nGRPO-level single-shot performance without compromising multi-shot performance (see Appendix\nA.2 for other domains), addressing a long-standing downside to RL-posttraining.\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\nk\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nPass@k Accuracy\nMATH500\nOurs\nBase\nGRPO\nFigure 5: Pass@k performance on MATH500. We plot the pass@k accuracy (correct if at least one of k\nsamples is accurate) of power sampling (ours) and RL (GRPO) relative to the base model (Qwen2.5-Math-\n7B). Our performance curve is strictly better than both GRPO and the base model, and our pass rate at high k\nmatches the base model, demonstrating sustained generation diversity.\nThe effect of power distributions. The two most important hyperparameters for power sampling\nare the choice of α and the number of MCMC (resampling) steps during sequence generation\n10\n\nFigure 6: Effect of hyperparameters on power sampling. Left: We plot MATH500 accuracy across model\nfamilies for various values of α. Right: We plot the increase in accuracy of power sampling on Qwen models\nas the number of MCMC steps increases.\nNMCMC. At the extremes, choosing α = 1.0 samples from the base model directly, while taking\nα →∞has the effect of deterministically accepting any resampled sequence that strictly increases\nthe likelihood. Of course, even though higher base model likelihoods correlate with better reasoning\n(Figure 4), directly optimizing for likelihood is not necessarily optimal for reasoning, suggesting an\nideal intermediate value of α.\nIn Figure 6, we display MATH500 accuracies across various values of α and find that an intermediate\nα = 4.0 outperforms other values, as expected. Noticeably, the accuracies of power sampling remain\nrelatively stable beyond α ≥2.0, suggesting that power sampling in practice is relatively robust to\nthe choice of α.\nTest-time scaling with MCMC steps. On the other hand, NMCMC toggles the inference-time com-\npute expended by our algorithm, providing a natural axis for test-time scaling. In Section 4.3 we\nraised the notion of a mixing time, or the number of MCMC steps required before adequately sam-\npling from the target distribution. In our case, we expect that the fewer MCMC steps we take, the\nfurther our algorithm samples from the target pα.\nWe plot performance dependence on NMCMC in Figure 6 and notice a steady increase in accuracy\nuntil NMCMC = 10, beyond which accuracy remains roughly stable (not plotted). The accuracy\ndifference from using fewer MCMC steps is noticeable but no more than 3-4% between NMCMC = 2\nand NMCMC = 10. However, the jump in accuracy by using at least two steps as opposed to none is\nsubstantial (3-4%).\nWe can even compute the total amount of tokens generated by our method relative to running GRPO.\nFrom (12), our sampler generates\n1\n4B · NMCMCT times as many tokens as standard inference to gen-\nerate a sequence of length T. Plugging in our experimental parameters NMCMC = 10, T = 679 (our\naverage output length for MATH500) and B = 192, running inference with power sampling incurs\na multiplier of 8.84× the number of tokens as running standard inference. Since GRPO generates\nmultiple rollouts per example during training, our method incurs roughly the same inference cost as\none epoch of GRPO training, assuming 8 rollouts per sample with identical dataset sizes. Typically\nthough, one GRPO epoch is still more expensive as it uses 16 rollouts and a training set that is larger\nthan MATH500.\n6\nConclusion\nIn this work, we present an algorithm that samples directly from a base model without any additional\ntraining or access to an external signal, achieving a single-shot reasoning performance that is on par\nwith, and sometimes even better than, that of a state-of-the-art RL-posttraining algorithm. We use the\ndiscussion of RL distribution sharpening to motivate defining the power distribution as a valuable\ntarget distribution for reasoning. Although exact power distribution sampling is intractable, we\nemploy classic MCMC techniques alongside the sequential structure of autoregressive generation to\ndefine our power sampling algorithm, which demonstrates strong empirical performance.\n11\n\nOur results suggest that base model capabilities are underutilized at sampling time and point towards\na close relationship between high likelihood regions of the base model and strong reasoning capabil-\nities. Employing additional compute at sampling-time with a stronger understanding of base model\ncapabilites offers a promising direction for expanding the scope of reasoning beyond verifiability.\n7\nAcknowledgments\nA.K. would like to thank the Paul and Daisy Soros Foundation, NDSEG Fellowship, and Kempner\nInstitute for their support.\nReferences\nDaya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu,\nShirong Ma, Peiyi Wang, Xiao Bi, et al.\nDeepseek-r1: Incentivizing reasoning capability in\nLLMs via reinforcement learning. arXiv preprint arXiv:2501.12948, 2025. 1, 3, 10\nJingcheng Hu, Yinmin Zhang, Qi Han, Daxin Jiang, Xiangyu Zhang, and Heung-Yeung Shum.\nOpen-reasoner-zero: An open source approach to scaling up reinforcement learning on the base\nmodel. arXiv preprint arXiv:2503.24290, 2025. 1, 3\nDan Hendrycks, Collin Burns, Saurav Kadavath, Akul Arora, Steven Basart, Eric Tang, Dawn Song,\nand Jacob Steinhardt. 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Hashimoto.\nLength-controlled\nalpacaeval: A simple way to debias automatic evaluators. arXiv preprint arXiv:2404.04475, 2024.\nURL https://arxiv.org/abs/2404.04475. 8\n15\n\nA\nAppendix\nA.1\nAdditional Theoretical Discussion\nIn this section, we provide a stronger formalization of the phenomenon that power sampling down-\nweights tokens that trap outputs in low-likelihood futures while low-temperature sampling does not.\nProposition 2 (Informal). Power sampling upweights tokens with small support but high likelihood\ncompletions, while low-temperature sampling upweights tokens with large support but low likeli-\nhood completions.\nDefinition 2. For the rest of this section, fix a prefix x0:t−1. We say that xt has marginal weight ε\nunder the conditional next-token distribution if P\nx>t p(x0, . . . , xt, . . . xT ) = ε.\nWe consider a simplified model of the “critical window” or “pivotal token” phenomenon [Li et al.,\n2025, Abdin et al., 2024], which refers to intermediate tokens that strongly influence the quality of\nthe final generation. We differentiate between pivotal tokens that lead to high-likelihood futures vs.\nlow-likelihood ones.\nDefinition 3. At one extreme, a pivotal token maximally induces a high-likelihood completion if it\nplaces its entire marginal weight ε on one future (singular support); i.e., for only one choice of x > t\nis p(x0, . . . , xt, . . . , xT ) nonzero. We call such a token a positive pivotal token.\nDefinition 4. At the other extreme, a pivotal token minimizes the likelihood of any future if its entire\nmarginal weight ε is uniformly distributed across N future completions. In other words, there exist\nN completions x > t such that p(x0, . . . , xt, . . . , xT ) are all nonzero with likelihood\nε\nN . We call\nsuch a token a negative pivotal token.\nOur simplified model of high and low-likelihood futures examines when positive pivotal tokens are\nfavored over negative pivotal tokens under a given sampling distribution. In particular, we show that\npower sampling can upweight a positive pivotal token over a negative one even if the latter has a\nhigher marginal weight, whereas low-temperature sampling always upweights the negative pivotal\ntoken in such a scenario.\nOf course, whenever a positive pivotal token has higher marginal weight, both power sampling and\nlow-temperature sampling will upweight it.\nProposition 3. Let xt be a positive pivotal token with marginal weight ε, and let x′\nt be a negative\npivotal token with marginal weight ε′ and support N. Then if\nε′\nN 1−1/α < ε < ε′,\n(14)\nthe future likelihood of xt is higher than any future likelihood of x′\nt. Moreover, power sampling\nupweights xt over x′\nt while low-temperature sampling upweights x′\nt over xt.\nProof. Since α ≥1, it follows that\nε′\nN 1−1/α > ε′\nN\n(15)\nand thus ε > ε′\nN , establishing that the future completion likelihood of xt is greater than that of x′\nt\n(i.e. the assignment of positive and negative pivotal tokens is consistent).\nNow, if ε < ε′, then under the low-temperature distribution, the relative marginal weights on xt and\nx′\nt are εα and ε′α, so the probability of choosing xt is downweighted relative to x′\nt. However, for the\npower distribution, the relative marginal weights are ppow(xt|x<t) = εα and ppow(x′\nt|x<t) =\nε′α\nN α−1 .\nThen, as long as εα >\nε′α\nN α−1 ⇐⇒ε >\nε′\nN 1−1/α , token xt will be upweighted relative to token x′\nt.\nIn other words, the marginal weight on xt can be less than the mass on x′\nt under p, but if the\ncompletion for xt has higher likelihood than any individual completion for x′\nt, power sampling\nfavors xt over x′\nt.\nA.2\nPass@k Accuracies over Multiple Domains\nIn this section, we plot the pass@k performance of power sampling, GRPO, and the base model\n(Qwen2.5-Math-7B) over MATH500, GPQA, and HumanEval to demonstrate that our sampling\n16\n\nalgorithm is highly performant at both single-shot and multi-shot reasoning while maintaining re-\nsponse diversity. Power sampling is plotted with α = 4.0 for MATH500 and GPQA and α = 1.67\nfor HumanEval (this temperature exhibits slightly better results at earlier k). In all cases, both in-\ndomain and out-of-domain for GRPO, power sampling has near universally better performance than\nboth GRPO and the base model in pass@k for k > 1 and matches, if not exceeds, the base model\nupper bound at large k.\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\nk\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nPass@k Accuracy\nMATH500\nOurs\nBase\nGRPO\nFigure 7: Pass@k performance on MATH500 (Qwen2.5-Math-7B).\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\nk\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\nPass@k Accuracy\nHumanEval\nOurs\nBase\nGRPO\nFigure 8: Pass@k performance on HumanEval (Qwen2.5-Math-7B).\nOne thing to note about these plots is that the loss in diversity varies noticeably from benchmark\nto benchmark. MATH500 and GPQA clearly show that GRPO has a significantly lower pass@k\nperformance and diversity even for smaller k, while on HumanEval GRPO exhibits better pass@k\n17\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\nk\n0.3\n0.4\n0.5\n0.6\n0.7\n0.8\n0.9\n1.0\nPass@k Accuracy\nGPQA\nOurs\nBase\nGRPO\nFigure 9: Pass@k performance on GPQA (Qwen2.5-Math-7B).\nthan the base model until k = 16. We speculate this might be due to the fact that while MATH500\nand GPQA are graded on a “singular” answer, HumanEval is based on completing code where\nmultiple solutions might be accepted, indicating a reduction in diversity may emerge but at much\nhigher k.\nA.3\nMore Qualitative Examples\nCan you solve the following math problem? Write\n3\n20 as a decimal. Please reason step by step,\nand put your final answer within \\boxed{{}}. Correct answer: 0.15\nSystem\nAnswer\nOurs\nTo convert the fraction\n3\n20 to a decimal, we can follow these steps:\n1. Identify the numerator and the denominator of the fraction. In this case, the numerator\nis 3 and the denominator is 20. 2. Divide the numerator by the denominator. In this case,\nwe divide 3 by 20. 3. Perform the division. The result is 0.15.\nTherefore, the fraction\n3\n20 as a decimal is 0.15 .\nGRPO\nTo convert the fraction\n3\n20 to a decimal, we can follow these steps:\n1. Understand the fraction: The fraction\n3\n20 means 3 divided by 20.\n2. Perform the division: We need to divide 3 by 20. Since 3 is less than 20, we will add a\ndecimal point and zeros to 3 to make the division possible.\n3. Set up the division: Write 3 as 3.000 (adding three zeros for precision).\n4. Divide: - 20 goes into 30 once (since 20 × 1 = 20). - Subtract 20 from 30 to get a\nremainder of 10. - Bring down the next 0 to make it 100. - 20 goes into 100 five times\n(since 20 × 5 = 100). - Subtract 100 from 100 to get a remainder of 0.\n5. Combine the results: The quotient from the division is 0.15.\nTherefore, the decimal representation of\n3\n20 is 0.15 .\nTable 3: Comparison on MATH500: Qwen2.5-Math-7B. We present an example where our method and GRPO\nare correct.\n18\n\nThe Fib4 number sequence is a sequence similar to the Fibonacci sequence defined as: fib4(0)=0,\nfib4(1)=0, fib4(2)=2, fib4(3)=0, and for n ≥4, fib4(n) = fib4(n-1) + fib4(n-2) + fib4(n-3) +\nfib4(n-4). Write a function to efficiently compute the n-th element without recursion.\nSystem\nCompletion\nPassed\nOurs\n# Base\ncases\nif n == 0:\nreturn 0\nelif n == 1:\nreturn 0\nelif n == 2:\nreturn 2\nelif n == 3:\nreturn 0\n# Initialize\nthe\nfirst\nfour\nelements\nfib4_values = [0, 0, 2, 0]\n# Compute\nthe\nrest\nof the\nsequence\niteratively\nfor i in range (4, n+1):\nnext_value = ( fib4_values [i -1] + fib4_values [\ni -2] +\nfib4_values [i -3] + fib4_values [\ni -4])\nfib4_values .append(next_value )\n# Return\nthe n-th\nelement\nreturn\nfib4_values [n]\ntrue\nGRPO\na, b, c, d = 0, 0, 2, 0\nfor _ in range(n):\na, b, c, d = b, c, d, a + b + c + d\nreturn d\nfalse\nTable 4: HumanEval comparison on Phi-3.5-mini-instruct.\n19\n\nCan you solve the following math problem? What integer n satisfies 0 ≤n < 18 and\nn ≡−11213141\n(mod 18) ?\nPlease reason step by step, and put your final answer within \\boxed{{}}. Correct answer: 13\nMethod\nAnswer\nOurs\nFirst, we perform the division of 11213141 by 18:\n11213141 ÷ 18 ≈622952.277777 . . .\nThe integer part of the quotient is 622952. Now, we multiply 622952 by 18 to find the\nproduct:\n622952 × 18 = 11213136\nNext, we subtract this product from 11213141 to find the remainder:\n11213141 −11213136 = 5\nSo, the remainder when 11213141 is divided by 18 is 5. This means:\n11213141 ≡5\n(mod 18)\nTherefore:\n−11213141 ≡−5\n(mod 18)\nSince we need a non-negative remainder, we convert −5 to a positive equivalent by adding\n18:\n−5 + 18 = 13\nThus:\n−11213141 ≡13\n(mod 18)\nThe integer n that satisfies 0 ≤n < 18 and n ≡−11213141 (mod 18) is: 13 .\nGRPO\nStep 1: Find the remainder when 11213141 is divided by 18. We can use the property of\nmodular arithmetic that states a ≡b (mod m) if and only if a mod m = b mod m.\nSo, we need to compute 11213141 mod 18.\nTo simplify this, we can use the fact that the remainder of a number when divided by 18 is\nthe same as the remainder of the sum of its digits when divided by 18 (although this is not\nalways the most efficient method, it can be used here for simplicity).\nFirst, let’s sum the digits of 11213141:\n1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 = 14.\nNow, we need to find the remainder when 14 is divided by 18. Since 14 is less than 18, the\nremainder is simply 14. Therefore,\n11213141 ≡14\n(mod 18).\nStep 2: Find the remainder when −11213141 is divided by 18. Since 11213141 ≡14\n(mod 18), it follows that\n−11213141 ≡−14\n(mod 18).\nWe need to convert −14 to a positive remainder between 0 and 17. To do this, we add 18\nto −14:\n−14 + 18 = 4.\nTherefore,\n−11213141 ≡4\n(mod 18).\nThe integer n that satisfies 0 ≤n < 18 and n ≡−11213141 (mod 18) is 4 .\nTable 5: MATH500 comparison between our sampling algorithm and GRPO for Qwen2.5-Math-7B. Here is an\nexample where GRPO gets an incorrect answer, while our sampling algorithm succeeds. Our sample answer\nuses a distinct method altogether.\n20\n"},"cleaned":{"kind":"string","value":"Reasoning with Sampling: Your Base Model is Smarter Than You Think Aayush Karan1, Yilun Du1 1Harvard University Website Code Abstract Frontier reasoning models have exhibited incredible capabilities across a wide array of disciplines, driven by posttraining large language models (LLMs) with reinforcement learning (RL). However, despite the widespread success of this paradigm, much of the literature has been devoted to disentangling truly novel behaviors that emerge during RL but are not present in the base models. In our work, we approach this question from a different angle, instead asking whether comparable reasoning capabilites can be elicited from base models at inference time by pure sampling, without any additional training. Inspired by Markov chain Monte Carlo (MCMC) techniques for sampling from sharpened distributions, we propose a simple iterative sampling algorithm leveraging the base models' own likelihoods. Over different base models, we show that our algorithm offers substantial boosts in reasoning that nearly match and even outperform those from RL on a wide variety of single-shot tasks, including MATH500, HumanEval, and GPQA. Moreover, our sampler avoids the collapse in diversity over multiple samples that is characteristic of RL-posttraining. Crucially, our method does not require training, curated datasets, or a verifier, suggesting broad applicability beyond easily verifiable domains. 1 Introduction Reinforcement learning (RL) has become the dominant paradigm for enhancing the reasoning capabilities of large language models (LLMs) [Guo et al., 2025, Hu et al., 2025]. Equipped with a reward signal that is typically automatically verifiable, popular RL techniques have been successfully applied to posttrain frontier models, leading to sizeable performance gains in domains like math, coding, and science [Hendrycks et al., 2021, Li et al., 2022, Rein et al., 2024]. Despite the widespread empirical success of RL for LLMs, a large body of literature has centered around the following question: are the capabilities that emerge during RL-posttraining fundamentally novel behaviors that are not present in the base models? This is the question of distribution sharpening [He et al., 2025, Shao et al., 2025, Yue et al., 2025]: that is, whether the posttrained distribution is simply a \"sharper\" version of the base model distribution, instead of placing mass on reasoning traces the base model is unlikely to generate. Several works point towards the difficulty in learning new capabilities with RL-posttraining. He et al. [2025], Song et al. [2025] compare the pass@k (multi-shot) scores of base models with posttrained models, finding that for large k, base models actually outperform while the latter suffer from degraded generation diversity. In such cases, RL appears to redistribute pass@k performance to single-shot performance at the expense of multi-shot reasoning. Yue et al. [2025] also notes that the reasoning traces post-RL are tightly concentrated at high likelihoods/confidences under the base model, seemingly drawing from existing high-likelihood capabilities. We illustrate this point in our own experiments in Figure 4. Regardless, the advantage of RL-posttraining for single-shot reasoning has remained, as of yet, undeniable. Preprint. Under review. 16 Oct 2025 MATH500 HumanEval GPQA 10 20 30 40 50 60 70 80 Accuracy (%) 49.6 32.9 27.8 78.5 53.7 39.9 74.8 57.3 38.9 Reasoning AlpacaEval2.0 0.5 1.0 1.5 2.0 2.5 3.0 Score 1.61 2.38 2.88 General Base (Qwen2.5-Math-7B) GRPO (RL) Ours (Training-free) Figure 1: Our sampling algorithm can match and outperform RL-posttraining. Left: we compare our sampling algorithm (ours) against the base model (base) and RL-posttraining (GRPO) on three verifiable reasoning tasks (MATH500, HumanEval, GPQA). Right: we compare them on an unverifiable general task (AlpacaEval2.0). Our algorithm achieves comparable performance to GRPO within the posttraining domain (MATH500) but can outperform on out-of-domain tasks such as HumanEval and AlpacaEval. In this paper, we present a surprising result: sampling directly from the base model can achieve single-shot reasoning capabilites on par with those from RL. We propose a sampling algorithm for base models that leverages additional compute at inference time, achieving single-shot performance that nearly matches RL-posttraining on in-domain reasoning tasks and can even outperform on out-of-domain reasoning tasks. Furthermore, we observe that generation diversity does not degrade with our sampler; in fact, our pass@k (multi-shot) performance strongly outperforms RL. We benchmark specifically against Group Relative Policy Optimization (GRPO), which is the standard RL algorithm for enhancing LLM reasoning [Shao et al., 2024]. Crucially, our algorithm is training-free, dataset-free, and verifier-free, avoiding some of the inherent weaknesses of RL methods including extensive hyperparameter sweeps to avoid training instabilities, the need to curate a diverse and expansive posttraining dataset, and the lack of guaranteed access to a ground truth verifier/reward signal [Prabhudesai et al., 2025]. Our contributions can be summarized as follows: i) We introduce the power distribution as a useful sampling target for reasoning tasks. Since it can be explicitly specified with a base LLM, no additional training is required. ii) We further introduce an approximate sampling algorithm for the power distribution using a Markov chain Monte Carlo (MCMC) algorithm that iteratively resamples token subsequences according to their base model likelihoods. iii) We empirically demonstrate the effectiveness of our algorithm over a range of models (Qwen2.5Math-7B, Qwen2.5-7B, Phi-3.5-mini-instruct) and reasoning tasks (MATH500, HumanEval, GPQA, AlpacaEval 2.0). Our results show that sampling directly from the base model can achieve results on par with GRPO. In fact, for some out-of-domain tasks, our algorithm consistently outperforms the RL baseline. Moreover, over multiple samples, we avoid the collapse in diversity afflicting RL-posttraining, achieving the best of both worlds in terms of single-to-few-shot reasoning capabilities as well as sample diversity. Our results collectively illustrate that existing base models are much more capable at single-shot reasoning than current sampling methods reveal. 2 2 Related Works Reinforcement learning for LLMs. RL has been instrumental in posttraining LLMs. Early on, RL with human feedback (RLHF) [Ouyang et al., 2022] was developed as a technique to align LLMs with human preferences using a trained reward model. Recently, RL with verifiable rewards (RLVR) has emerged as a powerful new posttraining technique, where many works [Guo et al., 2025, Lambert et al., 2024, Hu et al., 2025, Zeng et al., 2025] discovered that a simple, end-of-generation reward given by an automated verifier could substantially enhance performance on difficult reasoning tasks in mathematics and coding. The Group Relative Policy Optimization (GRPO) algorithm was at the center of these advances [Shao et al., 2024]. Building off of this success, many subsequent works have examined using reward signals derived from internal signals such as self-entropy [Zhao et al., 2025], confidence [Prabhudesai et al., 2025], and even random rewards [Shao et al., 2025]. Similar to these works, our paper examines base model likelihoods as a mechanism for improving reasoning performance, but crucially, our technique is training-free. Autoregressive MCMC sampling with LLMs. Prior works have explored integrating classic MCMC techniques with autoregressive sampling. Many settings including red-teaming, promptengineering, and personalized generation can be framed as targeting sampling from the base LLM distribution but tilted towards an external reward function. Zhao et al. [2024] proposes learning intermediate value functions that are used in a Sequential Monte Carlo (SMC) framework [Chopin, 2004], where multiple candidate sequences are maintained and updated according to their expected future reward. Similarly, Faria et al. [2024] proposes a Metropolis-Hastings (MH) algorithm, which instead of maintaining multiple candidates performs iterative resampling, again updating according to expected reward. Methodologically, our sampling algorithm is most similar to this latter work, but the crucial difference is that our target sampling distribution is completely specified by the base LLM, avoiding the need for an external reward. Annealed sampling for diffusion. In the statistical physics and Monte Carlo literature, sampling from pα is known as sampling from an annealed, or tempered, distribution [Neal, 1998] and has inspired a new wave of interest within the diffusion community. Indeed, in traditional MCMC sampling, annealing is used as a way to avoid mode-collapse during sampling and more accurately sample from complex multimodal distributions [Łatuszy ́nski et al., 2025]. This has re-emerged as inference-time sampling methods for diffusion that aim to steer a pretrained model towards \"tilted distributions\" [Du et al., 2023, Kim et al., 2025, Karan et al., 2025, Wang et al., 2025, Kong et al., 2025, Zhang et al., 2025]. Where traditional RL techniques exhibit mode collapse, applications in the physical sciences [Sambridge, 2014] require multimodal sampling. To this end, works such as Du et al. [2023], Wang et al. [2025], Kim et al. [2025] construct sequences of annealed distributions to ease the transition from base diffusion distribution to tilted distribution. Other works [Skreta et al., 2025, Xu et al., 2025] intentionally target sampling from pα for α > 1 as a means of generating higher quality samples from the base diffusion model, which is particularly popular for generating more designable proteins [Geffner et al., 2025]. 3 Preliminaries Let X be a finite vocabulary of tokens, and let X T denote the set of finite sequences of tokens x0:T = (x0, x1, . . . , xT ), where xi ∈X for all i and T ∈Z≥0 is some nonnegative integer. For convenience, for a given t, let x t = (xt+1, . . . , xT ), with similar definitions for x≤t and x≥t. In general, x refers to a token sequence x0:T , where T is implicitly given. Then an LLM defines a distribution p over token sequences X T by autoregressively learning the conditional token distributions p(xt|x p(x′) =⇒ p(x)α p(x′)α > p(x) p(x′) (α ∈[1, ∞]), (2) it follows that exponentiating p increases the relative weight on higher likelihood sequences (x) while decreasing the relative weight on lower likelihood ones (x′) (see Figure 2 for a visualization). A related but well-known sharpening strategy is low-temperature sampling [Wang et al., 2020], which exponentiates the conditional next-token distributions at each step: ptemp(xt|x0 . . . xt-1) = p(xt|xt-1 . . . x0)α P x′ t∈X p(x′ t|xt-1 . . . x0)α , (3) where the temperature is τ = 1/α. A common misconception is that sampling with (3) over T tokens is equivalent to sampling from pα; however, this is false in a subtle yet crucial way, as we illuminate in the following. Proposition 1. Low-temperature sampling does not sample from the power distribution pα. Proof. We show that the associated conditional next-token distributions are distinct at each timestep t. The conditional distribution on xt for pα is given by ppow(xt|x0 . . . xt-1) = P x>t p(x0, . . . , xt, . . . , xT )α P x≥t p(x0, . . . , xt, . . . , xT )α . (4) Using Bayes rule p(xt|xt-1 . . . x0) = p(x0, . . . , xt) p(x0, . . . , xt-1) = P x>t p(x0, . . . , xt, . . . , xT ) P x≥t p(x0, . . . , xt, . . . , xT ), (5) we can rewrite the low-temperature marginal (3) as ptemp(xt|x0 . . . xt-1) = P x>t p(x0, . . . , xt, . . . , xT ) α P x′ t P x>t p(x0, . . . , xt, . . . , xT ) α . (6) Ignoring normalizations for clarity, the relative weight on token xt for sampling from pα is given by a sum of exponents ppow(xt|x t p(x0, . . . , xt, . . . , xT )α. (7) 4 Meanwhile, the relative weight for low-temperature sampling is given by an exponent of sums ptemp(xt|x t p(x0, . . . , xt, . . . , xT ) !α . (8) Since the relative weights of next-token prediction are distinct for each sampling strategy, it follows that the joint distribution over seqeunces must also be distinct for each sampler. Hence, the distribution on sequences given by low-temperature sampling is not the same as the one given by pα. One intuitive way to understand this difference is that low-temperature sampling does not account for how exponentiation sharpens the likelihoods of \"future paths\" at time step t, instead \"greedily\" averaging all these future likelihoods (exponent of sums (8)). On the other hand, sampling from pα inherently accounts for future completions as it exponentiates all future paths (sum of exponents (7)) before computing the weights for next-token prediction. This has the following consequence: Observation 1. The power distribution upweights tokens with few but high likelihood future paths, while low-temperature sampling upweights tokens with several but low likelihood completions. Example 1. We can observe this phenomenon with a simple example. Let us consider the token vocabulary X = {a, b} and restrict our attention to two-token sequences (x0, x1): aa, ab, ba, bb. Let p(aa) = 0.00, p(ab) = 0.40, p(ba) = 0.25, p(bb) = 0.25, so that p(x0 = a) = 0.40, p(x0 = b) = 0.50. Let α = 2.0. Under pα, we have ppow(x0 = a) ∝0.002 + 0.402 = 0.160, ppow(x0 = b) ∝0.252 + 0.252 = 0.125, so pα prefers sampling a over b. Under low-temperature sampling, ptemp(x0 = a) ∝(0.00 + 0.40)2 = 0.160, ptemp(x0 = b) ∝(0.25 + 0.25)2 = 0.250, preferring sampling b over a. If pα samples x0 = a, there is only one future path with likelihood 0.40. If ptemp samples x0 = b, there are two future paths ba, bb, but either choice has likelihood 0.25. In other words, even though a has lower conditional likelihood under both p and ptemp, pα upweights a and samples the highest likelihood two-token sequence. b has many future paths contributing to a higher likelihood under p and ptemp, but leads to low likelihood sequences. We provide a stronger formalization of this phenomenon in Appendix A.1. Thus, sampling from pα encourages sampling tokens which have fewer but higher likelihood \"future paths\", as opposed to tokens with several lower likelihood completions. This type of behavior is immensely valuable for reasoning tasks. For example, choosing \"wrong\" tokens that have high average likelihoods but trap outputs in low likelihood individual futures are examples of critical windows or pivotal tokens [Li et al., 2025, Abdin et al., 2024], a phenomenon where a few tokens are highly influential in the correctness of language model outputs. In fact, sharp critical windows have been shown to correlate strongly with reasoning failures [Li et al., 2025]. Instead, embedded in sampling from the power distribution is an implicit bias towards planning for future high likelihood tokens. 4.2 The Metropolis-Hastings Algorithm Now that we have seen how sampling from pα can in theory assist the underlying LLM's ability to reason, our aim now turns towards proposing an algorithm to accurately sample from it. Given an LLM p, we have access to the values pα over any sequence length; however, these values are unnormalized. Direct sampling from the true probabilities requires normalizing over all sequences (x0, . . . , xT ) ∈X T , which is computationally intractable. To get around this, we invoke a Markov Chain Monte Carlo (MCMC) algorithm known as Metropolis-Hastings (MH) [Metropolis et al., 1953], which targets exactly what we want: approximate sampling from an unnormalized probability distribution. The MH algorithm constructs a 5 Figure 3: Illustrating Metropolis-Hastings with random resampling. A random index t is selected and a new candidate is generated by resampling. Based on the relative likelihoods, the candidate is accepted or rejected, and the process repeats. Markov chain of sample sequences (x0, x1, . . . , xn) using an arbitrary proposal distribution q(x|xi) to select the next candidate xi+1. With probability A(x, xi) = min 1, pα(x) · q(xi|x) pα(xi) · q(x|xi) , (9) candidate x is accepted as xi+1; otherwise, MH sets xi+1 = xi. This algorithm is especially convenient as it only requires the relative weights given by pα (as the normalization weights in A cancel) and works with any generic but tractable sampler q with minimal restrictions. Remarkably, for large enough n, this process converges to sampling from the target distribution pα under the following (quite minimal) conditions on the proposal distribution [Neal, 1993]: Definition 1. The proposal distribution q is irreducible if for any set X with nonzero mass under the target distribution pα, q has nonzero probability of eventually sampling from X. The proposal is aperiodic if the induced chain of samples does not return to the same sample after a fixed interval number of steps. Thus, we must simply ensure that our proposal distribution satisfies irreducibility and aperiodicity, and Metropolis-Hastings takes care of the rest. On a practical level, we would also like both q(x|xi) and its reverse q(xi|x) to be easily computable. Consider the following family of random resampling proposal distributions (see Figure 3). Let pprop be a proposal LLM. With uniform probability 1 T , select a random t ∈[1, T] and resample the sequence starting at index t using pprop. Then the transition likelihood q(x|xi) is simply the likelihood of the resampling. Note that at each candidate selection step, we have a nonzero probability of transitioning between any two sequences x, x′ ∈X, since with some probability we can always resample as early as the beginning of x. This ensures our proposal distribution is both irreducible and aperiodic. Moreover, q(xi|x) is easy to calculate by symmetry, since we can treat xi as a resampled version of x. With the flexibility endowed by Metropolis-Hastings, we can choose the proposal LLM pprop to be any LLM with any sampling strategy (e.g., low-temperature sampling). 4.3 Power Sampling with Autoregressive MCMC A direct implementation of Metropolis-Hastings for LLMs would involve initializing with a sampled token sequence of length T, subsequently generating new candidates of length T with (9) over many, many iterations. This process is computationally expensive, however, due to the repeated, full sequence inference calls to the LLM. In fact, the main downside to MCMC algorithms in practice is the potential for an exponential mixing time [Gheissari et al., 2017], where a poor choice of initialization or proposal distribution can result in an exponentially large number of samples required before convergence to the target distribution. This problem is exacerbated if the sample space has high dimensionality [Bandeira et al., 2022, Schmidler and Woodard, 2013], which is precisely exhibited by the sequence space of tokens X T , especially for long sequences/large values of T. To remedy this, we propose an algorithm that leverages the sequential structure of autoregressive sampling. We define a series of intermediate distributions which we progressively sample from, until converging to the target distribution pα. In particular, samples from one intermediate distribution initiate a Metropolis-Hastings process for the next, helping avoid pathological initializations. 6 Algorithm 1: Power Sampling for Autoregressive Models Input : base p; proposal pprop; power α; length T Hyperparams: block size B; MCMC steps NMCMC Output : (x0, . . . , xT ) ∼pα 1 Notation: Define the unnormalized intermediate target πk(x0:kB) ∝p(x0:kB)α. 2 for k ←0 to ⌈T B ⌉-1 do 3 Given prefix x0:kB, we wish to sample from πk+1. Construct initialization x0 by extending autoregressively with pprop: x(0) t ∼pprop xt | x 1. Moreover, our performance curve supersedes that of the base model until finally converging in performance. In particular, we are able to achieve GRPO-level single-shot performance without compromising multi-shot performance (see Appendix A.2 for other domains), addressing a long-standing downside to RL-posttraining. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 k 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pass@k Accuracy MATH500 Ours Base GRPO Figure 5: Pass@k performance on MATH500. We plot the pass@k accuracy (correct if at least one of k samples is accurate) of power sampling (ours) and RL (GRPO) relative to the base model (Qwen2.5-Math7B). Our performance curve is strictly better than both GRPO and the base model, and our pass rate at high k matches the base model, demonstrating sustained generation diversity. The effect of power distributions. The two most important hyperparameters for power sampling are the choice of α and the number of MCMC (resampling) steps during sequence generation 10 Figure 6: Effect of hyperparameters on power sampling. Left: We plot MATH500 accuracy across model families for various values of α. Right: We plot the increase in accuracy of power sampling on Qwen models as the number of MCMC steps increases. NMCMC. At the extremes, choosing α = 1.0 samples from the base model directly, while taking α →∞has the effect of deterministically accepting any resampled sequence that strictly increases the likelihood. Of course, even though higher base model likelihoods correlate with better reasoning (Figure 4), directly optimizing for likelihood is not necessarily optimal for reasoning, suggesting an ideal intermediate value of α. In Figure 6, we display MATH500 accuracies across various values of α and find that an intermediate α = 4.0 outperforms other values, as expected. Noticeably, the accuracies of power sampling remain relatively stable beyond α ≥2.0, suggesting that power sampling in practice is relatively robust to the choice of α. Test-time scaling with MCMC steps. On the other hand, NMCMC toggles the inference-time compute expended by our algorithm, providing a natural axis for test-time scaling. In Section 4.3 we raised the notion of a mixing time, or the number of MCMC steps required before adequately sampling from the target distribution. In our case, we expect that the fewer MCMC steps we take, the further our algorithm samples from the target pα. We plot performance dependence on NMCMC in Figure 6 and notice a steady increase in accuracy until NMCMC = 10, beyond which accuracy remains roughly stable (not plotted). The accuracy difference from using fewer MCMC steps is noticeable but no more than 3-4% between NMCMC = 2 and NMCMC = 10. However, the jump in accuracy by using at least two steps as opposed to none is substantial (3-4%). We can even compute the total amount of tokens generated by our method relative to running GRPO. From (12), our sampler generates 1 4B · NMCMCT times as many tokens as standard inference to generate a sequence of length T. Plugging in our experimental parameters NMCMC = 10, T = 679 (our average output length for MATH500) and B = 192, running inference with power sampling incurs a multiplier of 8.84× the number of tokens as running standard inference. Since GRPO generates multiple rollouts per example during training, our method incurs roughly the same inference cost as one epoch of GRPO training, assuming 8 rollouts per sample with identical dataset sizes. Typically though, one GRPO epoch is still more expensive as it uses 16 rollouts and a training set that is larger than MATH500. 6 Conclusion In this work, we present an algorithm that samples directly from a base model without any additional training or access to an external signal, achieving a single-shot reasoning performance that is on par with, and sometimes even better than, that of a state-of-the-art RL-posttraining algorithm. We use the discussion of RL distribution sharpening to motivate defining the power distribution as a valuable target distribution for reasoning. Although exact power distribution sampling is intractable, we employ classic MCMC techniques alongside the sequential structure of autoregressive generation to define our power sampling algorithm, which demonstrates strong empirical performance. 11 Our results suggest that base model capabilities are underutilized at sampling time and point towards a close relationship between high likelihood regions of the base model and strong reasoning capabilities. Employing additional compute at sampling-time with a stronger understanding of base model capabilites offers a promising direction for expanding the scope of reasoning beyond verifiability. 7 Acknowledgments A.K. would like to thank the Paul and Daisy Soros Foundation, NDSEG Fellowship, and Kempner Institute for their support. References Daya Guo, Dejian Yang, Haowei Zhang, Junxiao Song, Ruoyu Zhang, Runxin Xu, Qihao Zhu, Shirong Ma, Peiyi Wang, Xiao Bi, et al. 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Length-controlled alpacaeval: A simple way to debias automatic evaluators. arXiv preprint , 2024. URL https://arxiv.org/abs/2404.04475. 8 15 A Appendix A.1 Additional Theoretical Discussion In this section, we provide a stronger formalization of the phenomenon that power sampling downweights tokens that trap outputs in low-likelihood futures while low-temperature sampling does not. Proposition 2 (Informal). Power sampling upweights tokens with small support but high likelihood completions, while low-temperature sampling upweights tokens with large support but low likelihood completions. Definition 2. For the rest of this section, fix a prefix x0:t-1. We say that xt has marginal weight ε under the conditional next-token distribution if P x>t p(x0, . . . , xt, . . . xT ) = ε. We consider a simplified model of the \"critical window\" or \"pivotal token\" phenomenon [Li et al., 2025, Abdin et al., 2024], which refers to intermediate tokens that strongly influence the quality of the final generation. We differentiate between pivotal tokens that lead to high-likelihood futures vs. low-likelihood ones. Definition 3. At one extreme, a pivotal token maximally induces a high-likelihood completion if it places its entire marginal weight ε on one future (singular support); i.e., for only one choice of x > t is p(x0, . . . , xt, . . . , xT ) nonzero. We call such a token a positive pivotal token. Definition 4. At the other extreme, a pivotal token minimizes the likelihood of any future if its entire marginal weight ε is uniformly distributed across N future completions. In other words, there exist N completions x > t such that p(x0, . . . , xt, . . . , xT ) are all nonzero with likelihood ε N . We call such a token a negative pivotal token. Our simplified model of high and low-likelihood futures examines when positive pivotal tokens are favored over negative pivotal tokens under a given sampling distribution. In particular, we show that power sampling can upweight a positive pivotal token over a negative one even if the latter has a higher marginal weight, whereas low-temperature sampling always upweights the negative pivotal token in such a scenario. Of course, whenever a positive pivotal token has higher marginal weight, both power sampling and low-temperature sampling will upweight it. Proposition 3. Let xt be a positive pivotal token with marginal weight ε, and let x′ t be a negative pivotal token with marginal weight ε′ and support N. Then if ε′ N 1-1/α ε′ N (15) and thus ε > ε′ N , establishing that the future completion likelihood of xt is greater than that of x′ t (i.e. the assignment of positive and negative pivotal tokens is consistent). Now, if ε ε′α N α-1 ⇐⇒ε > ε′ N 1-1/α , token xt will be upweighted relative to token x′ t. In other words, the marginal weight on xt can be less than the mass on x′ t under p, but if the completion for xt has higher likelihood than any individual completion for x′ t, power sampling favors xt over x′ t. A.2 Pass@k Accuracies over Multiple Domains In this section, we plot the pass@k performance of power sampling, GRPO, and the base model (Qwen2.5-Math-7B) over MATH500, GPQA, and HumanEval to demonstrate that our sampling 16 algorithm is highly performant at both single-shot and multi-shot reasoning while maintaining response diversity. Power sampling is plotted with α = 4.0 for MATH500 and GPQA and α = 1.67 for HumanEval (this temperature exhibits slightly better results at earlier k). In all cases, both indomain and out-of-domain for GRPO, power sampling has near universally better performance than both GRPO and the base model in pass@k for k > 1 and matches, if not exceeds, the base model upper bound at large k. 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 k 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pass@k Accuracy MATH500 Ours Base GRPO Figure 7: Pass@k performance on MATH500 (Qwen2.5-Math-7B). 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 k 0.3 0.4 0.5 0.6 0.7 0.8 Pass@k Accuracy HumanEval Ours Base GRPO Figure 8: Pass@k performance on HumanEval (Qwen2.5-Math-7B). One thing to note about these plots is that the loss in diversity varies noticeably from benchmark to benchmark. MATH500 and GPQA clearly show that GRPO has a significantly lower pass@k performance and diversity even for smaller k, while on HumanEval GRPO exhibits better pass@k 17 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 k 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0 Pass@k Accuracy GPQA Ours Base GRPO Figure 9: Pass@k performance on GPQA (Qwen2.5-Math-7B). than the base model until k = 16. We speculate this might be due to the fact that while MATH500 and GPQA are graded on a \"singular\" answer, HumanEval is based on completing code where multiple solutions might be accepted, indicating a reduction in diversity may emerge but at much higher k. A.3 More Qualitative Examples Can you solve the following math problem? Write 3 20 as a decimal. Please reason step by step, and put your final answer within }. Correct answer: 0.15 System Answer Ours To convert the fraction 3 20 to a decimal, we can follow these steps: 1. Identify the numerator and the denominator of the fraction. In this case, the numerator is 3 and the denominator is 20. 2. Divide the numerator by the denominator. In this case, we divide 3 by 20. 3. Perform the division. The result is 0.15. Therefore, the fraction 3 20 as a decimal is 0.15 . GRPO To convert the fraction 3 20 to a decimal, we can follow these steps: 1. Understand the fraction: The fraction 3 20 means 3 divided by 20. 2. Perform the division: We need to divide 3 by 20. Since 3 is less than 20, we will add a decimal point and zeros to 3 to make the division possible. 3. Set up the division: Write 3 as 3.000 (adding three zeros for precision). 4. Divide: - 20 goes into 30 once (since 20 × 1 = 20). - Subtract 20 from 30 to get a remainder of 10. - Bring down the next 0 to make it 100. - 20 goes into 100 five times (since 20 × 5 = 100). - Subtract 100 from 100 to get a remainder of 0. 5. Combine the results: The quotient from the division is 0.15. Therefore, the decimal representation of 3 20 is 0.15 . Table 3: Comparison on MATH500: Qwen2.5-Math-7B. We present an example where our method and GRPO are correct. 18 The Fib4 number sequence is a sequence similar to the Fibonacci sequence defined as: fib4(0)=0, fib4(1)=0, fib4(2)=2, fib4(3)=0, and for n ≥4, fib4(n) = fib4(n-1) + fib4(n-2) + fib4(n-3) + fib4(n-4). Write a function to efficiently compute the n-th element without recursion. System Completion Passed Ours # Base cases if n == 0: return 0 elif n == 1: return 0 elif n == 2: return 2 elif n == 3: return 0 # Initialize the first four elements fib4_values = [0, 0, 2, 0] # Compute the rest of the sequence iteratively for i in range (4, n+1): next_value = ( fib4_values [i -1] + fib4_values [ i -2] + fib4_values [i -3] + fib4_values [ i -4]) fib4_values .append(next_value ) # Return the n-th element return fib4_values [n] true GRPO a, b, c, d = 0, 0, 2, 0 for _ in range(n): a, b, c, d = b, c, d, a + b + c + d return d false Table 4: HumanEval comparison on Phi-3.5-mini-instruct. 19 Can you solve the following math problem? What integer n satisfies 0 ≤n < 18 and n ≡-11213141 (mod 18) ? Please reason step by step, and put your final answer within }. Correct answer: 13 Method Answer Ours First, we perform the division of 11213141 by 18: 11213141 ÷ 18 ≈622952.277777 . . . The integer part of the quotient is 622952. Now, we multiply 622952 by 18 to find the product: 622952 × 18 = 11213136 Next, we subtract this product from 11213141 to find the remainder: 11213141 -11213136 = 5 So, the remainder when 11213141 is divided by 18 is 5. This means: 11213141 ≡5 (mod 18) Therefore: -11213141 ≡-5 (mod 18) Since we need a non-negative remainder, we convert -5 to a positive equivalent by adding 18: -5 + 18 = 13 Thus: -11213141 ≡13 (mod 18) The integer n that satisfies 0 ≤n < 18 and n ≡-11213141 (mod 18) is: 13 . GRPO Step 1: Find the remainder when 11213141 is divided by 18. We can use the property of modular arithmetic that states a ≡b (mod m) if and only if a mod m = b mod m. So, we need to compute 11213141 mod 18. To simplify this, we can use the fact that the remainder of a number when divided by 18 is the same as the remainder of the sum of its digits when divided by 18 (although this is not always the most efficient method, it can be used here for simplicity). First, let's sum the digits of 11213141: 1 + 1 + 2 + 1 + 3 + 1 + 4 + 1 = 14. Now, we need to find the remainder when 14 is divided by 18. Since 14 is less than 18, the remainder is simply 14. Therefore, 11213141 ≡14 (mod 18). Step 2: Find the remainder when -11213141 is divided by 18. Since 11213141 ≡14 (mod 18), it follows that -11213141 ≡-14 (mod 18). We need to convert -14 to a positive remainder between 0 and 17. To do this, we add 18 to -14: -14 + 18 = 4. Therefore, -11213141 ≡4 (mod 18). The integer n that satisfies 0 ≤n < 18 and n ≡-11213141 (mod 18) is 4 . Table 5: MATH500 comparison between our sampling algorithm and GRPO for Qwen2.5-Math-7B. Here is an example where GRPO gets an incorrect answer, while our sampling algorithm succeeds. Our sample answer uses a distinct method altogether. 20"}}},{"rowIdx":86,"cells":{"id":{"kind":"string","value":"2510.14896"},"raw":{"kind":"string","value":"Leveraging Multimodal LLM Descriptions of Activity for Explainable\nSemi-Supervised Video Anomaly Detection\nFurkan Mumcu1,†\nMichael J. Jones2,∗\nAnoop Cherian3,∗\nYasin Yilmaz4,†\n†University of South Florida\n∗Mitsubishi Electric Research Laboratories (MERL)\n1furkan@usf.edu\n2mjones@merl.com\n3cherian@merl.com\n4yasiny@usf.edu\nAbstract\nExisting\nsemi-supervised\nvideo\nanomaly\ndetection\n(VAD) methods often struggle with detecting complex\nanomalies involving object interactions and generally\nlack explainability.\nTo overcome these limitations,\nwe propose a novel VAD framework leveraging Multi-\nmodal Large Language Models (MLLMs). Unlike previ-\nous MLLM-based approaches that make direct anomaly\njudgments at the frame level, our method focuses on\nextracting and interpreting object activity and interac-\ntions over time.\nBy querying an MLLM with visual\ninputs of object pairs at different moments, we gener-\nate textual descriptions of the activity and interactions\nfrom nominal videos. These textual descriptions serve\nas a high-level representation of the activity and inter-\nactions of objects in a video. They are used to detect\nanomalies during test time by comparing them to tex-\ntual descriptions found in nominal training videos. Our\napproach inherently provides explainability and can be\ncombined with many traditional VAD methods to further\nenhance their interpretability. Extensive experiments on\nbenchmark datasets demonstrate that our method not\nonly detects complex interaction-based anomalies effec-\ntively but also achieves state-of-the-art performance on\ndatasets without interaction anomalies.\n1. Introduction\nVideo anomaly detection (VAD) has emerged as an ac-\ntive research field due to its significant applications in\nsecurity and public safety. The rapid growth of surveil-\nlance video data makes it impractical for humans to\nmonitor thoroughly.\nConsequently, VAD algorithms\nplay a vital role in identifying unusual events in video\nfootage, enabling human operators to focus their atten-\ntion on reviewing these flagged incidents. In this paper,\nwe focus on the formulation in which nominal videos\n(also called training videos) containing only normal ac-\ntivities in a particular scene are provided for learning a\nmodel. The goal is to identify when and where anoma-\nlous activities happen within test videos of the same\nscene. Along with other papers [3, 44], we refer to this\nversion of the problem as semi-supervised VAD because\nof the availability of normal, but not anomalous, training\nvideo. In Table 1, we summarize the properties of some\nrecent papers in this area to highlight the differences be-\ntween our work and other recent approaches.\nExisting semi-supervised video anomaly detection\nmethods suffer from several weaknesses. Recent studies\nstruggle to provide explainability for detected anoma-\nlies. While some existing work [36, 37] explored more\nexplainable approaches by interpreting the input fea-\ntures, they do not offer direct, textual explanations.\nFurthermore, the ComplexVAD dataset [28] introduced\ninteraction-based anomalies and demonstrated that de-\ntecting these anomalies is even more challenging than\nidentifying the simpler anomalies presented in earlier\ndatasets. Hence, we propose a new method to address\nthe drawbacks present in existing works, including the\nlack of focus on complex anomalies and explainabil-\nity. By utilizing a Large Language Model-based detec-\ntion approach, we introduce an effective video anomaly\ndetection framework that excels at identifying complex\nanomalies and features built-in explainability.\nLarge Language Models, especially in their multi-\nmodal forms (MLLMs), have demonstrated strong ca-\npabilities in interpreting and accurately describing com-\nplex situations from both text and images. GPT-4 with\nvision [1], for example, can analyze visual inputs and\nproduce detailed, contextually relevant descriptions. Ex-\nisting work has explored the use of MLLMs for video\nanomaly detection. Most of these methods are designed\nfor a multi-scene, weakly supervised setting. While such\napproaches are effective for detecting a common set of\nanomalies across multiple scenes, they fall short in iden-\ntifying scene-specific anomalies, which are central to\nthe semi-supervised, single-scene VAD task. To address\nthis limitation, we propose a novel MLLM-based frame-\nwork that: (1) considers object interactions in addition\nto individual object activities, (2) leverages an MLLM\n1\narXiv:2510.14896v1 [cs.CV] 16 Oct 2025\n\nMethod\nYear\nFocus\nFeatures\nTask Objective\nMLLM Interaction Explain-\nSemi\nScene\nUsage Anomalies\nability\nSupervised Specific\nVadCLIP [47]\n2023\nUsing CLIP for multi-scene weakly supervised VAD\nG#\n#\n#\n#\n#\nVera [50]\n2025\nOptimizing prompts for MLLM-based weakly supervised VAD\n \n#\n \n#\n#\nHolmes-VAU [53] 2025\nMLLM finetuning for anomaly understanding across multi-scene\n \n#\n \n#\n#\nAnomalyRuler [46] 2024 Scene-specific rule generation and frame-level descriptions via MLLMs\n \n#\nG#\n \n \nVLAVAD [16]\n2024\nUsing frame level MLLM descriptions for semi-supervised VAD\n \n#\n#\n \n \nEVAL [36]\n2023\nUsing object-level features for explainable VAD\n#\n#\nG#\n \n \nT-EVAL [37]\n2024\nObject-level, tracklet-based explainable VAD\n#\n#\nG#\n \n \nScene-Graph [28]\n2025\nExemplar and scene graph for interaction-based VAD\n#\n \nG#\n \n \nOurs\n2026\nObject-level MLLM descriptions for interaction-based VAD\n \n \n \n \n \nTable 1. Summary of properties and differences of some recent work in VAD. Our work is the only one designed for semi-\nsupervised, scene-specific VAD that handles interaction anomalies and provides textual explanations of anomalies.\nto generate high-level descriptions used in an exemplar-\nbased model that can be easily extended without retrain-\ning deep networks for each new scene, and (3) enables\nboth spatial and temporal localization of anomalies.\nIn our method, we adopt an object-level feature ex-\ntraction strategy similar to [28], using an object detector\nand tracker. We then employ an MLLM agent to de-\nscribe object activities and interactions by querying it\nwith pairs of frames, separated in time by a fixed in-\nterval, to capture temporal information. A key novelty\nis using these MLLM-generated textual descriptions di-\nrectly as features for modeling normal activity. By re-\nmoving redundant descriptions (Section 3.2), we con-\nstruct a final set of textual exemplars representing nom-\ninal videos. This exemplar set is then used to detect\nanomalies: descriptions from anomalous interactions\nare expected to deviate from nominal ones. In summary,\nour contributions are:\n• We introduce the first MLLM-based approach for\nvideo anomaly detection specifically designed to iden-\ntify complex anomalies caused by object interactions.\n• We propose a novel use of MLLMs for VAD: instead\nof directly deciding the presence of an anomaly, as\ndone in prior work, our method first derives a repre-\nsentation (using MLLM descriptions) of what is nor-\nmal in a scene and then identifies anomalies based on\ndeviations from this normal representation.\n• We demonstrate that our method offers built-in ex-\nplainability. Furthermore, we show that it can be in-\ntegrated with many existing VAD approaches to en-\nhance their interpretability.\n• We evaluate our method on multiple benchmark\ndatasets and show that it achieves state-of-the-art per-\nformance.\n2. Related Work\nMany different formulations of the video anomaly detec-\ntion (VAD) problem have been studied in recent years:\nsemi-supervised [13, 15, 19, 42], weakly supervised\n[38], unsupervised [51], training-free [52], single-scene\n[32] and multi-scene [22]. Each of these formulations\nhas applicability in different scenarios. This paper fo-\ncuses on the single-scene, semi-supervised version of\nvideo anomaly detection because it corresponds to the\nmost common use case that we have observed in the real-\nworld; namely, that of a surveillance camera at a partic-\nular location in the world in which video of normal ac-\ntivity is abundant but video containing anomalies is rare\nand expensive to collect. Furthermore, what is anoma-\nlous in one scene may be normal in another. This along\nwith the fact that location-dependent anomalies (e.g. a\npedestrian may be anomalous in some locations but not\nothers) are common in the single-scene version of VAD\nbut do not occur in the multi-scene version of VAD im-\nplies that the multi-scene version of VAD is not a gen-\neralization of single-scene VAD [32]. Solutions for one\nversion generally do not work well on the other version.\nEarly work on the single-scene, semi-supervised ver-\nsion of VAD includes many papers based on frame re-\nconstruction [13, 15] or frame prediction [19, 42]. One\nbig disadvantage of such approaches is that they require\ntraining a deep network on video from each new scene.\nTo avoid this disadvantage, others [7, 8, 28, 30, 31, 36,\n37] used an exemplar-based model that extracts a set of\nrepresentative features from normal video but does not\nrequire any deep network training for each scene. We\nalso use this idea in our work.\n2.1. MLLMs in Video Anomaly Detection\nRecently, researchers have explored ways of taking ad-\nvantage of large language models or vision language\nmodels for VAD. These approaches, however, vary sig-\nnificantly depending on the VAD problem formulation.\nWeakly-Supervised\nand\nTraining-Free\nVAD.\nMany approaches to VAD that use an MLLM try to\nuse its built-in knowledge of what is normal instead\nof learning normality from normal training video\n[23, 50, 52]. A disadvantage of such approaches is that\nthey cannot handle scenarios such as a boxing gym\nin which activity that is typically rare and anomalous\n(fighting) is actually very common. Another important\ndifference between our work and most other MLLM-\n2\n\nbased approaches to VAD is that others have focused\non the weakly supervised, multi-scene version of VAD\n[17, 23, 45, 50, 52].\nSemi-Supervised VAD. Compared to the weakly-\nsupervised setting, the capabilities of MLLMs are un-\nderstudied in the semi-supervised setting. Wu et al. [46]\naddresses semi-supervised, single-scene VAD, but with\na fundamentally different methodology. Their approach\nuses an MLLM to first generate a set of textual rules\nfrom normal video and then applies these rules during\ninference.\nIn contrast, our method directly compares\nMLLM-generated text descriptions between normal and\ntest videos without an intermediate rule-generation step.\nFurthermore, their model processes full video frames,\nwhereas we employ an object-centric approach that an-\nalyzes regions around objects and their pairs. This key\ndesign choice allows our method to model object inter-\nactions and perform both spatial and temporal localiza-\ntion, while their approach is limited to temporal local-\nization only.\nLi and Jiang [16] also focuses on semi-supervised\nsingle scene anomaly detection and proposes an MLLM-\nbased method. Like our method, they first detect and\ntrack objects and then query an MLLM to yield high-\nlevel descriptions of objects and their activity. However,\nthey do not model object interactions, and do not use an\nexemplar-based model.\nIn real-world scenarios, many anomalies are charac-\nterized by an unusual interaction between two objects.\nTherefore, one of the important aspects of our method is\nits modeling of object interactions in the scene to iden-\ntify anomalous interactions. There have been a few other\nmethods that also modeled interactions [5, 9, 28, 39, 40].\nWhile there are many differences among these meth-\nods, one of the main differences compared to our current\nwork is that none of them take advantage of MLLMs. In\nour approach, the textual descriptions of object activity\nand interactions provided by an MLLM yield powerful\ncues for a high-level understanding of a scene.\n3. Our Approach\nWe propose a novel method to detect anomalies in video.\nThe pipeline of our method (which we call MLLM-\nEVAD, for MLLM-based Explainable Video Anomaly\nDetection) is depicted in Figure 1. The main idea is\nto use a set of textual descriptions of the activity seen\nin normal training video of a scene as the model of\nnormality. These textual descriptions are obtained us-\ning an MLLM. Anomalies in testing video are found by\ncomparing textual descriptions of the activity in testing\nframes with the set of normal descriptions. A test de-\nscription that is dissimilar from all normal descriptions\nindicates an anomaly. The initial step of our method\nis detecting and tracking objects followed by identify-\ning pairs of objects that are in close spatial proximity\nand therefore likely to be interacting. Then, for each\nframe and a future frame offset by a fixed time interval,\nwe generate crops around pairs of interacting objects as\nwell as single objects that are not close to other objects.\nThe crops are given to an MLLM to obtain natural lan-\nguage descriptions of objects’ activities. These descrip-\ntions are embedded into a vector space using a sentence\nembedding model. By applying an exemplar selection\nalgorithm to all of the sentence embeddings found in\nnominal training video, we construct representative sets\nof embeddings for both object pairs and single objects.\nAnomaly detection is performed by comparing sentence\nembeddings extracted from testing frames to the exem-\nplar sets, and assigning anomaly scores based on their\ndistance to the most similar exemplar.\n3.1. Object Detection and Tracking\nIn our proposed method, we want to build a model of\nwhat objects do and how they interact in video contain-\ning normal activity. This requires detecting objects and\ntracking them across frames in training videos. We also\nfind pairs of objects that are likely to be interacting ac-\ncording to their spatial proximity.\nA video V is a collection of M frames {Fi}M\ni=1, such\nthat V = [F1, F2, ..., FM]. Each frame Fi is sent to the\nobject detector O, which returns X number of detected\nobjects. The object detector provides, for each object\no, the location l = (xo, yo) which is the x and y coor-\ndinates of the center of the object, b = (wo, ho) which\nis the width and height of the bounding box for the ob-\nject, and class id c. The output of the object detector\nis then O(F) = [o1, o2, ..., oX], where each object oi\nis represented by oi = [b, c, l]. After detecting objects\nin a frame, they are then tracked using an object tracker.\nEach detected object o is sent to the object tracker, which\nreturns x and y coordinates for that object in the subse-\nquent frames. In our method, we track objects for 30\nframes. Therefore, for every object, we acquire the tra-\njectory θ = {(x1, y1), (x2, y2), ..., (x30, y30)}.\nNext, we pair objects according to their spatial prox-\nimity in the frame. To determine which objects should\nbe paired, we need to calculate the 3D distances between\neach pair of objects. To calculate the 3D distance we\nneed to derive the 3D coordinates of the object locations\nby estimating a pseudo-depth since we do not have ac-\ncess to actual depth measurements. Given two objects\no1 and o2, we have 2D coordinates l1 = (x1, y1) and\nl2 = (x2, y2). Then we define a relative depth, z, be-\ntween two objects by taking the absolute difference of y\nvalues such that z = |y1 −y2|. This estimate of pseudo-\ndepth assumes that objects are resting on the ground\nplane and the ground plane is farther from the camera\n3\n\nObject Detection and \nTracking\nMLLM\nVisual Input: Crop of two objects \nenclosed by bounding boxes, from \ntwo frames at time t and t+30.\n.\n.\n.\nText Encoder\nAnswer sentence r: \nTwo \npeople \nare \nwalking side by side \non \nthe \ncrosswalk \nwithout interacting.\nX\nEmbedding of r\nExemplar set of embeddings\nTraining mode:\nIf score is greater than the \nthreshold add embedding of r to \nexemplar set.\nInference mode:\nUse score for \nanomaly detection.\n.\n.\n.\nPrompt: \nBriefly \ndescribe what the \npeople \nin \nthe \nenclosed regions of \nthese \nimages \nare \ndoing. \nThe \ntwo \nimages were taken \none second apart.\nScore calculation:\nEq. (1) & Eq. (2)\nVideo Frames\nFigure 1.\nThe pipeline of our method (MLLM-EVAD) for\nframe to textual description generation with the help of an ob-\nject detector, object tracker and MLLM. Please see the text for\nmore explanation.\nthe closer it is to the top of the image. The 3D distance\nd can then be calculated by taking the Euclidean distance\nbetween 3D coordinates (x1, y1, z) and (x2, y2, 0). Any\nobjects that have a 3D distance d smaller than a prede-\ntermined threshold h are paired. Due to applying the\nthreshold, not every single object is necessarily con-\nnected to another object, which leads to having single\nobjects in addition to object pairs.\n3.2. Generating Textual Descriptions\nAt the end of the object detection and tracking stage, for\neach frame Ft, we obtain a set of object pairs and a set\nof single objects. For each of these, we generate two\ncropped regions that capture the relevant spatial area for\nboth the current frame Ft and a future frame Ft+∆(e.g.,\n∆= 30), assuming a static camera (as depicted in Fig-\nure 1). This ensures that both crops contain the same\nbackground and object regions, allowing for consistent\ncomparisons across time. We provide the details of the\ncropping process in the Appendix. Red rectangles are\ndrawn around each object in the cropped images using\nthe bounding boxes detected by the object detector. The\nrectangles indicate to the MLLM which objects are be-\ning referred to in the prompt.\nWe then obtain a sentence r that describes the inter-\naction between objects o1 and o2 by querying an MLLM\nagent with the corresponding image crops Ct and Ct+∆,\nalong with the user prompt: “Briefly describe what the\n<object name>s in the enclosed regions of these im-\nages are doing. The two images were taken one second\napart.” For single-object cases, the prompt is adjusted\nto: “Briefly describe what the <object name> in the\nenclosed region of these images is doing. The two im-\nages were taken one second apart.” In both prompts, the\n<object name> placeholder is replaced by the class la-\nbel c obtained during feature selection.\nAn additional “system” prompt is also used with the\nMLLM: “You will be provided with two frames from a\nvideo and asked to describe what objects that are indi-\ncated by bounding boxes in the video are doing. Your\ntask is to answer the query in a simple sentence. If there\nis any interaction between the indicated objects, a de-\nscription of the interaction should be given.”\nThe MLLM-generated responses are then used in\nboth the model building and anomaly detection stages.\n3.3. Model Building\nWe follow a similar exemplar selection strategy as intro-\nduced in recent works [28, 30, 36, 37]. For all frames\nin a video, we collect all textual descriptions generated\nfrom object pairs into one set and all textual descriptions\ngenerated from single objects into another set. Then for\neach of the sets we run an exemplar selection algorithm\nwhich selects a subset of the elements of the set such\nthat no two members of the subset are near each other\naccording to a distance function. The intuition behind\nexemplar selection is to simply remove redundant (or\nnearly redundant) elements from the set leaving behind\na compact, representative subset of exemplars.\nVari-\nous distance functions between sentences can be used\nto compare textual descriptions. In most of our experi-\nments we use Sentence-BERT [33], but we also test us-\ning BLEU [29] and METEOR [2].\nGiven a set S, the exemplar selection algorithm pro-\nceeds as follows:\n(1) Initialize the exemplar set to\nNULL. (2) Add the first element of S to the exemplar\nset. (3) For each subsequent element of S, find its dis-\ntance to the nearest instance in the exemplar set. If this\ndistance is above a threshold, th, then add the element\nto the exemplar set.\nSince we obtain a textual description r for each ob-\nject pair or single object, we transform these descrip-\ntions into sentence embeddings using a Natural Lan-\nguage Processing (NLP) model N, such that Ir = N(r).\nThe distance D between two textual descriptions, r1 and\nr2, is then computed as:\nD(r1, r2) = 1 −cos (Ir1, Ir2)\n(1)\nwhere cos(·, ·) denotes the cosine similarity between the\ncorresponding embeddings. Using the distance function\nD and the described exemplar selection algorithm, we\n4\n\nconstruct two separate exemplar sets: Epair and Esingle,\ncontaining sentence embeddings of textual descriptions\nfor object pairs and single objects, respectively.\n3.4. Anomaly Detection in Test Video\nAfter constructing the exemplar sets from nominal\nvideos, the next step is to detect anomalies in a test video\nfrom the same scene.\nSimilar to the model-building\nphase, we extract textual descriptions and encode them\ninto sentence embeddings Ir. The anomaly score for\na given object pair (or single object) is then computed\nbased on its similarity to the corresponding exemplar set.\nSpecifically, the score is defined as:\nscore(r) = 1 −max (cos (Ir, e)) ,\ne ∈Epair or Esingle\n(2)\ndepending on whether the input is an object pair or a sin-\ngle object. A higher score indicates a greater deviation\nfrom the nominal descriptions and is thus more likely to\nbe anomalous.\n3.5. Combination with Previous Methods\nWhile the textual descriptions generated by an MLLM\nprovide powerful high-level features for modeling the\nnormal activity in a scene and for detecting anomalies in\nthe scene, some anomalies require more detailed, fine-\ngrained features for detection. For this reason, it can\nbe advantageous to combine our method with previous\nmethods that use more detailed features such as object\nsizes and trajectories. Combining with another method\nis especially easy to do if the other method is also object-\nbased and uses an exemplar-based model.\nIn our experiments, we show results when combining\nour method separately with two other methods; namely,\nthe scene-graph method of Mumcu et al. [28] and the\ntracklet method of Singh et al [37].\nOur method of modeling interactions was inspired by\nthe scene-graph method of Mumcu et al. [28]. Their\nmethod builds a scene graph for each frame where each\nobject is a node and objects that are nearby and, there-\nfore possibly interacting, are connected by an edge. The\nattributes stored at each node are the object’s location,\nsize, object class, trajectory and skeletal pose.\nThen\nan exemplar-based model is created for every connected\npair of nodes across all scene graphs in all nominal\nframes. A separate exemplar-based model is also created\nfor all singleton nodes across all scene graphs in all nom-\ninal frames. To combine our method with theirs, we can\nsimply add the MLLM-generated textual descriptions to\neach connected pair of nodes and to each singleton node.\nThe distance function used to compare nodes takes the\nmaximum over all attribute distances including the dis-\ntance between textual descriptions.\nIn the case of the tracklet method of Singh et al. [37],\nobjects in each frame are also detected and tracked. Sim-\nilar to Mumcu et al., for each object, a set of attributes\nare stored including the location, size, object class and\ntrajectory.\nAn exemplar-based model is then learned\nfrom all of the objects found in the nominal training\nvideos using a distance function for comparing objects\nthat takes the maximum over each attribute distance. To\ncombine our method with theirs, we again only need\nto add our MLLM-generated textual descriptions as an-\nother attribute associated with each object. In this case,\nobject interactions are not modeled. We use this combi-\nnation to test on datasets that do not contain anomalous\nobject interactions.\n4. Experiments\nIn our experiments, we use the following configurations:\nDatasets: We evaluate MLLM-EVAD on three video\nanomaly detection benchmark datasets which are all\npublicly available under a CC-BY-SA-4.0 license. Com-\nplexVAD [28] is a large-scale dataset with a total of\nover 3.6 million frames, with 2,069,941 frames for train-\ning and 1,611,497 for testing. It features interaction-\nbased anomalies and serves as our main focus, since our\nmethod leverages descriptions of object activity and in-\nteractions for video anomaly detection. The scene fea-\ntures a crosswalk, pedestrian sidewalks, and a two-lane\nstreet, which are heavily used. It demonstrates interac-\ntion anomalies such as a person leaving an object on the\nground, a person sitting on a car, or a dog walking with-\nout a person or leash, where the individual objects are\nnormal, but their interactions are anomalous.\nIn addition, we evaluate performance on the Avenue\nand Street Scene datasets. Avenue [21] contains short\nsurveillance videos of a campus walkway, with anoma-\nlies such as running, throwing objects, or walking in\nthe wrong direction. Street Scene [30] includes longer\nvideos of a city street environment, where anomalies\ninvolve vehicles or pedestrians behaving unusually, in-\ncluding cars driving on the wrong side of the road,\npedestrians crossing at non-designated areas, or bikers\nriding on the sidewalk. Avenue contains 30,652 frames,\nsplit nearly evenly between 15,328 training frames and\n15,324 testing frames. Street Scene is a larger dataset\nwith a total of 203,257 frames, of which 56,847 are used\nfor training and 146,410 for testing.\nImplementation Details: We use Detectron2 [48]\nand ByteTrack [54] as the object detector and object\ntracker in our method.\nSentence-BERT [33] (using\nthe pretrained weights, ”all-MiniLM-L6-v2”) is used as\nthe text embedding model to encode activity descrip-\ntions.\nInitially, GPT-4o [14] was used as the multi-\nmodal MLLM agent. However, with the recent introduc-\ntion of Gemma 3 [41], we re-conducted the experiments\n5\n\nMethod\nRBDC\nTBDC\nFrame\nMemAE [12]\n0.05\n0.0\n58.0\nEVAL [36]\n10.0\n62.0\n54.0\nAnomalyRuler [46]\nN/A\nN/A\n56.0\nScene-Graph[28]\n19.0\n64.0\n60.0\nMLLM-EVAD\n24.0\n68.0\n61.0\nScene-Graph +\nMLLM-EVAD\n25.0\n70.0\n63.0\nTable 2. Results on the ComplexVAD dataset. Each entry is\nthe area under the curve (AUC) as a percentage for the three\nbenchmark methods using the RBDC, TBDC and Frame-Level\nevaluation criteria. Bold face indicates the best score and blue\nindicates second best.\nMethod\nRBDC\nTBDC\nFrame\nAuto-encoder [13]\n0.3\n2.0\n61.0\nDictionary method [21]\n1.6\n10.0\n48.0\nFlow baseline [30]\n11.0\n52.0\n51.0\nFG Baseline [30]\n21.0\n53.0\n61.0\nEVAL [36]\n24.3\n64.5\nN/A\nContextual GMM [49]\n34.0\n62.5\n67.0\nT-EVAL [37]\n30.9\n72.9\n66.0\nT-EVAL+\nMLLM-EVAD\n31.1\n73.5\n66.8\nTable 3. Results on the Street Scene dataset using the area\nunder the curve (AUC) (as a percentage) for the RBDC and\nTBDC evaluation criteria for different methods. Bold face in-\ndicates the best score and blue indicates second best.\non ComplexVAD using Gemma 3. Hence, the results\non ComplexVAD are reported with both Gemma 3 and\nGPT-4o, while the results on the other datasets are re-\nported with GPT-4o. We choose a threshold th = 0.65\nfor exemplar selection that resulted in a modest num-\nber of total exemplars selected. From past work that\nused exemplar-based models, this threshold mainly ef-\nfects model size and has a small effect on test accuracy\n[36, 37]. We provide additional implementation details\nin the Appendix.\nEvaluation Criteria: We use the Region-Based De-\ntection Criterion (RBDC) and the Track-Based Detec-\ntion Criterion (TBDC) as proposed in [30] as our pri-\nmary evaluation criteria and report the area under the\ncurve (AUC) for false positive rates per frame from 0 to\n1. We also report frame-level AUC [24] scores. As high-\nlighted in previous works [30] frame-level AUC only\nevaluates temporal accuracy and disregards spatial lo-\ncalization of anomalies.\nRBDC and TBDC measure\na method’s capacity to accurately identify anomalous\nspatio-temporal regions within a given video sequence.\nMethod\nRBDC\nTBDC\nFrame\nSiamese[31]\n41.2\n78.6\n87.2\nHybrid[20]\n41.1\n86.2\n91.1\nBG agnostic [11]\n65.1\n66.9\n92.3\nHybrid[20] +\nPCAB[34]\n62.3\n89.3\n90.9\nBG agnostic[11]\n+ PCAB[34]\n66.0\n64.9\n92.9\nSSMTL++v1 [4]\n40.9\n82.1\n93.7\nSSMTL++v2 [4]\n47.8\n85.2\n91.6\nAnomalyRuler [46]\nN/A\nN/A\n89.7\nEVAL[36]\n68.2\n87.6\n86.0\nT-EVAL[37]\n67.5\n89.7\n88.0\nT-EVAL +\nMLLM-EVAD\n68.9\n90.1\n88.4\nTable 4. Results on the CUHK Avenue dataset using the area\nunder the curve (AUC) as a percentage for the RBDC and\nTBDC evaluation criteria for different methods. Bold face in-\ndicates the best score and blue indicates second best.\n4.1. Quantitative Results\nThe results of our method on ComplexVAD, as well\nas those of EVAL [36], MemAE [12], AnomalyRuler\n[46], and Scene-Graph [28], using the three evalua-\ntion criteria described above, are reported in Table 2.\nWe also provide the result for the combination of our\nmethod and Scene-Graph on the ComplexVAD dataset.\nOur MLLM-EVAD method outperforms the next best\nmethod (Scene-Graph) by 5, 4 and 1 percentage points\nin RBDC, TBDC, and frame-level evaluations, respec-\ntively.\nWe achieve the best result by combining the\nScene-Graph method with ours, reaching 25%, 70%,\nand 63% in RBDC, TBDC, and frame-level evaluations,\nrespectively.\nAnomalyRuler, a recent MLLM-based\nmethod [46], achieves 54% frame-level AUC. However,\nwe were not able to obtain results for the RBDC and\nTBDC criteria, as this method does not support spatial\nlocalization of anomalies. The MemAE method does\nvery poorly for the two criteria that measure spatial lo-\ncalization. This implies that the regions of an image that\nMemAE predicts as anomalous are usually normal.\nFor the experiments on Avenue and Street Scene,\nwe combine our method with the best-performing ex-\nisting exemplar-based method, namely Tracklet EVAL\n[37]. These datasets include many anomalies that need\nmore fine-grained attributes (such as the direction and\nspeed of travel for people and cars), so the high-level\ntextual descriptions used by MLLM-EVAD are not suf-\nficient on their own to detect all of the anomalies. We\nshow that our textual descriptions do add an important\nnew attribute, however, since, when combined with the\nmore fine-grained attributes of Tracklet EVAL, the re-\nsults with the MLLM textual descriptions improve ac-\ncuracy for both datasets across all three evaluation cri-\n6\n\nThe person is \ncrouching down \non the ground.\nThe person is walking \nacross the grass.\nThe person is holding a \nphone up and \nappearing to take a \npicture or video while \nbeing pushed in a large \nbox by another person.\nThe person is \nwalking along the \nsidewalk.\nAnomaly\nClosest \nExemplar\nAnomaly\nClosest \nExemplar\nThe two people are walking \nacross the crosswalk, with the \nperson on the right appearing to \nstumble or jump.\nTwo people are walking \nacross the crosswalk on the \nstreet.\nThe person on the bicycle is \nbeing pushed forward by \nanother person who is riding \na scooter.\nThe person is walking while \nthe cyclist is riding the \nbicycle alongside them.\nFigure 2. Examples of anomalies in the ComplexVAD dataset that are detected by our method. In each box, the top row shows two\nframes of the anomalous video clip followed by the textual description generated by Gemma 3. The second row shows two frames\nfrom the closest matching exemplar to the anomaly. The textual description for the closest exemplar is also shown. By contrasting\nthe anomalous description with the closest normal description, it is clear why the testing video clip was found to be anomalous.\nteria. As shown in Table 4, on the Avenue dataset, the\nTracklet-EVAL + MLLM-EVAD method improves the\nstate-of-the-art for both the RBDC and TBDC criteria.\nOn Street Scene (results shown in Table 3), the state-of-\nthe-art is surpassed for the TBDC criterion. Since the\nframe-level criterion does not measure spatial localiza-\ntion accuracy which is the focus of these datasets, we\nwould argue that the slightly-below SOTA frame-level\nnumbers are not representative of the true accuracy of\nthese methods. We should note that we were not able\nto exactly reproduce the results reported for Tracklet\nEVAL [37] on the Avenue dataset, most likely due to\ndifferent parameters, so we report the results we could\nreproduce for that method and use that version in com-\nbination with MLLM-EVAD for the results on the com-\nbined method.\n4.2. Qualitative Results: Explainability\nIn addition to the quantitative results, we present qualita-\ntive results to demonstrate the explainability capabilities\nof our method. The combination of using an MLLM\nagent and an exemplar selection strategy enables intu-\nitive explanations for detected anomalies by providing\ndescriptions of both the anomaly and the most simi-\nlar event from the nominal training video.\nIn Figure\n2, we demonstrate four detected anomalies from Com-\nplexVAD along with their textual descriptions and the\ntextual descriptions of the closest exemplars, as gener-\nated by our method. For example, one detected anomaly\nis described as ”The person is crouching down on the\nground.” The closest matching exemplar from the nom-\ninal set is ”The person is walking across the grass.”\nThe action of crouching deviates from the typical behav-\nior observed in the training data. This difference high-\nlights how our method leverages semantic understanding\nto identify uncommon activity, offering an interpretable\nexplanation for the anomaly.\nAnother example involves an anomaly described as\n”The person is holding a phone up and appearing to\ntake a picture or video while being pushed in a large\nbox by another person.” The closest matching exemplar\nfrom the nominal set is ”The person is walking along\nthe sidewalk.”, which is, in fact, the most common ac-\ntivity involving a person in this dataset. The anomalous\nevent, a person being pushed in a box, does not occur in\nthe nominal video and represents an unusual and com-\nplex behavior. This example further demonstrates our\nmethod’s ability to detect rare interaction patterns that\ndeviate from typical activities in the scene.\nIn addi-\ntion, these examples illustrate how our method also pro-\nvides reasonable explanations for detected anomalies.\nBy leveraging descriptive comparisons with nominal ex-\nemplars, our method enables enhanced and explainable\nvideo anomaly detection.\n4.3. Ablation Study\nWe conduct two ablation studies by modifying the de-\nfault settings of our method. In the first ablation study,\ninstead of using sentence embeddings and cosine dis-\nsimilarity as the distance metric, we use the raw sen-\ntences and compute distances using BLEU [29] and ME-\nTEOR [2]. In Table 5, we compare the results of cosine\ndistance over Sentence-BERT embeddings with those\nobtained using BLEU and METEOR, based on the de-\n7\n\nMethod\nRBDC\nTBDC\nFrame\nSentence-BERT\n19.0\n67.0\n59.0\nBLEU\n17.0\n69.0\n58.0\nMETEOR\n16.0\n69.0\n57.0\nTable 5. The table reports results on the ComplexVAD datasets\nusing the area under the curve (AUC) as a percentage for the\nRBDC, TBDC and Frame-Level evaluation criteria for the our\nmethod using 3 different sentence distance functions.\nMLLM\nRBDC\nTBDC\nFrame\nGPT-4o\n19.0\n67.0\n59.0\nGemma 3\n24.0\n68.0\n61.0\nTable 6. The table reports the performance of our proposed\nmethod on the ComplexVAD datasets using the area under the\ncurve (AUC) as a percentage for the RBDC, TBDC and Frame-\nLevel evaluation criteria with different MLLM agents GPT-4o\nand Gemma 3.\nscriptions generated by our method using GPT-4o as the\nMLLM agent. The results show that Sentence-BERT\nembeddings yield slightly higher accuracy in the RBDC\nand frame-level criteria while BLEU and METEOR per-\nform slightly better on the TBDC criterion. Sentence-\nBERT is the preferred similarity function considering its\ngreater efficiency due to embeddings which can be saved\nand only require a dot product for comparison.\nIn the second ablation study, we compare our\nmethod’s performance on ComplexVAD using different\nMLLM agents, namely Gemma 3 and GPT-4o. In Table\n6, we show that Gemma 3 achieves better performance,\nwith 24%, 68%, and 61% in the RBDC, TBDC, and\nframe-level criteria, respectively, while GPT-4o yields\n19%, 67%, and 59%. We attribute this difference to the\nfact that Gemma 3 generates more detailed and descrip-\ntive sentences compared to GPT-4o. This level of detail\nis particularly important for detecting interaction-based\nanomalies, where subtle contextual cues are crucial, and\nlikely contributed to the observed performance gap. We\nprovide a comparison of the descriptions generated by\nGemma 3 and GPT-4o in the Appendix.\n5. Limitations and Future Directions\nWhile our framework shows promising results, we iden-\ntify several limitations that pave the way for future re-\nsearch.\nFirst, our implementation relies on powerful\nbut computationally expensive multimodal models like\nGemma 3. Their high inference latency poses a chal-\nlenge for real-time applications.\nRecent studies have\nshown that smaller, task-specific models can perform ex-\nceptionally well on specialized tasks [25, 26, 35, 43].\nWe will explore fine-tuning such models for single\nscenes, specifically for describing object activities and\ninteractions.\nSecond, a primary contribution of our method is\nits semantic explainability, but quantitatively evaluating\nthis feature is difficult due to a lack of suitable bench-\nmarks. Current datasets with textual ground truth are\ndesigned for multi-scene or weakly-supervised tasks and\ndo not fit the semi-supervised, single-scene VAD prob-\nlem [10, 53, 55]. Therefore, a key future direction is the\ndevelopment of new datasets with ground-truth textual\ndescriptions for scene-specific anomalies.\nFinally, our framework currently leverages a robust\ntraditional object detector, which is well-suited for sce-\nnarios with known object classes. To further general-\nize our approach, a promising future direction is the in-\ntegration of open-vocabulary object detection methods\n[6, 18, 27]. By enabling the recognition of an open set of\nobject categories, this advancement would significantly\nbroaden our system’s applicability, allowing it to ana-\nlyze and describe a more diverse range of anomalies in\nunconstrained, real-world environments.\n6. Conclusions\nIn this work, we presented a novel video anomaly de-\ntection framework that leverages multimodal Large Lan-\nguage Models to detect and explain complex anomalies.\nBy extracting object-level activity and interactions and\nconverting them into textual descriptions, our method\nmoves beyond traditional pixel level modeling and in-\ntroduces a language-based, interpretable layer for video\nanomaly detection.\nUnlike many prior MLLM-based\nmethods that rely on direct frame-level judgments, our\napproach builds a set of nominal textual descriptions\nand identifies anomalies through language-based com-\nparison, offering both accuracy and explainability.\nWe demonstrate that our method outperforms exist-\ning approaches on ComplexVAD which is a challeng-\ning benchmark specifically designed to test interaction-\nbased anomalies. In addition our method also improves\nof the state-of-the-art results on standard datasets like\nAvenue and Street Scene when combined with the fine-\ngrained method Tracklet EVAL [37].\nThrough abla-\ntion studies, we showed the importance of using sen-\ntence embeddings and detailed MLLM-generated de-\nscriptions. Notably, we found that Gemma 3 provided\nmore informative descriptions than GPT-4o, contribut-\ning to higher detection accuracy.\nBeyond strong quantitative results, our method of-\nfers clear and interpretable explanations for detected\nanomalies, highlighting its potential for real world de-\nployment in safety critical surveillance applications. 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In Proceedings of the Eu-\nropean Conference on Computer Vision (ECCV), 2022.\n5\n[55] Liyun Zhu, Lei Wang, Arjun Raj, Tom Gedeon, and Chen\nChen. Advancing video anomaly detection: A concise\nreview and a new dataset. Advances in Neural Informa-\ntion Processing Systems, 37:89943–89977, 2024. 8\n11\n\nLeveraging Multimodal LLM Descriptions of Activity for Explainable\nSemi-Supervised Video Anomaly Detection\nSupplementary Material\n1. Compute Resources\nAll of our experiments were run on a cluster of eight\n80 GB NVIDIA A100 GPUs. A single 80 GB GPU\nwould be sufficient and multiple GPUs were used only\nto reduce the total compute time by processing multiple\nvideo sequences in parallel.\nOur experiments that used GPT-4o accessed it\nthrough an API via the cloud.\n2. Cropping Frames\nFor each selected object pair o1 and o2, we first com-\npute a single bounding box that tightly encloses both ob-\njects in a given frame. Let the individual bounding boxes\nfor the two objects in frame Ft be denoted as bboxa =\n(x1a, y1a, x2a, y2a) and bboxb = (x1b, y1b, x2b, y2b),\nwhere (x1, y1) represents the top-left corner and (x2, y2)\nthe bottom-right corner of a bounding box. These are\nmerged into a single bounding box bbox1 by taking the\nelement-wise minimum for the top-left corner and max-\nimum for the bottom-right corner:\nbbox1 =\n\u0000min(x1a, x1b), min(y1a, y1b),\nmax(x2a, x2b), max(y2a, y2b)\n\u0001\n(S1)\nWe apply the same procedure to obtain bbox2 in the\nfuture frame Ft+∆, using the tracked locations of the\nsame object pair.\nLet bbox1\n=\n(x1,1, y1,1, x2,1, y2,1) denote the\nmerged bounding box of the selected object pair in frame\nFt, and let bbox2 = (x1,2, y1,2, x2,2, y2,2) denote the\nmerged bounding box of the same objects tracked in the\nfuture frame Ft+∆. A unified bounding box that covers\nboth time steps is computed as:\nbboxmerged =\n\u0000min(x1,1, x1,2), min(y1,1, y1,2),\nmax(x2,1, x2,2), max(y2,1, y2,2)\n\u0001\n(S2)\nLet (x1, y1) and (x2, y2) represent the top-left and\nbottom-right corners of this merged bounding box. Let\nwmin and hmin denote half of the minimum desired crop\nwidth and height, respectively.\nIn our case, we set\nwmin = 240 and hmin = 135, corresponding to a min-\nimum crop size of 480 × 270 pixels. This size was cho-\nsen as a practical minimum that ensures sufficient con-\ntext around objects while keeping computational over-\nhead low. In our experiments, we found that a resolution\nof 480 × 270 served as a reliable lower bound, provid-\ning adequate spatial detail for capturing object activities\nwithout reducing the quality of the LLM generated de-\nscriptions. While larger crops may be used when avail-\nable, we observed this setting to be a consistent and ef-\nfective baseline across different datasets.\nWe then compute the crop dimensions as\nw = max\n\u0010\n|x2−x1|\n2\n, wmin\n\u0011\n,\nh = max\n\u0010\n|y2−y1|\n2\n, hmin\n\u0011\n(S3)\nThe crop boundaries are determined as:\nxmin = max(x1 −w, 0),\nymin = max(y1 −h, 0),\nxmax = min(x2 + w, W),\nymax = min(y2 + h, H)\n(S4)\nwhere W and H denotes the width and height of the full\nframe. The final cropped region Ct is extracted from the\ncurrent frame Ft as:\nCt = Ft[ymin : ymax, xmin : xmax]\nTo ensure both temporal frames share the same spa-\ntial context, we apply the exact same crop coordinates to\nthe future frame Ft+∆:\nCt+∆= Ft+∆[ymin : ymax, xmin : xmax]\nThis guarantees that both cropped frames contain the\nsame background and object regions, allowing for con-\nsistent comparisons across time. In the case of a single\nobject, the same procedure is applied with one excep-\ntion: instead of merging two object bounding boxes in\neach frame, we directly use the bounding box of the sin-\ngle object in frame Ft and its tracked location in Ft+∆.\nThe rest of the cropping process, including the computa-\ntion of the unified bounding box across time, minimum\nsize enforcement, and cropping, is performed exactly as\ndescribed above.\n3. Comparison of Gemma 3 and GPT-4o Ac-\ntivity Descriptions\nIn Figure S1, we share activity descriptions generated by\nGemma 3 and GPT-4o for given images and the promt:\n”Briefly describe what the <object name>s in the en-\nclosed regions of these images are doing. The two im-\nages were taken one second apart.”\n12\n\nGemma 3\nGPT-4o\nThe person is \ncrouching down on \nthe ground.\nThe person in the \nenclosed region is \nbending over and \nappears to remain in the \nsame position between \nthe two images\nThe person is \nholding a phone up \nand appearing to \ntake a picture or \nvideo while being \npushed in a large \nbox by another \nperson.\nIn the images, the person \nstanding seems to be \npushing or interacting \nwith a cart or box that the \nseated person appears to \nbe inside or using. The \nstanding person \nmaintains their position \nor posture relative to the \ncart between the images.\nThe person in the \nblue shirt and the \nperson on the bike \nare interacting, as \nthe person in blue \nappears to be \npreventing the \ncyclist from moving \nforward.\nThe person on the \nbicycle is being \npushed forward by \nanother person who \nis riding a scooter.\nThe person is \nwalking across \nthe crosswalk \nwhile looking at \ntheir phone, and a \ncyclist is passing \nby them on the \nright.\nThe person in the \nenclosed region is \nstopping to allow \na cyclist to pass \nby.\nIn the first image, the \nperson in the black \nshorts and dark hair \nappears to be jumping \nor falling next to the \nbald person. In the \nsecond image, the \nperson in the black \nshorts and dark hair is \nstanding up straight, \nwhile the bald person \nremains standing. \nThere is no apparent \ninteraction between \nthem.\nThe two people are \nwalking across the \ncrosswalk, with the \nperson on the right \nappearing to stumble \nor jump.\nFigure S1. The activity descriptions generated by Gemma 3 and GPT-4o with given images.\nWe have observed that Gemma 3 often gives more\nconcise descriptions with fewer unnecessary details such\nas the color of a person’s clothes. Moreover, Gemma 3\nsometimes includes important details that GPT-4o omits,\nsuch as the fact that a person is using his or her phone.\n13\n"},"cleaned":{"kind":"string","value":"Leveraging Multimodal LLM Descriptions of Activity for Explainable Semi-Supervised Video Anomaly Detection Furkan Mumcu1,† Michael J. Jones2,∗ Anoop Cherian3,∗ Yasin Yilmaz4,† † ∗Mitsubishi Electric Research Laboratories (MERL) Abstract Existing semi-supervised video anomaly detection (VAD) methods often struggle with detecting complex anomalies involving object interactions and generally lack explainability. To overcome these limitations, we propose a novel VAD framework leveraging Multimodal Large Language Models (MLLMs). Unlike previous MLLM-based approaches that make direct anomaly judgments at the frame level, our method focuses on extracting and interpreting object activity and interactions over time. By querying an MLLM with visual inputs of object pairs at different moments, we generate textual descriptions of the activity and interactions from nominal videos. These textual descriptions serve as a high-level representation of the activity and interactions of objects in a video. They are used to detect anomalies during test time by comparing them to textual descriptions found in nominal training videos. Our approach inherently provides explainability and can be combined with many traditional VAD methods to further enhance their interpretability. Extensive experiments on benchmark datasets demonstrate that our method not only detects complex interaction-based anomalies effectively but also achieves state-of-the-art performance on datasets without interaction anomalies. 1. Introduction Video anomaly detection (VAD) has emerged as an active research field due to its significant applications in security and public safety. The rapid growth of surveillance video data makes it impractical for humans to monitor thoroughly. Consequently, VAD algorithms play a vital role in identifying unusual events in video footage, enabling human operators to focus their attention on reviewing these flagged incidents. In this paper, we focus on the formulation in which nominal videos (also called training videos) containing only normal activities in a particular scene are provided for learning a model. The goal is to identify when and where anomalous activities happen within test videos of the same scene. Along with other papers [3, 44], we refer to this version of the problem as semi-supervised VAD because of the availability of normal, but not anomalous, training video. In Table 1, we summarize the properties of some recent papers in this area to highlight the differences between our work and other recent approaches. Existing semi-supervised video anomaly detection methods suffer from several weaknesses. Recent studies struggle to provide explainability for detected anomalies. While some existing work [36, 37] explored more explainable approaches by interpreting the input features, they do not offer direct, textual explanations. Furthermore, the ComplexVAD dataset [28] introduced interaction-based anomalies and demonstrated that detecting these anomalies is even more challenging than identifying the simpler anomalies presented in earlier datasets. Hence, we propose a new method to address the drawbacks present in existing works, including the lack of focus on complex anomalies and explainability. By utilizing a Large Language Model-based detection approach, we introduce an effective video anomaly detection framework that excels at identifying complex anomalies and features built-in explainability. Large Language Models, especially in their multimodal forms (MLLMs), have demonstrated strong capabilities in interpreting and accurately describing complex situations from both text and images. GPT-4 with vision [1], for example, can analyze visual inputs and produce detailed, contextually relevant descriptions. Existing work has explored the use of MLLMs for video anomaly detection. Most of these methods are designed for a multi-scene, weakly supervised setting. While such approaches are effective for detecting a common set of anomalies across multiple scenes, they fall short in identifying scene-specific anomalies, which are central to the semi-supervised, single-scene VAD task. To address this limitation, we propose a novel MLLM-based framework that: (1) considers object interactions in addition to individual object activities, (2) leverages an MLLM 1 16 Oct 2025 Method Year Focus Features Task Objective MLLM Interaction ExplainSemi Scene Usage Anomalies ability Supervised Specific VadCLIP [47] 2023 Using CLIP for multi-scene weakly supervised VAD G# # # # # Vera [50] 2025 Optimizing prompts for MLLM-based weakly supervised VAD # # # Holmes-VAU [53] 2025 MLLM finetuning for anomaly understanding across multi-scene # # # AnomalyRuler [46] 2024 Scene-specific rule generation and frame-level descriptions via MLLMs # G# VLAVAD [16] 2024 Using frame level MLLM descriptions for semi-supervised VAD # # EVAL [36] 2023 Using object-level features for explainable VAD # # G# T-EVAL [37] 2024 Object-level, tracklet-based explainable VAD # # G# Scene-Graph [28] 2025 Exemplar and scene graph for interaction-based VAD # G# Ours 2026 Object-level MLLM descriptions for interaction-based VAD Table 1. Summary of properties and differences of some recent work in VAD. Our work is the only one designed for semisupervised, scene-specific VAD that handles interaction anomalies and provides textual explanations of anomalies. to generate high-level descriptions used in an exemplarbased model that can be easily extended without retraining deep networks for each new scene, and (3) enables both spatial and temporal localization of anomalies. In our method, we adopt an object-level feature extraction strategy similar to [28], using an object detector and tracker. We then employ an MLLM agent to describe object activities and interactions by querying it with pairs of frames, separated in time by a fixed interval, to capture temporal information. A key novelty is using these MLLM-generated textual descriptions directly as features for modeling normal activity. By removing redundant descriptions (Section 3.2), we construct a final set of textual exemplars representing nominal videos. This exemplar set is then used to detect anomalies: descriptions from anomalous interactions are expected to deviate from nominal ones. In summary, our contributions are: • We introduce the first MLLM-based approach for video anomaly detection specifically designed to identify complex anomalies caused by object interactions. • We propose a novel use of MLLMs for VAD: instead of directly deciding the presence of an anomaly, as done in prior work, our method first derives a representation (using MLLM descriptions) of what is normal in a scene and then identifies anomalies based on deviations from this normal representation. • We demonstrate that our method offers built-in explainability. Furthermore, we show that it can be integrated with many existing VAD approaches to enhance their interpretability. • We evaluate our method on multiple benchmark datasets and show that it achieves state-of-the-art performance. 2. Related Work Many different formulations of the video anomaly detection (VAD) problem have been studied in recent years: semi-supervised [13, 15, 19, 42], weakly supervised [38], unsupervised [51], training-free [52], single-scene [32] and multi-scene [22]. Each of these formulations has applicability in different scenarios. This paper focuses on the single-scene, semi-supervised version of video anomaly detection because it corresponds to the most common use case that we have observed in the realworld; namely, that of a surveillance camera at a particular location in the world in which video of normal activity is abundant but video containing anomalies is rare and expensive to collect. Furthermore, what is anomalous in one scene may be normal in another. This along with the fact that location-dependent anomalies (e.g. a pedestrian may be anomalous in some locations but not others) are common in the single-scene version of VAD but do not occur in the multi-scene version of VAD implies that the multi-scene version of VAD is not a generalization of single-scene VAD [32]. Solutions for one version generally do not work well on the other version. Early work on the single-scene, semi-supervised version of VAD includes many papers based on frame reconstruction [13, 15] or frame prediction [19, 42]. One big disadvantage of such approaches is that they require training a deep network on video from each new scene. To avoid this disadvantage, others [7, 8, 28, 30, 31, 36, 37] used an exemplar-based model that extracts a set of representative features from normal video but does not require any deep network training for each scene. We also use this idea in our work. 2.1. MLLMs in Video Anomaly Detection Recently, researchers have explored ways of taking advantage of large language models or vision language models for VAD. These approaches, however, vary significantly depending on the VAD problem formulation. Weakly-Supervised and Training-Free VAD. Many approaches to VAD that use an MLLM try to use its built-in knowledge of what is normal instead of learning normality from normal training video [23, 50, 52]. A disadvantage of such approaches is that they cannot handle scenarios such as a boxing gym in which activity that is typically rare and anomalous (fighting) is actually very common. Another important difference between our work and most other MLLM2 based approaches to VAD is that others have focused on the weakly supervised, multi-scene version of VAD [17, 23, 45, 50, 52]. Semi-Supervised VAD. Compared to the weaklysupervised setting, the capabilities of MLLMs are understudied in the semi-supervised setting. Wu et al. [46] addresses semi-supervised, single-scene VAD, but with a fundamentally different methodology. Their approach uses an MLLM to first generate a set of textual rules from normal video and then applies these rules during inference. In contrast, our method directly compares MLLM-generated text descriptions between normal and test videos without an intermediate rule-generation step. Furthermore, their model processes full video frames, whereas we employ an object-centric approach that analyzes regions around objects and their pairs. This key design choice allows our method to model object interactions and perform both spatial and temporal localization, while their approach is limited to temporal localization only. Li and Jiang [16] also focuses on semi-supervised single scene anomaly detection and proposes an MLLMbased method. Like our method, they first detect and track objects and then query an MLLM to yield highlevel descriptions of objects and their activity. However, they do not model object interactions, and do not use an exemplar-based model. In real-world scenarios, many anomalies are characterized by an unusual interaction between two objects. Therefore, one of the important aspects of our method is its modeling of object interactions in the scene to identify anomalous interactions. There have been a few other methods that also modeled interactions [5, 9, 28, 39, 40]. While there are many differences among these methods, one of the main differences compared to our current work is that none of them take advantage of MLLMs. In our approach, the textual descriptions of object activity and interactions provided by an MLLM yield powerful cues for a high-level understanding of a scene. 3. Our Approach We propose a novel method to detect anomalies in video. The pipeline of our method (which we call MLLMEVAD, for MLLM-based Explainable Video Anomaly Detection) is depicted in Figure 1. The main idea is to use a set of textual descriptions of the activity seen in normal training video of a scene as the model of normality. These textual descriptions are obtained using an MLLM. Anomalies in testing video are found by comparing textual descriptions of the activity in testing frames with the set of normal descriptions. A test description that is dissimilar from all normal descriptions indicates an anomaly. The initial step of our method is detecting and tracking objects followed by identifying pairs of objects that are in close spatial proximity and therefore likely to be interacting. Then, for each frame and a future frame offset by a fixed time interval, we generate crops around pairs of interacting objects as well as single objects that are not close to other objects. The crops are given to an MLLM to obtain natural language descriptions of objects' activities. These descriptions are embedded into a vector space using a sentence embedding model. By applying an exemplar selection algorithm to all of the sentence embeddings found in nominal training video, we construct representative sets of embeddings for both object pairs and single objects. Anomaly detection is performed by comparing sentence embeddings extracted from testing frames to the exemplar sets, and assigning anomaly scores based on their distance to the most similar exemplar. 3.1. Object Detection and Tracking In our proposed method, we want to build a model of what objects do and how they interact in video containing normal activity. This requires detecting objects and tracking them across frames in training videos. We also find pairs of objects that are likely to be interacting according to their spatial proximity. A video V is a collection of M frames {Fi}M i=1, such that V = [F1, F2, ..., FM]. Each frame Fi is sent to the object detector O, which returns X number of detected objects. The object detector provides, for each object o, the location l = (xo, yo) which is the x and y coordinates of the center of the object, b = (wo, ho) which is the width and height of the bounding box for the object, and class id c. The output of the object detector is then O(F) = [o1, o2, ..., oX], where each object oi is represented by oi = [b, c, l]. After detecting objects in a frame, they are then tracked using an object tracker. Each detected object o is sent to the object tracker, which returns x and y coordinates for that object in the subsequent frames. In our method, we track objects for 30 frames. Therefore, for every object, we acquire the trajectory θ = {(x1, y1), (x2, y2), ..., (x30, y30)}. Next, we pair objects according to their spatial proximity in the frame. To determine which objects should be paired, we need to calculate the 3D distances between each pair of objects. To calculate the 3D distance we need to derive the 3D coordinates of the object locations by estimating a pseudo-depth since we do not have access to actual depth measurements. Given two objects o1 and o2, we have 2D coordinates l1 = (x1, y1) and l2 = (x2, y2). Then we define a relative depth, z, between two objects by taking the absolute difference of y values such that z = |y1 -y2|. This estimate of pseudodepth assumes that objects are resting on the ground plane and the ground plane is farther from the camera 3 Object Detection and Tracking MLLM Visual Input: Crop of two objects enclosed by bounding boxes, from two frames at time t and t+30. . . . Text Encoder Answer sentence r: Two people are walking side by side on the crosswalk without interacting. X Embedding of r Exemplar set of embeddings Training mode: If score is greater than the threshold add embedding of r to exemplar set. Inference mode: Use score for anomaly detection. . . . Prompt: Briefly describe what the people in the enclosed regions of these images are doing. The two images were taken one second apart. Score calculation: Eq. (1) & Eq. (2) Video Frames Figure 1. The pipeline of our method (MLLM-EVAD) for frame to textual description generation with the help of an object detector, object tracker and MLLM. Please see the text for more explanation. the closer it is to the top of the image. The 3D distance d can then be calculated by taking the Euclidean distance between 3D coordinates (x1, y1, z) and (x2, y2, 0). Any objects that have a 3D distance d smaller than a predetermined threshold h are paired. Due to applying the threshold, not every single object is necessarily connected to another object, which leads to having single objects in addition to object pairs. 3.2. Generating Textual Descriptions At the end of the object detection and tracking stage, for each frame Ft, we obtain a set of object pairs and a set of single objects. For each of these, we generate two cropped regions that capture the relevant spatial area for both the current frame Ft and a future frame Ft+∆(e.g., ∆= 30), assuming a static camera (as depicted in Figure 1). This ensures that both crops contain the same background and object regions, allowing for consistent comparisons across time. We provide the details of the cropping process in the Appendix. Red rectangles are drawn around each object in the cropped images using the bounding boxes detected by the object detector. The rectangles indicate to the MLLM which objects are being referred to in the prompt. We then obtain a sentence r that describes the interaction between objects o1 and o2 by querying an MLLM agent with the corresponding image crops Ct and Ct+∆, along with the user prompt: \"Briefly describe what the s in the enclosed regions of these images are doing. The two images were taken one second apart.\" For single-object cases, the prompt is adjusted to: \"Briefly describe what the in the enclosed region of these images is doing. The two images were taken one second apart.\" In both prompts, the placeholder is replaced by the class label c obtained during feature selection. An additional \"system\" prompt is also used with the MLLM: \"You will be provided with two frames from a video and asked to describe what objects that are indicated by bounding boxes in the video are doing. Your task is to answer the query in a simple sentence. If there is any interaction between the indicated objects, a description of the interaction should be given.\" The MLLM-generated responses are then used in both the model building and anomaly detection stages. 3.3. Model Building We follow a similar exemplar selection strategy as introduced in recent works [28, 30, 36, 37]. For all frames in a video, we collect all textual descriptions generated from object pairs into one set and all textual descriptions generated from single objects into another set. Then for each of the sets we run an exemplar selection algorithm which selects a subset of the elements of the set such that no two members of the subset are near each other according to a distance function. The intuition behind exemplar selection is to simply remove redundant (or nearly redundant) elements from the set leaving behind a compact, representative subset of exemplars. Various distance functions between sentences can be used to compare textual descriptions. In most of our experiments we use Sentence-BERT [33], but we also test using BLEU [29] and METEOR [2]. Given a set S, the exemplar selection algorithm proceeds as follows: (1) Initialize the exemplar set to NULL. (2) Add the first element of S to the exemplar set. (3) For each subsequent element of S, find its distance to the nearest instance in the exemplar set. If this distance is above a threshold, th, then add the element to the exemplar set. Since we obtain a textual description r for each object pair or single object, we transform these descriptions into sentence embeddings using a Natural Language Processing (NLP) model N, such that Ir = N(r). The distance D between two textual descriptions, r1 and r2, is then computed as: D(r1, r2) = 1 -cos (Ir1, Ir2) (1) where cos(·, ·) denotes the cosine similarity between the corresponding embeddings. Using the distance function D and the described exemplar selection algorithm, we 4 construct two separate exemplar sets: Epair and Esingle, containing sentence embeddings of textual descriptions for object pairs and single objects, respectively. 3.4. Anomaly Detection in Test Video After constructing the exemplar sets from nominal videos, the next step is to detect anomalies in a test video from the same scene. Similar to the model-building phase, we extract textual descriptions and encode them into sentence embeddings Ir. The anomaly score for a given object pair (or single object) is then computed based on its similarity to the corresponding exemplar set. Specifically, the score is defined as: score(r) = 1 -max (cos (Ir, e)) , e ∈Epair or Esingle (2) depending on whether the input is an object pair or a single object. A higher score indicates a greater deviation from the nominal descriptions and is thus more likely to be anomalous. 3.5. Combination with Previous Methods While the textual descriptions generated by an MLLM provide powerful high-level features for modeling the normal activity in a scene and for detecting anomalies in the scene, some anomalies require more detailed, finegrained features for detection. For this reason, it can be advantageous to combine our method with previous methods that use more detailed features such as object sizes and trajectories. Combining with another method is especially easy to do if the other method is also objectbased and uses an exemplar-based model. In our experiments, we show results when combining our method separately with two other methods; namely, the scene-graph method of Mumcu et al. [28] and the tracklet method of Singh et al [37]. Our method of modeling interactions was inspired by the scene-graph method of Mumcu et al. [28]. Their method builds a scene graph for each frame where each object is a node and objects that are nearby and, therefore possibly interacting, are connected by an edge. The attributes stored at each node are the object's location, size, object class, trajectory and skeletal pose. Then an exemplar-based model is created for every connected pair of nodes across all scene graphs in all nominal frames. A separate exemplar-based model is also created for all singleton nodes across all scene graphs in all nominal frames. To combine our method with theirs, we can simply add the MLLM-generated textual descriptions to each connected pair of nodes and to each singleton node. The distance function used to compare nodes takes the maximum over all attribute distances including the distance between textual descriptions. In the case of the tracklet method of Singh et al. [37], objects in each frame are also detected and tracked. Similar to Mumcu et al., for each object, a set of attributes are stored including the location, size, object class and trajectory. An exemplar-based model is then learned from all of the objects found in the nominal training videos using a distance function for comparing objects that takes the maximum over each attribute distance. To combine our method with theirs, we again only need to add our MLLM-generated textual descriptions as another attribute associated with each object. In this case, object interactions are not modeled. We use this combination to test on datasets that do not contain anomalous object interactions. 4. Experiments In our experiments, we use the following configurations: Datasets: We evaluate MLLM-EVAD on three video anomaly detection benchmark datasets which are all publicly available under a CC-BY-SA-4.0 license. ComplexVAD [28] is a large-scale dataset with a total of over 3.6 million frames, with 2,069,941 frames for training and 1,611,497 for testing. It features interactionbased anomalies and serves as our main focus, since our method leverages descriptions of object activity and interactions for video anomaly detection. The scene features a crosswalk, pedestrian sidewalks, and a two-lane street, which are heavily used. It demonstrates interaction anomalies such as a person leaving an object on the ground, a person sitting on a car, or a dog walking without a person or leash, where the individual objects are normal, but their interactions are anomalous. In addition, we evaluate performance on the Avenue and Street Scene datasets. Avenue [21] contains short surveillance videos of a campus walkway, with anomalies such as running, throwing objects, or walking in the wrong direction. Street Scene [30] includes longer videos of a city street environment, where anomalies involve vehicles or pedestrians behaving unusually, including cars driving on the wrong side of the road, pedestrians crossing at non-designated areas, or bikers riding on the sidewalk. Avenue contains 30,652 frames, split nearly evenly between 15,328 training frames and 15,324 testing frames. Street Scene is a larger dataset with a total of 203,257 frames, of which 56,847 are used for training and 146,410 for testing. Implementation Details: We use Detectron2 [48] and ByteTrack [54] as the object detector and object tracker in our method. Sentence-BERT [33] (using the pretrained weights, \"all-MiniLM-L6-v2\") is used as the text embedding model to encode activity descriptions. Initially, GPT-4o [14] was used as the multimodal MLLM agent. However, with the recent introduction of Gemma 3 [41], we re-conducted the experiments 5 Method RBDC TBDC Frame MemAE [12] 0.05 0.0 58.0 EVAL [36] 10.0 62.0 54.0 AnomalyRuler [46] N/A N/A 56.0 Scene-Graph[28] 19.0 64.0 60.0 MLLM-EVAD 24.0 68.0 61.0 Scene-Graph + MLLM-EVAD 25.0 70.0 63.0 Table 2. Results on the ComplexVAD dataset. Each entry is the area under the curve (AUC) as a percentage for the three benchmark methods using the RBDC, TBDC and Frame-Level evaluation criteria. Bold face indicates the best score and blue indicates second best. Method RBDC TBDC Frame Auto-encoder [13] 0.3 2.0 61.0 Dictionary method [21] 1.6 10.0 48.0 Flow baseline [30] 11.0 52.0 51.0 FG Baseline [30] 21.0 53.0 61.0 EVAL [36] 24.3 64.5 N/A Contextual GMM [49] 34.0 62.5 67.0 T-EVAL [37] 30.9 72.9 66.0 T-EVAL+ MLLM-EVAD 31.1 73.5 66.8 Table 3. Results on the Street Scene dataset using the area under the curve (AUC) (as a percentage) for the RBDC and TBDC evaluation criteria for different methods. Bold face indicates the best score and blue indicates second best. on ComplexVAD using Gemma 3. Hence, the results on ComplexVAD are reported with both Gemma 3 and GPT-4o, while the results on the other datasets are reported with GPT-4o. We choose a threshold th = 0.65 for exemplar selection that resulted in a modest number of total exemplars selected. From past work that used exemplar-based models, this threshold mainly effects model size and has a small effect on test accuracy [36, 37]. We provide additional implementation details in the Appendix. Evaluation Criteria: We use the Region-Based Detection Criterion (RBDC) and the Track-Based Detection Criterion (TBDC) as proposed in [30] as our primary evaluation criteria and report the area under the curve (AUC) for false positive rates per frame from 0 to 1. We also report frame-level AUC [24] scores. As highlighted in previous works [30] frame-level AUC only evaluates temporal accuracy and disregards spatial localization of anomalies. RBDC and TBDC measure a method's capacity to accurately identify anomalous spatio-temporal regions within a given video sequence. Method RBDC TBDC Frame Siamese[31] 41.2 78.6 87.2 Hybrid[20] 41.1 86.2 91.1 BG agnostic [11] 65.1 66.9 92.3 Hybrid[20] + PCAB[34] 62.3 89.3 90.9 BG agnostic[11] + PCAB[34] 66.0 64.9 92.9 SSMTL++v1 [4] 40.9 82.1 93.7 SSMTL++v2 [4] 47.8 85.2 91.6 AnomalyRuler [46] N/A N/A 89.7 EVAL[36] 68.2 87.6 86.0 T-EVAL[37] 67.5 89.7 88.0 T-EVAL + MLLM-EVAD 68.9 90.1 88.4 Table 4. Results on the CUHK Avenue dataset using the area under the curve (AUC) as a percentage for the RBDC and TBDC evaluation criteria for different methods. Bold face indicates the best score and blue indicates second best. 4.1. Quantitative Results The results of our method on ComplexVAD, as well as those of EVAL [36], MemAE [12], AnomalyRuler [46], and Scene-Graph [28], using the three evaluation criteria described above, are reported in Table 2. We also provide the result for the combination of our method and Scene-Graph on the ComplexVAD dataset. Our MLLM-EVAD method outperforms the next best method (Scene-Graph) by 5, 4 and 1 percentage points in RBDC, TBDC, and frame-level evaluations, respectively. We achieve the best result by combining the Scene-Graph method with ours, reaching 25%, 70%, and 63% in RBDC, TBDC, and frame-level evaluations, respectively. AnomalyRuler, a recent MLLM-based method [46], achieves 54% frame-level AUC. However, we were not able to obtain results for the RBDC and TBDC criteria, as this method does not support spatial localization of anomalies. The MemAE method does very poorly for the two criteria that measure spatial localization. This implies that the regions of an image that MemAE predicts as anomalous are usually normal. For the experiments on Avenue and Street Scene, we combine our method with the best-performing existing exemplar-based method, namely Tracklet EVAL [37]. These datasets include many anomalies that need more fine-grained attributes (such as the direction and speed of travel for people and cars), so the high-level textual descriptions used by MLLM-EVAD are not sufficient on their own to detect all of the anomalies. We show that our textual descriptions do add an important new attribute, however, since, when combined with the more fine-grained attributes of Tracklet EVAL, the results with the MLLM textual descriptions improve accuracy for both datasets across all three evaluation cri6 The person is crouching down on the ground. The person is walking across the grass. The person is holding a phone up and appearing to take a picture or video while being pushed in a large box by another person. The person is walking along the sidewalk. Anomaly Closest Exemplar Anomaly Closest Exemplar The two people are walking across the crosswalk, with the person on the right appearing to stumble or jump. Two people are walking across the crosswalk on the street. The person on the bicycle is being pushed forward by another person who is riding a scooter. The person is walking while the cyclist is riding the bicycle alongside them. Figure 2. Examples of anomalies in the ComplexVAD dataset that are detected by our method. In each box, the top row shows two frames of the anomalous video clip followed by the textual description generated by Gemma 3. The second row shows two frames from the closest matching exemplar to the anomaly. The textual description for the closest exemplar is also shown. By contrasting the anomalous description with the closest normal description, it is clear why the testing video clip was found to be anomalous. teria. As shown in Table 4, on the Avenue dataset, the Tracklet-EVAL + MLLM-EVAD method improves the state-of-the-art for both the RBDC and TBDC criteria. On Street Scene (results shown in Table 3), the state-ofthe-art is surpassed for the TBDC criterion. Since the frame-level criterion does not measure spatial localization accuracy which is the focus of these datasets, we would argue that the slightly-below SOTA frame-level numbers are not representative of the true accuracy of these methods. We should note that we were not able to exactly reproduce the results reported for Tracklet EVAL [37] on the Avenue dataset, most likely due to different parameters, so we report the results we could reproduce for that method and use that version in combination with MLLM-EVAD for the results on the combined method. 4.2. Qualitative Results: Explainability In addition to the quantitative results, we present qualitative results to demonstrate the explainability capabilities of our method. The combination of using an MLLM agent and an exemplar selection strategy enables intuitive explanations for detected anomalies by providing descriptions of both the anomaly and the most similar event from the nominal training video. In Figure 2, we demonstrate four detected anomalies from ComplexVAD along with their textual descriptions and the textual descriptions of the closest exemplars, as generated by our method. For example, one detected anomaly is described as \"The person is crouching down on the ground.\" The closest matching exemplar from the nominal set is \"The person is walking across the grass.\" The action of crouching deviates from the typical behavior observed in the training data. This difference highlights how our method leverages semantic understanding to identify uncommon activity, offering an interpretable explanation for the anomaly. Another example involves an anomaly described as \"The person is holding a phone up and appearing to take a picture or video while being pushed in a large box by another person.\" The closest matching exemplar from the nominal set is \"The person is walking along the sidewalk.\", which is, in fact, the most common activity involving a person in this dataset. The anomalous event, a person being pushed in a box, does not occur in the nominal video and represents an unusual and complex behavior. This example further demonstrates our method's ability to detect rare interaction patterns that deviate from typical activities in the scene. In addition, these examples illustrate how our method also provides reasonable explanations for detected anomalies. By leveraging descriptive comparisons with nominal exemplars, our method enables enhanced and explainable video anomaly detection. 4.3. Ablation Study We conduct two ablation studies by modifying the default settings of our method. In the first ablation study, instead of using sentence embeddings and cosine dissimilarity as the distance metric, we use the raw sentences and compute distances using BLEU [29] and METEOR [2]. In Table 5, we compare the results of cosine distance over Sentence-BERT embeddings with those obtained using BLEU and METEOR, based on the de7 Method RBDC TBDC Frame Sentence-BERT 19.0 67.0 59.0 BLEU 17.0 69.0 58.0 METEOR 16.0 69.0 57.0 Table 5. The table reports results on the ComplexVAD datasets using the area under the curve (AUC) as a percentage for the RBDC, TBDC and Frame-Level evaluation criteria for the our method using 3 different sentence distance functions. MLLM RBDC TBDC Frame GPT-4o 19.0 67.0 59.0 Gemma 3 24.0 68.0 61.0 Table 6. The table reports the performance of our proposed method on the ComplexVAD datasets using the area under the curve (AUC) as a percentage for the RBDC, TBDC and FrameLevel evaluation criteria with different MLLM agents GPT-4o and Gemma 3. scriptions generated by our method using GPT-4o as the MLLM agent. The results show that Sentence-BERT embeddings yield slightly higher accuracy in the RBDC and frame-level criteria while BLEU and METEOR perform slightly better on the TBDC criterion. SentenceBERT is the preferred similarity function considering its greater efficiency due to embeddings which can be saved and only require a dot product for comparison. In the second ablation study, we compare our method's performance on ComplexVAD using different MLLM agents, namely Gemma 3 and GPT-4o. In Table 6, we show that Gemma 3 achieves better performance, with 24%, 68%, and 61% in the RBDC, TBDC, and frame-level criteria, respectively, while GPT-4o yields 19%, 67%, and 59%. We attribute this difference to the fact that Gemma 3 generates more detailed and descriptive sentences compared to GPT-4o. This level of detail is particularly important for detecting interaction-based anomalies, where subtle contextual cues are crucial, and likely contributed to the observed performance gap. We provide a comparison of the descriptions generated by Gemma 3 and GPT-4o in the Appendix. 5. Limitations and Future Directions While our framework shows promising results, we identify several limitations that pave the way for future research. First, our implementation relies on powerful but computationally expensive multimodal models like Gemma 3. Their high inference latency poses a challenge for real-time applications. Recent studies have shown that smaller, task-specific models can perform exceptionally well on specialized tasks [25, 26, 35, 43]. We will explore fine-tuning such models for single scenes, specifically for describing object activities and interactions. Second, a primary contribution of our method is its semantic explainability, but quantitatively evaluating this feature is difficult due to a lack of suitable benchmarks. Current datasets with textual ground truth are designed for multi-scene or weakly-supervised tasks and do not fit the semi-supervised, single-scene VAD problem [10, 53, 55]. Therefore, a key future direction is the development of new datasets with ground-truth textual descriptions for scene-specific anomalies. Finally, our framework currently leverages a robust traditional object detector, which is well-suited for scenarios with known object classes. To further generalize our approach, a promising future direction is the integration of open-vocabulary object detection methods [6, 18, 27]. By enabling the recognition of an open set of object categories, this advancement would significantly broaden our system's applicability, allowing it to analyze and describe a more diverse range of anomalies in unconstrained, real-world environments. 6. Conclusions In this work, we presented a novel video anomaly detection framework that leverages multimodal Large Language Models to detect and explain complex anomalies. By extracting object-level activity and interactions and converting them into textual descriptions, our method moves beyond traditional pixel level modeling and introduces a language-based, interpretable layer for video anomaly detection. Unlike many prior MLLM-based methods that rely on direct frame-level judgments, our approach builds a set of nominal textual descriptions and identifies anomalies through language-based comparison, offering both accuracy and explainability. We demonstrate that our method outperforms existing approaches on ComplexVAD which is a challenging benchmark specifically designed to test interactionbased anomalies. In addition our method also improves of the state-of-the-art results on standard datasets like Avenue and Street Scene when combined with the finegrained method Tracklet EVAL [37]. Through ablation studies, we showed the importance of using sentence embeddings and detailed MLLM-generated descriptions. Notably, we found that Gemma 3 provided more informative descriptions than GPT-4o, contributing to higher detection accuracy. Beyond strong quantitative results, our method offers clear and interpretable explanations for detected anomalies, highlighting its potential for real world deployment in safety critical surveillance applications. We believe this framework opens new directions for integrating multimodal large language models into video understanding tasks and sets the stage for future research in explainable and high level language-based video anomaly detection. 8 References [1] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. Gpt-4 technical report. arXiv preprint , 2023. 1 [2] Satanjeev Banerjee and Alon Lavie. Meteor: An automatic metric for mt evaluation with improved correlation with human judgments. In Proceedings of the ACL Workshop on Intrinsic and Extrinsic Evaluation Measures for Machine Translation and/or Summarization, 2005. 4, 7 [3] Mohammad Baradaran and Robert Bergevin. 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A single 80 GB GPU would be sufficient and multiple GPUs were used only to reduce the total compute time by processing multiple video sequences in parallel. Our experiments that used GPT-4o accessed it through an API via the cloud. 2. Cropping Frames For each selected object pair o1 and o2, we first compute a single bounding box that tightly encloses both objects in a given frame. Let the individual bounding boxes for the two objects in frame Ft be denoted as bboxa = (x1a, y1a, x2a, y2a) and bboxb = (x1b, y1b, x2b, y2b), where (x1, y1) represents the top-left corner and (x2, y2) the bottom-right corner of a bounding box. These are merged into a single bounding box bbox1 by taking the element-wise minimum for the top-left corner and maximum for the bottom-right corner: bbox1 = min(x1a, x1b), min(y1a, y1b), max(x2a, x2b), max(y2a, y2b) (S1) We apply the same procedure to obtain bbox2 in the future frame Ft+∆, using the tracked locations of the same object pair. Let bbox1 = (x1,1, y1,1, x2,1, y2,1) denote the merged bounding box of the selected object pair in frame Ft, and let bbox2 = (x1,2, y1,2, x2,2, y2,2) denote the merged bounding box of the same objects tracked in the future frame Ft+∆. A unified bounding box that covers both time steps is computed as: bboxmerged = min(x1,1, x1,2), min(y1,1, y1,2), max(x2,1, x2,2), max(y2,1, y2,2) (S2) Let (x1, y1) and (x2, y2) represent the top-left and bottom-right corners of this merged bounding box. Let wmin and hmin denote half of the minimum desired crop width and height, respectively. In our case, we set wmin = 240 and hmin = 135, corresponding to a minimum crop size of 480 × 270 pixels. This size was chosen as a practical minimum that ensures sufficient context around objects while keeping computational overhead low. In our experiments, we found that a resolution of 480 × 270 served as a reliable lower bound, providing adequate spatial detail for capturing object activities without reducing the quality of the LLM generated descriptions. While larger crops may be used when available, we observed this setting to be a consistent and effective baseline across different datasets. We then compute the crop dimensions as w = max |x2-x1| 2 , wmin , h = max |y2-y1| 2 , hmin (S3) The crop boundaries are determined as: xmin = max(x1 -w, 0), ymin = max(y1 -h, 0), xmax = min(x2 + w, W), ymax = min(y2 + h, H) (S4) where W and H denotes the width and height of the full frame. The final cropped region Ct is extracted from the current frame Ft as: Ct = Ft[ymin : ymax, xmin : xmax] To ensure both temporal frames share the same spatial context, we apply the exact same crop coordinates to the future frame Ft+∆: Ct+∆= Ft+∆[ymin : ymax, xmin : xmax] This guarantees that both cropped frames contain the same background and object regions, allowing for consistent comparisons across time. In the case of a single object, the same procedure is applied with one exception: instead of merging two object bounding boxes in each frame, we directly use the bounding box of the single object in frame Ft and its tracked location in Ft+∆. The rest of the cropping process, including the computation of the unified bounding box across time, minimum size enforcement, and cropping, is performed exactly as described above. 3. Comparison of Gemma 3 and GPT-4o Activity Descriptions In Figure S1, we share activity descriptions generated by Gemma 3 and GPT-4o for given images and the promt: \"Briefly describe what the s in the enclosed regions of these images are doing. The two images were taken one second apart.\" 12 Gemma 3 GPT-4o The person is crouching down on the ground. The person in the enclosed region is bending over and appears to remain in the same position between the two images The person is holding a phone up and appearing to take a picture or video while being pushed in a large box by another person. In the images, the person standing seems to be pushing or interacting with a cart or box that the seated person appears to be inside or using. The standing person maintains their position or posture relative to the cart between the images. The person in the blue shirt and the person on the bike are interacting, as the person in blue appears to be preventing the cyclist from moving forward. The person on the bicycle is being pushed forward by another person who is riding a scooter. The person is walking across the crosswalk while looking at their phone, and a cyclist is passing by them on the right. The person in the enclosed region is stopping to allow a cyclist to pass by. In the first image, the person in the black shorts and dark hair appears to be jumping or falling next to the bald person. In the second image, the person in the black shorts and dark hair is standing up straight, while the bald person remains standing. There is no apparent interaction between them. The two people are walking across the crosswalk, with the person on the right appearing to stumble or jump. Figure S1. The activity descriptions generated by Gemma 3 and GPT-4o with given images. We have observed that Gemma 3 often gives more concise descriptions with fewer unnecessary details such as the color of a person's clothes. Moreover, Gemma 3 sometimes includes important details that GPT-4o omits, such as the fact that a person is using his or her phone. 13"}}},{"rowIdx":87,"cells":{"id":{"kind":"string","value":"2510.14892"},"raw":{"kind":"string","value":"A Comprehensive Framework for Efficient Court Case\nManagement and Prioritization\nShubham Varma\nshubham.varma@somaiya.edu\nAnanya Warior\nananya.warior@somaiya.edu\nDipti Pawade\ndiptipawade@somaiya.edu\nAvani Sakhapara\navanisakhapara@somaiya.edu\nAbstract\nThe Indian judicial system faces a critical challenge with approximately 52 [1]\nmillion pending cases, causing significant delays that impact socio-economic stabil-\nity. This study proposes a cloud-based software framework to classify and priori-\ntize court cases using algorithmic methods based on parameters such as severity of\ncrime committed, responsibility of parties involved, case filing dates, previous hear-\ning’s data, priority level (e.g., Urgent, Medium, Ordinary) provided as input, and rel-\nevant Indian Penal Code (IPC), Code of Criminal Procedure (CrPC), and other legal\nsections (e.g., Hindu Marriage Act, Indian Contract Act). Cases are initially entered\nby advocates on record or court registrars, followed by automated hearing date al-\nlocation that balances fresh and old cases while accounting for court holidays and\nleaves. The system streamlines appellate processes by fetching data from histori-\ncal case databases. Our methodology integrates algorithmic prioritization, a robust\nnotification system, and judicial interaction, with features that allow judges to view\ndaily case counts and their details. Simulations demonstrate that the system can pro-\ncess cases efficiently, with reliable notification delivery and positive user satisfaction\namong judges and registrars. Future iterations will incorporate advanced machine\nlearning for dynamic prioritization, addressing critical gaps in existing court case\nmanagement systems to enhance efficiency and reduce backlogs.\n1\nIntroduction\nWith approximately 52 [1] million cases pending in India, the judicial system faces se-\nvere delays that hinder economic activities, burden law enforcement, and erode public\ntrust in timely justice delivery. Traditional case management systems, often manual or\nsemiautomated, struggle to handle this volume, necessitating innovative technological\nsolutions. This study proposes a software framework that leverages algorithmic priori-\ntization to classify and manage court cases efficiently. By automating hearing date allo-\ncations, balancing fresh and old cases, and integrating input from judges, registrars, and\nadvocates, the system aims to revolutionize judicial processes. The framework operates\nwithin a cloud-based environment, ensuring scalability and accessibility. We simulated\n10,000 dummy cases to demonstrate the system’s potential to process cases efficiently\nand achieve high user satisfaction. The future scope will incorporate dynamic machine\nlearning models for real-time prioritization, ensuring adaptability to evolving judicial\nneeds. This research fills a critical gap by combining AI-driven prioritization with eth-\nical considerations and compatibility with existing systems such as eCourt Services [1],\nas highlighted in recent studies on AI in judicial systems [9, 10]. Figure 1 illustrates the\nworkflow of the system.\n1\narXiv:2510.14892v1 [cs.CY] 16 Oct 2025\n\nFigure 1: Workflow of the System\n2\nLiterature Survey\nThe inefficiencies of traditional court case management have been extensively studied,\nunderscoring the need for technological interventions. Rao et al. [2] advocate for digitiz-\ning legal records to improve transparency and reduce corruption, a fundamental prin-\nciple of our cloud-based framework. Their work highlights how digital records improve\naccess to case data, a feature our system leverages through MySQL database integration.\nAmofah [3] explores electronic court case management systems (eCCMS), demonstrating\ntheir ability to streamline judicial processes through automation, such as automated\ncase tracking. However, these systems lack AI-driven dynamic prioritization, a gap our\nresearch addresses. Pattnaik et al. [4] identify critical success factors for efficient court\nmanagement in India, including stakeholder collaboration and technological adoption,\nwhich align with our system’s design for judicial interaction and cloud-based scalability.\nChan [5] examines case management in Chinese-speaking regions, noting the influence\nof traditional legal structures on efficiency, while Ali and Varol [6] propose mobile so-\nlutions to bridge communication gaps, supporting our notification system. Tahura [7]\nexplores caseflow management to reduce backlogs, emphasizing structured scheduling,\nand Ambos [8] highlights prosecutorial discretion based on case gravity, informing our\nprioritization parameters.\nRecent advancements in artificial intelligence (AI) have transformed judicial systems\nglobally. Thomson Reuters Institute [9] reports that AI tools enable court employees\nto focus on human interactions and reduce bias, citing Palm Beach County’s AI-driven\ndocument processing system as an example of high efficiency. IBM [10] notes that ju-\ndicial systems, such as those in Germany, are adopting AI to manage vast datasets and\naccelerate case resolution, aligning with our goal of reducing India’s case backlog. The\nInternational Journal for Court Administration [11] discusses a study in the Netherlands\nusing AI to support judges in traffic violations cases, demonstrating the potential of AI\nin decision support. These studies underscore the feasibility of AI in judicial contexts,\nsupporting the approach of our system.\n2.1\nEthical Considerations\nThe integration of AI in judicial systems raises significant ethical concerns, particularly\naround bias and fairness. Judicature [12] emphasizes that AI must be used responsibly to\n2\n\navoid perpetuating biases, such as overprioritizing certain case types. The American As-\nsociation for the Advancement of Science (AAAS) [15] highlights the importance of fair-\nness, accountability, and transparency (FAccT) in AI systems, noting that biases can arise\nfrom datasets, design, or deployment. Above the Law [13] reports a case where a trial\ncourt relied on AI-generated fake caselaw, underscoring the need for human oversight.\nOur system mitigates these risks by incorporating expert-validated weights for prioriti-\nzation parameters and planning regular audits to ensure equitable outcomes, aligning\nwith ethical guidelines from the Council of Europe [11]. Additionally, we address poten-\ntial biases by involving legal experts in weight validation and ensuring transparency in\nthe prioritization process.\n2.2\nGaps Addressed\nExisting systems, such as those discussed by Rao et al. [2] and Amofah [3], focus on dig-\nitization and automation but lack AI-driven dynamic prioritization tailored to India’s\njudicial context. Our framework addresses this gap by integrating a regression-based\nmachine learning model for real-time weight adjustment, ensuring both efficiency and\nfairness. Unlike mobile solutions proposed by Ali and Varol [6] , our system offers a\ncomprehensive cloud-based platform that integrates with existing judicial infrastruc-\nture like eCourts Services, enhancing scalability and accessibility.\n3\nMethodology\n3.1\nData Retrieval and Preprocessing\nCase data is entered by advocates on record or court registrars into a MySQL database\nand preprocessed by categorizing cases into distinct datasets (e.g., Family, Criminal, Civil).\nThis segmentation enables custom prioritization strategies for different types of case,\nensuring flexibility and scalability. Preprocessing includes cleaning data, standardizing\nformats, and extracting relevant features such as filing dates, priority levels, and appli-\ncable legal sections (e.g., IPC, CrPC, Hindu Marriage Act).\n3.2\nCase Prioritization Algorithm\nThe core of our system is the case prioritization algorithm, which assigns a weighted\nscore to each case to determine its urgency and scheduling priority. The algorithm, de-\ntailed in Algorithm 1, processes the cases as follows:\n3\n\nAlgorithm 1 Case Prioritization Algorithm\n1: Input: Case data (case ID, type, filing date, severity, priority level, legal sections),\ncourt holidays, judge leaves, disposed cases database\n2: Output: Prioritized case list with assigned hearing dates\n3: Initialize MySQL database connection\n4: Retrieve pending case data from MySQL database\n5: for each case in pending cases do\n6:\nif case is an appeal then\n7:\nFetch relevant data from disposed cases database\n8:\nend if\n9:\nCategorize case into type (Criminal, Family, Civil)\n10:\nExtract features: case age (from filing date and previous hearings), severity, pri-\nority level (Urgent, Medium, Ordinary), legal sections (e.g., IPC 302, Hindu Marriage\nAct Section 13)\n11:\nCompute weight using regression-based ML model\n▷Model considers case age,\nseverity, priority level, and legal sections; IPC/CrPC weights validated by advocates\nand judges\n12:\nAssign priority based on weight\n13: end for\n14: Sort cases by weight in descending order\n15: Initialize scheduling with daily limit (100 cases: 50 fresh, 50 old)\n16: for each day in scheduling period do\n17:\nif day is not a court holiday or judge leave then\n18:\nAssign hearing dates to top-weighted cases (up to 50 fresh, 50 old)\n19:\nend if\n20: end for\n21: Export prioritized case list to database\n22: Send notifications (SMS/email) to stakeholders with hearing dates\n23: Periodically review low-priority cases\n24: if case pendency exceeds threshold then\n25:\nIncrease case priority\n26: end if\n27: Update ML model with case outcomes for dynamic weight adjustment\n28: Return: Prioritized case list with hearing dates\nThe algorithm retrieves all pending case data from a MySQL database, including details\nsuch as case ID, type, filing date, severity, priority level (e.g., Urgent, Medium, Ordinary)\nprovided by advocates or registrars, and relevant legal sections (e.g., IPC 302, Hindu\nMarriage Act Section 13). For appealed cases, it fetches data from the disposed cases\ndatabase. The data is preprocessed to categorize cases into categories such as Criminal,\nFamily, or Civil, enabling customized prioritization strategies. For each case, a weight\nis calculated using a regression-based machine learning model, which considers factors\nsuch as the age of the case (based on filing and previous hearing dates), severity (e.g.,\nhigh for murder, low for minor disputes), the input priority level, and applicable legal\nsections. The weights for the IPC and CrPC sections were determined through consul-\ntations with multiple advocates and judges to ensure legal accuracy and fairness. The\ncalculated weight determines the hearing date, with higher weights leading to earlier\nscheduling. Hearing dates are allocated while considering a fixed daily case limit (e.g.,\n100 cases, with 50 fresh and 50 old cases) and a predefined list of court holidays and\nadditional leaves entered into the system. Notifications are sent to stakeholders via SMS\n4\n\nor email to inform them of the assigned hearing dates. To prevent low-priority cases\nfrom being indefinitely delayed, the system periodically checks for pendency, and if a\ndefined threshold is exceeded, it increases the case’s priority by multiplying it with a\nfixed multiplier. The machine learning model dynamically adjusts the weights based on\nthe historical case outcomes, achieving high precision in the simulations, as shown in\nFigure 2.\nFigure 2: Case Prioritization Flowchart\n3.3\nNotification System and Judicial Integration\nPost-prioritization, an automated notification system sends SMS/email alerts to stake-\nholders regarding hearing dates. Judges and registrars access a prioritized case list via\na secure web interface, which also displays the number of cases scheduled per day, fa-\ncilitating efficient scheduling and case management. The interface is designed for com-\npatibility with existing systems like e-Courts Services [1].\n4\nSystem Architecture\nThe system architecture, illustrated in Figure 3, orchestrates a seamless flow of oper-\nations to prioritize and manage court cases within a secure cloud-based environment.\nThe process is structured into distinct stages, ensuring scalability, efficiency, and integra-\ntion with judicial workflows, including case entry by advocates or registrars, handling\nappeals, balanced case scheduling, and holiday considerations.\nThe process begins with case data entry, where advocates on record or court registrars\ninput case details, such as case ID, type, filing date, severity, priority level (e.g., Urgent,\nMedium, Ordinary), and relevant legal sections (e.g., IPC, CrPC, Hindu Marriage Act),\ninto a MySQL database. This initial step ensures that all relevant case information is\ncaptured accurately for subsequent processing.\nThe system initializes necessary configurations and establishes a secure connection to\nthe MySQL database. This step ensures that the system is fully prepared to access and\nprocess case data reliably, setting the foundation for subsequent operations.\nNext, the system retrieves all pending case data from the MySQL database, including\nthe details entered by advocates or registrars. For cases that have gone to appeal, the\n5\n\nsystem fetches relevant data from the disposed cases database, ensuring that historical\ncase information is available for consistent prioritization in higher courts.\nThe retrieved data is then preprocessed to categorize cases into distinct groups, such\nas Criminal, Family, or Civil. This categorization enables the system to apply tailored\nprioritization strategies, accommodating the unique characteristics of each case type,\nincluding appealed cases, and enhancing scalability for diverse judicial needs.\nFor each case, the system employs a regression-based machine learning model to calcu-\nlate a weight that determines its hearing date. The model evaluates factors such as the\nage of the case (calculated from filing and previous hearing dates), the severity of the of-\nfense (e.g., high for murder under IPC 302, low for minor civil disputes), the priority level\nprovided as an input parameter by advocates or registrars, and applicable legal sections\n(e.g., IPC, CrPC, Hindu Marriage Act). Weights for IPC and CrPC sections were determined\nthrough consultations with multiple advocates and judges to ensure legal accuracy. The\ncalculated weight guides the allocation of hearing dates, with higher weights leading to\nearlier scheduling. The system schedules a fixed number of cases per day, for example,\n100 cases, with 50 fresh cases and 50 old (next hearing) cases, to balance judicial work-\nload. Hearing dates are assigned while accounting for a predefined list of court holidays\nand any additional leaves entered into the system, ensuring accurate and conflict-free\nscheduling.\nThe prioritized case list is exported to a secure database for record keeping, ensuring\nthat all scheduling decisions, including those for fresh and old cases, are systematically\nstored. This step maintains data integrity and supports traceability for both pending and\nappealed cases.\nNotifications are then sent to stakeholders, including judges, lawyers, and litigants, via\nSMS or email, informing them of the assigned hearing dates. For appealed cases, notifi-\ncations also include relevant higher court details, ensuring all parties are informed and\nprepared.\nTo prevent low-priority cases from being indefinitely delayed, the system periodically\nreviews the time since the last hearing for each case. If a case, including those under\nappeal, has been pending beyond a predefined threshold, its priority is increased by\nmultiplying it with a fixed multiplier to ensure timely resolution, promoting fairness\nacross all case types.\nFor cases involving appeals, the system streamlines the process by retrieving data from\nthe disposed cases database, applying the same prioritization process (weight calcula-\ntion based on input priority and other factors, and date allocation with the fixed case\nlimit), and notifying higher courts. This ensures consistency and efficiency in handling\nappeals, maintaining seamless integration with the judicial hierarchy.\nThe system continuously monitors its performance by measuring execution times for\neach operation, ensuring efficiency and identifying potential bottlenecks. Performance\ndata is logged to support ongoing optimization, applicable to both pending and appealed\ncases.\nJudges access the prioritized case list, including appealed cases, through a secure web\ninterface, which presents cases in order of priority and displays the number of cases\nscheduled per day, helping judges manage their daily workload effectively. Judges can\nmake decisions, such as scheduling the next hearing or disposing of a case. When a\njudge specifies a time frame for the next hearing (e.g., ’after 15 days’), the system sched-\nules the hearing as soon as possible after that period, subject to the fixed case limit and\n6\n\nholiday/leave constraints, and updates the database accordingly; if a case is disposed, it\nis marked as resolved, stored in the disposed cases database, and the outcome is used to\nupdate the machine learning model, enhancing its accuracy for future prioritizations,\nincluding for appeals.\nThis structured flow, as shown in Figure 3, ensures that the system efficiently manages\nlarge case volumes, including appealed cases, while balancing fresh and old cases, re-\nspecting holidays and leaves, and integrating seamlessly with existing judicial platforms\nlike eCourts Services, providing a robust solution for India’s judicial backlog.\nFigure 3: System Architecture for Court Case Management and Prioritization\n5\nResults\n5.1\nIllustrative Case Study\nTo demonstrate the system’s functionality, we present a synthetic case study with five\nsimulated cases from our 10,000-case dataset (Table 1). These cases vary in type, filing\ndate, severity, priority level, and applicable legal sections, showcasing the algorithm’s\nability to prioritize effectively. Weights for IPC and CrPC sections were validated through\nconsultations with multiple advocates and judges to ensure legal accuracy.\n7\n\nTable 1: Illustrative Case Study\nCase\nID\nType\nFiling Date\nSeverity\nLegal Sections\nPriority\nWeight No\nof\nHear-\nings\nHearing\nDate\n001\nCriminal\n01/01/2024\nHigh\nIPC 302, 34\nUrgent\n0.95\n4\n15/07/2025\n002\nCivil\n01/02/2025\nLow\nIndian\nCon-\ntract\nAct,\n1872,\nSection\n73\nOrdinary\n0.45\n0\n30/08/2025\n003\nFamily\n01/06/2024\nMedium\nHindu\nMar-\nriage\nAct,\n1955,\nSection\n13\nMedium\n0.70\n3\n01/08/2025\n004\nCriminal\n01/02/2024\nHigh\nIPC 420, 406\nUrgent\n0.90\n5\n20/07/2025\n005\nCriminal\n01/04/2025\nMedium\nIPC 323, CrPC\n200\nOrdinary\n0.50\n2\n15/09/2025\nNarrative\n• Case 001: A murder case filed on 01/01/2024, with high severity and an input pri-\nority of Urgent, governed by IPC Section 302 (murder) and Section 34 (acts done by\nseveral persons in furtherance of common intention). With four hearings held and\nits high severity validated by advocate and judge consultations, it has a weight of\n0.95, scheduling it for 15/07/2025, ensuring swift justice for a serious crime.\n• Case 002: A civil case (e.g., breach of contract) filed on 01/02/2025, with low severity\nand an input priority of Ordinary, governed by Section 73 of the Indian Contract\nAct, 1872 (compensation for loss). With no hearings yet and its recent filing and\nlow severity, it yields a weight of 0.45, scheduling it for 30/08/2025, reflecting its\nlower urgency.\n• Case 003: A family law case (e.g., divorce) filed on 01/06/2024, with medium sever-\nity and an input priority of Medium, governed by Section 13 of the Hindu Marriage\nAct, 1955 (grounds for divorce). With three hearings held and moderate pendency\nand severity, it results in a weight of 0.70, scheduling it for 01/08/2025, balancing\nurgency and recency.\n• Case 004: A cheating case filed on 01/02/2024, with high severity and an input pri-\nority of Urgent, governed by IPC Section 420 (cheating) and Section 406 (criminal\nbreach of trust). With five hearings held and its high severity validated, it yields\na weight of 0.90, scheduling it for 20/07/2025, addressing a significant criminal of-\nfense promptly.\n• Case 005: A case involving voluntarily causing hurt filed on 01/04/2025, with medium\nseverity and an input priority of Ordinary, governed by IPC Section 323 (voluntar-\nily causing hurt) and CrPC Section 200 (examination of complainant). With two\nhearings held and its recent filing and medium severity, it results in a weight of\n0.50, scheduling it for 15/09/2025, as it is less severe than murder or cheating.\nThis case study illustrates how the system prioritizes cases based on severity, pendency,\ninput priority level, and applicable legal sections, validated by legal experts, ensuring\ncritical cases are addressed promptly while managing diverse case types.\n8\n\n5.2\nSystem Performance Metrics\nSimulations with 10,000 dummy cases demonstrated that the system can process cases\nefficiently, with reliable notification delivery and positive feedback from judges and reg-\nistrars regarding usability.\n5.3\nMachine Learning Model Performance\nThe regression-based machine learning model used for dynamic weight adjustment achieved\nan F1 score of 99.7%, precision of 99.8%, and recall of 99.4% in simulated tests, indicating\nhigh accuracy and reliability in prioritizing cases.\n6\nDiscussion and Conclusion\nThe proposed framework significantly enhances judicial efficiency by automating case\nprioritization and scheduling. By integrating algorithmic approaches, a robust notifi-\ncation system, and a dynamic feedback loop, the system addresses challenges like case\nbacklogs and delayed hearings. The illustrative case study demonstrates its practical\napplicability, prioritizing urgent cases like murder (IPC 302, 34) while ensuring fairness\nacross civil and family cases. The system’s compatibility with existing platforms like\neCourts Services [1] ensures seamless integration into India’s judicial infrastructure.\n6.1\nEthical Implications\nAI-driven prioritization risks biases, such as over-prioritizing criminal cases, which could\ndisadvantage civil or family cases. Our system mitigates this through expert-validated\nweights, informed by consultations with advocates and judges, and regular audits, align-\ning with ethical guidelines from the Council of Europe [11]. Above the Law [13] high-\nlights risks of AI-generated inaccuracies, underscoring the need for human oversight,\nwhich our system incorporates through judicial interaction.\n6.2\nReal-World Feasibility\nThe system is designed for compatibility with existing platforms like eCourts Services,\nusing standardized MySQL databases and secure cloud storage. Pilot testing in a regional\ncourt will validate its effectiveness, addressing potential challenges like data migration\nand user training.\n6.3\nFuture Work\nFuture iterations will prioritize real-world deployment, integrating advanced machine\nlearning to enhance case management. Deep learning models, such as transformers, will\nimprove feature extraction from legal texts, identifying attributes like case complexity\nand urgency from documents in Table 1. Reinforcement learning will enable adaptive\nprioritization by optimizing schedules based on court data, such as judge availability\nand case backlog, improving efficiency.\nThe system will also expand to handle multilingual case data, as demonstrated by Clio\n[14] for California courts, to enhance accessibility in diverse regions. NLP techniques,\nincluding cross-lingual embeddings, will process languages like Hindi and English, ad-\ndressing challenges in legal terminology translation. Additionally, predictive analytics\n9\n\nwill forecast case pendency using data like hearing counts and severity, aiding resource\nallocation. Piloting in high-volume courts will test scalability, with collaboration from\nlegal experts ensuring judicial alignment and data security, ultimately streamlining pro-\ncesses and improving access to justice.\nReferences\n[1] eCourts Services. https://ecourts.gov.in/ecourts_home/\n[2] Rao, A. P., et al. (2018). Court Case Management System.\n[3] Amofah, L. R. (2017). Electronic court case management system (for law court com-\nplex).\n[4] Pattnaik, P. N., et al. (2018). Mapping Critical Success Factors in Efficient Court Man-\nagement. International Journal of Law and Management, 60(2).\n[5] Chan, P. C. H. (2018). Framing the structure of the court system in the perspective of\ncase management. Peking University Law Journal, 6(1), 55–79.\n[6] Ali, C. T., & Varol, A. (2020). Design and Implementation of a Simple Online Court\nCase Management System Based on the Android Platform. IEEE.\n[7] Tahura, U. S. (2013). Case Management: A Magic LAMP in Reducing Case Backlogs.\nBangladesh Law Journal, 13.\n[8] Ambos, K. (2018). Office of the Prosecutor: Policy Paper on Case Selection and Priori-\ntisation. International Legal Materials, 57(6), 1131–1145.\n[9] Thomson Reuters Institute. (2024). Humanizing Justice:\nThe transformational\nimpact\nof\nAI\nin\ncourts.\nhttps://www.thomsonreuters.com/en-us/posts/\nai-in-courts/humanizing-justice/\n[10] IBM. (2025). Judicial systems are turning to AI to help manage vast quantities of\ndata and expedite case resolution. https://www.ibm.com/case-studies/blog/\njudicial-systems-are-turning-to-ai-to-help-manage-its-vast-quantities-of-dat\n[11] International Journal for Court Administration. (2020). Courts and Artificial Intelli-\ngence. https://iacajournal.org/articles/10.36745/ijca.343\n[12] Judicature.\n(2024).\nAI\nin\nthe\nCourts:\nHow\nWorried\nShould\nWe\nBe?\nhttps://judicature.duke.edu/articles/\nai-in-the-courts-how-worried-should-we-be/\n[13] Above\nthe\nLaw.\n(2025).\nTrial\nCourt\nDecides\nCase\nBased\nOn\nAI-Hallucinated\nCaselaw.\nhttps://abovethelaw.com/2025/07/\ntrial-court-decides-case-based-on-ai-hallucinated-caselaw/\n[14] Clio. (2025). AI in the Courtroom: Opportunities and Challenges. https://www.\nclio.com/resources/ai-for-lawyers/ai-in-courtrooms/\n[15] American Association for the Advancement of Science (AAAS). Artificial In-\ntelligence and the Courts:\nMaterials for Judges. https://www.aaas.org/ai2/\nprojects/law/judicialpapers\n10\n"},"cleaned":{"kind":"string","value":"A Comprehensive Framework for Efficient Court Case Management and Prioritization Shubham Varma Ananya Warior Dipti Pawade Avani Sakhapara Abstract The Indian judicial system faces a critical challenge with approximately 52 [1] million pending cases, causing significant delays that impact socio-economic stability. This study proposes a cloud-based software framework to classify and prioritize court cases using algorithmic methods based on parameters such as severity of crime committed, responsibility of parties involved, case filing dates, previous hearing's data, priority level (e.g., Urgent, Medium, Ordinary) provided as input, and relevant Indian Penal Code (IPC), Code of Criminal Procedure (CrPC), and other legal sections (e.g., Hindu Marriage Act, Indian Contract Act). Cases are initially entered by advocates on record or court registrars, followed by automated hearing date allocation that balances fresh and old cases while accounting for court holidays and leaves. The system streamlines appellate processes by fetching data from historical case databases. Our methodology integrates algorithmic prioritization, a robust notification system, and judicial interaction, with features that allow judges to view daily case counts and their details. Simulations demonstrate that the system can process cases efficiently, with reliable notification delivery and positive user satisfaction among judges and registrars. Future iterations will incorporate advanced machine learning for dynamic prioritization, addressing critical gaps in existing court case management systems to enhance efficiency and reduce backlogs. 1 Introduction With approximately 52 [1] million cases pending in India, the judicial system faces severe delays that hinder economic activities, burden law enforcement, and erode public trust in timely justice delivery. Traditional case management systems, often manual or semiautomated, struggle to handle this volume, necessitating innovative technological solutions. This study proposes a software framework that leverages algorithmic prioritization to classify and manage court cases efficiently. By automating hearing date allocations, balancing fresh and old cases, and integrating input from judges, registrars, and advocates, the system aims to revolutionize judicial processes. The framework operates within a cloud-based environment, ensuring scalability and accessibility. We simulated 10,000 dummy cases to demonstrate the system's potential to process cases efficiently and achieve high user satisfaction. The future scope will incorporate dynamic machine learning models for real-time prioritization, ensuring adaptability to evolving judicial needs. This research fills a critical gap by combining AI-driven prioritization with ethical considerations and compatibility with existing systems such as eCourt Services [1], as highlighted in recent studies on AI in judicial systems [9, 10]. Figure 1 illustrates the workflow of the system. 1 16 Oct 2025 Figure 1: Workflow of the System 2 Literature Survey The inefficiencies of traditional court case management have been extensively studied, underscoring the need for technological interventions. Rao et al. [2] advocate for digitizing legal records to improve transparency and reduce corruption, a fundamental principle of our cloud-based framework. Their work highlights how digital records improve access to case data, a feature our system leverages through MySQL database integration. Amofah [3] explores electronic court case management systems (eCCMS), demonstrating their ability to streamline judicial processes through automation, such as automated case tracking. However, these systems lack AI-driven dynamic prioritization, a gap our research addresses. Pattnaik et al. [4] identify critical success factors for efficient court management in India, including stakeholder collaboration and technological adoption, which align with our system's design for judicial interaction and cloud-based scalability. Chan [5] examines case management in Chinese-speaking regions, noting the influence of traditional legal structures on efficiency, while Ali and Varol [6] propose mobile solutions to bridge communication gaps, supporting our notification system. Tahura [7] explores caseflow management to reduce backlogs, emphasizing structured scheduling, and Ambos [8] highlights prosecutorial discretion based on case gravity, informing our prioritization parameters. Recent advancements in artificial intelligence (AI) have transformed judicial systems globally. Thomson Reuters Institute [9] reports that AI tools enable court employees to focus on human interactions and reduce bias, citing Palm Beach County's AI-driven document processing system as an example of high efficiency. IBM [10] notes that judicial systems, such as those in Germany, are adopting AI to manage vast datasets and accelerate case resolution, aligning with our goal of reducing India's case backlog. The International Journal for Court Administration [11] discusses a study in the Netherlands using AI to support judges in traffic violations cases, demonstrating the potential of AI in decision support. These studies underscore the feasibility of AI in judicial contexts, supporting the approach of our system. 2.1 Ethical Considerations The integration of AI in judicial systems raises significant ethical concerns, particularly around bias and fairness. Judicature [12] emphasizes that AI must be used responsibly to 2 avoid perpetuating biases, such as overprioritizing certain case types. The American Association for the Advancement of Science (AAAS) [15] highlights the importance of fairness, accountability, and transparency (FAccT) in AI systems, noting that biases can arise from datasets, design, or deployment. Above the Law [13] reports a case where a trial court relied on AI-generated fake caselaw, underscoring the need for human oversight. Our system mitigates these risks by incorporating expert-validated weights for prioritization parameters and planning regular audits to ensure equitable outcomes, aligning with ethical guidelines from the Council of Europe [11]. Additionally, we address potential biases by involving legal experts in weight validation and ensuring transparency in the prioritization process. 2.2 Gaps Addressed Existing systems, such as those discussed by Rao et al. [2] and Amofah [3], focus on digitization and automation but lack AI-driven dynamic prioritization tailored to India's judicial context. Our framework addresses this gap by integrating a regression-based machine learning model for real-time weight adjustment, ensuring both efficiency and fairness. Unlike mobile solutions proposed by Ali and Varol [6] , our system offers a comprehensive cloud-based platform that integrates with existing judicial infrastructure like eCourts Services, enhancing scalability and accessibility. 3 Methodology 3.1 Data Retrieval and Preprocessing Case data is entered by advocates on record or court registrars into a MySQL database and preprocessed by categorizing cases into distinct datasets (e.g., Family, Criminal, Civil). This segmentation enables custom prioritization strategies for different types of case, ensuring flexibility and scalability. Preprocessing includes cleaning data, standardizing formats, and extracting relevant features such as filing dates, priority levels, and applicable legal sections (e.g., IPC, CrPC, Hindu Marriage Act). 3.2 Case Prioritization Algorithm The core of our system is the case prioritization algorithm, which assigns a weighted score to each case to determine its urgency and scheduling priority. The algorithm, detailed in Algorithm 1, processes the cases as follows: 3 Algorithm 1 Case Prioritization Algorithm 1: Input: Case data (case ID, type, filing date, severity, priority level, legal sections), court holidays, judge leaves, disposed cases database 2: Output: Prioritized case list with assigned hearing dates 3: Initialize MySQL database connection 4: Retrieve pending case data from MySQL database 5: for each case in pending cases do 6: if case is an appeal then 7: Fetch relevant data from disposed cases database 8: end if 9: Categorize case into type (Criminal, Family, Civil) 10: Extract features: case age (from filing date and previous hearings), severity, priority level (Urgent, Medium, Ordinary), legal sections (e.g., IPC 302, Hindu Marriage Act Section 13) 11: Compute weight using regression-based ML model ▷Model considers case age, severity, priority level, and legal sections; IPC/CrPC weights validated by advocates and judges 12: Assign priority based on weight 13: end for 14: Sort cases by weight in descending order 15: Initialize scheduling with daily limit (100 cases: 50 fresh, 50 old) 16: for each day in scheduling period do 17: if day is not a court holiday or judge leave then 18: Assign hearing dates to top-weighted cases (up to 50 fresh, 50 old) 19: end if 20: end for 21: Export prioritized case list to database 22: Send notifications (SMS/email) to stakeholders with hearing dates 23: Periodically review low-priority cases 24: if case pendency exceeds threshold then 25: Increase case priority 26: end if 27: Update ML model with case outcomes for dynamic weight adjustment 28: Return: Prioritized case list with hearing dates The algorithm retrieves all pending case data from a MySQL database, including details such as case ID, type, filing date, severity, priority level (e.g., Urgent, Medium, Ordinary) provided by advocates or registrars, and relevant legal sections (e.g., IPC 302, Hindu Marriage Act Section 13). For appealed cases, it fetches data from the disposed cases database. The data is preprocessed to categorize cases into categories such as Criminal, Family, or Civil, enabling customized prioritization strategies. For each case, a weight is calculated using a regression-based machine learning model, which considers factors such as the age of the case (based on filing and previous hearing dates), severity (e.g., high for murder, low for minor disputes), the input priority level, and applicable legal sections. The weights for the IPC and CrPC sections were determined through consultations with multiple advocates and judges to ensure legal accuracy and fairness. The calculated weight determines the hearing date, with higher weights leading to earlier scheduling. Hearing dates are allocated while considering a fixed daily case limit (e.g., 100 cases, with 50 fresh and 50 old cases) and a predefined list of court holidays and additional leaves entered into the system. Notifications are sent to stakeholders via SMS 4 or email to inform them of the assigned hearing dates. To prevent low-priority cases from being indefinitely delayed, the system periodically checks for pendency, and if a defined threshold is exceeded, it increases the case's priority by multiplying it with a fixed multiplier. The machine learning model dynamically adjusts the weights based on the historical case outcomes, achieving high precision in the simulations, as shown in Figure 2. Figure 2: Case Prioritization Flowchart 3.3 Notification System and Judicial Integration Post-prioritization, an automated notification system sends SMS/email alerts to stakeholders regarding hearing dates. Judges and registrars access a prioritized case list via a secure web interface, which also displays the number of cases scheduled per day, facilitating efficient scheduling and case management. The interface is designed for compatibility with existing systems like e-Courts Services [1]. 4 System Architecture The system architecture, illustrated in Figure 3, orchestrates a seamless flow of operations to prioritize and manage court cases within a secure cloud-based environment. The process is structured into distinct stages, ensuring scalability, efficiency, and integration with judicial workflows, including case entry by advocates or registrars, handling appeals, balanced case scheduling, and holiday considerations. The process begins with case data entry, where advocates on record or court registrars input case details, such as case ID, type, filing date, severity, priority level (e.g., Urgent, Medium, Ordinary), and relevant legal sections (e.g., IPC, CrPC, Hindu Marriage Act), into a MySQL database. This initial step ensures that all relevant case information is captured accurately for subsequent processing. The system initializes necessary configurations and establishes a secure connection to the MySQL database. This step ensures that the system is fully prepared to access and process case data reliably, setting the foundation for subsequent operations. Next, the system retrieves all pending case data from the MySQL database, including the details entered by advocates or registrars. For cases that have gone to appeal, the 5 system fetches relevant data from the disposed cases database, ensuring that historical case information is available for consistent prioritization in higher courts. The retrieved data is then preprocessed to categorize cases into distinct groups, such as Criminal, Family, or Civil. This categorization enables the system to apply tailored prioritization strategies, accommodating the unique characteristics of each case type, including appealed cases, and enhancing scalability for diverse judicial needs. For each case, the system employs a regression-based machine learning model to calculate a weight that determines its hearing date. The model evaluates factors such as the age of the case (calculated from filing and previous hearing dates), the severity of the offense (e.g., high for murder under IPC 302, low for minor civil disputes), the priority level provided as an input parameter by advocates or registrars, and applicable legal sections (e.g., IPC, CrPC, Hindu Marriage Act). Weights for IPC and CrPC sections were determined through consultations with multiple advocates and judges to ensure legal accuracy. The calculated weight guides the allocation of hearing dates, with higher weights leading to earlier scheduling. The system schedules a fixed number of cases per day, for example, 100 cases, with 50 fresh cases and 50 old (next hearing) cases, to balance judicial workload. Hearing dates are assigned while accounting for a predefined list of court holidays and any additional leaves entered into the system, ensuring accurate and conflict-free scheduling. The prioritized case list is exported to a secure database for record keeping, ensuring that all scheduling decisions, including those for fresh and old cases, are systematically stored. This step maintains data integrity and supports traceability for both pending and appealed cases. Notifications are then sent to stakeholders, including judges, lawyers, and litigants, via SMS or email, informing them of the assigned hearing dates. For appealed cases, notifications also include relevant higher court details, ensuring all parties are informed and prepared. To prevent low-priority cases from being indefinitely delayed, the system periodically reviews the time since the last hearing for each case. If a case, including those under appeal, has been pending beyond a predefined threshold, its priority is increased by multiplying it with a fixed multiplier to ensure timely resolution, promoting fairness across all case types. For cases involving appeals, the system streamlines the process by retrieving data from the disposed cases database, applying the same prioritization process (weight calculation based on input priority and other factors, and date allocation with the fixed case limit), and notifying higher courts. This ensures consistency and efficiency in handling appeals, maintaining seamless integration with the judicial hierarchy. The system continuously monitors its performance by measuring execution times for each operation, ensuring efficiency and identifying potential bottlenecks. Performance data is logged to support ongoing optimization, applicable to both pending and appealed cases. Judges access the prioritized case list, including appealed cases, through a secure web interface, which presents cases in order of priority and displays the number of cases scheduled per day, helping judges manage their daily workload effectively. Judges can make decisions, such as scheduling the next hearing or disposing of a case. When a judge specifies a time frame for the next hearing (e.g., 'after 15 days'), the system schedules the hearing as soon as possible after that period, subject to the fixed case limit and 6 holiday/leave constraints, and updates the database accordingly; if a case is disposed, it is marked as resolved, stored in the disposed cases database, and the outcome is used to update the machine learning model, enhancing its accuracy for future prioritizations, including for appeals. This structured flow, as shown in Figure 3, ensures that the system efficiently manages large case volumes, including appealed cases, while balancing fresh and old cases, respecting holidays and leaves, and integrating seamlessly with existing judicial platforms like eCourts Services, providing a robust solution for India's judicial backlog. Figure 3: System Architecture for Court Case Management and Prioritization 5 Results 5.1 Illustrative Case Study To demonstrate the system's functionality, we present a synthetic case study with five simulated cases from our 10,000-case dataset (Table 1). These cases vary in type, filing date, severity, priority level, and applicable legal sections, showcasing the algorithm's ability to prioritize effectively. Weights for IPC and CrPC sections were validated through consultations with multiple advocates and judges to ensure legal accuracy. 7 Table 1: Illustrative Case Study Case ID Type Filing Date Severity Legal Sections Priority Weight No of Hearings Hearing Date 001 Criminal 01/01/2024 High IPC 302, 34 Urgent 0.95 4 15/07/2025 002 Civil 01/02/2025 Low Indian Contract Act, 1872, Section 73 Ordinary 0.45 0 30/08/2025 003 Family 01/06/2024 Medium Hindu Marriage Act, 1955, Section 13 Medium 0.70 3 01/08/2025 004 Criminal 01/02/2024 High IPC 420, 406 Urgent 0.90 5 20/07/2025 005 Criminal 01/04/2025 Medium IPC 323, CrPC 200 Ordinary 0.50 2 15/09/2025 Narrative • Case 001: A murder case filed on 01/01/2024, with high severity and an input priority of Urgent, governed by IPC Section 302 (murder) and Section 34 (acts done by several persons in furtherance of common intention). With four hearings held and its high severity validated by advocate and judge consultations, it has a weight of 0.95, scheduling it for 15/07/2025, ensuring swift justice for a serious crime. • Case 002: A civil case (e.g., breach of contract) filed on 01/02/2025, with low severity and an input priority of Ordinary, governed by Section 73 of the Indian Contract Act, 1872 (compensation for loss). With no hearings yet and its recent filing and low severity, it yields a weight of 0.45, scheduling it for 30/08/2025, reflecting its lower urgency. • Case 003: A family law case (e.g., divorce) filed on 01/06/2024, with medium severity and an input priority of Medium, governed by Section 13 of the Hindu Marriage Act, 1955 (grounds for divorce). With three hearings held and moderate pendency and severity, it results in a weight of 0.70, scheduling it for 01/08/2025, balancing urgency and recency. • Case 004: A cheating case filed on 01/02/2024, with high severity and an input priority of Urgent, governed by IPC Section 420 (cheating) and Section 406 (criminal breach of trust). With five hearings held and its high severity validated, it yields a weight of 0.90, scheduling it for 20/07/2025, addressing a significant criminal offense promptly. • Case 005: A case involving voluntarily causing hurt filed on 01/04/2025, with medium severity and an input priority of Ordinary, governed by IPC Section 323 (voluntarily causing hurt) and CrPC Section 200 (examination of complainant). With two hearings held and its recent filing and medium severity, it results in a weight of 0.50, scheduling it for 15/09/2025, as it is less severe than murder or cheating. This case study illustrates how the system prioritizes cases based on severity, pendency, input priority level, and applicable legal sections, validated by legal experts, ensuring critical cases are addressed promptly while managing diverse case types. 8 5.2 System Performance Metrics Simulations with 10,000 dummy cases demonstrated that the system can process cases efficiently, with reliable notification delivery and positive feedback from judges and registrars regarding usability. 5.3 Machine Learning Model Performance The regression-based machine learning model used for dynamic weight adjustment achieved an F1 score of 99.7%, precision of 99.8%, and recall of 99.4% in simulated tests, indicating high accuracy and reliability in prioritizing cases. 6 Discussion and Conclusion The proposed framework significantly enhances judicial efficiency by automating case prioritization and scheduling. By integrating algorithmic approaches, a robust notification system, and a dynamic feedback loop, the system addresses challenges like case backlogs and delayed hearings. The illustrative case study demonstrates its practical applicability, prioritizing urgent cases like murder (IPC 302, 34) while ensuring fairness across civil and family cases. The system's compatibility with existing platforms like eCourts Services [1] ensures seamless integration into India's judicial infrastructure. 6.1 Ethical Implications AI-driven prioritization risks biases, such as over-prioritizing criminal cases, which could disadvantage civil or family cases. Our system mitigates this through expert-validated weights, informed by consultations with advocates and judges, and regular audits, aligning with ethical guidelines from the Council of Europe [11]. Above the Law [13] highlights risks of AI-generated inaccuracies, underscoring the need for human oversight, which our system incorporates through judicial interaction. 6.2 Real-World Feasibility The system is designed for compatibility with existing platforms like eCourts Services, using standardized MySQL databases and secure cloud storage. Pilot testing in a regional court will validate its effectiveness, addressing potential challenges like data migration and user training. 6.3 Future Work Future iterations will prioritize real-world deployment, integrating advanced machine learning to enhance case management. Deep learning models, such as transformers, will improve feature extraction from legal texts, identifying attributes like case complexity and urgency from documents in Table 1. Reinforcement learning will enable adaptive prioritization by optimizing schedules based on court data, such as judge availability and case backlog, improving efficiency. The system will also expand to handle multilingual case data, as demonstrated by Clio [14] for California courts, to enhance accessibility in diverse regions. NLP techniques, including cross-lingual embeddings, will process languages like Hindi and English, addressing challenges in legal terminology translation. Additionally, predictive analytics 9 will forecast case pendency using data like hearing counts and severity, aiding resource allocation. Piloting in high-volume courts will test scalability, with collaboration from legal experts ensuring judicial alignment and data security, ultimately streamlining processes and improving access to justice. References [1] eCourts Services. https://ecourts.gov.in/ecourts_home/ [2] Rao, A. P., et al. (2018). Court Case Management System. [3] Amofah, L. R. (2017). Electronic court case management system (for law court complex). [4] Pattnaik, P. N., et al. (2018). Mapping Critical Success Factors in Efficient Court Management. International Journal of Law and Management, 60(2). [5] Chan, P. C. H. (2018). Framing the structure of the court system in the perspective of case management. Peking University Law Journal, 6(1), 55-79. [6] Ali, C. T., & Varol, A. (2020). Design and Implementation of a Simple Online Court Case Management System Based on the Android Platform. IEEE. [7] Tahura, U. S. (2013). Case Management: A Magic LAMP in Reducing Case Backlogs. Bangladesh Law Journal, 13. [8] Ambos, K. (2018). Office of the Prosecutor: Policy Paper on Case Selection and Prioritisation. International Legal Materials, 57(6), 1131-1145. [9] Thomson Reuters Institute. (2024). Humanizing Justice: The transformational impact of AI in courts. https://www.thomsonreuters.com/en-us/posts/ ai-in-courts/humanizing-justice/ [10] IBM. (2025). Judicial systems are turning to AI to help manage vast quantities of data and expedite case resolution. https://www.ibm.com/case-studies/blog/ judicial-systems-are-turning-to-ai-to-help-manage-its-vast-quantities-of-dat [11] International Journal for Court Administration. (2020). Courts and Artificial Intelligence. https://iacajournal.org/articles/10.36745/ijca.343 [12] Judicature. (2024). AI in the Courts: How Worried Should We Be? https://judicature.duke.edu/articles/ ai-in-the-courts-how-worried-should-we-be/ [13] Above the Law. (2025). Trial Court Decides Case Based On AI-Hallucinated Caselaw. https://abovethelaw.com/2025/07/ trial-court-decides-case-based-on-ai-hallucinated-caselaw/ [14] Clio. (2025). AI in the Courtroom: Opportunities and Challenges. https://www. clio.com/resources/ai-for-lawyers/ai-in-courtrooms/ [15] American Association for the Advancement of Science (AAAS). Artificial Intelligence and the Courts: Materials for Judges. https://www.aaas.org/ai2/ projects/law/judicialpapers 10"}}},{"rowIdx":88,"cells":{"id":{"kind":"string","value":"2510.14894"},"raw":{"kind":"string","value":"Secure Sparse Matrix Multiplications and their Applications\nto Privacy-Preserving Machine Learning\nMarc Damie\nm.f.d.damie@utwente.nl\nUniversity of Twente, The Netherlands\nInria, France\nFlorian Hahn\nUniversity of Twente, The Netherlands\nAndreas Peter\nUniversity of Oldenburg, Germany\nJan Ramon\nInria, France\nAbstract\nTo preserve privacy, multi-party computation (MPC) enables executing Machine Learning\n(ML) algorithms on secret-shared or encrypted data. However, existing MPC frameworks\nare not optimized for sparse data. This makes them unsuitable for ML applications involving\nsparse data, e.g., recommender systems or genomics. Even in plaintext, such applications\ninvolve high-dimensional sparse data, that cannot be processed without sparsity-related\noptimizations due to prohibitively large memory requirements.\nSince matrix multiplication is central in ML algorithms, we propose MPC algorithms to\nmultiply secret sparse matrices. On the one hand, our algorithms avoid the memory issues\nof the “dense” data representation of classic secure matrix multiplication algorithms. On the\nother hand, our algorithms can significantly reduce communication costs (some experiments\nshow a factor 1000) for realistic problem sizes.\nWe validate our algorithms in two ML\napplications in which existing protocols are impractical.\nAn important question when developing MPC algorithms is what assumptions can be made.\nIn our case, if the number of non-zeros in a row is a sensitive piece of information then a\nshort runtime may reveal that the number of non-zeros is small. Existing approaches make\nrelatively simple assumptions, e.g., that there is a universal upper bound to the number\nof non-zeros in a row. This often doesn’t align with statistical reality, in a lot of sparse\ndatasets the amount of data per instance satisfies a power law. We propose an approach\nwhich allows adopting a safe upper bound on the distribution of non-zeros in rows/columns\nof sparse matrices.\n1\nIntroduction\nMPC protocols are cryptographic primitives allowing to compute functions on secret inputs. Over the years,\nthey have been used in many application domains such as privacy-preserving ML (PPML) Mohassel &\nZhang (2017) or telemetry Asharov et al. (2022) to address confidentiality and privacy concerns. However,\nexisting protocols are impractical for many real-world applications, especially when the applications involve\nhigh-dimensional sparse data, i.e., data with a large majority of zero values.\nStoring large sparse datasets in a dense format (one value per cell) demands excessive memory. On real-world\nsparse data, this “dense storage” often becomes prohibitive; making computation infeasible due to memory\n1\narXiv:2510.14894v1 [cs.CR] 16 Oct 2025\n\nconstraints. Moreover, dense representations lead to inefficient linear algebra operations, as most of the\ncomputational effort is wasted processing zeros.\nOver the last 50 years, algorithms to process plaintext sparse data efficiently have been proposed Buluç et al.\n(2011); Duff (1977). Since 2002, dedicated software libraries Duff et al. (2002) popularized them. Sparse\ndata is present in many ML applications but also in other fields, including graph algorithms. Our paper\nfocuses on PPML applications, but our results apply to any application involving private sparse data.\nSeveral MPC frameworks enable training ML models on secret-shared data Koti et al. (2021); Lu et al.\n(2023); Mohassel & Rindal (2018); Mohassel & Zhang (2017); Patra & Suresh (2020); Wagh et al. (2019;\n2021). These algorithms make use of existing MPC frameworks such as MP-SPDZ Keller (2020). However,\nnone of these frameworks provide algorithms optimized for sparse data.\nThe scalability limitations (especially memory issues) of dense plaintext algorithms transfer naturally to dense\nMPC algorithms. Thus, MPC frameworks need dedicated sparse algorithms for ML applications like\nrecommender systems. These frameworks are not fundamentally incompatible with sparse data; there is\nsimply no suitable secure sparse algorithm in the literature.\nRelated works\nThere exist protocols to sum sparse vectors securely, including sorting-based Asharov\net al. (2022); Bell et al. (2022), private histograms Bell et al. (2022), or private-union-based Beguier et al.\n(2021).\nA few papers Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) studied secure sparse multiplica-\ntions in a two-party setting. However, these protocols require the data owners to actively participate because\none of the computation parties must know the plaintext sparse data. Hence, they do not support more than\ntwo data owners simultaneously, which is too limiting, as modern ML applications involve thousands of data\nowners.\nTypical ML applications require a more generic MPC setup: outsourced training.\nIn this setting (see\nFigure 1), the data owners share their secret data with a group of computation servers and disconnect.\nThis distinction between input parties and computation parties enables supporting an arbitrary number of\ndata owners, contrary to existing works Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019).\nAppendix B details why the adaptation of these works either results in an inefficient construction or is not\nstraightforward. The secure multiplication of sparse matrices in the outsourced setting is then a significant\nopen problem. Our work will focus on this outsourced setting; providing sparse multiplications compatible\nwith any MPC framework.\nSome information-theory works Egger et al. (2023); Xhemrishi et al. (2022) studied the multiplication of a\nsecret-shared matrix with a public matrix. Due to the public input, their setting differs from classic MPC\nwhere all inputs are secret. Finally, Nayak et al. Nayak et al. (2015) presented secure graph computations (a\nproblem closely related to sparse operations due to the sparse adjacency matrices), but they do not present\nmultiplication protocols.\nOur contributions\nWe present two secure algorithms to compute matrix multiplications on secret-\nshared sparse data. Our algorithms avoid the memory issues of dense matrix multiplications on sparse\ndata (e.g., in an experiment from Section 5: 19TB using dense multiplications vs.\n60GB using ours).\nMoreover, our algorithms reduce the communication costs up to a factor 1000, and we implement two\nML applications that are impractical with existing secure algorithms. Third, as we highlight that any efficient\nMPC protocol (i.e., not only ours) on sparse data requires public knowledge about the sparsity, we leverage\nthe properties of real-world sparse data to minimize this mandatory public knowledge and obtain it\nin a privacy-preserving manner.\nNotations\nLet nnz(x) (resp. nnz(X)) refer to the number of non-zero values in a vector x (resp. matrix\nX). Let [[a]] refer to a share of a secret-shared value a. In our algorithms, any operation involving secret-\nshared values corresponds to a secure variant; e.g., [[a]] + [[b]] is the secure addition of a and b and [[sort(l)]]\nis the oblivious sort of l. Secure algorithms are executed jointly by all shareholders.\n2\n\nFigure 1: Outsourced MPC setting\n2\nPreliminaries\n2.1\nThreat model\nThreat modeling describes the expected behavior of the agents involved in an MPC protocol: an honest agent\nfollows the protocol, an honest-but-curious agent follows the protocol and attempts to infer private informa-\ntion passively, and a malicious agent does not necessarily follow the protocol and can actively disrupt the\nprotocol to obtain private information. MPC works distinguish security against a majority of honest agents\nfrom security against a majority of dishonest agents. “Dishonest agent” is an umbrella term encompassing\nmalicious and honest-but-curious agents (depending on the security setup considered).\nWe analyze our algorithms under honest and dishonest majority. We focus on honest-but-curious security,\nbut our algorithms rely on oblivious shuffling and sorting with known maliciously secure variants Hamada\net al. (2014); Laur et al. (2011).\nWe leave for future work the extension to malicious security and the\nsubsequent security proofs.\nIn line with related works, our threat model excludes poisoning attacks where malicious data owners forge\ninput data so the protocol reveal private information.\nWe assume the data owners provide well-formed\nsecret-shared inputs.\n2.2\nOutsourced setting\nOur contribution focuses on an “outsourced” computation setup (popular in recent PPML works Hegde et al.\n(2021); Mohassel & Zhang (2017); Wagh et al. (2019; 2021)): each data owner shares their secret data with\na group of computation servers. These servers build a secret-shared matrix [[X]] containing all data owners’\ninputs (e.g., one row per data owner1) and perform secure computations on this matrix. For example, the\nservers can execute a protocol to train an ML model [[θ]] on the dataset X, as in Figure 1. In this setting,\nthe data owners share their secret data and disconnect. Outsourced protocols supports an arbitrary number\nof data owners because it separates input parties (i.e., data owners) from computation parties.\nContrary to many PPML works that support a two-party Mohassel & Zhang (2017); Hegde et al. (2021)\nor three-party Wagh et al. (2019; 2021) outsourced setups, our work support an unbounded number of\ncomputation parties. Nevertheless, secret-sharing protocols induce a quadratic communication cost in the\nnumber of parties. This quadratic cost makes outsourcing necessary to support thousands of data owners\nlike in ML applications.\nAll existing secure sparse multiplications Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019)\nsupport a non-outsourced setting. They presented two-party protocols assuming the computation parties are\n1This is sometimes referred to as “horizontal” data partitioning in PPML literature.\n3\n\nalso data owners (⇒two data owners maximum) because they require one of the computation parties to\nknow the plaintext sparse data. Such a setting is too limiting for ML applications as they typically involve\nmany data owners.\nAdapting existing protocols to avoid designing new ones? The adaptation is not easy because a\nplaintext knowledge is often essential to these protocols. Appendix B details why the adaptation of Chen\net al. (2021); Cui et al. (2021); Schoppmann et al. (2019) to the outsourced setting either results in an\ninefficient construction or is not straightforward.\n2.3\nMPC primitives\nSecret sharing Shamir (1979) enables sharing a secret value x between multiple parties. Each shareholder\nj holds a share [[x]]j, revealing nothing about x.\nThe shareholders can reconstruct the secret value by\ngathering their shares. Several secret sharing schemes exist, e.g., additive shares are finite field elements\ndrawn randomly such that: P\nj[[x]]j = x. Many protocols Evans et al. (2018) enable processing secret-shared\nvalues; from arithmetic operations to complex operations (e.g., sorting).\nLike related works in private data analysis Araki et al. (2021); Asharov et al. (2022), our algorithms rely\nheavily on oblivious shuffling and sorting.\n“Oblivious sorting” designates MPC protocols for sort-\ning secret-shared values without revealing information about the values. Sorting networks Hamada et al.\n(2014) are straightforward oblivious sorting solutions, as their execution is data-independent. Their cost\nis O(m log2 m) for a list of size m. The oblivious radix sort Hamada et al. (2014) is the state-of-the-art\nhonest-majority protocol, with its optimal complexity of O(m log m) (and low constant factors). Recent\npapers optimized this protocol in a three-party setting Araki et al. (2021); Asharov et al. (2022).\n“Oblivious shuffling” protocols randomly permute lists of secret values without revealing the permutation.\nLaur et al. Laur et al. (2011) proposed an honest-majority protocol with linear costs, used in many recent\nprotocols Araki et al. (2021); Asharov et al. (2022); Hamada et al. (2014). A recent work Lu & Kate (2023)\nalso described dishonest-majority protocols, but with supra-linear costs.\n2.4\nMatrix multiplication\nLet Z be the result of the multiplication between the n × m matrix X and the m × p matrix Y . Z is a n × p\nmatrix defined as follows:\nZij =\nm\nX\nk=1\nxik · ykj, ∀i ∈{1 . . . n}, j ∈{1 . . . p}\n(1)\nThis formula holds for all matrices.\nHowever, the implementation may differ depending on the type of\nmatrices (e.g., vectors, diagonal matrices, or triangular matrices). Our work focuses on optimized algorithms\nfor sparse matrices. Sparse algorithms avoid “useless” multiplications and only multiply the non-zero values.\nIndeed, if the matrix multiplication is implemented naively (i.e., triple nested loop), the algorithm would\ncompute many scalar multiplications with a null result. Hence, sparse algorithms leverage the matrix sparsity\nto have a cost inverse proportional to the sparsity.\nSoftware libraries usually implement dense matrix multiplication via a triple nested loop, resulting in an\nO(nmp) cost. While this cost has long been considered the lower bound for dense matrix multiplication,\nseveral works described algorithms with a sub-cubic cost Coppersmith & Winograd (1987). Dumas et al.\nDumas et al. (2019) described an MPC protocol to execute a matrix multiplication with sub-cubic cost. These\nalgorithms have constant factors limiting their practical interest (even in plaintext), so software libraries favor\nalgorithms with a cubic cost.\n2.5\nDense matrix multiplication on secret-shared data\nThe cubic-cost matrix multiplication (i.e., the naive implementation of Equation (1)) can easily be imported\nto MPC. However, recent works Chen et al. (2020); Mohassel & Rindal (2018); Mohassel & Zhang (2017);\nMono & Güneysu (2023); Patra & Suresh (2020); Wagh et al. (2021) described more efficient protocols under\n4\n\nhonest or dishonest majority assumptions. These works reduce the communication costs, but the computation\ncost remains asymptotically cubic.\nMoreover, the storage costs remain asymptotically equivalent to the\nplaintext algorithm, so they cannot support larger matrices than the plaintext dense algorithm.\nDishonest majority\nBeaver’s multiplication triplets are a popular building block to speed up secure\nmultiplications via preprocessing steps Evans et al. (2018). Mohassel and Zhang Mohassel & Zhang (2017)\ngeneralized this concept to matrices: matrix multiplication triplets. These triplets make the communication\ncosts proportional to the input and output size. For example, the multiplication of an n × m matrix with\nan m × p matrix requires O(nm + mp + np) communications (instead of O(nmp) with the naive algorithm).\nThis approach reduces the asymptotic cost for matrix-matrix multiplications, but the communication costs of\nthe vector-vector and matrix-vector multiplications remains asymptotically equal to the naive costs. Matrix\nmultiplication triplets with malicious security were later described Chen et al. (2020); Mono & Güneysu\n(2023).\nHonest majority\nUsing the honest majority assumption, several recent works Koti et al. (2021); Lu et al.\n(2023); Mohassel & Rindal (2018); Patra & Suresh (2020); Wagh et al. (2021) designed protocols with a\ncommunication cost proportional to the output size. Hence, a square matrix multiplication only requires\nO(n2) communications, and a vector multiplication O(1) communications. Before these recent protocols, De\nHoogh Hoogh (2012) had already described a matrix multiplication with equivalent costs based on Shamir’s\nsecret sharing (SSS). The recent protocols Koti et al. (2021); Lu et al. (2023); Mohassel & Rindal (2018);\nWagh et al. (2021) are more complex to understand than the SSS-based algorithm of Hoogh (2012) because\nthey rely on share conversion protocols. Moreover, these protocols only support 3 parties, while the SSS-\nbased solution supports an arbitrary number of parties.\nNevertheless, the SSS-based solution only has\nhonest-but-curious security, while Koti et al. (2021); Lu et al. (2023); Patra & Suresh (2020); Wagh et al.\n(2021) support malicious security.\nTo understand the gain provided by the honest-majority assumption, let us detail the optimized vector mul-\ntiplication under SSS Hoogh (2012). In SSS, each secret-shared value is represented using a polynomial of\ndegree t. To multiply two secret-shared scalars in SSS, the parties locally multiply their local shares (obtain-\ning a polynomial of degree 2t) and then execute a degree reduction protocol (via a “resharing” protocol).\nFor dense vector multiplication, we can delay the degree reduction and add all local multiplication results\ntogether. The addition is possible because all local multiplication results are shares of equal polynomial\ndegree (i.e., 2t). Then, a single degree reduction is done on the sum result.\n3\nSparse data in PPML\n3.1\nSparse data representation\nThe term “sparse data” refers to data containing a large proportion of zero values. If stored using the default\nmatrix format, the zeros occupy a significant memory space. When the data contains a sufficient number of\nzeros, sparse data representations can significantly reduce the storage space.\nSeveral sparse data representations exist Buluç et al. (2011), each with specific algorithms to perform oper-\nations on the data. Depending on the data representations, some operations are more efficient than others.\nIn other words, there is no perfect sparse data representation.\nWe use the tuple representation Buluç et al. (2011): each sparse vector is a list of non-zero tuples:\nt = (i, vi) with i the coordinate in the vector and vi the non-zero value.\nLet coord(t) = i denote the\ncoordinate of the non-zero and val(t) = vi its value.\nOn the one hand, we refer to algorithms supporting sparse data as “sparse algorithm”, a common abuse of\nterminology. On the other hand, we use the term “dense algorithms” to refer to classic algorithms.\n5\n\n3.2\nLarger datasets are sparser\nOptimizing dense algorithms, a dead end\nTo process sparse data, one may consider to use dense\nmatrix multiplication, with more memory. Unfortunately, it would require tremendous amount of memory.\nSecure dense algorithms have the same asymptotic memory footprint and hence run into the same memory\nissue. There is a limit to the scalability of dense algorithms. Even if extreme amounts of memory would be\navailable, dense secure multiplications would still fail to scale because their communication cost is typically\nat least linear in the size of the dense input.\nA workaround to avoid this memory issue could be to partition the matrix in sub-matrices and perform dense\nmultiplications on these sub-matrices. This approach is effective in classic linear algebra to handle large dense\ndatasets and could (to some extent) reduce memory consumption. However, it would still require storing\nthe product as a dense matrix; the memory benefit will not be significant. Moreover, it adds significant\ncosts for repeated sparse-to-dense conversions. Finally, this partitioning approach would not reduce the\ncommunication costs of MPC protocols that would remain correlated to the full matrix size.\nTo sum up, sparse algorithms are popular in plaintext because there is no other efficient solutions to process\nhigh-dimensional sparse datasets. In plaintext, dense matrix multiplications usually have a sparse equivalent\n(e.g., NumPy for dense and SciPy for sparse in Python).\nTo cover all possible ML applications, MPC\nframeworks should similarly have sparse algorithms. The first motivation of sparse algorithms is to avoid the\nmemory issues caused by dense algorithms. Furthermore, they often provide significant performance gains.\nSparsity-size correlation\nReal-world sparse datasets show an interesting phenomenon: the sparsity is\ncorrelated to the matrix size. Indeed, larger sparse datasets have in average a larger sparsity. Thus, the\nlarger datasets are, the more beneficial the sparse algorithms become.\nTo understand this phenomenon, let us take the example of recommender systems on a marketplace (e.g.,\nAmazon): the matrix quantifies the interaction of each consumer with each product. The dataset is sparse\nbecause each consumer only interacts with a small subset of all possible products. If the marketplace increases\nits number of products by a factor 1000, the consumer will not consume 1000 times more products. Even if\nthe consumption increases a bit, existing datasets show that it does not follow the matrix growth; i.e., the\nsparsity increases. The real-world datasets that we arbitrarily selected from Li et al. (2017) satisfy the same\ntrend: MovieLens 1M has 95% sparsity for 1.7K columns, Yahoo Movie has 99.7% sparsity for 12K columns,\nBookcrossing has 99.99% sparsity for 340K columns, and flicker has 99.999% sparsity for 500K columns.\nTheoretical point of view\nThe correlation between sparsity and matrix size is observable analytically.\nSparse datasets can often be represented as graphs: e.g., a recommendation dataset is a bipartite graph\nlinking n customers to m products. Also, note that graphs can be represented as sparse dataset via their\nadjacency matrix.\nThe literature Aiello et al. (2001) showed that, in many application domains, graph data follows a “power\nlaw” distribution. In particular, if we take a random vertex v ∈V (G) of a graph G, then the probability\ndistribution of the degree d(v) of v is given by:\nf(x) = P(d(v) = x) = x−γ\nZ(γ)\n(2)\nfor some γ > 1 where Z(γ) = P∞\ni=1 i−γ is a normalization constant. Typically, we have 2 ≤γ ≤3 Adamic\net al. (2001), but application domains exist where other values of γ are adequate. Practical applications may\ndeviate a bit from (2), especially for small degrees, as this equation is mainly intended to describe asymptotic\nbehavior.\nIn a dataset with n rows (i.e, vertices) where every vertex has degree at most m, one can use a corrected\nnormalization constant Z(γ, m) = Pm\ni=1 i−γ. Remember that, in our case, the degree of a vertex corresponds\n6\n\nto the number of non-zeros in a row. The expected degree of a vertex is:\nEv[d(v)] =\nm\nX\ni=1\ni1−γ\nZ(γ, m)\nWe can bound Z(γ, m) ≥1+\nR ∞\n1\nx−γdx = 1+1/1(1−γ) and Pm\ni=1 i1−γ <\nR ∞\n1\nx−γdx = (1−m2−γ)/(γ−2) <\n1/(2 −γ). Therefore, we can bound the average degree by a constant K not depending on n or m. We can\nrewrite the number of edges in graph G as |E| = nK = O(n). Hence, the sparsity of the adjacency matrix\nof the graph G decreases with n:\n|E|\nnm = O(m−1)\n(3)\nThis equation is the theoretical illustration of the phenomenon: larger (sparse) datasets are sparser.\n3.3\nPublic knowledge in secure sparse multiplications\nOther secure sparse algorithms Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) assume some\npublic knowledge. For example, Schoppmann et al. Schoppmann et al. (2019) assumes “that the sparsity\nmetric of the input is public.” The definition of this “sparsity metric” varies in each of their algorithm: the\ntotal number of non-zeros, or the number of non-zero rows or columns. In our paper, this public “sparsity\nmetric” is the number of non-zeros per row.\nPublic knowledge could be interpreted as a limiting assumption, but public knowledge is mandatory to\ndesign efficient and secure sparse algorithms. In sparse algorithms, the performance gains (in memory,\ncommunication, and computation) are correlated to the sparsity. Thus, it is necessary to know the sparsity\nto expect any performance gain.\nAnyway, sparsity knowledge is often necessary to ML practitioners. ML applications can be seen\nas a pipeline: exploratory data analysis (to choose an appropriate ML model), ML training, and model\ndeployment. Data analysis on sparse dataset necessarily includes a sparsity estimation; often essential to\nchoose an appropriate model. Hence, the exploratory data analysis would provide this public knowledge,\nbefore the sparse matrix multiplications.\nFinally, Section 6 proposes some protocols to minimize the public knowledge and Section 7 proposes\nprotocols to obtain the minimized knowledge in a privacy-preserving manner.\n4\nSecure sparse multiplications\nThis section presents two secure sparse matrix multiplications: matrix-vector and matrix-matrix. Existing\nsecure sparse multiplications are incompatible with the outsourced setting, so we can only compare to the\ndense baselines. We assume the bit length and number of computation parties are O(1), so our asymptotic\ncomplexities are only in function of the matrix size.\nOur algorithms rely on secure additions, multiplications, comparisons, oblivious shuffling, and oblivious\nsorting Hamada et al. (2014). On the one hand, sorting-network-based oblivious sorting only requires secure\nadditions, multiplications, and comparisons. On the other hand, we can implement oblivious shuffling using\noblivious sorting: generate a random secret-shared value for each element in the input list and obliviously\nsort the list based on these random values. Our algorithms are then compatible with all secret-sharing\nschemes supporting basic arithmetic and comparison operations. However, for optimal performances, one\nshould implement our algorithms using honest-majority sorting/shuffling Hamada et al. (2014); Laur et al.\n(2011).\nAppendix A sketches a security proof for our algorithms.\n7\n\nAlgorithm 1 Secure sparse vector multiplication\nInput: [[x]] and [[y]] are two secret-shared sparse vectors\nPublic knowledge: Number of non-zero elements per vector\n1: function SparseVectorMult([[x]], [[y]])\n2:\n[[z]] ←Concatenate([[x]], [[y]])\n3:\n[[z]] ←[[sort(z)]]\n▷Asc. sort on the non-zero coordinate.\n4:\n[[s]] ←0\n5:\nfor i ←1 to nnz(x) + nnz(y) −1 do\n6:\n[[obliv. if]] [[coord(zi)]] = [[coord(zi+1)]]\n7:\n[[s]] ←[[s]] + [[val(zi)]] × [[val(zi+1)]]\n8:\nreturn [[s]]\n▷a scalar\n4.1\nToy protocol: vector-vector\nWe propose to start with a simple vector multiplication. Since dense vector multiplications have an O(1)\ncommunication cost (under honest majority), we know in advance that it is unbeatable. Modern ML appli-\ncations essentially involve matrix-vector and matrix-matrix multiplications, so vector multiplication is not a\nrelevant focus. However, this toy protocol provides simple intuitions to understand how to build sorting-based\nsparse multiplications.\nWe want to multiply two n-dimensional sparse vectors x and y, represented as tuple lists. Alg. 1 presents\nour solution. First, we concatenate the two tuple lists. Note that this concatenation is trivial (and) secure\nbecause the sparse vectors x and y are lists of shares. Thus, the concatenation requires no communication;\nthe shareholders simply concatenate locally their list of shares. Then, we sort (obliviously) the resulting list\nby coordinate. If two consecutive tuples have equal coordinates, we multiply their values and sum all the\nmultiplication results to build the vector multiplication value.\nWe use the notation [[obliv. if]] to describe conditional structures on secret-shared values. The presence of\nconditional branches may seem contrary to the obliviousness goal. Indeed, revealing the executed branch\nwould leak information about the secret-shared value. However, conditional structures on secret-shared values\nare implemented so that the executed branch remains secret. All the [[obliv. if]] instructions are implemented\nvia a combination of arithmetic and comparison operations. For example, we implement the lines 6 to 7 of\nAlg. 1 using the following expression: [[s]] ←[[s]]+([[coord(zi)]] = [[coord(zi+1)]])×[[val(zi)]]×[[val(zi+1)]]; with\nthe secure comparison [[coord(zi)]] = [[coord(zi+1)]] outputting [[1]] or [[0]]. Such instructions are less readable,\nso we keep conditional structures in our algorithms with a dedicated notation. Such conditional statements\nare recurrent in MPC literature: e.g., Asharov et al. Asharov et al. (2022) used conditional statements in\ntheir heavy hitters protocol.\nAlg. 1 costs O((nnz(x)+nnz(y)) log(nnz(x)+nnz(y))) communications and computations in O(log(nnz(x)+\nnnz(y))) rounds. The sorting step induces the superlinear complexity. The multiplication loop is paralleliz-\nable, so it only costs O(nnz(x) + nnz(y)) in O(1) rounds. In comparison, the dense multiplication has O(1)\ncommunication costs under honest majority, and an O(n log n) cost under dishonest majority. Moreover, our\nsparse algorithm significantly reduces the memory footprint under both threat models.\n4.2\nMatrix-vector\nWe want to multiply an n × m matrix X to an m-dimensional vector y. Our work assumes the public\nknowledge motivated in Section 3.3: per-row sparsity. Thus, we can adapt the sparse matrix representation\nand group the non-zero elements by rows: a sparse matrix would be a list of sparse vectors, one per row.\nAn intuitive solution is to compute a vector-vector multiplication on each matrix row. This approach is\nadequate for dense multiplications, but inefficient for sparse matrices. Indeed, this naive extension induces a\nlinear dependency on the number of rows: O((nnz(X)+n·nnz(y)) log(nnz(X)+n·nnz(y))) communications\nand computations. The term n · nnz(y) comes from the “replication” of y for each row of X. The vector y\n8\n\nAlgorithm 2 Secure sparse matrix-vector multiplication\nInput: [[X]] a secret-shared sparse matrix, [[y]] a secret-shared sparse vector\nPublic knowledge: Number of non-zeros per matrix/vector\nNotation: the function RowCoord() (resp. ColCoord()) returns the first/row (resp. second/column) coor-\ndinate of a non-zero tuple.\n1: function SparseMatVectMult([[X]], [[y]])\n2:\n[[z]] ←X\n3:\nfor ([[j]], [[v]]) in [[y]] do\n4:\nAppend tuple (coord : ([[⊥]], [[j]]), val : [[v]]) to [[z]]\n5:\n[[z]] ←[[sort(z)]]\n▷Asc. sort on the 2nd and then 1st coordinates (with ⊥< 1)\n6:\nprevY ←(coord : ([[⊥]], [[⊥]]), val : [[⊥]])\n7:\nfor k ←1 to (nnz(X) + nnz(y)) do\n8:\n[[obliv. if]] [[ColCoord(prevY )]] = [[ColCoord(zk)]]\n9:\n[[val(zk)]] ←[[val(zk)]] × [[val(prevY )]]\n10:\nelse\n11:\n[[val(zk)]] ←[[0]]\n12:\n[[obliv. if]] [[RowCoord(zk)]] = [[−1]]\n13:\n[[prevY ]] ←[[zk]]\n14:\n[[val(zk)]] ←[[⊥]]\n15:\nDrop the second coordinate from all tuples in [[z]]\n16:\n[[z]] ←[[sort(z)]]\n▷Asc. sort on the remaining coord.\n17:\nAggEqualCoord([[z]])\n18:\nPlaceholderRemoval([[z]])\n19:\nreturn [[z]]\n▷sparse vector stored as a list of non-zeros\nAlgorithm 3 Aggregation of tuples with equal coordinates\nInput: [[Z]], a sorted list of secret shared tuples\n1: procedure AggEqualCoord([[Z]])\n2:\nfor k ←1 to length([[Z]]) −1 do\n3:\n[[obliv. if]] [[coord(Zk)]] = [[coord(Zk+1)]]\n4:\n[[coord(Zk)]] ←[[⊥]]\n5:\n[[val(Zk+1)]] ←[[val(Zk+1)]] + [[val(Zk)]]\nis sorted independently for each row, which induces this inefficiency. To sum up, we need a dedicated sparse\nmatrix-vector multiplication because the naive extension is inefficient.\nAlg. 2 presents a more optimized solution avoiding the linear dependency on n. The intuition behind this\noptimization is to build a tuple list sorted as follows [yj, xi1,j, xi2,j, . . . yj′, xi3,j′ . . .]. In this list, we group\nthe elements related to the same column, starting with the element from vector y. We then fix yj, iterate\nover all x∗,j, and multiply it with yj until we reach the next yj′.\nOnce we finish the vector component multiplications, we sort the multiplication results according to the first\ncoordinate to aggregate all the results with equal coordinates and build the result. Every time two values\nare aggregated, the first value is replaced by a placeholder. Alg. 3 details this aggregation step.\nAfter the aggregation (Line 17 of Alg. 2), we remove the placeholders using Alg. 4. To avoid revealing\ninformation about the non-zero coordinates, Alg. 4 uses the “shuffle-and-reveal” trick frequently used in\nsecure sparse sum Asharov et al. (2022); Bell et al. (2022).\nAlg. 2 costs O((nnz(X) + nnz(y)) log(nnz(X) + nnz(y))) communications and computations, but it has a\nlinear round complexity due to the multiplication loop (lines 7 to 14 of Alg. 2) and the aggregation step\n9\n\nAlgorithm 4 Placeholder removal\nInput: [[Z]], a list of secret shared tuples\n1: procedure PlaceholderRemoval([[Z]])\n2:\n[[Z]] ←[[shuffle(Z)]]\n3:\nfor k ←1 to length([[Z]]) do\n4:\nis_placeholder ←reveal([[coord(Zk)]] = ⊥)\n5:\nif is_placeholder\n6:\nRemove the k-th tuple from [[Z]]\nMajority\nAlgorithm\nComm.\nComp.\nStorage\nDishonest\nDense [Naive alg.], Chen\net al. (2020); Mohassel &\nZhang\n(2017);\nMono\n&\nGüneysu (2023)\nO(nm)\nO(nm)\nSparse (ours: Alg. 2)\nO(nnz∗· log(nnz∗))\nO(nnz∗)\nHonest\nDense Hoogh (2012); Koti\net al. (2021); Lu et al.\n(2023); Mohassel & Rindal\n(2018); Wagh et al. (2021)\nO(n)\nO(nm)\nO(nm)\nSparse (ours: Alg. 2)\nO(nnz∗· log(nnz∗))\nO(nnz∗)\nTable 1: Complexities of the sparse and dense matrix-vector multiplications. Notations: nnz∗= nnz(X) +\nnnz(y), nnz(X) ≪n × m, nnz(y) ≪m.\n(Alg. 3). Appendix C presents recursive algorithms taking inspiration from the logarithmic-round secure\nmaximum (notably implemented in MPyC Schoenmakers (2018)) in order to build a multiplication loop and\nan aggregation step with logarithmic round complexity.\nTable 1 compares the complexities of our sparse algorithm to the dense baselines. On the one hand, honest-\nmajority dense multiplication is no longer constant in communications, so our algorithm is advantageous for\nhighly sparse matrices. On the other hand, the memory footprint is more critical in this operation. Several\nreal-world sparse matrices do not fit in memory using a dense format (see example application in Section 5),\neven in plaintext! Our sparse algorithm is then mandatory when memory overflowing prevents\nfrom using dense algorithms.\n4.3\nMatrix-matrix\nWe want to multiply two sparse matrices. Such multiplications are used in ML algorithms for specific compu-\ntations, especially correlation matrices (i.e., X⊤X). Other ML algorithms also require sparse-dense matrix\nmultiplication (e.g., to multiply a sparse input matrix with a dense parameter matrix). Even in plaintext,\nsparse-dense multiplications require specific algorithms differing from those of sparse-sparse multiplications\n(i.e., the focus of our paper). Thus, we leave sparse-dense multiplications for future works. This subsection\nthen emphasizes the computation X⊤X, which is a key sparse-sparse operation in ML.\nTo compute X⊤X, we multiply an m × n sparse matrix to an n × m matrix. With our public knowledge,\nwe know the per-column sparsity of the first matrix (because X is transposed), and per-row sparsity of the\nsecond. Alg. 5 generalizes this problem: multiplying an n × m sparse matrix X to an m × p sparse matrix\nY , with per-column sparsity knowledge for X and a per-row knowledge for Y . We use the notation X(k)\n(resp. Y {k}) to refer to the k-th column (resp. row) of X (resp. Y ) stored as a sparse vector.\nWhen implementing a matrix-matrix multiplication naively, it is common to design a triple nested loop\nmultiplying: each row i of X is multiplied to each column j of Y . The third loop is the vector multiplication\non each k-th element of the vectors. Linear algebra libraries usually leverage a simple optimization to speed\n10\n\nAlgorithm 5 Secure sparse matrix multiplication\nInput: [[X]] and [[Y ]] are two secret-shared sparse matrices\nPublic knowledge: Number of non-zeros per column in X (with X(k) a column), number of non-zeros per\nrow in Y (with Y {k} a row)\n1: function SparseMatMult([[X]], [[Y ]])\n2:\n[[Z]] ←∅\n3:\nfor k ←1 to m do\n4:\nfor i ←1 to length([[X(k)]]) do\n5:\nfor j ←1 to length([[Y {k}]]) do\n6:\n[[ri]] ←[[coord(X(k)\ni\n)]]\n▷Row coordinate\n7:\n[[rj]] ←[[coord(Y {k}\nj\n)]]\n▷Column coordinate\n8:\n[[rv]] ←[[val(X(k)\ni\n)]] × [[val(Y {k}\nj\n)]]\n9:\nAppend a tuple (([[ri]], [[rj]]), [[rv]]) to [[Z]]\n10:\n// Non-zero tuple: coordinates then value\n11:\n[[Z]] ←[[sort(Z)]]\n▷Asc. sort on the coordinates.\n12:\nAggEqualCoord([[Z]])\n13:\nPlaceholderRemoval([[Z]])\n14:\nreturn [[Z]]\n▷sparse matrix stored as a list of non-zeros\nMajority\nAlgorithm\nComm.\nComp.\nStorage\nHonest\nDense\nChen\net\nal.\n(2020);\nMohassel\n&\nZhang (2017); Mono &\nGüneysu (2023)\nO(nm + mp + np)\nO(nmp)\nO(nm + mp + np)\nSparse (ours: Alg. 5)\nO(MinMult · log(MinMult))\nO(MinMult)\nDishonest\nDense Hoogh (2012); Lu\net al. (2023); Mohassel\n& Rindal (2018); Wagh\net al. (2021)\nO(np)\nO(nmp)\nO(nm + mp + np)\nSparse (ours: Alg. 5)\nO(MinMult · log(MinMult))\nO(MinMult)\nTable 2: Complexities of the sparse and dense matrix-matrix multiplications.\nNotations: MinMult = Pm\nk=0 nnz(X(k)) · nnz(Y {k}), nnz(X) ≪n × m, nnz(Y ) ≪m × p.\nup the dense matrix multiplication: swapping the loops on i and k. This optimization significantly speeds\nup the implementation by reducing the memory access costs Leiserson (2018).\nOur sparse algorithm reuses a similar intuition to design a fast algorithm and avoid performing a series of\nvector multiplications. Our algorithms have two main steps: (1) compute all the individual scalar multipli-\ncations, (2) aggregate them using oblivious sorting. In matrix-matrix multiplications, all values in the k-th\ncolumn of X must be multiplied by all values in the k-th row of Y . Our public knowledge provides direct\naccess to the k-th column of X and the k-th row of Y , so we can compute the individual scalar multiplications\nby iterating on the non-zeros of X(k) and Y {k}. Alg. 5 reuses Alg. 3 and 4 for aggregation and placeholder\nremoval.\nTable 2 compares our sparse algorithm to the dense algorithms. The notation MinMult = P\nk nnz(X(k)) ·\nnnz(Y {k}) denotes the number of non-zero scalar multiplications required by the matrix multiplication. It\nrepresents an intuitive lower bound: the best plaintext algorithm is O(MinMult), with no optimality proof.\nOur algorithm only adds a logarithmic factor compared to the plaintext sparse algorithm Buluç et al. (2011).\n11\n\n5\nExperimental analysis\nThis section compares the dense algorithms to our sparse algorithms under three sparsity levels: 99%, 99.9%,\n99.99% of zero values. These thresholds are recurrent in sparse data applications (see real-world datasets\nmentioned in Section 3.2). The dense algorithm is, by definition, insensitive to the sparsity level, so we\nprovide one curve for the dense algorithm and one per sparsity level for our sparse algorithm.\nOur comparison focuses on communication costs and memory usage. The communication costs corresponds\nto the total communication exchanged between the servers during the protocol.\nThe memory usage is\nrepresented thanks to the memory overflows: we scale up our experiments until the algorithms cannot be\nexecuted due to a lack of memory. We use red crosses to represent these memory overflows on the plots.\nFor each multiplication type (i.e., matrix-vector and matrix-matrix), we also implement an example use case\nbuilt upon it. Each example use case relies on a real-world dataset to demonstrate the impracticality of\ndense algorithms on simple ML applications involving sparse data.\n5.1\nExperimental setup\nWe execute our experiments on a server with 188GB of RAM and an Intel Xeon CPU. We use the MPyC\nframework Schoenmakers (2018) to simulate 3-party protocols.\nWe perform our experiments under an honest-majority assumption because this setting provides the best\ndense baselines (see Tables 1 and 2). More specifically, we chose the SSS-based matrix multiplication Hoogh\n(2012) as dense baseline instead of the other honest-majority protocols Koti et al. (2021); Lu et al. (2023);\nMohassel & Rindal (2018); Wagh et al. (2021) for two reasons. First, the SSS-based matrix multiplication\nhas smaller constant factors; e.g., it only requires the communication of a single ring element, while Koti\net al. (2021); Lu et al. (2023); Mohassel & Rindal (2018); Wagh et al. (2021) require multiple communication\nrounds. Second, the SSS-based protocol supports an arbitrary number of parties (like our sparse multiplica-\ntions), while Koti et al. (2021); Lu et al. (2023); Mohassel & Rindal (2018); Wagh et al. (2021) are 3-party\nprotocols.\nImplementation details\nWe use 64-bit (secret-shared) fixed-point numbers, with a 32-bit fractional part.\nFloating-point arithmetic is still poorly scalable in MPC protocols. MPC works usually rely on fixed-point\narithmetic Kelkar et al. (2022) to avoid performance bottlenecks. The minor precision loss due to fixed-point\narithmetic is often acceptable for ML applications Kelkar et al. (2022).\nWe use the protocol by Laur et al. Laur et al. (2011) for oblivious shuffling and the oblivious radix sort\nHamada et al. (2014) for sorting. The first step of the oblivious radix sort is a secure bit decomposition. As\nthis operation is expensive, we accelerate the process using pre-computations. Our algorithms assume the\ndata owners pre-compute and share a bit decomposition of their non-zero coordinates. Hence, they share a\nbit array instead of a single integer per non-zero coordinate.\n5.2\nMatrix-vector\nFigure 2a compares the dense and sparse matrix-vector multiplications for a square matrix. Note that our\nalgorithm with 99.99% sparsity has a negligible cost when the number of rows is small because the high\nsparsity leads to empty sparse vectors. Overall, the dense algorithm is less expensive than our algorithm.\nThe main takeaway is the memory issue. Indeed, while the dense algorithm is less expensive, our\nalgorithm supports larger matrices (i.e., the first motivation of sparse algorithms). For example, the dense\nalgorithm (contrary to our sparse algorithm with 99.99% sparsity) triggers memory overflows for matrices\nwith more than 10K columns. As developed in Section 3.2, there is no (dense) solution to avoid memory\nissues (19TB of memory would be needed).\nOne may argue that our algorithm also has some memory overflows. To analyze these results, we must\nremind the results from Section 3.2: larger datasets become sparser. Understanding this phenomenon is\n12\n\n(a) Matrix-vector (with a square matrix)\n(b) Matrix-matrix (operation: X⊤X).\nFigure 2: Comparison of the secure matrix multiplications under honest majority: dense algorithm vs. our\nsparse algorithm with three sparsity levels.\nessential to interpret Figure 2a: the memory overflows illustrated in this figure should not impact most\nreal-world datasets because of their correlation between sparsity and matrix sizes.\nOur figures represent the costs for fixed sparsity rates, but large matrices are unlikely to have a low sparsity\nrate. For example, real-world datasets with 10K columns Li et al. (2017) can have much more than 99%\nsparsity; e.g., with 10K columns, Yahoo Movie dataset already has a 99.7% sparsity! Thus, using real-world\nsparse data, our algorithm should not be affected by such memory issues, because larger datasets are sparser.\nMore generally, any algorithm (even sparse) has memory issues when the data is large enough.\nSparse\nalgorithms simply support larger sparse matrices than dense algorithms. In practice, the correlation between\nsparsity and matrix size even amplifies the gains provided by sparse algorithms.\nExample use case: Recommender system\nMatrix-vector is a core component of some recommender\nsystems. This example use case builds a simple recommender system in which a client sends a (secret-shared)\nitem identifier and receives a list of similar items. We use the Bookcrossing dataset: contains ratings\nfrom 279K users on 340K books, resulting in 99.998% zeros in the dataset (1K users as training dataset).\nEach row corresponds to a user, and each column to a book.\nOur recommender system relies on a nearest neighbors algorithm Aggarwal (2016). This model has no\ntraining step: the inference is computed on the training dataset, and requires two matrix-vector multiplica-\ntions. This algorithm computes the similarity between the book the client chose and all the other books in\nthe database. Then, the system returns the identifiers of the k most similar books.\nThe dense algorithm triggers a memory overflow because it cannot store the training data using a dense\nmatrix. The sparse algorithm takes, in average, 48 minutes. The dense algorithm is impractical while\nour sparse algorithms support this application. Our runtime could be further reduced using MPC\noptimizations (e.g., precomputed comparisons Sun & Makri (2025)).\n5.3\nMatrix-matrix\nFigure 2b shows the communication costs for the multiplication X⊤X. This experiment considers a varying\nnumber of columns and a fixed number of rows (100 rows).\nThe sparse algorithm brings a significant\ncommunication reductions compared to the dense multiplication: ×100 gain for 99.9% sparsity and a ×1000\ngain for 99.99% sparsity.\nThe cost reduction is much higher for matrix-matrix multiplication than for\nmatrix-vector because the matrix-matrix multiplication complexity has a quadratic dependency on sparsity.\n13\n\nBesides the performance gain, our algorithm supports much larger matrices: the dense algorithm\ntriggers a memory overflow for matrices with more than 1K, while our sparse algorithm scales up to 1M\ncolumns. Figure 2b shows no memory overflow for our algorithm because we stopped experiments exceeding\n10 hours.\nExample use case: Access control\nThis multiplication can be used to build an ML-based access control\nsystem. This example use case trains an ML model deciding whether an access request is suspicious. Access\ndata can be particularly privacy-sensitive, e.g., the access log of medical databases can indirectly leak patient\ninformation. As medical access logs are sensitive, there is no publicly available dataset, so we use use the\nAmazon access control dataset2; that should have similar properties as more sensitive datasets.\nThe Amazon dataset contains 32.7K samples, each containing the user’s properties, the service properties,\nand the access decision (granted or not). Each property is a categorical value that we transform into binary\narrays using the one-hot encoding. This encoded dataset has 15K features with 99.95% sparsity.\nOur access control relies on Linear Discriminant Analysis. The training requires computing: (1) the\nproportion of each class (i.e., access decision), (2) the mean vector of each class, and (3) the covariance\nmatrix of the whole dataset. The most expensive operation is, by far, the covariance matrix estimation.\nNote that the covariance matrix is necessarily dense. As Cov(X) = 1\nnX⊤X −¯x⊤¯x, we store the covariance\nin two parts: the sparse product X⊤X and the dense mean vector ¯x. This trick provides an efficient storage\nof the covariance matrix. This covariance matrix is used at inference time via a matrix-vector multiplication\n(e.g., Cov(X) · y =\n1\nnX⊤X · y −¯x⊤¯x · y). Thus, this compact representation does not change the model\ninference algorithm and avoids a potential memory overflow due to the high-dimensional dense matrix.\nWe implement this application using the dense and sparse matrix multiplications. We use the first 10K\nsamples as training dataset.\nAs expected from our previous benchmark, the dense algorithm triggers a\nmemory overflow during the covariance computation. The sparse algorithm completes the training algorithm\nin 5 hours. Hence, our sparse algorithm provides a satisfying solution to this problem, while the dense\nalgorithm is impractical.\n6\nMinimizing the public knowledge\nSection 3.3 highlighted the need for sparsity-related public knowledge in secure sparse algorithms. While\nnecessary, this information disclosure might be problematic in some specific use cases in which revealing the\nexact per-row sparsity is too sensitive. For such cases, we propose three techniques to minimize the public\nknowledge necessary to our algorithms.\n6.1\nRow anonymization\nBy default, our algorithms assume the per-row sparsity to be public.\nThis information means that the\nnumber of non-zeros from each data owner is public (but the position and value of the non-zeros remain\nsecret). To enhance their privacy, the data owner can submit their shares via an anonymization layer (e.g.,\nTor network). Machine learning algorithms typically assume that training instances are independently and\nidentically drawn and don’t extract information from the order of the instances (rows). Permuting the rows\nof the input matrix hence does not substantially change the input.\nThanks to the anonymization layer, the servers would be unable to link a data owner to a number of non-\nzeros. This privacy enhancement also reduces the public knowledge: instead of the per-row sparsity, the\ndistribution of the per-row sparsity is public.\nThis difference is subtle, but it is important for privacy: instead of individual information, a “collective”\ninformation is public. Indeed, this public knowledge provides information about the whole population, but\nnot about an individual.\n2https://github.com/pyduan/amazonaccess/tree/master/data\n14\n\nFigure 3: Per-row sparsity distribution in Bookcrossing Ziegler et al. (2005)\nML practitioners usually need this sparsity distribution information, before performing any ML training.\nSparsity is essential information to choose an appropriate ML model. Thus, it is realistic to assume that the\ndistribution of the per-row sparsity is public.\n6.2\nMax-row padding\nAs motivated above, the per-row sparsity distribution is acceptable in practical ML applications. However, if\nneeded, we can further reduce the public knowledge and only reveal an upper bound on the per-row sparsity.\nTo only assume this upper bound as public knowledge, the input sparse matrix must be padded to this\nmaximum per-row sparstity. To pad a row, the data owner represents some of the zero elements as non-zeros\nanyway (i.e., creating dummy non-zeros), such that all rows have the pre-specified number of non-zeros.\nComputing the maximum per-row sparsity over all rows in MPC requires O(n log n) communications (in\nO(log n) rounds) for a matrix of n rows.\n6.3\nMatrix templating\nThe main drawback of max-row padding is its sensitivity to the maximum number of non-zeros per row.\nSparse datasets usually contain only a few rows with many non-zeros. Most rows contain much fewer non-\nzeros than the maximum. For example, Figure 3 shows the per-row sparsity distribution on the Bookcrossing\ndataset. While the maximum number of non-zeros per row is nearly 104, 99% of the rows have less than 100\nnon-zeros and 90% of the rows have less than 10! Thus, padding the rows until a global maximum number\nof non-zeros would add many unnecessary dummy non-zeros to the dataset.\nUp until now, all our techniques and algorithms were oblivious to the number of rows owned by each data\nowner: they support all possible settings from data owners owning a single row to settings with data owners\nowning a sub-matrix. Note that the input size (i.e., number of rows) of each data owner is usually public in\nMPC. If each data owner owns a sub-matrix, we can propose a more subtle padding technique avoiding the\nshortcomings of max-row padding.\nInstead of padding each row to a fixed maximum, we propose to build a “matrix template”, a safe upper\nbound for the distribution of number of non-zeros per row, for which we are sure the real data matrix will fit\nin it. Let us call ˆF(d) = P∞\ni=d ˆf(i) the number of rows in the real dataset having d non-zeros or more. Let us\ndescribe what we call a template for a matrix Y : the template divides Y (horizontally) into K sub-matrices:\nY1, . . . , YK, with respectively n1, . . . , nK rows. Each row in Yk must have nnzk non-zero elements, and we\n15\n\nhave nnz1 ≤nnz2 ≤. . . nnzK. A matrix template is then defined by T = {(n1, nnz1), . . . , (nK, nnzK)}. A real\ndataset will fit in the template (after a permutation of the rows, see Section 6.2) if ∀i ∈[K] : ˆF(nnzi + 1) ≤\nPK\nj=i+1 nj.\nTo set the thresholds {nnz1, . . . , nnzK}, we propose to consider the quantiles: 0.25, 0.5, 0.75, 0.9, 0.99, 1,\nbecause they capture nicely the power-law distribution of sparse data.\nIn other words,\nˆF(nnz1) =\n0.25, ˆF(nnz2) = 0.5, . . . , ˆF(nnzK) = 1.\nA data owner P owning a dataset Y can fit Y into a template T by padding Y according to T. First, they\nsort (increasingly) the rows of Y based on their number of non-zeros. Remember that the order of the rows\ndoes not matter in ML applications. Second, they pad the first n1 rows of Y to have nnz1 non-zeros. They\nrepeat this padding process for the next n2 rows with nnz2 non-zeros, etc. until the last nK rows with\nnnzK non-zeros. Thanks to the sub-matrices, only YK is padded to have the maximum number of non-zeros,\nthe rest of the rows can have much fewer non-zeros. Note that matrix templating with K = 1 has the\nsame output as max-row padding. Similarly, the higher K is, the smaller the overhead because the matrix\ntemplate becomes more accurate.\nWe envision four privacy-preserving approaches to obtain such matrix templates. First, one may know the\npopulation distribution, i.e., the probability f(d) of drawing a row with d non-zeros in the process generating\nthe data. Second, without such domain knowledge one may have seen a public dataset D from which one\ncould draw statistics and approximate the population distribution by inferring ˆFD(d) ≥F(d), before one has\nto process the sensitive data. This typically leads to quite accurate results. Third, one could start from the\nreal data provided as input and use differential privacy to release safe upper bounds ˆFDP (d) ≥ˆF(d). This\nstrategy leading to looser bounds to protect privacy. Fourth, we can compute it using MPC-based quantile\nestimation.\nSection 7 details these private estimations of the matrix template.\n6.4\nOverhead comparison\nRow padding and matrix templating induce some overhead due to the dummy non-zeros they add. Figure\n4 compares the memory footprint of several alternatives: dense multiplication, no knowledge minimization,\nrow anonymization, max-row padding, and matrix templating (using quantiles as in Section 6.3).\nThe\nmemory footprint is a simple proxy to understand the impact of knowledge minimization techniques on our\nalgorithms’ related communication and computation costs (see also our tables in Section 4). We base the\ncomparison on four real-world datasets: SMS Spam Gómez Hidalgo et al. (2006), Amazon Access Control,\nBookcrossing Ziegler et al. (2005), and MovieLens Harper & Konstan (2016). For the matrix templating,\nour experiments simulate the MPC-based estimation with 20 data owners.\nOn the one hand, (as expected) row anonymization has no effect on the memory footprint. On the other\nhand, padding rows to a global maximum induces a small overhead on the Spam and the Access control\ndatasets, while it induces significant overhead on the Bookcrossing and MovieLens datasets. The overhead\non Bookcrossing and MovieLens makes the sparse matrix nearly as large as the dense matrix... in such case,\nthe sparse algorithms would have no interest.\nOur matrix templating significantly reduces this overhead.\nFor example, on MovieLens, adding dummy\nnon-zeros via matrix templating only doubles the memory footprint (vs. nearly ×100 for max-row padding).\nAs mentioned earlier, the overhead could be even further reduced if we reveal more than 5 quantiles.\nFinally, on the Access Control dataset, all knowledge minimization techniques have an equal memory foot-\nprint. In this dataset, all rows have by design the same number of non-zeros (i.e., naturally padded), so\nmax-row padding and matrix templating does not need to add any dummy non-zero.\n7\nPrivate estimation of the public knowledge\nAs explained in Section 6, there are several ways to construct a matrix template, i.e., to find upper bounds\non the number of non-zeros in the rows in certain quantiles of the data. While this matrix template limits\n16\n\nFigure 4: Matrix storage costs for different techniques of public knowledge minimization on four real-world\ndatasets\nthe prior knowledge necessary, we can further enhance privacy by estimating it using privacy-preserving\ntechniques. This section details three techniques to estimate privately matrix templates: based on MPC,\nbased on Differential Privacy, and on the population distribution.\n7.1\nMPC-based\nFirst, each data owner i shares a value [[nnzj]] for each row j they own. Then, the servers compute securely\nthe quantiles on the list [[[nnz1]], . . . , [[nnzn]]]. To compute them, the servers simply need to sort obliviously\nthe list and to obtain the elements on indices: ⌊0.25n⌋, ⌊0.5n⌋, ⌊0.75n⌋, ⌊0.9n⌋, ⌊0.99n⌋, n.\nThese quantiles provide an approximate template: some submatrices may not fit perfectly into it because\neach data owner has a slightly different row sparsity. For example, data owner i may have 50% of rows with\nless than 13 non-zeros while data owner j has 50% with less than 17 non-zeros. If the MPC protocol outputs\n15 as quantile 0.5, data owner i can pad its rows, but data owner j has to remove some non-zeros to fit the\ntemplate.\nTo finalize our template, we compute a scaling factor ensuring that all sub-matrices fit in it.\nLet\n{[\nnnz1 . . . [\nnnzK} refer to the parameters from the previous approximate template.\nBased on the imper-\nfect template, each data owner i computes a scaling factor αi ≥1 such that: for any row Xj owned by data\nowner i, nnzj ≤αi[\nnnzk ([\nnnzk being the padding parameter corresponding to Xj in the template). Then,\nthe data owners share their respective [[αi]] and the servers compute α using a secure maximum protocol on\nthese secret-shared values. We can now deduce the final templates: nnzk = α[\nnnzk, for k ∈{1 . . . K}.\nBased the values {nnz1, . . . , nnzK} and ni number of rows by data owner i (usually public in MPC), we can\ninfer the dimensions of their padded matrix: nnz1 non-zeros in their first ⌊0.25ni⌋rows, nnz2 in their next\n⌊(0.5 −0.25)ni⌋rows, ..., and nnz6 non-zeros in their last ⌊(1 −0.99)ni⌋rows.\nUsing matrix templating, the public knowledge is then reduced to a few quantiles from the\nper-row sparsity distribution.\n17\n\n7.2\nBased on Differential Privacy\nLet [n] be the set of first n integers. Let X = {xi}n\ni=1 be a set of numbers identically and independently\ndrawn from an unknown but fixed probability distribution DX. Let ˆf(X, t) = |{i ∈[n] | xi = t}| be the\nempirical probability distribution (histogram) of X and ˆF(X, t) = |{i ∈[n] | xi ≥t}| be the empirical\ncumulative distribution function.\nConsider the PPML setting where each data owner has exactly one subset X(j), with X = ∪k\nj=1X(j). Then,\nthe data owners can compute a secret share of ˆF(X, t) by doing computations linear in n to first locally\ncompute the partial counts ˆF(X(j), t) and then summing these partial counts by MPC.\nIf the numbers in X are sensitive, revealing ˆF(X, t) for some values of t may be undesirable. A classic\napproach is to use differential privacy. In particular, two sets X and X′ are adjacent if the sets differ in only\none element. Next, a mechanism M is ϵ-differentially private (DP) if for all pairs of adjacent X and X′, and\nfor all possible subsets Y of the output space of M there holds that\n\f\f\f\flog\n\u0012P(M(X) ⊆Y )\nP(P(X′) ⊆Y )\n\u0013\f\f\f\f ≤ϵ\nIn contrast to classic DP mechanisms which add zero-mean noise, in this section we are interested in finding\na DP upper bound U ˆ\nF of ˆF.\nLet us first consider finding a DP upper bound to a single number y = ˆF(X, t). We define M+\nϵ,δ(y) =\ny −1\nϵ log(2δ) + Lap\n\u0000 1\nϵ\n\u0001\n, where Lap(b)(y) = exp(−|y|/b)/2b is the Laplace distribution.\nLemma 1. M+\nϵ,δ is ϵ-DP and with probability 1 −δ there holds y ≤M+\nϵ,δ(y).\nProof. For λ > 0, the probability that Lap(1/ϵ) ≤−λ is\nZ −λ\n−∞\nexp(−tϵ)ϵ/2 dt = 1\n2 exp(−λϵ)\nSet λ = −log(2δ)/ϵ, then the probability that Lap(1/ϵ) ≤−λ is δ. Hence, the probability that M+\nϵ,δ(y) =\ny + λ + Lap(1/ϵ) < y is δ.\nNext, let us consider the problem of finding a DP vector (ui)l\ni=1 such that ∀i ∈[l] : ˆF(X, ti) ≤ui with high\nprobability. A well-known result on differential privacy describing the relation between the privacy cost ϵ and\nthe needed amount of noise for histograms and similar data structures was first introduced by Dwork Dwork\net al. (2010). The main idea is that for two adjacent datasets X and X′, the sequence ( ˆF(X, ti)−ˆF(X′, ti))l\ni=1\nstarts with 0 or more zeros, followed by a series of 0 or more +1 or −1 values, followed by 0 or more zeros.\nThis is a consequence of the assumption that X and X′ differ in only one element, say xT and x′\nT , and hence\nthis causes the sequence to only be different at the thresholds ti between the smallest and the largest of\nxT and x′\nT . As a consequence, the series ( ˆF(X, ti))l\ni=1 and ( ˆF(X′, ti))l\ni=1 correlate strongly and a strategy\nadding correlated noise gives favorable results (low variance and low ϵ). In particular, let us assume l = 2L\nis a power of 2 (else, we can just add a few thresholds ti and benefit from determining DP upper bounds to\nˆF(X, ti) for these thresholds). For i ∈[l] and j + 1 ∈[L], let bi,j give the bit decomposition of i −1 in the\nsense that i −1 = PL−1\nj=0 2jbi,j.\nLet T = (ti)l\ni=1 be a set of thresholds. For i ∈[L] and j + 1 ∈[2i−1], let\nηi,j = Lap((L + 1)/ϵ) be Laplace random variable. Then, for j + 1 ∈[L] and k ∈[2j] we define\nM+l\nϵ,δ(X, ti) = ˆF(X, ti) + L(L + 1)\nϵ\nlog\n\u0012L(L + 1)\n2δ\n\u0013\n+\nL−1\nX\nj=0\nηj+1,⌊(i−1)/2j⌋\nLemma 2. Let δ = λLe−λ\nL!\nThe complete sequence\n\u0010\nM+l\nϵ,δ(X, ti)\n\u0011l\ni=1 is ϵ-DP and with probability at least\n1 −δ there holds ∀i ∈[l] : ˆF(X, ti) ≤M+l\nϵ,δ(X, ti).\n18\n\nFigure 5: Distribution of row non-zero counts, and DP safe upper bounds for variable template block sizes\nProof. That the sequence\n\u0010\nM+l\nϵ,δ(X, ti)\n\u0011l\ni=1 is ϵ-DP follows directly from a result by Dwork (Dwork et al.,\n2010, thm 4.1).\nSimilarly to Lemma 1.\nFor the second part of the statement, we need to analyze the\nprobability that\nL−1\nX\nj=0\nηj+1,⌊(i−1)/2j⌋≥−L(L + 1)\nϵ\nlog\n\u0012L(L + 1)\n2δ\n\u0013\nThe lefthand side is a sum of L independent Laplace distributions Lap((L + 1)/ϵ), for each the probability\nthat Lap\n\u0000 L+1\nϵ\n\u0001\n≤−L+1\nϵ\nlog\n\u0010\nL(L+1)\n2δ\n\u0011\nis not larger than δ\nL. Hence, the probability that this equality holds\nfor all L terms is δ.\nFigure 5 plots for the Bookcrossing dataset the real cumulative distribution of non-zeros per row, in particular\nthe function value for x gives how many books were rated by x or more users. As can be seen, a few books\nwere rated by a large number of users and most books were only rated by a few users. The figure also shows\nseveral curves where DP statistics of the empirical cumulative distribution function were used to publish\nupper bounds for the number of non-zero slots needed per row. The several curves use a different number of\nrows per block in the template matrix (the quantiles discussed in Section 6). We use equal-sized blocks, i.e.,\nthe uninformed prior. In reality it is likely for most datasets (and also this one) we could perform better by\ntaking smaller blocks for small x and larger blocks for larger x.\nWe used 0.01-differential privacy in Figure 5, and the upper bounds are guaranteed to give sufficient space\nin the template for the data with probability at least 1 −106. In case the data owners would in the end\nsend rows not fitting in the template, the servers would need to abort the protocol and restart. In practice,\nmachine learning algorithms usually use values for ϵ larger than 0.01.\nAs one can see, revealing in a DP way more quantiles induces a higher privacy cost and requires adding more\nnoise, which in turns results in a higher upper bound. Too many template partitions (and hence quantiles)\nwill attempt to unnecessarily closely model the data and suffer from the differential privacy protection, while\ntoo few quantiles will insufficiently take into account the decrease of the function, e.g., here on the left of the\nplot one can see that the first block always needs to provide sufficient non-zeros to fit the most rated book.\nIn contrast to Figure 3 for the same dataset, Figure 5 uses a linear scale, making the memory footprint more\nvisual. In particular, the memory footprint is the area under the curve of a certain strategy. It seems that\nin this case blocks of 64 rows is the best among the three options shown.\n19\n\n(a) Linear scale\n(b) Logarithmic scale\nFigure 6: Distribution of row non-zero counts, and upper bounds based on knowledge of the population\ndistribution\n7.3\nBased on the population distribution\nThis subsection considers the situation where either the population distribution of the non-zero counts of\nrows is known publicly, or one has a public sample from which one can compute statistics.\nAssume that we know the population distribution f where f(d) is the probability that a row has d non-zeros,\nand F(d) is the probability that a row has d or more non-zeros. In that case, drawing rows identically and\nindependently (as is needed for a sound statistical analysis) implies that we expect in a sample on average\na fraction F(d) of rows to require d or more non-zeros, and the standard deviation on this estimate is\np\nF(d)(1 −F(d))/n where n is the total number of rows. We can from this compute the probability δ(λ)\nthat the real fraction of rows with more than d non-zeros is larger than F(d) + λ for λ > 0.\nIn case one doesn’t know the population distribution but has a sample from which one can estimate these\nstatistics, a similar approach can be followed applying the appropriate formulas, e.g., the standard deviation\nof a sample of size s is\np\np(1 −p)/(s −1) with p the measured probability.\nFigure 6a plots the number of non-zeros in the template matrix for every row in a way similar to Figure\n5 for the Bookcrossing dataset. As there is no differential privacy cost, we can take the size of a template\nblock to be a single row. The several curves use 5, 10 and 20 standard deviations, i.e., going up to a high\nlevel of confidence that the real data will fit in the template matrix if the data is drawn according to the\npopulation distribution (or the same distributon as the data from which statistics were taken). In the plot,\nall curves coincide because much less margin is needed compared to the differential privacy based strategy.\nTo see the difference better, the same plot is shown in Figure 6b in logarithmic scale.\n8\nConclusion\nOur paper introduced algorithms based on oblivious sorting to multiply secret-shared sparse matrices. Our\nalgorithms are compatible with a more generic MPC setup (i.e., outsourced computations) than\nexisting works on secure sparse matrix multiplications. Our algorithms avoid memory issues present in\ndense secure multiplications by leveraging the data sparsity. Our experiments show communication cost\nreductions up to ×1000 compared to dense matrix multiplications. We also implemented two real-world\nML applications highlighting that our sparse algorithms support applications impractical with\nexisting protocols. Besides matrix multiplications, our work also introduces various (sparse) data analysis\nalgorithms commonly used by ML practitioners. 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ISSN 2299-0984. URL https://petsymposium.org/popets/2021/\npopets-2021-0011.php.\nMarvin Xhemrishi, Rawad Bitar, and Antonia Wachter-Zeh. Distributed Matrix-Vector Multiplication with\nSparsity and Privacy Guarantees. In 2022 IEEE International Symposium on Information Theory (ISIT),\n2022. doi: 10.1109/ISIT50566.2022.9834805.\nCai-Nicolas Ziegler, Sean M. McNee, Joseph A. Konstan, and Georg Lausen. Improving recommendation\nlists through topic diversification. In Proceedings of the 14th international conference on World Wide Web,\n2005. ISBN 978-1-59593-046-0. doi: 10.1145/1060745.1060754.\n23\n\nA\nSecurity proof sketch\nThis appendix sketches the security proof of our algorithms. We provide proof sketches in the honest-but-\ncurious model using the real-ideal paradigm Evans et al. (2018).\nWe can define the ideal functionality for the sparse vector multiplication Fη\nVectMult as follows:\n• Input: [[xa]] = {([[i1,a]], [[v1,a]]) , . . . ([[iη,a]], [[vη,a]])} and [[xb]] (defined similarly)\n• Output: a secret-shared scalar [[s]] such that: s = P\nj,k∈{1...η} vj,a · vk,b · δ(ij,a, ik,b) (with δ the\nKroenecker delta).\nThis functionality has a property η. This property represents the protocol public knowledge in our functional-\nity. As motivated in Section 3.3, public knowledge is necessary to have efficient secure sparse multiplications.\nWe sketch a proof showing that Alg. 1 securely realizes the ideal functionality Fη\nVectMult. Our proof relies\nessentially on the composition theorem for the honest-but-curious model (Theorem 7.3.3 of Goldreich (2009)).\nCertain subprotocols we use have been proven to be secure and correct in the honest-but-curious model. By\nusing the composition theorem, we can rely on their security and only prove the security of our overall\nprotocol in which these subprotocols are embedded.\nThe correctness of Alg. 1 can be verified by an easy arithmetic exercise.\nSecurity of the sub-protocols The first statement in Alg.\n1 is a concatenation that is performed locally\nwith no communication and hence does not need to be simulated. The second statement is an oblivious\nsort. The proof depends on the sub-protocol chosen for instantiation. This choice is highly influenced by\nthe threat model, so we consider the Batcher’s sort for dishonest majority and the oblivious radix sort for\nhonest majority (i.e., the two threat models studied in our paper). On the one hand, the Batcher’s sort only\nrequires comparisons and arithmetic operations. It is then secure because the arithmetic and comparison\noperations on secret-shared values are trivially secure in the honest-but-curious model Hoogh (2012). On the\nother hand, the oblivious radix sort is proven secure in Hamada et al. (2014). After the sorting statement,\nwe have a loop. This loop statement reveals no information because it uses η, a property of Fη\nVectMult (i.e.,\na public value). Finally, the loop contains a conditional statement. As highlighted in Section 4, we use\nthe notation [[obliv. if]] to maximize the readability. This conditional statement would be implemented as\nfollows: [[s]] ←[[s]] + ([[coord(zi)]] = [[coord(zi+1)]]) × [[val(zi)]] × [[val(zi+1)]], with [[coord(zi)]] = [[coord(zi+1)]]\noutputting [[1]] or [[0]]. The arithmetic operations are trivially secure in the honest-but-curious model. For\nthe secure equality, we can use the secure protocol proposed by Toft Toft (2007). Hence, this oblivious\nconditional statement is secure because it relies on secure comparisons and arithmetic operations. Note that\nsuch oblivious conditional statements are recurrent in MPC protocols (e.g., in private heavy hitters Asharov\net al. (2022)).\nSecurity of the overall protocol\nSince all the sub-protocols are individually secure, according to the\nComposition Theorem, Alg. 1 preserves the input secrecy in the honest-but-curious model.\nMatrix-vector and matrix-matrix\nWe do not develop the proof sketch for our sparse matrix-vector\n(Alg. 2) and matrix-matrix (Alg. 5) multiplications, but the proof is very similar. As our sparse vector\nmultiplication, these two algorithms use oblivious sorting, comparisons, arithmetic operations, and oblivious\nconditional structures. For all these operations, the security would be proven as sketched for the vector mul-\ntiplication. However, they also use two additional sub-protocols: AggEqualCoord and PlaceholderRemoval.\nThe AggEqualCoord relies on an oblivious conditional structure and arithmetic operations. The proof is then\ntrivial, because we can express the conditional structure using a combination of comparisons and arithmetic\noperators.\nThe PlaceholderRemoval uses a “shuffle-and-reveal” trick: shuffle a list of secret-shared values and reveal\nwhich of them are placeholders. Other security papers Hamada et al. (2014) used similar “shuffle-and-reveal”\ntricks in protocols with proven security. In our algorithms, revealing the number of placeholders indirectly\n24\n\nreveals the number of non-zero elements in the output matrix.\nHowever, this information is public by\ndefinition because it is contained in the output.\nExtension to malicious security\nOur security proofs hold in the honest-but-curious model. Malicious\nsecurity would require a verifiable secret-sharing scheme and dedicated proofs for the security in the malicious\nmodel. We remind that our main sub-protocols have known maliciously secure variant (especially oblivious\nsorting Asharov et al. (2022); Hamada et al. (2014) and oblivious shuffling Asharov et al. (2022); Laur et al.\n(2011)) that would be the basis of a malicious security proof. We leave this extension for future work.\nB\nAdapting existing secure sparse multiplication to the outsourced setting\nOur work introduces new protocols to perform secure sparse multiplications in an outsourced setting. While\nwe introduce the first outsourced secure sparse multiplications, existing works Chen et al. (2021); Cui et al.\n(2021); Schoppmann et al. (2019) had already described secure protocols, but in a non-outsourced setting.\nInstead of introducing new protocols, one may wonder whether the existing protocols can be adapted to the\noutsourced setting. This appendix details why such an adaptation of Chen et al. (2021); Cui et al. (2021);\nSchoppmann et al. (2019) is not straightforward and can result in inefficient constructions.\nAdapting Chen et al. (2021); Cui et al. (2021)\nCui et al. and Chen et al. described sparse-dense\nmatrix multiplications where one party P1 knows the plaintext sparse matrix X and the other party P2\nknows the plaintext dense matrix Y . In both protocols, the dense matrix Y is homomorphically encrypted\nand the first party P1 performs a dense multiplication between their plaintext sparse matrix X and the\nencrypted dense matrix Y it receives from P2. In Chen et al. Chen et al. (2021), the encrypted matrix Y is\nentirely transmitted to the first party. In Cui et al. Cui et al. (2021), the parties use a private information\nretrieval to avoid transmitting the whole matrix.\nDespite their differences, these two protocols cannot be adapted to the outsourced setting for the same\nreasons. First, the homomorphic encryption step is complex in an outsourced setting because there is no\nevident key dealer (contrary to the two-data-owner setup). Second, these protocols are possible because P1\nknows the position of all non-zero values in X. Indeed, P1 performs a multiplication between the plaintext\nsparse matrix and the encrypted dense matrix. This operation is possible because homomorphically encrypted\nvalues can be trivially multiplied by plaintext constants. If these non-zero positions are hidden, this simple\nmultiplication is no longer possible. There is then no straightforward solution to adapt the approach of Chen\net al. (2021); Cui et al. (2021) to the outsourced setting.\nThird, these protocols only support sparse-dense multiplications: the matrix Y must be dense, or at least\nbe converted into a dense representation before being homomorphically encrypted. Otherwise, P1 may learn\nwhich values of Y are zero. As a result, the memory needed to store Y could blow up with several orders of\nmagnitude, and so could the communication and computation costs.\nAdapting Schoppmann et al. Schoppmann et al. (2019)\nThey adopted a different approach than\nChen et al. (2021); Cui et al. (2021): they introduce a low-level protocol called “ROOM” (Read-Only\nOblivious Map) upon which they built their matrix multiplications. The ROOM protocol performs a secure\nlook-up in a secret key-value store. This protocol guarantees that neither the key-value store, nor the look-up\nquery are revealed. They introduce three instanciations of the ROOM protocol: BasicROOM, CircuitROOM,\nand PolyROOM\nThey presented all their functionality in a two-data-owner setup with the query issued by the first party and\nthe key-value store owned by the second. However, we can easily write these functionalities for an outsourced\nsetup.\nOut of the three ROOM instanciations, only CircuitROOM is easily adaptabled to the outsourced setting\nbecause it relies on oblivious sorting and oblivious shuffling. Since our protocols also rely on sorting and\nshuffling, this observation may lead the reader to two questions: (1) What are the differences between\nthe CircuitROOM-based sparse multiplications and our protocols? (2) What are the implications of these\ndifferences on efficiency?\n25\n\nThey use the ROOM protocol to extract a dense sub-matrix from a sparse matrix. In contrast, we use\nsorting and shuffling for the overall sparse matrix multiplication. Besides this key algorithmic difference,\nefficient ROOM-based protocols in the outsourced setting is not straightforward. The following paragraphs\nstudy each matrix multiplication types to identify the algorithmic and efficiency differences.\nTo multiply two sparse vectors x and y, Schoppmann et al. Schoppmann et al. (2019) first use CircuitROOM\nto extract a dense vector of size nnz(x) from y (i.e., corresponding to the non-zero values of x). This operation\ncosts O((nnz(x) + nnz(y)) log(nnz(x) + nnz(y))) communications.\nTheir protocol then performs a dense\nvector multiplication between the non-zero values of x and the extracted dense vector from y. This overall\ncommunication cost is O((nnz(x)+nnz(y)) log(nnz(x)+nnz(y))), as in our sparse vector multiplication (Alg.\n1). However, their protocol has larger constant factors: while our protocol only requires an oblivious sort,\nCircuitROOM performs both an oblivious sort and oblivious shuffle. Their protocol even requires a dense\nvector multiplication after CircuitROOM.\nTo multiply a sparse matrix X with a sparse vector y, Schoppmann et al. Schoppmann et al. (2019) call\nthe ROOM protocol once for each data owner. In their protocol, each data owner shares a dense sub-matrix\ncontaining its non-zero columns (i.e., a column with at least one non-zero value). Hence, their protocol call\nthe matrix-vector multiplication (and the underlying ROOM protocol) for each matrix of non-zero columns;\nin other words, for each data owner. On the contrary, our protocol can group the non-zeros from all data\nowners and process them all at the same time.\nIn their paper, this design choice is not an issue because there is only two data owners. However, in an\noutsourced setup, each matrix row can be owned by a different data owner. Hence, the matrix multiplication\nwould perform one vector multiplication per matrix row. Their communication cost is then O((nnz(X) + n ·\nnnz(y)) log(nnz(X) + nnz(y))), which is worse than our cost, by a linear factor n. Indeed, Section 4.2\npresents a non-trivial matrix-vector multiplication avoiding this linear factor (i.e., Alg. 2). Since ML datasets\ncan contain thousands or even millions of rows (e.g., 340K in the Bookcrossing dataset), the CircuitROOM-\nbased matrix-vector multiplication would be inefficient compared to our Alg. 2 (and even compared to the\ndense multiplication).\nTo compute a correlation matrix X⊤X, Schoppmann et al. Schoppmann et al. (2019) require, in addition to\nROOM, another sub-protocol called \"Scatter\". Unfortunately, their protocol described in the non-outsourced\nsetting cannot be adapted easily to an outsourced setting because it relies on oblivious transfer, a purely\ntwo-party protocol with a party having access to the plaintext secret. Hence, we cannot adapt their sparse\nmatrix-matrix multiplication to the outsourced setting.\nTo sum up, the adaptation of the CircuitROOM-based sparse multiplication is either inefficient (for vector-\nvector and matrix-vector multiplications) or not straightforward (for matrix-matrix multiplications).\nC\nAvoiding the linear number of rounds\nThis appendix presents optimized algorithms to avoid the linear round complexity in Alg. 2 and 5. In Alg.\n2, this round complexity has two sources: the aggregation step (Alg. 3) and the multiplication loop: lines 7\nto 14. In Alg. 5, the round complexity is only caused by the aggregation step.\nTo avoid this round complexity, we take inspiration from the recursive secure maximum implemented in\nmany MPC libraries (including MPyC). If implemented naively (i.e., scanning the list elements one by one),\nthe round complexity of the maximum function would be linear. MPC libraries rely on a recursive algorithm\nto reach a logarithmic complexity. Alg. 6 details this algorithm. The intuition behind it is to represent the\nlist using a tree. Then, each sub-tree recursively computes its maximum and “propagate” this value toward\nthe root.\nWe reuse the same recursive structure to build our optimized algorithms with logarithmic round complexity.\nAlg. 7 presents the optimized aggregation algorithm. Alg. 10 presents the optimized multiplication loop.\nDue to their recursive structure, these algorithms are more convoluted than their naive alternatives, but\nthese optimizations are necessary to respect the round complexities presented in the main text for the sparse\n26\n\nAlgorithm 6 Secure maximum via a recursive function\nInput: [[V ]], an unsorted list of values.\n1: function RecursiveMax([[V ]])\n2:\nn ←length([[V]])\n3:\nif n = 1\n4:\nreturn [[V1]]\n5:\nInitialize [[Roots]], a list of ⌈n/2⌉secret-shared values\n6:\nfor k ←1 to ⌈n/2⌉do\n7:\n[[obliv. if]] [[V2k]] > [[V2k+1]]\n8:\n[[Rootsk]] ←[[V2k]]\n9:\nelse\n10:\n[[Rootsk]] ←[[V2k+1]]\n11:\nreturn RecursiveMax(Roots)\nAlgorithm 7 Optimized aggregation of tuples with equal coordinates\nInput: [[Z]], a sorted list of n secret-shared tuples.\nPublic knowledge: None\n1: function OptimizedAggEqualCoord([[Z]])\n2:\n// Pre-processing: O(1) rounds\n3:\nInitialize list [[Children]] with ⌈n/4⌉empty elements\n4:\nfor k ←1 to ⌈n/4⌉do\n5:\nAggIfEqual(Z4k, Z4k+1)\n6:\nAggIfEqual(Z4k+1, Z4k+2)\n7:\nAggIfEqual(Z4k+2, Z4k+3)\n8:\n[[Childrenk]] ←([[Z4k]], [[Z4k+1]], [[Z4k+2]], [[Z4k+3]])\n9:\n// Online phase: O(log n) rounds\n10:\n[[Z]] ←RecProp([[Children]])\n11:\n// Offline post-processing\n12:\nfor k ←1 to ⌈n/4⌉do\n13:\n([[Z4k]], [[Z4k+1]], [[Z4k+2]], [[Z4k+3]]) ←[[Childrenk]]\n14:\nfor k ←1 to n do\n15:\n[[obliv. if]] [[val(Zk)]] ̸= [[0]]\n16:\n[[val(Zk)]] ←[[⊥]]\n17:\nreturn [[Z]]\nmatrix-vector and matrix-matrix multiplications.\nThese two algorithms are similar and reuse the same\nintuitions; only a few minor operations differ. Hence, this appendix only discusses Alg. 7 in detail.\nOur algorithm builds a binary tree from the non-zero tuples. Each leaf contains four non-zero tuples, and the\ninternal nodes represent some sub-lists of non-zero tuples. Each internal node stores four non-zero tuples:\n“minimum tuple” of the left (child) sub-tree, “maximum tuple” of the left sub-tree, “minimum tuple” of the\nright sub-tree, and “maximum tuple” of the right sub-tree. We compare the tuples based on their coordinate,\nso the maximum tuple is the tuple with the highest coordinates.\nTo understand the intuition behind Alg. 7, it is necessary to understand the role of the recursion. Let\nus assume we have two sub-lists on which the aggregation is already completed, and let us understand the\nnecessary steps to obtain a whole list with the aggregation completed. To know whether a sub-list must\nbe updated, we only need to compare the “maximum” tuple of the left sub-list to the “minimum” tuple\n27\n\nAlgorithm 8 Sub-functions used in Alg. 7\n1: procedure AggIfEqual([[tup1]], [[tup2]])\n2:\n[[obliv. if]] [[coord(tup1)]] = [[coord(tup2)]]\n3:\n[[val(tup2)]] ←[[val(tup2)]] + [[val(tup1)]]\n4:\n[[val(tup1)]] ←[[0]]\n5: function UpProp([[Child1]], [[Child2]])\n6:\n[[Root]] ←(MinLeft([[Child1]]), MaxRight([[Child1]]), MinLeft([[Child2]]), MaxRight([[Child2]]))\n7:\nAggIfEqual(MinLeft([[Root]]), MaxLeft([[Root]]))\n8:\nAggIfEqual(MaxLeft([[Root]]), MinRight([[Root]]))\n9:\nAggIfEqual(MinRight([[Root]]), MaxRight([[Root]]))\n10:\nreturn [[Root]]\n11: procedure DownProp([[Child]], [[NewMin]], [[NewMax]])\n12:\nMinLeft([[Child]]) ←[[NewMin]]\n13:\nAggIfEqual(MinLeft([[Root]]), MaxLeft([[Root]]))\n14:\nAggIfEqual(MaxLeft([[Root]]), MinRight([[Root]]))\n15:\nMaxRight([[Child]]) ←[[NewMax]]\nAlgorithm 9 Generic recursive propagation used in Alg. 7 and 10\nInput: [[Children]], a list of 4-element tuples. UpProp and DownProp implementations vary depending on\nthe algorithm using this recursive approach.\n1: function RecProp([[Children]])\n2:\nn ←length([[Children]])\n3:\nif n = 1\n4:\nreturn [[Children]]\n5:\nfor k ←1 to ⌈n/2⌉do\n6:\n[[Rootsk]] ←UpProp([[Children2k]], [[Children2k+1]])\n7:\n[[Roots]] ←RecProp([[Roots]])\n8:\nfor k ←1 to ⌈n/2⌉do\n9:\nDownProp([[Children2k]], MinLeft([[Rootsk]]), MaxLeft([[Rootsk]]))\n10:\nDownProp([[Children2k+1]], MinRight([[Rootsk]]), MaxRight([[Rootsk]]))\n11:\nreturn [[Children]]\nof the right sub-list. This claim holds because our aggregation algorithm takes as input a sorted list of\nnon-zero tuples. If these two tuples share the same coordinate, the “maximum” tuple of the left list must be\naggregated to the “minimum” tuple in the right list. Otherwise, no operation is necessary.\nContrary to the secure maximum algorithm, Alg. 7 passes through the tree twice: from leaves to root, then\nfrom root to leaves. The first phase identifies which consecutive sub-trees share minimum-maximum with\nequal coordinates. The second phase propagates the aggregated values to the leaves. Alg. 9 describes this\nrecursive propagation. The implementation of the “upward” and “downward” propagations of Alg. 7 are\ngiven in Alg. 8.\nAs each node stores four non-zero tuples, Alg. 7 implicitly assumes that the input list is a multiple of 4 in\nlength. If not, we can pad the list with placeholders without impacting the rest of the algorithm.\nSince we rely on a binary-tree approach, the complexity of this algorithm is O(m log m) with m the number\nof leaves. This algorithm reduces the round complexity to O(log m) because we can parallelize the operations\nat each tree level.\n28\n\nAlgorithm 10 Optimized multiplication loop for Alg. 2\nInput: [[Z]], a sorted list of n secret-shared tuples.\nPublic knowledge: None\n1: function OptimizedMultLoop([[Z]])\n2:\n// Pre-processing O(1) rounds\n3:\nV1 ←[[val(Z1)]]\n4:\nfor k ←2 to n do\n5:\n[[obliv. if]] [[coord(Zk)]] = [[coord(Zk−1)]]\n6:\nVk ←[[val(Zk)]]\n7:\nelse\n8:\nVk ←[[⊥]]\n9:\nInitialize list [[Children]] with ⌈n/4⌉empty elements\n10:\nfor k ←1 to ⌈n/4⌉do\n11:\nReplaceIfNull(V4k, V4k+1)\n12:\nReplaceIfNull(V4k+1, V4k+2)\n13:\nReplaceIfNull(V4k+2, V4k+3)\n14:\n[[Childrenk]] ←([[V4k]], [[V4k+1]], [[V4k+2]], [[V4k+3]])\n15:\n// Online phase: O(log n) rounds\n16:\n[[Children]] ←RecProp([[Children]])\n17:\n// Post-processing: O(1) rounds\n18:\nfor k ←1 to ⌈n/4⌉do\n19:\n([[V4k]], [[V4k+1]], [[V4k+2]], [[V4k+3]]) ←[[Childrenk]]\n20:\nfor k ←1 to n do\n21:\n[[obliv. if]] [[Vk]] ̸= [[⊥]]\n22:\n[[val(Zk)]] ←[[val(Zk)]] × [[Vk]]\n23:\nreturn [[Z]]\n24: procedure ReplaceIfNull([[vold]], [[vnew]])\n25:\n[[obliv. if]] [[vold]] = [[⊥]]\n26:\n[[vold]] ←[[vnew]]\n27: function UpProp([[Child1]], [[Child2]])\n28:\n[[Root]] ←(MinLeft([[Child1]]), MaxRight([[Child1]]), MinLeft([[Child2]]), MaxRight([[Child2]]))\n29:\nReplaceIfNull(MinLeft([[Root]]), MaxLeft([[Root]]))\n30:\nReplaceIfNull(MaxLeft([[Root]]), MinRight([[Root]]))\n31:\nReplaceIfNull(MinRight([[Root]]), MaxRight([[Root]]))\n32:\nreturn [[Root]]\n33: procedure DownProp([[Child]], [[NewMin]], [[NewMax]])\n34:\nReplaceIfNull(MinLeft([[Child]]), [[NewMin]])\n35:\nReplaceIfNull(MaxLeft([[Child]]), [[NewMin]])\n36:\nReplaceIfNull(MinRight([[Child]]), [[NewMin]])\n37:\nMaxRight([[Child]]) ←[[NewMax]]\nThe optimized multiplication loop (Alg. 10) reuses the same intuition as the optimized aggregation. The\nmain difference is that we do not want to aggregate some values but replicate some values. We want to\nbuild a vector containing one multiplicative value for each non-zero tuple. The algorithm starts by creating\na vector of placeholders with a few multiplicative values. Then, we use our recursive propagation to replace\neach placeholder with the closest (non-placeholder) value on the left. Once this vector is built, we multiply\nour non-zero elements by their corresponding multiplicative value. This element-wise vector multiplication\nrequires one communication round.\n29\n\nContrary to Alg. 7, the recursive part of Alg. 10 work on a vector of scalar instead of working on a vector of\ntuples. This difference slightly simplifies the value propagation but requires more pre- and post-processing\nthan Alg. 7. Hence, Alg. 10 also uses the recursive Alg. 9 but with different “upward” and “downward”\npropagations described in Alg. 10. As Alg. 7, this optimized multiplication loop has a logarithmic round\ncomplexity.\n30\n"},"cleaned":{"kind":"string","value":"Secure Sparse Matrix Multiplications and their Applications to Privacy-Preserving Machine Learning Marc Damie -party computation (MPC) enables executing Machine Learning (ML) algorithms on secret-shared or encrypted data. However, existing MPC frameworks are not optimized for sparse data. This makes them unsuitable for ML applications involving sparse data, e.g., recommender systems or genomics. Even in plaintext, such applications involve high-dimensional sparse data, that cannot be processed without sparsity-related optimizations due to prohibitively large memory requirements. Since matrix multiplication is central in ML algorithms, we propose MPC algorithms to multiply secret sparse matrices. On the one hand, our algorithms avoid the memory issues of the \"dense\" data representation of classic secure matrix multiplication algorithms. On the other hand, our algorithms can significantly reduce communication costs (some experiments show a factor 1000) for realistic problem sizes. We validate our algorithms in two ML applications in which existing protocols are impractical. An important question when developing MPC algorithms is what assumptions can be made. In our case, if the number of non-zeros in a row is a sensitive piece of information then a short runtime may reveal that the number of non-zeros is small. Existing approaches make relatively simple assumptions, e.g., that there is a universal upper bound to the number of non-zeros in a row. This often doesn't align with statistical reality, in a lot of sparse datasets the amount of data per instance satisfies a power law. We propose an approach which allows adopting a safe upper bound on the distribution of non-zeros in rows/columns of sparse matrices. 1 Introduction MPC protocols are cryptographic primitives allowing to compute functions on secret inputs. Over the years, they have been used in many application domains such as privacy-preserving ML (PPML) Mohassel & Zhang (2017) or telemetry Asharov et al. (2022) to address confidentiality and privacy concerns. However, existing protocols are impractical for many real-world applications, especially when the applications involve high-dimensional sparse data, i.e., data with a large majority of zero values. Storing large sparse datasets in a dense format (one value per cell) demands excessive memory. On real-world sparse data, this \"dense storage\" often becomes prohibitive; making computation infeasible due to memory 1 16 Oct 2025 constraints. Moreover, dense representations lead to inefficient linear algebra operations, as most of the computational effort is wasted processing zeros. Over the last 50 years, algorithms to process plaintext sparse data efficiently have been proposed Buluç et al. (2011); Duff (1977). Since 2002, dedicated software libraries Duff et al. (2002) popularized them. Sparse data is present in many ML applications but also in other fields, including graph algorithms. Our paper focuses on PPML applications, but our results apply to any application involving private sparse data. Several MPC frameworks enable training ML models on secret-shared data Koti et al. (2021); Lu et al. (2023); Mohassel & Rindal (2018); Mohassel & Zhang (2017); Patra & Suresh (2020); Wagh et al. (2019; 2021). These algorithms make use of existing MPC frameworks such as MP-SPDZ Keller (2020). However, none of these frameworks provide algorithms optimized for sparse data. The scalability limitations (especially memory issues) of dense plaintext algorithms transfer naturally to dense MPC algorithms. Thus, MPC frameworks need dedicated sparse algorithms for ML applications like recommender systems. These frameworks are not fundamentally incompatible with sparse data; there is simply no suitable secure sparse algorithm in the literature. Related works There exist protocols to sum sparse vectors securely, including sorting-based Asharov et al. (2022); Bell et al. (2022), private histograms Bell et al. (2022), or private-union-based Beguier et al. (2021). A few papers Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) studied secure sparse multiplications in a two-party setting. However, these protocols require the data owners to actively participate because one of the computation parties must know the plaintext sparse data. Hence, they do not support more than two data owners simultaneously, which is too limiting, as modern ML applications involve thousands of data owners. Typical ML applications require a more generic MPC setup: outsourced training. In this setting (see Figure 1), the data owners share their secret data with a group of computation servers and disconnect. This distinction between input parties and computation parties enables supporting an arbitrary number of data owners, contrary to existing works Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019). Appendix B details why the adaptation of these works either results in an inefficient construction or is not straightforward. The secure multiplication of sparse matrices in the outsourced setting is then a significant open problem. Our work will focus on this outsourced setting; providing sparse multiplications compatible with any MPC framework. Some information-theory works Egger et al. (2023); Xhemrishi et al. (2022) studied the multiplication of a secret-shared matrix with a public matrix. Due to the public input, their setting differs from classic MPC where all inputs are secret. Finally, Nayak et al. Nayak et al. (2015) presented secure graph computations (a problem closely related to sparse operations due to the sparse adjacency matrices), but they do not present multiplication protocols. Our contributions We present two secure algorithms to compute matrix multiplications on secretshared sparse data. Our algorithms avoid the memory issues of dense matrix multiplications on sparse data (e.g., in an experiment from Section 5: 19TB using dense multiplications vs. 60GB using ours). Moreover, our algorithms reduce the communication costs up to a factor 1000, and we implement two ML applications that are impractical with existing secure algorithms. Third, as we highlight that any efficient MPC protocol (i.e., not only ours) on sparse data requires public knowledge about the sparsity, we leverage the properties of real-world sparse data to minimize this mandatory public knowledge and obtain it in a privacy-preserving manner. Notations Let nnz(x) (resp. nnz(X)) refer to the number of non-zero values in a vector x (resp. matrix X). Let [[a]] refer to a share of a secret-shared value a. In our algorithms, any operation involving secretshared values corresponds to a secure variant; e.g., [[a]] + [[b]] is the secure addition of a and b and [[sort(l)]] is the oblivious sort of l. Secure algorithms are executed jointly by all shareholders. 2 Figure 1: Outsourced MPC setting 2 Preliminaries 2.1 Threat model Threat modeling describes the expected behavior of the agents involved in an MPC protocol: an honest agent follows the protocol, an honest-but-curious agent follows the protocol and attempts to infer private information passively, and a malicious agent does not necessarily follow the protocol and can actively disrupt the protocol to obtain private information. MPC works distinguish security against a majority of honest agents from security against a majority of dishonest agents. \"Dishonest agent\" is an umbrella term encompassing malicious and honest-but-curious agents (depending on the security setup considered). We analyze our algorithms under honest and dishonest majority. We focus on honest-but-curious security, but our algorithms rely on oblivious shuffling and sorting with known maliciously secure variants Hamada et al. (2014); Laur et al. (2011). We leave for future work the extension to malicious security and the subsequent security proofs. In line with related works, our threat model excludes poisoning attacks where malicious data owners forge input data so the protocol reveal private information. We assume the data owners provide well-formed secret-shared inputs. 2.2 Outsourced setting Our contribution focuses on an \"outsourced\" computation setup (popular in recent PPML works Hegde et al. (2021); Mohassel & Zhang (2017); Wagh et al. (2019; 2021)): each data owner shares their secret data with a group of computation servers. These servers build a secret-shared matrix [[X]] containing all data owners' inputs (e.g., one row per data owner1) and perform secure computations on this matrix. For example, the servers can execute a protocol to train an ML model [[θ]] on the dataset X, as in Figure 1. In this setting, the data owners share their secret data and disconnect. Outsourced protocols supports an arbitrary number of data owners because it separates input parties (i.e., data owners) from computation parties. Contrary to many PPML works that support a two-party Mohassel & Zhang (2017); Hegde et al. (2021) or three-party Wagh et al. (2019; 2021) outsourced setups, our work support an unbounded number of computation parties. Nevertheless, secret-sharing protocols induce a quadratic communication cost in the number of parties. This quadratic cost makes outsourcing necessary to support thousands of data owners like in ML applications. All existing secure sparse multiplications Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) support a non-outsourced setting. They presented two-party protocols assuming the computation parties are 1This is sometimes referred to as \"horizontal\" data partitioning in PPML literature. 3 also data owners (⇒two data owners maximum) because they require one of the computation parties to know the plaintext sparse data. Such a setting is too limiting for ML applications as they typically involve many data owners. Adapting existing protocols to avoid designing new ones? The adaptation is not easy because a plaintext knowledge is often essential to these protocols. Appendix B details why the adaptation of Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) to the outsourced setting either results in an inefficient construction or is not straightforward. 2.3 MPC primitives Secret sharing Shamir (1979) enables sharing a secret value x between multiple parties. Each shareholder j holds a share [[x]]j, revealing nothing about x. The shareholders can reconstruct the secret value by gathering their shares. Several secret sharing schemes exist, e.g., additive shares are finite field elements drawn randomly such that: P j[[x]]j = x. Many protocols Evans et al. (2018) enable processing secret-shared values; from arithmetic operations to complex operations (e.g., sorting). Like related works in private data analysis Araki et al. (2021); Asharov et al. (2022), our algorithms rely heavily on oblivious shuffling and sorting. \"Oblivious sorting\" designates MPC protocols for sorting secret-shared values without revealing information about the values. Sorting networks Hamada et al. (2014) are straightforward oblivious sorting solutions, as their execution is data-independent. Their cost is O(m log2 m) for a list of size m. The oblivious radix sort Hamada et al. (2014) is the state-of-the-art honest-majority protocol, with its optimal complexity of O(m log m) (and low constant factors). Recent papers optimized this protocol in a three-party setting Araki et al. (2021); Asharov et al. (2022). \"Oblivious shuffling\" protocols randomly permute lists of secret values without revealing the permutation. Laur et al. Laur et al. (2011) proposed an honest-majority protocol with linear costs, used in many recent protocols Araki et al. (2021); Asharov et al. (2022); Hamada et al. (2014). A recent work Lu & Kate (2023) also described dishonest-majority protocols, but with supra-linear costs. 2.4 Matrix multiplication Let Z be the result of the multiplication between the n × m matrix X and the m × p matrix Y . Z is a n × p matrix defined as follows: Zij = m X k=1 xik · ykj, ∀i ∈{1 . . . n}, j ∈{1 . . . p} (1) This formula holds for all matrices. However, the implementation may differ depending on the type of matrices (e.g., vectors, diagonal matrices, or triangular matrices). Our work focuses on optimized algorithms for sparse matrices. Sparse algorithms avoid \"useless\" multiplications and only multiply the non-zero values. Indeed, if the matrix multiplication is implemented naively (i.e., triple nested loop), the algorithm would compute many scalar multiplications with a null result. Hence, sparse algorithms leverage the matrix sparsity to have a cost inverse proportional to the sparsity. Software libraries usually implement dense matrix multiplication via a triple nested loop, resulting in an O(nmp) cost. While this cost has long been considered the lower bound for dense matrix multiplication, several works described algorithms with a sub-cubic cost Coppersmith & Winograd (1987). Dumas et al. Dumas et al. (2019) described an MPC protocol to execute a matrix multiplication with sub-cubic cost. These algorithms have constant factors limiting their practical interest (even in plaintext), so software libraries favor algorithms with a cubic cost. 2.5 Dense matrix multiplication on secret-shared data The cubic-cost matrix multiplication (i.e., the naive implementation of Equation (1)) can easily be imported to MPC. However, recent works Chen et al. (2020); Mohassel & Rindal (2018); Mohassel & Zhang (2017); Mono & Güneysu (2023); Patra & Suresh (2020); Wagh et al. (2021) described more efficient protocols under 4 honest or dishonest majority assumptions. These works reduce the communication costs, but the computation cost remains asymptotically cubic. Moreover, the storage costs remain asymptotically equivalent to the plaintext algorithm, so they cannot support larger matrices than the plaintext dense algorithm. Dishonest majority Beaver's multiplication triplets are a popular building block to speed up secure multiplications via preprocessing steps Evans et al. (2018). Mohassel and Zhang Mohassel & Zhang (2017) generalized this concept to matrices: matrix multiplication triplets. These triplets make the communication costs proportional to the input and output size. For example, the multiplication of an n × m matrix with an m × p matrix requires O(nm + mp + np) communications (instead of O(nmp) with the naive algorithm). This approach reduces the asymptotic cost for matrix-matrix multiplications, but the communication costs of the vector-vector and matrix-vector multiplications remains asymptotically equal to the naive costs. Matrix multiplication triplets with malicious security were later described Chen et al. (2020); Mono & Güneysu (2023). Honest majority Using the honest majority assumption, several recent works Koti et al. (2021); Lu et al. (2023); Mohassel & Rindal (2018); Patra & Suresh (2020); Wagh et al. (2021) designed protocols with a communication cost proportional to the output size. Hence, a square matrix multiplication only requires O(n2) communications, and a vector multiplication O(1) communications. Before these recent protocols, De Hoogh Hoogh (2012) had already described a matrix multiplication with equivalent costs based on Shamir's secret sharing (SSS). The recent protocols Koti et al. (2021); Lu et al. (2023); Mohassel & Rindal (2018); Wagh et al. (2021) are more complex to understand than the SSS-based algorithm of Hoogh (2012) because they rely on share conversion protocols. Moreover, these protocols only support 3 parties, while the SSSbased solution supports an arbitrary number of parties. Nevertheless, the SSS-based solution only has honest-but-curious security, while Koti et al. (2021); Lu et al. (2023); Patra & Suresh (2020); Wagh et al. (2021) support malicious security. To understand the gain provided by the honest-majority assumption, let us detail the optimized vector multiplication under SSS Hoogh (2012). In SSS, each secret-shared value is represented using a polynomial of degree t. To multiply two secret-shared scalars in SSS, the parties locally multiply their local shares (obtaining a polynomial of degree 2t) and then execute a degree reduction protocol (via a \"resharing\" protocol). For dense vector multiplication, we can delay the degree reduction and add all local multiplication results together. The addition is possible because all local multiplication results are shares of equal polynomial degree (i.e., 2t). Then, a single degree reduction is done on the sum result. 3 Sparse data in PPML 3.1 Sparse data representation The term \"sparse data\" refers to data containing a large proportion of zero values. If stored using the default matrix format, the zeros occupy a significant memory space. When the data contains a sufficient number of zeros, sparse data representations can significantly reduce the storage space. Several sparse data representations exist Buluç et al. (2011), each with specific algorithms to perform operations on the data. Depending on the data representations, some operations are more efficient than others. In other words, there is no perfect sparse data representation. We use the tuple representation Buluç et al. (2011): each sparse vector is a list of non-zero tuples: t = (i, vi) with i the coordinate in the vector and vi the non-zero value. Let coord(t) = i denote the coordinate of the non-zero and val(t) = vi its value. On the one hand, we refer to algorithms supporting sparse data as \"sparse algorithm\", a common abuse of terminology. On the other hand, we use the term \"dense algorithms\" to refer to classic algorithms. 5 3.2 Larger datasets are sparser Optimizing dense algorithms, a dead end To process sparse data, one may consider to use dense matrix multiplication, with more memory. Unfortunately, it would require tremendous amount of memory. Secure dense algorithms have the same asymptotic memory footprint and hence run into the same memory issue. There is a limit to the scalability of dense algorithms. Even if extreme amounts of memory would be available, dense secure multiplications would still fail to scale because their communication cost is typically at least linear in the size of the dense input. A workaround to avoid this memory issue could be to partition the matrix in sub-matrices and perform dense multiplications on these sub-matrices. This approach is effective in classic linear algebra to handle large dense datasets and could (to some extent) reduce memory consumption. However, it would still require storing the product as a dense matrix; the memory benefit will not be significant. Moreover, it adds significant costs for repeated sparse-to-dense conversions. Finally, this partitioning approach would not reduce the communication costs of MPC protocols that would remain correlated to the full matrix size. To sum up, sparse algorithms are popular in plaintext because there is no other efficient solutions to process high-dimensional sparse datasets. In plaintext, dense matrix multiplications usually have a sparse equivalent (e.g., NumPy for dense and SciPy for sparse in Python). To cover all possible ML applications, MPC frameworks should similarly have sparse algorithms. The first motivation of sparse algorithms is to avoid the memory issues caused by dense algorithms. Furthermore, they often provide significant performance gains. Sparsity-size correlation Real-world sparse datasets show an interesting phenomenon: the sparsity is correlated to the matrix size. Indeed, larger sparse datasets have in average a larger sparsity. Thus, the larger datasets are, the more beneficial the sparse algorithms become. To understand this phenomenon, let us take the example of recommender systems on a marketplace (e.g., Amazon): the matrix quantifies the interaction of each consumer with each product. The dataset is sparse because each consumer only interacts with a small subset of all possible products. If the marketplace increases its number of products by a factor 1000, the consumer will not consume 1000 times more products. Even if the consumption increases a bit, existing datasets show that it does not follow the matrix growth; i.e., the sparsity increases. The real-world datasets that we arbitrarily selected from Li et al. (2017) satisfy the same trend: MovieLens 1M has 95% sparsity for 1.7K columns, Yahoo Movie has 99.7% sparsity for 12K columns, Bookcrossing has 99.99% sparsity for 340K columns, and flicker has 99.999% sparsity for 500K columns. Theoretical point of view The correlation between sparsity and matrix size is observable analytically. Sparse datasets can often be represented as graphs: e.g., a recommendation dataset is a bipartite graph linking n customers to m products. Also, note that graphs can be represented as sparse dataset via their adjacency matrix. The literature Aiello et al. (2001) showed that, in many application domains, graph data follows a \"power law\" distribution. In particular, if we take a random vertex v ∈V (G) of a graph G, then the probability distribution of the degree d(v) of v is given by: f(x) = P(d(v) = x) = x-γ Z(γ) (2) for some γ > 1 where Z(γ) = P∞ i=1 i-γ is a normalization constant. Typically, we have 2 ≤γ ≤3 Adamic et al. (2001), but application domains exist where other values of γ are adequate. Practical applications may deviate a bit from (2), especially for small degrees, as this equation is mainly intended to describe asymptotic behavior. In a dataset with n rows (i.e, vertices) where every vertex has degree at most m, one can use a corrected normalization constant Z(γ, m) = Pm i=1 i-γ. Remember that, in our case, the degree of a vertex corresponds 6 to the number of non-zeros in a row. The expected degree of a vertex is: Ev[d(v)] = m X i=1 i1-γ Z(γ, m) We can bound Z(γ, m) ≥1+ R ∞ 1 x-γdx = 1+1/1(1-γ) and Pm i=1 i1-γ 0, the probability that Lap(1/ε) ≤-λ is Z -λ -∞ exp(-tε)ε/2 dt = 1 2 exp(-λε) Set λ = -log(2δ)/ε, then the probability that Lap(1/ε) ≤-λ is δ. Hence, the probability that M+ ε,δ(y) = y + λ + Lap(1/ε) 0. In case one doesn't know the population distribution but has a sample from which one can estimate these statistics, a similar approach can be followed applying the appropriate formulas, e.g., the standard deviation of a sample of size s is p p(1 -p)/(s -1) with p the measured probability. Figure 6a plots the number of non-zeros in the template matrix for every row in a way similar to Figure 5 for the Bookcrossing dataset. As there is no differential privacy cost, we can take the size of a template block to be a single row. The several curves use 5, 10 and 20 standard deviations, i.e., going up to a high level of confidence that the real data will fit in the template matrix if the data is drawn according to the population distribution (or the same distributon as the data from which statistics were taken). In the plot, all curves coincide because much less margin is needed compared to the differential privacy based strategy. To see the difference better, the same plot is shown in Figure 6b in logarithmic scale. 8 Conclusion Our paper introduced algorithms based on oblivious sorting to multiply secret-shared sparse matrices. Our algorithms are compatible with a more generic MPC setup (i.e., outsourced computations) than existing works on secure sparse matrix multiplications. Our algorithms avoid memory issues present in dense secure multiplications by leveraging the data sparsity. Our experiments show communication cost reductions up to ×1000 compared to dense matrix multiplications. We also implemented two real-world ML applications highlighting that our sparse algorithms support applications impractical with existing protocols. Besides matrix multiplications, our work also introduces various (sparse) data analysis algorithms commonly used by ML practitioners. 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Improving recommendation lists through topic diversification. In Proceedings of the 14th international conference on World Wide Web, 2005. ISBN 978-1-59593-046-0. 23 A Security proof sketch This appendix sketches the security proof of our algorithms. We provide proof sketches in the honest-butcurious model using the real-ideal paradigm Evans et al. (2018). We can define the ideal functionality for the sparse vector multiplication Fη VectMult as follows: • Input: [[xa]] = {([[i1,a]], [[v1,a]]) , . . . ([[iη,a]], [[vη,a]])} and [[xb]] (defined similarly) • Output: a secret-shared scalar [[s]] such that: s = P j,k∈{1...η} vj,a · vk,b · δ(ij,a, ik,b) (with δ the Kroenecker delta). This functionality has a property η. This property represents the protocol public knowledge in our functionality. As motivated in Section 3.3, public knowledge is necessary to have efficient secure sparse multiplications. We sketch a proof showing that Alg. 1 securely realizes the ideal functionality Fη VectMult. Our proof relies essentially on the composition theorem for the honest-but-curious model (Theorem 7.3.3 of Goldreich (2009)). Certain subprotocols we use have been proven to be secure and correct in the honest-but-curious model. By using the composition theorem, we can rely on their security and only prove the security of our overall protocol in which these subprotocols are embedded. The correctness of Alg. 1 can be verified by an easy arithmetic exercise. Security of the sub-protocols The first statement in Alg. 1 is a concatenation that is performed locally with no communication and hence does not need to be simulated. The second statement is an oblivious sort. The proof depends on the sub-protocol chosen for instantiation. This choice is highly influenced by the threat model, so we consider the Batcher's sort for dishonest majority and the oblivious radix sort for honest majority (i.e., the two threat models studied in our paper). On the one hand, the Batcher's sort only requires comparisons and arithmetic operations. It is then secure because the arithmetic and comparison operations on secret-shared values are trivially secure in the honest-but-curious model Hoogh (2012). On the other hand, the oblivious radix sort is proven secure in Hamada et al. (2014). After the sorting statement, we have a loop. This loop statement reveals no information because it uses η, a property of Fη VectMult (i.e., a public value). Finally, the loop contains a conditional statement. As highlighted in Section 4, we use the notation [[obliv. if]] to maximize the readability. This conditional statement would be implemented as follows: [[s]] ←[[s]] + ([[coord(zi)]] = [[coord(zi+1)]]) × [[val(zi)]] × [[val(zi+1)]], with [[coord(zi)]] = [[coord(zi+1)]] outputting [[1]] or [[0]]. The arithmetic operations are trivially secure in the honest-but-curious model. For the secure equality, we can use the secure protocol proposed by Toft Toft (2007). Hence, this oblivious conditional statement is secure because it relies on secure comparisons and arithmetic operations. Note that such oblivious conditional statements are recurrent in MPC protocols (e.g., in private heavy hitters Asharov et al. (2022)). Security of the overall protocol Since all the sub-protocols are individually secure, according to the Composition Theorem, Alg. 1 preserves the input secrecy in the honest-but-curious model. Matrix-vector and matrix-matrix We do not develop the proof sketch for our sparse matrix-vector (Alg. 2) and matrix-matrix (Alg. 5) multiplications, but the proof is very similar. As our sparse vector multiplication, these two algorithms use oblivious sorting, comparisons, arithmetic operations, and oblivious conditional structures. For all these operations, the security would be proven as sketched for the vector multiplication. However, they also use two additional sub-protocols: AggEqualCoord and PlaceholderRemoval. The AggEqualCoord relies on an oblivious conditional structure and arithmetic operations. The proof is then trivial, because we can express the conditional structure using a combination of comparisons and arithmetic operators. The PlaceholderRemoval uses a \"shuffle-and-reveal\" trick: shuffle a list of secret-shared values and reveal which of them are placeholders. Other security papers Hamada et al. (2014) used similar \"shuffle-and-reveal\" tricks in protocols with proven security. In our algorithms, revealing the number of placeholders indirectly 24 reveals the number of non-zero elements in the output matrix. However, this information is public by definition because it is contained in the output. Extension to malicious security Our security proofs hold in the honest-but-curious model. Malicious security would require a verifiable secret-sharing scheme and dedicated proofs for the security in the malicious model. We remind that our main sub-protocols have known maliciously secure variant (especially oblivious sorting Asharov et al. (2022); Hamada et al. (2014) and oblivious shuffling Asharov et al. (2022); Laur et al. (2011)) that would be the basis of a malicious security proof. We leave this extension for future work. B Adapting existing secure sparse multiplication to the outsourced setting Our work introduces new protocols to perform secure sparse multiplications in an outsourced setting. While we introduce the first outsourced secure sparse multiplications, existing works Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) had already described secure protocols, but in a non-outsourced setting. Instead of introducing new protocols, one may wonder whether the existing protocols can be adapted to the outsourced setting. This appendix details why such an adaptation of Chen et al. (2021); Cui et al. (2021); Schoppmann et al. (2019) is not straightforward and can result in inefficient constructions. Adapting Chen et al. (2021); Cui et al. (2021) Cui et al. and Chen et al. described sparse-dense matrix multiplications where one party P1 knows the plaintext sparse matrix X and the other party P2 knows the plaintext dense matrix Y . In both protocols, the dense matrix Y is homomorphically encrypted and the first party P1 performs a dense multiplication between their plaintext sparse matrix X and the encrypted dense matrix Y it receives from P2. In Chen et al. Chen et al. (2021), the encrypted matrix Y is entirely transmitted to the first party. In Cui et al. Cui et al. (2021), the parties use a private information retrieval to avoid transmitting the whole matrix. Despite their differences, these two protocols cannot be adapted to the outsourced setting for the same reasons. First, the homomorphic encryption step is complex in an outsourced setting because there is no evident key dealer (contrary to the two-data-owner setup). Second, these protocols are possible because P1 knows the position of all non-zero values in X. Indeed, P1 performs a multiplication between the plaintext sparse matrix and the encrypted dense matrix. This operation is possible because homomorphically encrypted values can be trivially multiplied by plaintext constants. If these non-zero positions are hidden, this simple multiplication is no longer possible. There is then no straightforward solution to adapt the approach of Chen et al. (2021); Cui et al. (2021) to the outsourced setting. Third, these protocols only support sparse-dense multiplications: the matrix Y must be dense, or at least be converted into a dense representation before being homomorphically encrypted. Otherwise, P1 may learn which values of Y are zero. As a result, the memory needed to store Y could blow up with several orders of magnitude, and so could the communication and computation costs. Adapting Schoppmann et al. Schoppmann et al. (2019) They adopted a different approach than Chen et al. (2021); Cui et al. (2021): they introduce a low-level protocol called \"ROOM\" (Read-Only Oblivious Map) upon which they built their matrix multiplications. The ROOM protocol performs a secure look-up in a secret key-value store. This protocol guarantees that neither the key-value store, nor the look-up query are revealed. They introduce three instanciations of the ROOM protocol: BasicROOM, CircuitROOM, and PolyROOM They presented all their functionality in a two-data-owner setup with the query issued by the first party and the key-value store owned by the second. However, we can easily write these functionalities for an outsourced setup. Out of the three ROOM instanciations, only CircuitROOM is easily adaptabled to the outsourced setting because it relies on oblivious sorting and oblivious shuffling. Since our protocols also rely on sorting and shuffling, this observation may lead the reader to two questions: (1) What are the differences between the CircuitROOM-based sparse multiplications and our protocols? (2) What are the implications of these differences on efficiency? 25 They use the ROOM protocol to extract a dense sub-matrix from a sparse matrix. In contrast, we use sorting and shuffling for the overall sparse matrix multiplication. Besides this key algorithmic difference, efficient ROOM-based protocols in the outsourced setting is not straightforward. The following paragraphs study each matrix multiplication types to identify the algorithmic and efficiency differences. To multiply two sparse vectors x and y, Schoppmann et al. Schoppmann et al. (2019) first use CircuitROOM to extract a dense vector of size nnz(x) from y (i.e., corresponding to the non-zero values of x). This operation costs O((nnz(x) + nnz(y)) log(nnz(x) + nnz(y))) communications. Their protocol then performs a dense vector multiplication between the non-zero values of x and the extracted dense vector from y. This overall communication cost is O((nnz(x)+nnz(y)) log(nnz(x)+nnz(y))), as in our sparse vector multiplication (Alg. 1). However, their protocol has larger constant factors: while our protocol only requires an oblivious sort, CircuitROOM performs both an oblivious sort and oblivious shuffle. Their protocol even requires a dense vector multiplication after CircuitROOM. To multiply a sparse matrix X with a sparse vector y, Schoppmann et al. Schoppmann et al. (2019) call the ROOM protocol once for each data owner. In their protocol, each data owner shares a dense sub-matrix containing its non-zero columns (i.e., a column with at least one non-zero value). Hence, their protocol call the matrix-vector multiplication (and the underlying ROOM protocol) for each matrix of non-zero columns; in other words, for each data owner. On the contrary, our protocol can group the non-zeros from all data owners and process them all at the same time. In their paper, this design choice is not an issue because there is only two data owners. However, in an outsourced setup, each matrix row can be owned by a different data owner. Hence, the matrix multiplication would perform one vector multiplication per matrix row. Their communication cost is then O((nnz(X) + n · nnz(y)) log(nnz(X) + nnz(y))), which is worse than our cost, by a linear factor n. Indeed, Section 4.2 presents a non-trivial matrix-vector multiplication avoiding this linear factor (i.e., Alg. 2). Since ML datasets can contain thousands or even millions of rows (e.g., 340K in the Bookcrossing dataset), the CircuitROOMbased matrix-vector multiplication would be inefficient compared to our Alg. 2 (and even compared to the dense multiplication). To compute a correlation matrix X⊤X, Schoppmann et al. Schoppmann et al. (2019) require, in addition to ROOM, another sub-protocol called \"Scatter\". Unfortunately, their protocol described in the non-outsourced setting cannot be adapted easily to an outsourced setting because it relies on oblivious transfer, a purely two-party protocol with a party having access to the plaintext secret. Hence, we cannot adapt their sparse matrix-matrix multiplication to the outsourced setting. To sum up, the adaptation of the CircuitROOM-based sparse multiplication is either inefficient (for vectorvector and matrix-vector multiplications) or not straightforward (for matrix-matrix multiplications). C Avoiding the linear number of rounds This appendix presents optimized algorithms to avoid the linear round complexity in Alg. 2 and 5. In Alg. 2, this round complexity has two sources: the aggregation step (Alg. 3) and the multiplication loop: lines 7 to 14. In Alg. 5, the round complexity is only caused by the aggregation step. To avoid this round complexity, we take inspiration from the recursive secure maximum implemented in many MPC libraries (including MPyC). If implemented naively (i.e., scanning the list elements one by one), the round complexity of the maximum function would be linear. MPC libraries rely on a recursive algorithm to reach a logarithmic complexity. Alg. 6 details this algorithm. The intuition behind it is to represent the list using a tree. Then, each sub-tree recursively computes its maximum and \"propagate\" this value toward the root. We reuse the same recursive structure to build our optimized algorithms with logarithmic round complexity. Alg. 7 presents the optimized aggregation algorithm. Alg. 10 presents the optimized multiplication loop. Due to their recursive structure, these algorithms are more convoluted than their naive alternatives, but these optimizations are necessary to respect the round complexities presented in the main text for the sparse 26 Algorithm 6 Secure maximum via a recursive function Input: [[V ]], an unsorted list of values. 1: function RecursiveMax([[V ]]) 2: n ←length([[V]]) 3: if n = 1 4: return [[V1]] 5: Initialize [[Roots]], a list of ⌈n/2⌉secret-shared values 6: for k ←1 to ⌈n/2⌉do 7: [[obliv. if]] [[V2k]] > [[V2k+1]] 8: [[Rootsk]] ←[[V2k]] 9: else 10: [[Rootsk]] ←[[V2k+1]] 11: return RecursiveMax(Roots) Algorithm 7 Optimized aggregation of tuples with equal coordinates Input: [[Z]], a sorted list of n secret-shared tuples. Public knowledge: None 1: function OptimizedAggEqualCoord([[Z]]) 2: // Pre-processing: O(1) rounds 3: Initialize list [[Children]] with ⌈n/4⌉empty elements 4: for k ←1 to ⌈n/4⌉do 5: AggIfEqual(Z4k, Z4k+1) 6: AggIfEqual(Z4k+1, Z4k+2) 7: AggIfEqual(Z4k+2, Z4k+3) 8: [[Childrenk]] ←([[Z4k]], [[Z4k+1]], [[Z4k+2]], [[Z4k+3]]) 9: // Online phase: O(log n) rounds 10: [[Z]] ←RecProp([[Children]]) 11: // Offline post-processing 12: for k ←1 to ⌈n/4⌉do 13: ([[Z4k]], [[Z4k+1]], [[Z4k+2]], [[Z4k+3]]) ←[[Childrenk]] 14: for k ←1 to n do 15: [[obliv. if]] [[val(Zk)]] ̸= [[0]] 16: [[val(Zk)]] ←[[⊥]] 17: return [[Z]] matrix-vector and matrix-matrix multiplications. These two algorithms are similar and reuse the same intuitions; only a few minor operations differ. Hence, this appendix only discusses Alg. 7 in detail. Our algorithm builds a binary tree from the non-zero tuples. Each leaf contains four non-zero tuples, and the internal nodes represent some sub-lists of non-zero tuples. Each internal node stores four non-zero tuples: \"minimum tuple\" of the left (child) sub-tree, \"maximum tuple\" of the left sub-tree, \"minimum tuple\" of the right sub-tree, and \"maximum tuple\" of the right sub-tree. We compare the tuples based on their coordinate, so the maximum tuple is the tuple with the highest coordinates. To understand the intuition behind Alg. 7, it is necessary to understand the role of the recursion. Let us assume we have two sub-lists on which the aggregation is already completed, and let us understand the necessary steps to obtain a whole list with the aggregation completed. To know whether a sub-list must be updated, we only need to compare the \"maximum\" tuple of the left sub-list to the \"minimum\" tuple 27 Algorithm 8 Sub-functions used in Alg. 7 1: procedure AggIfEqual([[tup1]], [[tup2]]) 2: [[obliv. if]] [[coord(tup1)]] = [[coord(tup2)]] 3: [[val(tup2)]] ←[[val(tup2)]] + [[val(tup1)]] 4: [[val(tup1)]] ←[[0]] 5: function UpProp([[Child1]], [[Child2]]) 6: [[Root]] ←(MinLeft([[Child1]]), MaxRight([[Child1]]), MinLeft([[Child2]]), MaxRight([[Child2]])) 7: AggIfEqual(MinLeft([[Root]]), MaxLeft([[Root]])) 8: AggIfEqual(MaxLeft([[Root]]), MinRight([[Root]])) 9: AggIfEqual(MinRight([[Root]]), MaxRight([[Root]])) 10: return [[Root]] 11: procedure DownProp([[Child]], [[NewMin]], [[NewMax]]) 12: MinLeft([[Child]]) ←[[NewMin]] 13: AggIfEqual(MinLeft([[Root]]), MaxLeft([[Root]])) 14: AggIfEqual(MaxLeft([[Root]]), MinRight([[Root]])) 15: MaxRight([[Child]]) ←[[NewMax]] Algorithm 9 Generic recursive propagation used in Alg. 7 and 10 Input: [[Children]], a list of 4-element tuples. UpProp and DownProp implementations vary depending on the algorithm using this recursive approach. 1: function RecProp([[Children]]) 2: n ←length([[Children]]) 3: if n = 1 4: return [[Children]] 5: for k ←1 to ⌈n/2⌉do 6: [[Rootsk]] ←UpProp([[Children2k]], [[Children2k+1]]) 7: [[Roots]] ←RecProp([[Roots]]) 8: for k ←1 to ⌈n/2⌉do 9: DownProp([[Children2k]], MinLeft([[Rootsk]]), MaxLeft([[Rootsk]])) 10: DownProp([[Children2k+1]], MinRight([[Rootsk]]), MaxRight([[Rootsk]])) 11: return [[Children]] of the right sub-list. This claim holds because our aggregation algorithm takes as input a sorted list of non-zero tuples. If these two tuples share the same coordinate, the \"maximum\" tuple of the left list must be aggregated to the \"minimum\" tuple in the right list. Otherwise, no operation is necessary. Contrary to the secure maximum algorithm, Alg. 7 passes through the tree twice: from leaves to root, then from root to leaves. The first phase identifies which consecutive sub-trees share minimum-maximum with equal coordinates. The second phase propagates the aggregated values to the leaves. Alg. 9 describes this recursive propagation. The implementation of the \"upward\" and \"downward\" propagations of Alg. 7 are given in Alg. 8. As each node stores four non-zero tuples, Alg. 7 implicitly assumes that the input list is a multiple of 4 in length. If not, we can pad the list with placeholders without impacting the rest of the algorithm. Since we rely on a binary-tree approach, the complexity of this algorithm is O(m log m) with m the number of leaves. This algorithm reduces the round complexity to O(log m) because we can parallelize the operations at each tree level. 28 Algorithm 10 Optimized multiplication loop for Alg. 2 Input: [[Z]], a sorted list of n secret-shared tuples. Public knowledge: None 1: function OptimizedMultLoop([[Z]]) 2: // Pre-processing O(1) rounds 3: V1 ←[[val(Z1)]] 4: for k ←2 to n do 5: [[obliv. if]] [[coord(Zk)]] = [[coord(Zk-1)]] 6: Vk ←[[val(Zk)]] 7: else 8: Vk ←[[⊥]] 9: Initialize list [[Children]] with ⌈n/4⌉empty elements 10: for k ←1 to ⌈n/4⌉do 11: ReplaceIfNull(V4k, V4k+1) 12: ReplaceIfNull(V4k+1, V4k+2) 13: ReplaceIfNull(V4k+2, V4k+3) 14: [[Childrenk]] ←([[V4k]], [[V4k+1]], [[V4k+2]], [[V4k+3]]) 15: // Online phase: O(log n) rounds 16: [[Children]] ←RecProp([[Children]]) 17: // Post-processing: O(1) rounds 18: for k ←1 to ⌈n/4⌉do 19: ([[V4k]], [[V4k+1]], [[V4k+2]], [[V4k+3]]) ←[[Childrenk]] 20: for k ←1 to n do 21: [[obliv. if]] [[Vk]] ̸= [[⊥]] 22: [[val(Zk)]] ←[[val(Zk)]] × [[Vk]] 23: return [[Z]] 24: procedure ReplaceIfNull([[vold]], [[vnew]]) 25: [[obliv. if]] [[vold]] = [[⊥]] 26: [[vold]] ←[[vnew]] 27: function UpProp([[Child1]], [[Child2]]) 28: [[Root]] ←(MinLeft([[Child1]]), MaxRight([[Child1]]), MinLeft([[Child2]]), MaxRight([[Child2]])) 29: ReplaceIfNull(MinLeft([[Root]]), MaxLeft([[Root]])) 30: ReplaceIfNull(MaxLeft([[Root]]), MinRight([[Root]])) 31: ReplaceIfNull(MinRight([[Root]]), MaxRight([[Root]])) 32: return [[Root]] 33: procedure DownProp([[Child]], [[NewMin]], [[NewMax]]) 34: ReplaceIfNull(MinLeft([[Child]]), [[NewMin]]) 35: ReplaceIfNull(MaxLeft([[Child]]), [[NewMin]]) 36: ReplaceIfNull(MinRight([[Child]]), [[NewMin]]) 37: MaxRight([[Child]]) ←[[NewMax]] The optimized multiplication loop (Alg. 10) reuses the same intuition as the optimized aggregation. The main difference is that we do not want to aggregate some values but replicate some values. We want to build a vector containing one multiplicative value for each non-zero tuple. The algorithm starts by creating a vector of placeholders with a few multiplicative values. Then, we use our recursive propagation to replace each placeholder with the closest (non-placeholder) value on the left. Once this vector is built, we multiply our non-zero elements by their corresponding multiplicative value. This element-wise vector multiplication requires one communication round. 29 Contrary to Alg. 7, the recursive part of Alg. 10 work on a vector of scalar instead of working on a vector of tuples. This difference slightly simplifies the value propagation but requires more pre- and post-processing than Alg. 7. Hence, Alg. 10 also uses the recursive Alg. 9 but with different \"upward\" and \"downward\" propagations described in Alg. 10. As Alg. 7, this optimized multiplication loop has a logarithmic round complexity. 30"}}},{"rowIdx":89,"cells":{"id":{"kind":"string","value":"2510.14895"},"raw":{"kind":"string","value":"Fast\nand\nfault-tolerant\nlogical\nmeasurements:\nAuxiliary hypergraphs and transversal surgery\nAlexander Cowtan1, Zhiyang He2, Dominic J. Williamson3,4, and Theodore J. Yoder5\n1Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK\n2Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA 02139, USA\n3School of Physics, The University of Sydney, NSW 2006, Australia\n4IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA\n5IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA\nQuantum code surgery is a promising technique to perform fault-tolerant\ncomputation on quantum low-density parity-check codes. Recent developments\nhave significantly reduced the space overhead of surgery.\nHowever, generic\nsurgery operations still require O(d) rounds of repeated syndrome extraction\nto be made fault-tolerant. In this work, we focus on reducing the time overhead\nof surgery. We first present a general set of conditions that ensure fault-tolerant\nsurgery operations can be performed with constant time overhead. This fast\nsurgery necessarily makes use of an auxiliary complex described by a hyper-\ngraph rather than a graph. We then introduce a concrete scheme called block\nreading, which performs transversal surgery across multiple code blocks. We\nfurther investigate surgery operations with intermediate time overhead, be-\ntween O(1) and O(d), which apply to quantum locally testable codes. Finally,\nwe establish a circuit equivalence between homomorphic measurement and hy-\npergraph surgery and derive bounds on the time overhead of generic logical\nmeasurement schemes. Overall, our results demonstrate that reducing the time\ncost of code surgery is not reliant on the quantum memory being single-shot.\nInstead it is chiefly the connectivity between a code and its measurement an-\ncilla system that determines the achievable measurement time overhead.\n1\nIntroduction\nQuantum error correction [1–5] is an essential building block in any fault-tolerant quan-\ntum computer that is capable of running large-scale applications. The surface code and\nother topological codes [6–10] have been extensively studied due to their rich mathemat-\nical structure and strong practical performance. Despite their many desirable properties,\ntopological codes incur an appreciable space overhead [11] when performing fault-tolerant\nquantum computation (FTQC). This motivates the study of quantum low-density parity\ncheck (QLDPC) codes, which can achieve better parameters, notably constant encoding\nrate, in both the asymptotic and near-term regimes [12]. QLDPC codes possess the prac-\ntically desirable LDPC property – each check (or detector [13]) involves a constant number\nAlexander Cowtan: akcowtan@gmail.com\n1\narXiv:2510.14895v1 [quant-ph] 16 Oct 2025\n\nof qubits and each qubit is in a constant number of checks. Following exciting advance-\nments in both theoretical code design [14–18] and experimental qubit connectivity [19–\n21], QLDPC codes are now competitive candidates to serve as low-overhead fault-tolerant\nquantum memories [22, 23].\nTo realize FTQC in low space overhead with QLDPC codes, we need to design schemes\nfor performing logical computation on their encoded qubits. This is a flourishing area of\nresearch with many recent works.\nPopular techniques include constant-depth unitary\ncircuits [23–36], resource state preparation and teleportation [37–46], and code switch-\ning [47–69]. Here, we focus on the technique of QLDPC code surgery, which has seen\nrapid recent developments [52–67] due to its general applicability. On a high level, code\nsurgery is a technique for performing Pauli measurement on logical qubits encoded in a\nstabiliser code by introducing an ancilla system, consisting of new qubits and stabiliser\ngenerators, to deform the code in such a way that the chosen logical operators are de-\ncomposed into products of (newly introduced) stabilisers. The deformation procedure can\nbe viewed as gauge fixing or weight reduction [70–74]. Code surgery can be made fault-\ntolerant by repeatedly measuring all new stabiliser generators for a sufficient number of\nrounds (typically taken to be the code distance d) to prevent timelike errors. After ob-\ntaining the logical measurement outcome, the ancillary qubits are individually measured\nout to return to the original code.\nUnfortunately, measuring all stabiliser generators for d rounds incurs a substantial time\noverhead when compared to protocols that use constant-depth gates. For some computa-\ntions this cost can be substantially ameliorated by using Pauli Based Computation [75] to\nreduce the number of gates that must be implemented [62, 76]. However, in the worst case\na large time overhead remains. While a growing number of code families are known to be\nsingle-shot decodable [77–83], only high-dimensional topological codes are known to admit\nsingle-shot surgery [60]. Overall, prior to this work we lacked a rigorous understanding\nof the time overhead required for the fault-tolerance of surgery operations, except that d\nrounds is typically sufficient.\nIn this work, we develop a theory of fast quantum code surgery, formulating the con-\nditions needed for surgery operations to be fault-tolerant with only a constant number of\nrounds of error correction. Notably, the base memory code does not need to be single-\nshot decodable. As our primary example, we present a protocol called block reading which\nenables transversal surgery across identical CSS code blocks in constant space and time\noverhead. We discuss our main results in Section 1.2, and discuss how our work relates to\nprior work in the literature, including independent works which appeared shortly before\nours [66–69], in Section 1.3.\n1.1\nSetting\nIn this work, we aim to preserve the LDPC property of the codes and detectors throughout,\nand highlight where this is not guaranteed. However, we ignore any further specific connec-\ntivity requirements of potential physical implementations. We assume a phenomenological\nPauli and measurement noise model in our notion of fault tolerance. We focus primarily\non asymptotics rather than specific constant factors. We do not concern ourselves with\ndecoders for block reading, which depend greatly on the initial CSS codeblocks. We leave\nthis important question to future work.\n2\n\n1.2\nSummary of main results\nTo discuss our main results, we briefly review the usual setup of generalized code surgery,\nand refer readers to Section 2 for a more thorough introduction.\nOne starts with, in\ngeneral, a collection of memory code blocks with encoded logical qubits, on which the\naim is to perform fault-tolerant logical Pauli measurements. One constructs an ancilla\nsystem of qubits and stabilisers, which can be specified by a hypergraph [56, 58]. One\nthen connects this ancilla system to the memory blocks, in the process modifying the\nstabilisers of the ancilla system and the memory blocks according to the connections.\nThe modified set of stabilisers is an abelian group, which specifies a stabiliser code (the\ndeformed code). Certain logical operators from the memory blocks become products of\nstabilisers in this deformed code, which means they are measured when we measure these\nnew stabilisers. By measuring the deformed code stabilisers for a sufficient number of\nrounds (typically, in prior work, d rounds, where d is the distance of the memory codes),\none can deduce the logical measurement results fault-tolerantly. Finally, one measures\nout the ancilla system to return to the original code. The operators measured depend\nlargely on the ancilla system (equivalently the hypergraph) and its connections to the\nmemory. This formulation is general enough to capture all prior works on code surgery,\nsee Section 3.2 of [62] for a brief review.\nIn this work, we augment the above formulation by associating a 4-term chain complex\nwith the hypergraph H = (V, E). Specifically, let C be a basis of cycles of H, where a\ncycle is a collections of hyperedges which include every vertex in V an even number of\ntimes. Let W be a set of O(1)-sized components of H, where a component is a collection\nof vertices which intersect every hyperedge on an even number of vertices.1 Then\nH• = FW\n2\nFV\n2\nFE\n2\nFC\n2\nδ2\nδ1\nδ0\nis a 4-term cochain complex, where the coboundary maps δi are specified by the inclusion\nrelations of W, V, E, C. Note that when H is a simple connected graph (i.e., every hyper-\nedge is a regular edge), the only component is the set of all vertices V . Typically when\ntreating cell complexes the above coboundary maps would instead be boundary maps: we\nflip this convention to suit our conventions in the main body.\nIn prior works on surgery the focus was on the 3-term complex without W, which\nfully specifies the ancilla system. By including W, we can now consider the 1-cosystolic\ndistance of H•, which is\nd•\n1 = min{|u| : u ∈ker(δ2)\\im(δ1)}.\nOur main result states that t ≥d hypergraph surgery operations, all satisfying certain\ndistance conditions, can be performed sequentially with each operation taking a constant\ntime, such that the overall protocol performs t distinct logical operations fault-tolerantly\nin O(t) time.\nTheorem 1.1 (Fast hypergraph surgery, informal statement of Theorem 5.3). Let Q,\nQ′, Q′′... be a set of CSS LDPC codeblocks, each with distance at least d. Let t ≥d and\nH•\n1, · · · , H•\nt be sparse chain complexes from hypergraphs which each define a Z-type surgery\noperation on the memory blocks, such that\n• All H•\ni have 1-cosystolic distances at least d, and\n1Note here that W is a set of our choice and is often not a basis of all components of H.\n3\n\n• The compacted code, which is a CSS code defined by the t surgery operations, has\ndistance at least d.\nThen the t surgery operations can be performed sequentially with O(1) time each. The\nphenomenological fault-distance of the protocol is at least d and the LDPC property is\npreserved throughout.\nWe elaborate on the definition of the compacted code as well as other subtle conditions\nin the main text, but note here that there exist simpler sufficient conditions to require\nof each surgery operation individually or of the original code that together ensure the\ndistance of the compacted code is large.\nImportantly, we consider the fault-tolerance of performing t ≥d surgery operations\nin time O(t), instead of performing one surgery operation in time O(1). This is because,\nstrictly speaking, performing one surgery operation in time O(1) is not fault-tolerant. We\ndid not assume that the memory code blocks are single-shot decodable, and the deformed\ncodes are not necessarily single-shot decodable. Therefore, the syndrome information from\nO(1) rounds of noisy measurements is insufficient for fault-tolerance.\nIn contrast, performing t operations in O(t) time generates enough syndrome informa-\ntion to ensure the whole protocol has phenomenological fault-distance at least d. In this\nwork, we refer to this as amortised constant time.\nThis theorem formulates general conditions on surgery operations which enable constant-\ntime-overhead logical gates.\nHypergraph surgeries that measure stabilisers and logical\noperators of a code are forced to incorporate its local structure, up to changing the choice\nof generators. In the remaining parts of this work, we study a concrete fast surgery op-\neration which we call block reading. In its simplest form, given two identical code blocks\nof a CSS QLDPC code Q, block reading acts transversally on all pairs of data and check\nqubits of the two blocks and measure all pairs of logical qubits in the Z ⊗Z (or similarly\nX ⊗X) basis. It does so by taking the auxiliary complex (or hypergraph) to be the code\nQ itself, which by assumption has 1-cosystolic distance at least d. This example can be\neasily generalized to act on more than two code blocks and to measure more sophisticated\nPauli product operators, including the checks of any CSS QLDPC code, transversally. We\nterm this protocol full block reading.\nTheorem 1.2 (Full block reading, informal statement of Theorem 3.12). Let Q, Q′, Q′′...\nbe a set of c identical Jn, k, dK CSS LDPC codeblocks. Then a set of sparse logical Z Pauli\noperators across Q, Q′, Q′′... can be measured in parallel using O(cn) additional qubits\nand amortised O(1) time overhead. Every measurement acts transversally on all k logical\nqubits in a given codeblock. The phenomenological fault-distance of the protocol is at least\nd, and the LDPC property is preserved throughout.\nTwo important remarks are in order.\nFirst and foremost, parallels can be drawn\nbetween block reading and Steane-style error correction, or more generally logical mea-\nsurements performed via ancillary code blocks and transversal CNOT gates. In terms\nof logical action, full block reading has the same effect as measurements via transversal\ngates, except with a code-deformation procedure in contrast to a unitary circuit. Notably,\nhowever, the latter protocol usually assumes the ancillary code block is fault-tolerantly\nprepared, which in general requires O(d) time overhead. The more subtle result of algo-\nrithmic fault-tolerance [84] showed that this need not be the case, and an ancillary code\nblock prepared in O(1) time can still be used for measurements via transversal CNOT,\nsuch that the whole computation is fault-tolerant even though this individual operation\nis not. Our result on full block reading is similar to algorithmic fault-tolerance, in that\n4\n\nwe prove d surgery operations can be performed fault-tolerantly in O(d) time despite the\nfact that a single operation performed in constant time is not necessarily fault-tolerant.\nNotably we introduce new surgery gadgets and develop a very different proof to Ref. [84].\nThis establishes a curious connection between transversal surgery and transversal gates.\nSecondly, it is instructive to consider the simplest case of full block reading, where there\nis a single memory block and we measure all logical qubits in the Z basis. This case mirrors\nSteane error correction, and showcases why a single fast hypergraph surgery operation is\nnot necessarily fault-tolerant: if it were, we would have constant space-time overhead\nstate preparation of arbitrary CSS QLDPC codes (see Remark 3.10)! Of course, full block\nreading with constant syndrome rounds applied to a non-single-shot code generates a frame\nerror akin to that of Steane error correction with an ancillary code block (under-)prepared\nwith constant syndrome rounds. This frame error is only eliminated in the subsequent\nO(d) time steps. Therefore, this is one of the reasons that performing either scheme in\nconstant time is not strictly speaking fault-tolerant.\nWith full block reading as a motivating example, we study more flexible fast hyper-\ngraph surgery operations by taking subcodes of the memory code and thickening them\nappropriately. We call this partial block reading, which acts transversally on all physical\nand logical qubits contained in the subcodes.\nTheorem 1.3 (Partial block reading, informal statement of Theorem 4.5 and Corol-\nlary 7.13). Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d.\nLet A be a subcode of each of the codeblocks Q, Q′, Q′′... with length nA and distance at\nleast d. Then a suitable hypergraph H can be constructed using O(nAd) additional qubits.\nWhen A has soundness ρA this bound can be reduced to O( nA\nρA ), assuming the existence of\na sparse cycle basis.\nThe precise notions of subcode, soundness and sparse cycle basis are defined in the\nmain text. As usual in the study of code surgery, the factor O(d) space overhead is a\nworst case upper bound. When the subcode has suitable expansion properties, this space\noverhead can be significantly reduced, all the way to O(1) in some cases. We expect the\nspace overhead of partial block reading to be highly optimizable in practice, a problem we\nleave to future work.\nBeyond constant time overhead surgery, we further study the case where the conditions\nof Theorem 1.1 are not satisfied, yet surgery can still be performed in lower than O(d)\ntime. We show that when the memory codes have soundness properties, then partial block\nreading can be performed with less than O(d) syndrome rounds (in amortisation) when\nthe subcode A has distance lower than d.\nTheorem 1.4 (Intermediate time hypergraph surgery, informal statement of Theorem 6.5).\nLet Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and Let t ≥d\nand H•\n1, · · · , H•\nt be sparse chain complexes from hypergraphs which each define a surgery\noperation on the memory blocks, such that\n• Each auxiliary complex H•\n1, · · · , H•\nt has 1-cosystolic distance di ≥\n1\nαi d,\n• The deformed code each has distance at least d, and\n• The original codes have constant soundness.\nThen the t surgery operations can be performed sequentially with ⌈αi⌉time each. The\nphenomenological fault-distance of the protocol is at least d and the LDPC property is\npreserved throughout.\n5\n\nOur hypergraph surgery constructions rely on homomorphic chain maps between the\nauxiliary complex and the memory. This fact draws strong parallel to the technique of\nhomomorphic measurement [42], which is a logical measurement scheme that employs\ntransversal gates and generalizes Steane error correction. We formalize this connection by\nproving a general circuit equivalence between surgery and homomorphic measurement.\nTheorem 1.5 (Informal statement of results in Section 9). Let f define a homomorphic\nmeasurement protocol.\nThen the circuit for f can be rewritten via ZX calculus to an\nequivalent hypergraph surgery protocol and vice versa.\nNevertheless, there are several subtleties, and this equivalence does not generally pre-\nserve the fault distance of the measurement protocols. At a high level, the data (check)\nqubits in an ancillary block for homomorphic measurement correspond to new check (data)\nqubits in the corresponding surgery protocol respectively. As a consequence, timelike er-\nrors in surgery protocols can be resolved by measuring stabilisers for multiple rounds.\nHowever, the corresponding spacelike errors on a homomorphic measurement ancilla code\nare more difficult to resolve in a similar manner. This is why surgery protocols tend to be\nmore flexible in addressing logical qubits: one can create a highly targeted ancilla system\nwith low measurement fault distance, and then boost that measurement fault distance by\nmeasuring for multiple rounds.\nFinally, we complement our constructions with a structural upper bound on the fault-\ndistance of any logical measurement scheme, including surgery, homomorphic measurement\nand other similar protocols. The idea for this bound is based on quantifying the smallest\nundetectable fault that flips the logical representatives being measured while avoiding any\nancilla system, see Proposition 9.3. This logical fault corresponds to a spacetime stabiliser\nin the fault-tolerant memory on the code without any logical measurement, that becomes a\nnontrivial logical fault when the code is measured. Here, we present a simplified statement\nof this general bound.\nTheorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quan-\ntum LDPC code by an auxiliary system requires connections to more than one logical\nrepresentative to maintain phenomenological fault distance Ω(d), unless the correct logical\nmeasurement outcome is known in advance.\nIn other words, reducing the time cost of code surgery is not reliant on the quantum\nmemory being single-shot, nor does fast logical measurement require transversal gates\nbetween codeblocks. Instead it is chiefly the degree of connectivity between a code and its\nmeasurement ancilla system that determines the achievable measurement time overhead.\n1.3\nRelated work\nCode surgery is an active topic of research, and our work has many connections to the\nliterature. Notably, Ref. [67] appeared on arXiv shortly before this paper. Our works were\ndeveloped independently, but share important results in common. We briefly discuss the\nareas of overlap and difference.\nMost importantly, Theorem 5 of Ref. [67] states that a surgery operation defined by\na sparse chain complex and a sparse chain homomorphism can be performed in constant\ntime overhead fault-tolerantly if the auxiliary complex has suitable expansion properties.\nThis is similar to our Theorem 1.1 (formally Theorem 5.3), with a few points of distinction:\n1. We study the fault-tolerance of performing t ≥d suitable surgery operations in\nO(t) time, and claim that the time overhead of each logical measurement is con-\nstant in amortisation. In contrast, Ref. [67] study the fault-tolerance of performing\n6\n\none suitable surgery operation in constant time. As we argued, this operation is\nnot fault-tolerant by itself, and Ref. [67] remedies this by assuming that the syn-\ndrome information on the memory is noiseless before and after the constant time\nsurgery operation (see their Section IIIC). This is a strict assumption in the context\nof single-shot operations – for generic quantum codes, producing reliable syndrome\ninformation requires O(d) rounds of syndrome measurement. In this work, we con-\nsider a protocol where we start and end with O(d) rounds of syndrome measurement\non the memory, and perform t ≥d hypergraph surgery operations in between. We\ndemonstrate that each hypergraph surgery operation generates a round of syndrome\ninformation on the memory blocks, and accumulating these rounds guarantees fault-\ntolerance of the full protocol.\n2. Ref. [67] asks for the auxiliary complex to have suitable expansion properties, which\nis a sufficient but not necessary condition for the deformed code to have high distance.\nFrom this perspective our requirements are less stringent. Notably, full block reading\nfor generic codes does not satisfy the expansion conditions in Ref. [67]. On the other\nhand, for partial block reading we prove that a similar expansion condition can\nsatisfy the second condition of Theorem 1.1.\nBoth Ref. [67] and our work show that the desired expansion can be achieved by\ntaking a tensor product with a repetition code.\nThe idea of using expansion to\nimprove deformed code distance, and the idea of boosting expansion with a repetition\ncode are both standard in the study of code surgery and weight reduction [52, 55,\n56, 58, 72].\nOur works are largely distinct and complimentary otherwise. As our primary examples\nwe study full and partial block reading (Theorems 1.2 and 1.3), which perform transversal\nmeasurement of all logical operators contained in a (sub)code. Ref. [67] provides construc-\ntive, addressable surgery operations, capable of measuring one or more logical operators,\nusing techniques including brute-force branching and auxiliary graph surgery [59, 61].\nRef. [67] considered concrete examples on abelian multi-cycle codes and demonstrated\npromising numerical simulation results. We further study the case of hypergraph surgery\nwith intermediate time overhead (Theorem 1.4), where the conditions of Theorem 1.1 are\nnot satisfied yet soundness of the memory codes enables fault-tolerant surgery in less than\nO(d) time.2 Our results on circuit-equivalence between surgery and homomorphic mea-\nsurement, Theorem 1.5, and on the connectivity requirements for fast logical measurement,\nTheorem 1.6, have no counterparts in Ref. [67].\nThere are several other recent independent works which share some overlap with our\nresults. Ref. [65] also considered the use of hypergraphs for code surgery. They explored\nrandomized constructions demonstrating promising practical performance. While they did\nnot explore the time overhead of these surgery operations, they stated a similar condition\nto boost the distance of the deformed code with expansion properties of the hypergraph\n(see their Lemma 2).\nIn parallel, Ref. [68] and [69] developed single-shot dimension jumping techniques3 for\nthree-dimensional homological product codes. Relatedly, Ref. [66] developed single-shot\ncode switching techniques for more general homological product codes.\nIn its general\nform, full block reading can be seen as measuring, for O(1) rounds, the stabilisers of a\n2Ref. [67] has a brief remark with a similar observation.\n3Dimension jump is a special form of code switching, where one switch from (multiple copies of) lower\ndimensional codes to (fewer copies) of higher dimensional codes and vice versa.\n7\n\nhomological product code: that of the memory code and the pattern matrix.4 Our results\nshow that this protocol can be combined with further hypergraph surgery protocols to be\nmade fault-tolerant. In comparison, Refs. [66, 68, 69] all proved stronger fault-tolerance\nguarantees in their specific settings. They also developed techniques to switch from the\nresultant homological product code to other codes, which enable a range of interesting\nlogical computation gadgets. We remark that the aforementioned works [65–69] all have\nother contributions not mentioned in our brief discussion above.\nConnecting to prior literature, while conventional quantum LDPC code surgery takes\nas input the support of a logical operator to be measured [52–56, 58], hypergraph surgery\nmust incorporate the local structure of the code on the region being measured. We focus\non the simplest choice of hypergraph corresponding to the subcode itself, namely block\nreading. This is similar to the CSS code surgery described in Ref. [64], as well as the finely-\ndevised sticking in Ref. [59]. Our work is also related to Ref. [60], in which single-shot\nsurgery with 3D and 4D toric codes was studied using the formalism of fault complexes.\nOur proofs in Appendix A use this formalism, and extend it to include deformations on\nspacetime volumes.\nWe are aware of upcoming work in which the time overhead of lattice surgery with\nsurface codes is reduced by using the complementary gap to estimate confidence in a\nsurgery protocol and stop early if the confidence is high [85]. Our work is grounded in a\ndifferent setting, concerned with phenomenological fault-distance and general CSS LDPC\ncodes, and focuses on reducing the time overhead asymptotically.\n1.4\nOrganization\nThis work is lengthy and our results involve many subtleties. For the sake of the reader,\nwe have deferred the majority of technical proofs to the appendices and aimed to build up\ncomplicated results from simpler ones. In Section 2, we review important background and\ndefinitions. In Section 3, we introduce full block reading as the motivating example, and\ndemonstrate fault-tolerance guarantees with amortised constant time overhead surgery\noperations.\nIn Section 4, we study partial block reading.\nIn Section 5, we generalize\nthis to hypergraph surgery. For readers familiar with the techniques of code surgery, we\nrecommend skimming Sections 3, 4 and 5 on an initial reading.\nIn Section 6, we study hypergraph surgery operations with intermediate time over-\nheads, between O(1) and O(d).\nThis type of surgery is enabled by quantum locally\ntestable codes with constant soundness. In Section 7, we define a novel generalisation of\nthe notions of both soundness and relative expansion from Ref. [57] which we call modular\nexpansion. We show that hypergraphs with modular expansion can be used to perform\nhypergraph surgery in lower space overhead. In Section 8 we demonstrate our block read-\ning schemes on a variety of examples. We show that our formalism captures fast surgery\nwith topological codes such as 4D toric codes, and we describe block reading with 2D\ntopological codes and bivariate bicycle codes [86]. Finally, in Section 9 we prove the cir-\ncuit equivalence between homomorphic measurement and surgery, and prove an upper\nbound on the phenomenological fault distance for generic logical measurement protocol\non quantum LDPC codes.\n4More generally the pattern complex, if we use full block reading for both X and Z basis transversal\nlogical measurements.\n8\n\n2\nPreliminaries\n2.1\nStabiliser codes and Tanner graphs\nDefinition 2.1. Let Pn be the Pauli group over n qubits. A qubit stabiliser code Q is\nspecified by an Abelian subgroup −1 /∈S ⊂Pn such that the codespace H is the mutual\n+1 eigenspace of S, i.e.\nU |ψ⟩= |ψ⟩\n∀U ∈S, |ψ⟩∈H.\nWe say S is the stabiliser group for Q.\nA qubit stabiliser code Q can be specified by a stabiliser check matrix\nH = [HX|HZ] ∈Fr×2n\n2\n, such that H\n \n0\n1\n1\n0\n!\nH⊺= 0,\nwhere a row [u|v] corresponds to a generator of the stabiliser group, and therefore check\non Q, iu·vX(u)Z(v), for u, v ∈Fn\n2.\nDefinition 2.2. A qubit CSS code Q is a qubit stabiliser code where the generators of S\ncan be split into two sets SX and SZ. SX contains Pauli products with terms drawn from\n{X, I} and SZ terms drawn from {Z, I}.\nThus there is a stabiliser check matrix H for Q such that\nH =\n \nHX\n0\n0\nHZ\n!\n,\nHXH⊺\nZ = 0.\nDefinition 2.3. A CSS low-density parity check (LDPC) code family is a family of CSS\ncodes such that the column and row weights of HX and HZ are bounded above by some\nconstant σ.\nLogical operators in a stabiliser code are Pauli products that commute with all the\nchecks of the code. As the stabiliser group is Abelian, all stabilisers are equivalent to the\ntrivial logical operator. The code distance d of Q is the minimum Pauli weight of all the\nnontrivial logical operator representatives.\nDefinition 2.4. A Tanner graph is a bipartite graph with a vertex for each qubit and each\ncheck. A qubit is connected to each check in which it participates with an edge labeled [1|0],\n[0|1], or [1|1] depending on whether the check acts on the qubit as X, Z, or Y .\nIn our convention, data qubits are black circles and checks are boxes. If a check only\nmeasures Z or X terms then we omit the edge label, and instead label the box with a Z\nor X. Therefore for a CSS code we have no edge labels, and all boxes contain either Z\nor X labels. In this case, the condition that all stabilisers commute is the same as saying\nthat the number of paths between every Z and X check is even.\nDepicting large Tanner graphs directly becomes unwieldy, so we use scalable notation5\nto make them more compact. Qubits are gathered into disjoint named sets Q0, Q1, Q2, ...,\nand the same for checks C0, C1, C2, ... . An edge between Qi and Cj is then labeled with\nthe stabiliser check matrix of Q restricted to Qi and lifted to Cj, so we have\n[CX|CZ] ∈F|Cj|×2|Qi|\n2\n.\n5Terminology borrowed from Ref. [87].\n9\n\n[HX|HZ]\n(a) A generic stabiliser code,\nX\nZ\nHX\nHZ\n(b) A CSS code,\nX\nZ\nX\nL\n∂1\n∂T\n1\n∂1\nI\nI\nI\nI\n(c) A hypergraph product code.\nFigure 1: Examples of scalable Tanner graphs.\nIf that check matrix is all-zeros then omit the edge entirely. If the checks are all of the\nsame type then we omit the all-zeros half of the check matrix; for example if Cj contains\nonly Z checks then we label the check box Z and the edge\n[CZ] ∈F|Cj|×|Qi|\n2\n.\nFor example, in Figure 1 we show some basic scalable Tanner graphs with qubit sets\ndrawn as circles and check sets drawn as boxes. We have (a) a generic stabiliser code, (b)\na CSS code, and (c) the hypergraph product [14] of a classical code C, with bits L and\ncheck matrix ∂1, by a repetition code R with blocksize 2 and check matrix\n\u0010\n1\n1\n\u0011\n.\nWe use scalable Tanner graphs to describe our procedures.\n2.2\nCSS codes and chain complexes\nThe well-known CSS code-homology correspondence relates quantum CSS codes to chain\ncomplexes, allowing for the study of various properties using the techniques of homological\nalgebra [17, 29, 53, 58, 64, 88, 89]. As we employ this formalism to study block reading,\nwe give a brief summary of this correspondence here.\nA based chain complex over F2,\nC• = · · ·\nCl\nCl−1\nCl−2\n· · ·\n∂l+1\n∂l\n∂l−1\n∂l−2\nis a Z-graded vector space L\ni∈Z Ci over F2 equipped with a basis at each degree i, and\nboundary operators ∂i : Ci →Ci−1 such that ∂i−1 ◦∂i = 0, ∀i ∈Z.\nEquivalently,\nim∂i+1 ⊆ker ∂i. The homology space at degree i is Hi(C•) = ker ∂i/im∂i+1.\nA CSS code must satisfy the property that HXH⊺\nZ = 0, and therefore we can view an\nJn, k, dK CSS code as a chain complex with 3 nonzero terms:\nC• = C2\nC1\nC0\n∂2\n∂1\nwhere explicitly at each degree we have\nC• = FmZ\n2\nFn\n2\nFmX\n2\nH⊺\nZ\nHX\nAn undetectable Z operator must have support on a vector v ∈ker HX, and so every such\noperator is of the form Z(v). Z stabilisers are elements of imH⊺\nZ. Thus equivalence classes\n10\n\nof Z logical operators are elements of H1(C•) = ker HX/imH⊺\nZ, the homology space of C•\nat degree 1. Hence k = dim H1(C•), as the number of logical qubits must be equal to the\nnumber of independent equivalence classes of Z logical operators. The Z-distance dZ of a\nCSS code is the weight of the smallest nontrivial Z operator, so dZ =\nmin\nv∈ker HX\\imH⊺\nZ\n|v|.\nDual to the notion of chain complex is a cochain complex,\nC• = · · ·\nCl\nCl+1\nCl+2\n· · ·\nδl−1\nδl\nδl+1\nδl+2\nwhich is defined similarly to a chain complex but with boundary operators transposed to\nbecome coboundary operators δi, over F2. The cohomology space is Hi(C•) = ker δi/imδi−1,\nand enjoys the isomorphism Hi(C•) ∼= Hi(C•).\nAs HXH⊺\nZ = 0, we also have HZH⊺\nX = 0, hence we can equally view a CSS code as a\ncochain complex with 3 terms:\nC• = FmX\n2\nFn\n2\nFmZ\n2\nH⊺\nX\nHZ\nThis view prioritises X operators, rather than Z operators. We then have k = dim H1(C•)\nand dX =\nmin\nu∈ker HZ\\imH⊺\nX\n|u|. Furthermore, d = min(dX, dZ) as the error types are detected\nindependently.\nA chain map, or homomorphism between chain complexes, is a linear map:\nf• : C• →D•\nwhich has a component fi : Ci →Di at each degree i, such that the following diagram\ncommutes,\n· · ·\nCl+1\nCl\nCl−1\n· · ·\n· · ·\nDl+1\nDl\nDl−1\n· · ·\n∂C\nl+1\nfl+1\n∂C\nl\nfl\nfl−1\n∂D\nl+1\n∂D\nl\n(1)\ni.e. ∂D\ni+1fi+1 = fi∂C\ni+1 ∀i ∈Z.\nA dual definition applies to a cochain map f•, which can be found by taking the\ntranspose of each map in the diagram.\nDefinition 2.5 (Subcode). Let a subcode of the CSS code defined by a chain complex C•\nbe a code defined by the chain complex A• with an injective chain map f• : A• →C•. We\nstipulate further that f• must take each data qubit in A• to a single data qubit in C•, and\nthe same for any Z or X checks in A•.6\nThe additional stipulation is made to maintain control over weights, as this is essential\nfor our analysis; if C• is LDPC, so is A•. Each matrix in the chain map f• is therefore a\npermutation matrix, up to some all-zero rows.\nWe can equally take the subcode in a dual manner, viewing the original CSS code via\nits cochain complex C• and stipulating that there is a similar cochain map g• : A• →C•.\nIn this work, it is always made clear from context whether the subcode is taken in the\nchain (Z) or cochain (X) sense.\n6In Ref. [53] this property of f• is called basis-preservation.\n11\n\n2.2.1\nTensor products\nThe tensor product of (co)chain complexes is inherited from the tensor product of Z-graded\nvector spaces. Namely,\nDefinition 2.6. [90, Sec. 2.7] Let C•, D• be chain complexes over F2. Define (C ⊗D)•\nwith components\n(C ⊗D)l =\nM\ni+j=l\nCi ⊗Dj\nwhere the latter tensor product is the normal tensor product of vector spaces. Differentials\nbetween components are given for ∂(C ⊗D)\nl\nby matrices\n \n∂C\ni ⊗I\nI ⊗∂D\nj\n!\nfor a given i, j, then stacked horizontally for each term i, j | i + j = l, and vertically for\neach i′, j′ | i′ + j′ = l −1. One can check that ∂(C ⊗D)\nl−1\n◦∂(C ⊗D)\nl\n= 0 (mod 2), as desired.\nLemma 2.7 (K¨unneth formula). [90, Thm 3.6.3]\nHl(C ⊗D) ∼=\nM\ni+j=l\nHi(C) ⊗Hj(D)\nThat is, the homology spaces factor through the tensor product conveniently. Both\nthe tensor product and K¨unneth formula can be applied dually to cochain complexes.\nBy virtue of the fact that we are working with finite-dimensional vector spaces with\nexplicit bases, we can strengthen the K¨unneth formula:\nLemma 2.8. [91, Prop. 1.13]\nHl(C ⊗D) =\nM\ni+j=l\nHi(C) ⊗Hj(D)7\nProof. We first observe that any basis element of Hi(C) ⊗Hj(D) for i + j = l, which is\nan equivalence class of elements in Ci ⊗Dj, is an element of Hl(C ⊗D), and that linearly\nindependent elements remain independent. Then by a counting argument, Hl(C ⊗D) =\nL\ni+j=l Hi(C) ⊗Hj(D).\nAs CSS codes can be viewed as (co)chain complexes, the tensor product can be used\nto construct new codes [14, 91].\n2.2.2\nMapping cones\nGiven a chain map f• : A• →C• over F2, the mapping cone of f• is defined to be the chain\ncomplex with spaces cone(f)i = Ci ⊕Ai−1 and maps ∂cone(f)\ni\n: cone(f)i →cone(f)i−1\ngiven by\nAi−1\nAi−2\nCi\nCi−1\nfi−1\n∂C\ni\n∂A\ni−1\n⊕\n⊕\n7Strictly speaking this is still an isomorphism, but the isomorphism is canonical.\n12\n\n∂cone(f)\ni\n=\n \n∂C\ni\nfi−1\n0\n∂A\ni−1\n!\n.\nThe homology of mapping cones can be studied using exact sequences and the Snake\nLemma [90, Lem. 1.3.2], see App. A.\nUsing chain maps and mapping cones, one can construct new quantum codes.\nIn\nparticular, Z logical measurement of a CSS code C• can be seen as constructing a particular\nauxiliary system A• and then using the chain map f• to build the deformed code cone(f•)\n[58]; X logical measurements are then mapping cones from cochain maps. This perspective\noriginates from the quantum weight reduction of Hastings [71, 72].\n2.3\nMetachecks and soundness\nTypically, quantum codes require O(d) rounds of error correction for every logical timestep\nin order to maintain fault-tolerance. This is because in addition to errors occurring on data\nqubits, measurement errors can occur on checks. As checks are not typically protected\nagainst measurement errors, one must measure repeatedly in order to diagnose such errors\nsuccessfully. This can be seen as constructing a spacetime code whose timelike component\nis a repetition code [60].\nThere are, however, quantum codes that do have protection against measurement\nerrors [77–82]. Although there are multiple forms of such protection, here we focus on\nthe approach that uses metachecks [78]. A metacheck is a set of stabilisers that has a\nproduct of outcomes known to be +1 in advance, even in the presence of faults on data\nqubits. Hence a metacheck is a detector [92] on the spacetime code, which detects solely\nmeasurement errors.\nIt is possible to construct codes with sufficient metacheck structures, such that the\nminimum number of faulty measurements in a single error-correction round required to\ncause a logical fault is the same as the code distance d. Such codes can be single-shot\ncodes, which in principle require only a single error-correction round against adversarial\nnoise [78]. Single-shot LDPC codes with metachecks require redundancy in their stabiliser\ngenerators and so can be expensive to construct, as having enough redundant checks to\nyield sufficient detectors for protection against up to d measurement errors can entail a\nlarger space overhead.\nGiven a scalable Tanner graph for a CSS code, we can incorporate metachecks with\nnew sets of vertices and edges:\nX\nZ\nHX\nHZ\nMZ\nMX\nwhere blue (red) diamonds represent metachecks on Z (X) checks respectively, and\nMZ (MX) is the F2-linear map taking a syndrome outcome to its corresponding set of\nmetachecks.\nEquivalently, metachecks can be viewed as additional terms in the (co)chain complex\nof a code:\nC• = C3\nC2\nC1\nC0\nC−1\n∂3\n∂2\n∂1\n∂0\nC• = C3\nC2\nC1\nC0\nC−1\nδ2\nδ1\nδ0\nδ−1\nwhere C3 = C3 represents metachecks on Z checks and C−1 = C−1 represents metachecks\non X checks.\n13\n\nIn scalable Tanner graph notation, maps extend outwards from data qubits, to checks\nand then metachecks. In (co)chain complexes, maps begin at the lowest or highest degree\nand point towards the higher or lower degree respectively.\nDefinition 2.9. [78] Let Q be a CSS code. The Z-single shot distance is\ndss\nZ =\nmin\nv∈ker MZ\\imHZ\n|v| =\nmin\nv∈ker δ2\\imδ1 |v|\nand is the weight of the smallest Z-measurement fault that is undetectable by Z-\nmetachecks and is not equivalent to an X fault on data qubits. Similarly,\ndss\nX =\nmin\nu∈ker MX\\imHX\n|u| =\nmin\nu∈ker ∂0\\im∂1\n|u|\nThe single-shot distance of Q is therefore dss = min(dss\nX, dss\nZ ), as the Z and X checks are\nindependent. If ker ∂0\\im∂1 = ∅then by convention we say that dss\nX = ∞, and similar for\ndss\nZ .\nLike logical operators, measurement faults can be partitioned into equivalence classes,\nin H2(C) and H0(C) for Z measurement errors and X measurement errors respectively.\nEven if the single-shot distance is at least the distance of the code, i.e.\ndss ≥d, it\nis still possible for a low-weight measurement fault to prevent fault-tolerant single-shot\nerror correction. This is because there could be a low-weight measurement fault which is\nequivalent to a fault on data qubits, but that fault is very large. Hence the measurement\nfault may be interpreted by the decoder as a large fault on data qubits, and a large,\nerroneous recovery operator applied. As a consequence, the next logical cycle would require\nonly a small fault on data qubits to cause a logical error.\nThis motivates the study of LDPC codes with good soundness [78, 93, 94], where\nevery valid small set of syndrome outcomes has a sufficiently small set of data qubit errors\nwhich produces them; thus a minimum distance decoder will not misinterpret a small\nmeasurement fault as a large data qubit fault.\nThere are several different definitions of soundness for both classical and quantum\ncodes. For classical codes, we adopt a simple combinatorial definition of local-testability\nfrom Ref. [95, Def. 11].\nDefinition 2.10 (Local testability). A binary linear code C is (ω, ρ)-locally testable if it\nhas a parity-check matrix H : Fn\n2 →Fm\n2 with rows of weight at most ω such that for any\nvector v ∈Fn\n2,\n1\nm|Hv| ≥ρ\nnd(v, C)\nwhere d(v, C) = minx∈C(|v + x|) and | · | is the Hamming weight. The values ω and ρ are\nthe locality and soundness of the code respectively.\nFor quantum stabiliser codes, we follow Ref. [96]. For an n-qubit quantum code Q, let\nQt = Span{(A1 ⊗· · · ⊗An) |ψ⟩: |ψ⟩∈Q, #{i ∈[n], Ai ̸= I} ≤t}\nbe the t-fattening of Q, which is the set of states with distance at most t from the codespace.\nThen define\nDQ =\nX\nt≥1\nt(ΠQt −ΠQt−1).\nwhere ΠQt is the projector onto the space Qt.\n14\n\nDefinition 2.11. A qubit stabiliser code Q with generators S1, · · · , Sm is locally testable\nwith locality ω and soundness ρ if all generators have weight at most ω and\n1\nm\nm\nX\ni=1\n1\n2(I −Si) ⪰ρ\nnDQ\nwhere A ⪰B means the operator A −B is positive semidefinite.\nLemma 2.12. [96, Fact 17] A quantum CSS code CSS(HX, HZ) with parity-check matri-\nces HX, HZ is quantum locally testable with soundness ρ if ker HX, ker HZ are classical\nlocally testable codes with soundness ρ. Conversely, if CSS(HX, HZ) has soundness ρ then\nker HX, ker HZ are classical locally testable codes with soundness at least ρ/2.\nRemark 2.13. Note that Def. 2.11 is subtly different from the version of soundness given\nin Ref. [78], and is closer to the definition of confinement in Ref. [82]. Intuitively, the\nversion in Ref. [78] stipulates that every valid small set of syndrome outcomes has a\nsufficiently small set of data qubit errors which produces them. Def. 2.11 and confinement\ninstead stipulate that every small set of data qubit errors produces a sufficiently large set of\nsyndrome outcomes. See Ref. [82] for more discussion about the distinction between these\ndefinitions. Def. 2.11 aligns more closely with the definition of soundness for classical\ncodes, so is preferable for our purposes.\nIt is well-known that a quantum code with both sufficiently high single-shot distance\nand sufficient soundness, of a certain type, can be decoded in a single-shot manner against\nadversarial noise using a minimum-weight decoder [78], and similar results apply too to\nconfinement [82]. We remark that the development of quantum locally testable codes with\ngood parameters is an ongoing line of research [97–100].\nIn summary, a metachecked code being single-shot is reliant on the decoder and noise\nmodel, in addition to other code properties than just the single-shot distance. For phe-\nnomenological adversarial noise, a minimum weight decoder is sufficient when the codes\nhave sufficient properties related to soundness [78, 82]. In this work we do not require the\nquantum memory to be single-shot, or even have high single-shot distance, but we make\nuse of the general framework of metachecks, soundness and their relation to homology.\n2.4\nQuantum code surgery\nGeneralised lattice surgery, or quantum code surgery, is a method of performing compu-\ntation by code deformation, see Section 3.2 of Ref. [62] for a brief review. In this work,\nwe presume that the codes are all stabiliser codes, although one can perform surgery with\nnon-Abelian codes [101–105]. We also focus on logical measurements by the introduction\nof an auxiliary system, while surgery can be performed in other ways to implement a larger\nclass of operations [53, 64, 106, 107].\nThose exceptions aside, all prior works in quantum LDPC code surgery are known\nto be unified by the construction of a measurement hypergraph [56, 58, 62].\nIn brief,\nfor a given initial quantum memory Q and logical measurement to be performed, we\nconstruct a hypergraph H(V, E), consisting of vertices V, edges E and cycles.8 Edges are\nassociated to new data qubits, and there are two types of new stabiliser checks: vertex\nchecks, whose measurement outcomes are used to infer the logical measurement outcome,\nand cycle checks, which are present to gauge-fix and thus prevent any new logical qubits\n8A cycle in a hypergraph is a collection of edges that contains every vertex an even number of times.\n15\n\nfrom appearing in the deformed code.9 The measurement hypergraph is connected to the\ninitial code Q. Specifically, a subset of vertices (new checks) are connected to qubits in\nQ in order to perform the logical measurement, while a subset of edges (new data qubits)\nare connected to checks in Q such that the deformed code commutes as a stabiliser code.\nDefinition 2.14. Let H be a hypergraph with edge-vertex incidence matrix G : F2E →F2V.\nA cycle basis is a full-rank matrix N : F2F →F2E such that imN = ker G. F is identified\nas a set of faces, or cycles, in H. We say that the cycle basis is sparse if N is a sparse\nmatrix.\nGiven a cycle basis of a measurement hypergraph, each cycle in the basis can be\nassociated to a new check, which fixes the gauge of the logical which would otherwise have\nsupport on the cycle. If the cycle basis is sparse then this can be done while maintaining\nthe LDPC property of the deformed code.\nThere are several methods of constructing such measurement hypergraphs [52–59, 61,\n62], which are described in more detail in Ref. [62, Sec. 3].\nThe most asymptotically\nefficient method currently known to measure a generic set of Pauli product operators\non disjoint logical qubits is by brute-force branching and gauging logical measurements\nwith universal adapters [61]. For an arbitrary collection of logically disjoint Pauli prod-\nuct measurements supported on t logical qubits, this scheme uses O\n\u0000tω(log t + log3 ω)\n\u0001\nancilla qubits, where ω ≥d is the maximum weight of the single logical Pauli representa-\ntives involved in the measurements, and d is the code distance. This is all done in time\nO(d) independent of t, while preserving the phenomenological fault distance and LDPC\nproperty.\nThis is a vast improvement to the original work on surgery with quantum LDPC\ncodes [52], which required O(dω) ancilla qubits to measure a single weight ω logical opera-\ntor representative10 and could not generally perform parallel measurements without losing\nthe LDPC property. However, despite these advances, all contemporary methods require\nO(d) rounds of error correction per logical cycle for generic codes in order to maintain\nfault-tolerance11. It also means that the study of LDPC code surgery does not, prior to\nthis work, cover the well-known cases of single-shot surgery with 3D and 4D topological\ncodes [60].\nThe primary reason for this stark contrast is that, in LDPC code surgery, the measure-\nment outcomes of new checks on vertices in the measurement hypergraph H are used to\ninfer logical measurement outcomes; and in prior works on LDPC code surgery these new\nchecks did not have any protection against measurement errors. When measuring a single\nlogical operator using gauging logical measurements or homological measurements [56, 58],\nfor example, the product of vertex check outcomes is interpreted as the logical measure-\nment outcome. Hence, were they measured for only one round, just a single measurement\nerror would flip the product, and constitute a logical error. As such, the checks must be\nmeasured for O(d) rounds in order to infer the correct logical measurement outcome.\nIn this work, we focus on reducing the number of rounds required by introducing\nmetachecks which protect against measurement errors in H. All of our protocols yield\nCSS-type surgery.\n9These new logical qubits could have low weight logical operators, and lower the dressed weights of\nother operators acting on logical qubits in the code [52], which would lower the fault-distance.\n10The representative was also required to be irreducible, meaning that it did not contain any other\nlogical representatives, a problem which was fully resolved in Refs [56, 58].\n11This assumes a single representative is measured; Ref. [56] described a method to decrease the time\noverhead to O( d\nm) via parallel surgery measurement of 2m −1 equivalent logical operators.\n16\n\nDefinition 2.15. A CSS-type code surgery is a surgery protocol which begins with a CSS\ncode memory, then performs code deformation such that the code remains CSS throughout.\nCSS-type code surgery is a limited subset of more general code surgery, as one cannot\nmeasure e.g. Y terms or products of the form X ⊗Z while preserving the CSS property.\nThus all product measurements of CSS-type each contain terms drawn from either {Z, I}\nor {X, I}. We remark that CSS surgery is easily extended to measure more general logical\nPauli operators on codes that are self-dual.\nLemma 2.16. Given any CSS code Q, measurement of Z logicals by a measurement\nhypergraph results in a deformed code with X-distance at least as high as the X-distance\nof Q, so long as all cycles are gauge-fixed or otherwise stabilisers.\nThis is an immediate corollary of Ref. [58, Thm. 1].\nA dual result follows for the\nZ-distance when performing measurement of X logicals.\n3\nFull block reading\nTo understand and reduce the time overhead of surgery operations, we first consider full\nblock reading, which is a surgery operation that acts transversally on all the physical\nqubits, and thereby logical qubits, of identical code blocks. The fact that one can perform\nblock reading with surface codes is a widely known folklore result, as these are in a sense\n‘transversal’ measurements between patches. Our results extend substantially beyond this\nsimple case. We start with the simple example of two blocks.\n3.1\nFull block reading on two blocks\nGiven two identical Jn, k, dK quantum memories, which are CSS LDPC codeblocks labelled\nQ and Q′,\nX\nZ\nHX\nHZ\nQ\nX\nZ\nHX\nHZ\nQ′\nwe can initialise a hypergraph H(V, E) between the two\nX\nZ\nHX\nHZ\nQ\nX\nZ\nHX\nHZ\nQ′\nZ\nI\nI\nI\nI\nG\nE\nV\n(2)\n17\n\nsuch that new data qubits are edges, new Z checks are vertices and H has the edge-vertex\nincidence matrix G ∈FV×E\n2\n. H has 1-to-1 maps from edges to X checks in Q and to X\nchecks in Q′, and similar for vertices. Thus in order for the checks of the deformed code\nto commute, we have G = H⊺\nX. Let D be this deformed code.\nLemma 3.1. Measuring the checks of D performs a Z ⊗Z measurement on each pair of\nlogical qubits (qi, q′\ni) for i ∈[k], where qi is a logical qubit in Q and q′\ni the same logical\nqubit in Q′.\nProof. Consider an arbitrary Z logical operator representative acting on qi, and call it\nZi = Z(vi). Let Z′\ni be the identical copy of Zi in Q′.\nBy definition, vi ∈ker(HX). Multiplying Zi by the bijective Z checks P in V does\nnot leave any support on E, as Pvi ∈ker(G⊺) = ker(HX), but does clean Zi into Q′. The\noperator which it is cleaned to is precisely Z′\ni, hence the two logicals are now stabiliser\nequivalent and so we have performed a measurement of Zi ⊗Z′\ni.\nNow because ker(G⊺) = ker(HX), the only operators which can be cleaned are of this\nform and so we have performed a Z⊗Z measurement on each pair of logical qubits between\nthe two codes.\nSpecifically, each set of Z checks in V in bijection with a logical Z operator in Q and\nQ′ measures the product of those operators. The logical measurement outcome is inferred\nas the product of the syndrome outcomes in that set.\nLemma 3.2. D has no additional logical qubits.\nProof. New logical qubits can only appear when performing measurement hypergraph\nsurgery if there are cycles in the hypergraph [56, 58], which do not have corresponding\nfaces, i.e. checks, to gauge-fix. In general if ker(G) ̸= 0 then there can exist a new X\nlogical operator with support on u ∈ker(G).\nThis hypergraph may have cycles, but all logical operators on these cycles are also\nproducts of X stabilisers from Q (or, alternatively and symmetrically, from Q′). In par-\nticular, given any element u ∈ker(G), multiply X(u) by the bijective X checks S in Q.\nBecause u ∈ker(G), Su ∈ker(H⊺\nX) and so the cleaned operator maps to 0 on data qubits\nin Q, hence is a stabiliser.\nWe shall presently prove that the code distance is preserved by the deformation, but\nin order to do so it is easier to use the language of cohomology.\nLemma 3.3. Let C• be the initial codeblock Q viewed as a cochain complex. Then the\ndeformed code D has the cochain complex\nD• = (C ⊗R)•\nwhere\nR• = R0\nR1\nP\n,\nR0 = F2\n2,\nR1 = F2,\nP =\n\u0010\n1\n1\n\u0011\n.\nProof.\n(C⊗R)• = C0 ⊗F2\n2\nC1 ⊗F2\n2 ⊕C0 ⊗F2\nC2 ⊗F2\n2 ⊕C1 ⊗F2\nC2 ⊗F2\nIn words,\n18\n\n• at the level of X checks we have two copies of C0, the X checks of Q,\n• at the level of data qubits we have two copies of C1 the data qubits of Q, and an\nadditional set of data qubits in bijection with the X checks of Q,\n• at the level of Z checks we have two copies of C2 and an additional set of Z checks\nin bijection with the data qubits of Q,\n• there is another term, which we ignore for now and consider only the first 3 terms.\nWe now check that the coboundary maps are correct.\nδ0\nC⊗R =\n \nH⊺\nX ⊗I\nI ⊗P\n!\n,\nδ1\nC⊗R =\n \nHZ ⊗I\n0\nI ⊗P\nH⊺\nX ⊗I\n!\nwhich we recognise as the check matrices, with appropriate transposes, from Eq. 2.\nUsing the perspective of cochains it is easy to verify the logical dimension using the\nK¨unneth formula. Moreover, by Lemma 2.8 we have the equation:\nH1(C ⊗R) = H1(C) ⊗H0(R) ⊕H0(C) ⊗H1(R)\nhence the set of X logical operators in D is given by representatives of\nH1(C) ⊗H0(R) ⊕H0(C) ⊗H1(R)\nwhich explicitly is\nker HZ/imH⊺\nX ⊗ker P ⊕C2/imHZ ⊗0 = ker HZ/imH⊺\nX ⊗ker P\nso every X logical operator belongs to a stabiliser equivalence class [ui, u′\ni] of the identical\nlogical operators ui, u′\ni in Q and Q′, or a nontrivial sum thereof.\nLemma 3.4. The distance of D is at least d.\nProof. As there are no new logical qubits, to prove the distance bound it is sufficient to\nensure that cleaning Z logicals cannot reduce their weight below d. The X-distance is\nguaranteed by Lemma 2.16.\nFor Z logicals, observe that multiplying any Z logical in Q by Z checks in V adds\nsupport to the corresponding data qubits in Q′ by a 1-to-1 map, and potentially adds\nsupport on E as well. Any Z logical which has support on both Q and Q′, say v and v′,\ncan in the worst case be cleaned to have weight |v + v′| on Q and Q′, considering both v\nand v′ to be in Fn\n2, and v + v′ is a valid logical of Q so must have weight at least d.\nAs a consequence of Lemmas 3.2 and 3.4 it is guaranteed by the arguments in Ref. [56]\nthat the block reading can be performed with fault distance d in d rounds of error cor-\nrection, as the procedure can be viewed as a parallel gauging logical measurement. All\nnew data qubits are initialised in |+⟩, and all checks are measured for d rounds, before\nmeasuring out the new data qubits in the X basis. We reduce this from d to O(1) in\nSection 3.3.\n19\n\n3.2\nFull block reading of Z type\nWe can extend block reading straightforwardly to acting on many codeblocks simultane-\nously. We start with a set of c identical Jn, k, dK LDPC codeblocks.\nDefinition 3.5 (Pattern Matrix). Let P : Fc\n2 →Fϕ\n2 be a full-rank matrix, which defines\nthe pattern of logical measurements to be performed between codeblocks by deforming to a\nnew code D. We call P the pattern matrix.\nEach row r of P corresponds to a set of parallel measurements to be performed. Each\nentry of 1 in r denotes that a codeblock is involved in that set of measurements, so if rj = 1\nfor column j then codeblock j is involved, and if rj = 0 then it is not. Then r specifies a\nmeasurement of P 1\ni ⊗P 2\ni ⊗P 3\ni · · · for every logical qubit i ∈[k] in codeblocks Q1, Q2, Q3...\nin the basis P j ∈{Z, I} depending on whether rj = 1 or 0.\nIf P is not full rank, and has some linearly dependent rows, then those rows correspond\nto logical measurements which are already performed by other rows and so are redundant.\nAdditionally, if any row has weight 1, with the 1 in column j, that measurement is equiv-\nalent to measuring out codeblock j in the Z basis. Hence for simplicity we assume that P\nis full rank, with no linearly dependent rows, and that there are no rows with weight 1.\nDefinition 3.6 (Z-type full block reading). A full block reading of Z type takes as input\nc CSS LDPC codes, each with cochain complex C•, and a pattern matrix P : Fc\n2 →Fϕ\n2.\nThe cochain complex for the deformed code D of the full block reading denoted by P is\ngiven by (C ⊗R)• where\nR• = Fc\n2\nFϕ\n2\nP\n,\nFirst we check that the definition above is consistent with the definition of a pattern\nmatrix. We have\n(C⊗R)• = C0 ⊗Fc\n2\nC1 ⊗Fc\n2 ⊕C0 ⊗Fϕ\n2\nC2 ⊗Fc\n2 ⊕C1 ⊗Fϕ\n2\nC2 ⊗Fϕ\n2\nwhere we currently ignore the last C2 ⊗Fϕ\n2 term. We can pick out\nC0 ⊗Fc\n2 →C1 ⊗Fc\n2 →C2 ⊗Fc\n2\nas the c original codeblocks. Then,\nδ0\nC⊗R =\n \nH⊺\nX ⊗I\nI ⊗P\n!\n,\nδ1\nC⊗R =\n \nHZ ⊗I\n0\nI ⊗P\nH⊺\nX ⊗I\n!\nas for the example in Eq. 2 but where P is now a more general matrix. The Z logical\noperators in D are then\nH1(C⊗R) = H1(C)⊗H0(R)⊕H0(C)⊗H1(R) = ker HX/imH⊺\nZ⊗Fc\n2/imP⊺⊕C0/imHX⊗ker P⊺\nwhere ker P⊺= 0 as P has no surplus rows, so\nH1(C ⊗R) = ker HX/imH⊺\nZ ⊗Fc\n2/imP⊺.\nGiven that our initial code had Z logical operators given by ker HX/imH⊺\nZ ⊗Fc\n2, we see\nthat those in imP⊺are now stabilisers, and so block reading has measured precisely the\nrows of P on each logical qubit. Furthermore, no new equivalence classes are present, so\nthere are no new logical qubits.\n20\n\nRemark 3.7. Let P be LDPC and each identical codeblock be LDPC. Then the deformed\ncode (C ⊗R)• during measurement is LDPC, and the procedure uses O(cn) ancilla qubits.\nThis is immediate from Definition 3.6.\nLemma 3.8. The deformed code has code distance at least d.\nProof. The X-distance follows from Lemma 2.16.\nThe Z logicals are given by H1(C ⊗R) = ker HX/imH⊺\nZ ⊗Fc\n2/imP⊺. We start with a\nZ logical ΛZ composed of different Z logicals v, v′, v′′, ... in each codeblock, at least one of\nwhich must have weight at least d and be a nontrivial Z logical. Applying a new Z check\nin V only cleans support from the codeblocks if the incident data qubits all have support\nin ΛZ; otherwise the check applies at least 1 new Z to a data qubit in a codeblock, and\nthe weight of ΛZ cannot be reduced below d in this way.\nIf we apply new Z checks in V to clean support from ccl codeblocks, in the worst case\nwe have support |v| + |v′| + |v′′| + · · · −ccl|v ∩v′ ∩v′′ ∩· · · | left in the codeblocks. But\n|v| + |v′| + |v′′| + · · · −ccl|v ∩v′ ∩v′′ ∩· · · | = |v ∪v′ ∪v′′ ∪· · · | −|v ∩v′ ∩v′′ ∩· · · |\n≥|v∆v′∆v′′∆· · · |\n= |v + v′ + v′′ + · · · | ≥d\nwhere ∆is the symmetric difference of sets.\nShould the row weights of P be too high, meaning that the weights of these logical\nmeasurements are high and so the degree of the Tanner graph is high, then one can intro-\nduce ancilla blocks to break those measurements up, making P sparse. This corresponds\nto adding new bits, i.e., columns to the check matrix P, such that a row can be split up\ninto multiple rows. The ancilla codeblocks used to reduce its weight can be initialised at\nthe same time as the code deformation procedure begins, and can be considered part of\nthe measurement hypergraph, where data qubits in ancilla blocks are edges, Z checks are\nvertices and X checks are cycles, as normal. Consequently, the data qubits are initialised\nin |+⟩. This can be thought of as thickening the hypergraph, see Figure 4 later on.\n3.3\nTime overhead of full block reading\nWe now return to our original example, Eq. 2 from Section 3.1. When computing the\ncochain complex in Lemma 3.3 we had an additional term (C ⊗R)3 = C2 ⊗F2 = C2,\nwhich corresponds to a collection of metachecks on the Z-checks of the deformed code.\nThus the full scalable Tanner graph for the logical measurement is\nX\nZ\nHX\nHZ\nQ\nX\nZ\nHX\nHZ\nQ′\nZ\nI\nI\nI\nI\nG\nE\nI\nI\nG\nV\n21\n\nwhere G = HZ and MZ = δ2\nC⊗R =\n\u0010\nI ⊗P\nHZ ⊗I\n\u0011\n=\n\u0010\nI ⊗P\nHZ\n\u0011\n.\nConsider first the case where measurement faults can occur only on checks in V. As\nG = HZ, if a measurement fault v /∈ker(HZ) then it is detected by the metachecks.\nIf v ∈ker(HZ) then either (a) it is equivalent to an X data qubit error on E or (b)\nit has weight on checks at least d.\n(b) follows from the fact that G = H⊺\nX, and the\nlowest weight vector in ker HZ\\H⊺\nX has by definition weight at least d. For (a), as all\nthe data qubits in E are initialised in the |+⟩basis, X errors leave the state invariant so\nthe logical measurement outcome is unaffected. Therefore to cause an incorrect logical\nmeasurement outcome, the measurement fault weight on V must be at least d, in the\ncase where measurement faults cannot happen elsewhere. Note further that any vector\nv ∈ker(G) representing an undetectable measurement fault on E, regardless of weight, is\nequivalent to an undetectable X data qubit fault on Q (or Q′), as multiplying by X faults\non bijective qubits leaves 0 on the Z checks in Q (or Q′).\nTo study the more general case it is fruitful to use the cohomology of (C ⊗R)•.\nH2(C ⊗R) = H2(C) ⊗H0(R) ⊕H1(C) ⊗H1(R),\nwhich is explicitly\nC2/imHZ ⊗ker P ⊕ker HZ/imH⊺\nX ⊗0 = C2/imHZ ⊗ker P.\nThus any Z measurement fault which is neither detected by metachecks nor equivalent to a\ndata qubit fault must be a linear combination of pairs of identical Z check faults in Q and\nQ′, which are not equivalent to X data qubit faults, and Z check faults which are equivalent\nto X data qubit faults. Note that the Z single shot distance dss\nZ of (C ⊗R)• can be very\nlow even in the more general case, as C2/imHZ can have low weight representatives.\nNevertheless, any measurement fault which includes Z checks in V must be able to be\ncleaned, by applying some subset of X errors to Q and Q′, so that it has support only on\nthe Z checks of Q and Q′. Only if the set of X errors which it is cleaned to is large does\nthe measurement fault cause a logical measurement fault.\nIn our analysis of the time overhead, we have first a weak result:\nProposition 3.9. Let Q, Q′, Q′′,...\nbe a set of identical CSS LDPC codeblocks with\ndistance d, and let P be a pattern matrix determining a full block reading.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and\nafter the full block reading procedure. Then full block reading performs logical measurement\nin O(1) rounds of syndrome measurements with fault-distance d.\nProof. See Appendix A.\nThe intuition is that in the timestep where the full block reading is performed, any\nmeasurement fault on the check qubits in V with weight < d which is undetectable by\nmetachecks is either (a) in imG and so does not affect the logical measurement outcome\nor (b) has a paired measurement fault on Z checks outside of V, in the original code,\nsuch that the measurement faults on the original code must extend to future and previous\ntimesteps to avoid detection.\nIn order to maintain the fault-distance of the entire procedure we must assume d\nrounds of syndrome measurements on the original code before and after block reading.\nThis is because there may be low weight faults on data qubits, Z checks and X checks\noutside of H which are not detected within 1 timestep, and so could lead to logical errors\nby connecting to timelike errors extending from the top or bottom boundaries. This is\nequivalent to d measurement rounds of ‘padding’ before and after block reading [108].\n22\n\nRemark 3.10 (State preparation). Proposition 3.9 does not allow one to generically use\nfull block reading on a single codeblock to prepare a logical |+⟩k or |0⟩k state in constant\ntime, because of the padding required to provably preserve fault-distance.\nRemark 3.11 (Frame change). At the end of the block reading protocol, the ancilla data\nqubits are measured out in the X-basis, and the X checks in the original code are un-\ndeformed back to their original incidence. These measurements do not commute with the\npreceding Z checks, and so have random outcomes, although products of these measurement\noutcomes which correspond to cycles in the hypergraph must be +1 in the absence of errors,\nas these are precisely the sets of X measurements which do commute with the preceding\nZ checks. Because each new data qubit is in 1-to-1 correspondence with an X check in\neach incident codeblock, the anticipated outcomes of X-checks in the original codeblocks\nare updated after the block reading protocol by taking a product with the outcomes of those\ndata qubit X measurements, inducing a frame change.\nWhen the X measurements on ancilla data qubits undergo faults, the frame is updated\nincorrectly. These measurement errors are therefore in bijection with X check errors on\nthe original code, detected by preceding and subsequent rounds of measurement.\nThe reason why preceding rounds of measurement can diagnose such frame errors is\nbecause in the measurement round where ancilla data qubits are measured out the product\nof such a measurement and its corresponding X-check in an original codeblock commutes\nwith the Z checks in the deformed code, and so must yield a +1 outcome in the absence\nof errors. Thus having a consistent history of checks before the frame change can detect\nframe errors.\nProposition 3.9 is at first not particularly helpful, because the overall time cost of\nthe procedure is still O(d). However, the proof extends to performing full block read-\nings sequentially without padding between logical measurement rounds, with each logical\nmeasurement taking O(1) rounds.\nTheorem 3.12. Let Ξ be a set of η full block reading procedures of Z-type such that no\nblock reading procedure in Ξ is a product of any others.\nThen all η logical measurement rounds of full block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the full set of procedures.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. See Appendix A.\nRemark 3.13. In the above statement, the assumption that the pattern matrix is full rank\nis for technical reasons related to our proof technique. We expect that this requirement can\nbe relaxed.\n3.4\nFull block reading of CSS-type\nFull block reading can be generalised to measure sets of commuting operators of the form\nZ ⊗Z ⊗· · · ⊗Z and X ⊗X ⊗· · · ⊗X simultaneously. This allows for fast code switching\ninto a transversally concatenated code, similar to a procedure in Ref. [66] and the basic\noperation in the protocol of Ref. [109].\nTo see this, let PZ be the pattern matrix for Z-type logical measurements. Recall that\nthe rows of PZ each specify a set of parallel Z-type measurements as per Definition 3.5.\nIn order for us to perform X-type block reading at the same time, the X-type logical\noperators to measure must commute with the Z-type logical operators. Thus each row s\n23\n\nof a PX pattern matrix, specifying a set of parallel X-type logical measurements, must\nsatisfy r · s for each row r of PZ, where · is the usual dot product on vectors over F2.\nTherefore, PZP⊺\nX = 0.\nOur pattern matrix is now upgraded to a pattern complex.\nDefinition 3.14 (CSS-type full block reading). A full block reading of CSS-type takes as\ninput c CSS LDPC codes, each with cochain complex C•, and a pattern cochain complex\nR•.\nThe cochain complex for the deformed code of the full block reading denoted by P is\ngiven by (C ⊗R)• where\nR• = Fθ\n2\nFc\n2\nFϕ\n2\nP⊺\nX\nPZ\nwith R−1 = Fθ\n2, R0 = Fc\n2 and R1 = Fϕ\n2, and we assume that both PZ and PX are full rank\nwith no redundant rows, and that all rows have weight greater than 1.\nAs the tensor product of cochain complexes is also a cochain complex, the checks in\nthe deformed code commute. We have\n(C ⊗R)−1 = C0 ⊗R−1\n(C ⊗R)0 = C1 ⊗R−1 ⊕C0 ⊗R0\n(C ⊗R)1 = C2 ⊗R−1 ⊕C1 ⊗R0 ⊕C0 ⊗R1\n(C ⊗R)2 = C2 ⊗R0 ⊕C1 ⊗R1\n(C ⊗R)3 = C2 ⊗R1\nand\nH1(C ⊗R) = C2/imHZ ⊗ker P⊺\nX ⊕ker HZ/imH⊺\nX ⊗ker PZ/imP⊺\nX ⊕C0 ⊗R1/imPZ\nwhich given that ker P⊺\nX = 0 and R1/imPZ = 0 reduces to\nH1(C ⊗R) = ker HZ/imH⊺\nX ⊗ker PZ/imP⊺\nX.\nA similar calculation gives\nH1(C ⊗R) = ker HX/imH⊺\nZ ⊗ker PX/imP⊺\nZ,\nso we see that logical Z operators in imP⊺\nZ, i.e. the row space of PZ, are now stabilisers,\nand the same for logical X operators in the row space of PX, for every logical qubit in the\ncodeblocks.\nThat the distance of the CSS-type block reading deformed code is at least d is imme-\ndiate by applying the arguments of Lemma 3.8 twice, first for the Z-type auxiliary system\nand then for the X-type.\nIn the CSS-type full block reading measurement protocol, all new data qubits included\nin the Z-type (X-type) measurements are initialised in |+⟩(|0⟩) states respectively. All\nsyndromes are measured for O(1) rounds, and then all new data qubits are measured out\nin the X (Z) bases respectively.\nRemark 3.15. Let R• be LDPC and each identical codeblock Q, Q′, Q”... be LDPC. Then\nthe deformed code (C ⊗R)• is LDPC.\n24\n\nLemma 3.16. Let Ξ be a set of η full block reading procedures of CSS-type such that no\nblock reading procedure in Ξ is a product of any others, and each of the measurements\ncommute.\nThen all η logical measurement rounds of full block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the full set of procedures.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. See Appendix A.\nIn Appendix B we also show that full block reading can be extended to perform non-\nCSS measurements, namely those with X ⊗Z and Y terms, on CSS codes, but that\nour constructions for these require specific structure on the codes, such as transversal\nS gates, so they add no extra computational power to full block reading with CSS-type\nmeasurements.\nFull block reading gives no addressability on logical qubits within a codeblock, as\nthe measurements are on all logical qubits, so the utility of full block reading is limited\nin the same manner as 1-to-1 transversal CNOT gates between CSS codes. The logical\ncomputational power of a full block reading is identical to initialising an ancilla block,\nthen applying transversal CNOTs, then measuring out the ancilla block.\nConsequently, it is unclear how useful full block reading would be for practical fault-\ntolerant computation.\nCompared to measurement using transversal gates and ancilla\ncodeblocks, full block reading requires fewer qubits overall, but full block reading also\nrequires more connections between the initial codeblocks and the auxiliary system.\nNevertheless, our analysis of full block reading sets the stage for the more sophisticated\noperations in the next sections. We first consider the case of reading a subcode, then\npresent our general result on fast hypergraph surgery.\n4\nPartial block reading\nReturning again to the example in Eq. 2, consider a different generalisation. We start\nwith two codeblocks Q and Q′, which may be different,\nX\nZ\nHX\nHZ\nQ\nX\nZ\nH′\nX\nH′\nZ\nQ′\n25\n\nand construct the measurement hypergraph H to connect only to a subcode of each, but\nwhere the subcodes are identical.\nX\nZ\nHX\nHZ\nQ\nX\nZ\nH′\nX\nH′\nZ\nQ′\nZ\nMl\nMr\nFl\nFr\nG\nE\nV\nThat is, up to permutation and the addition of all-zero rows, the matrices satisfy\nMl = Mr;\nFl = Fr.\nGiven an appropriate choice of subcode and measurement hypergraph, the measurements\nare only between those logical qubits whose logical representatives have connections to the\nhypergraph, and so are contained in the subcode. This is called a partial block reading.\nThe construction no longer yields a tensor product of cochain complexes, so we in-\nstead use mapping cones from Section 2.2.2 to describe partial block reading.\nAs the\nconstructions become quite complicated, we also refrain from studying multiple partial\nblock readings occurring simultaneously, unlike in the full block reading case. All par-\ntial block readings are assumed to be performed sequentially, albeit with only a constant\nnumber of syndrome rounds between them.\nDefinition 4.1. Let A be a subcode of Q, Q′, Q′′..., each of which have chain complexes\nC•, C′\n•, C′′\n• ..., such that there is a collection of injective chain maps f•, f′\n•, f′′\n• ... of the\nforms\nf• : A• →C•;\nf′\n• : A• →C′\n•;\nf′′\n• : A• →C′′\n• ...\nThen the partial block reading of Z type defined by {f•, f′\n•, f′′\n• ...} is\nD• = cone(h•);\nh• = f• + f′\n• + f′′\n• + ...\nExplicitly, D• is the chain complex\nA1\nA0\n0\nC2\nC1\nC0\nC′\n2\nC′\n1\nC′\n0\nC′′\n2\nC′′\n1\nC′′\n0\n...\n...\n...\n∂A\n1\n0\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n⊕\n26\n\nwhere labels for the differentials in each codeblock, and labels for the chain maps, have been\nsuppressed for visibility. Note that also A• generally possesses a nonzero A2 component.\nThis component corresponds to metachecks, so we ignore that term for now.\nIt is trivial to compute, using the Snake Lemma [90, Lem. 1.3.2], that the measured\nlogical qubits are those in each code which have Z logical representatives contained in the\nsubcode A•.\nAs a Tanner graph, we have\nX\nZ\nHX\nHZ\nQ\nX\nZ\nH′\nX\nH′\nZ\nQ′\nZ\nf0\nf⊺\n1\nG\nX\nZ\nH′′\nX\nH′′\nZ\nQ′′\nf′\n0\nf′\n1\n⊺\nf′′\n0\nf′′\n1\n⊺\nE\nV\n(3)\nfor a partial block reading between three codeblocks.\nObserve that the chain maps\nf•, f′\n•, f′′\n• determine the connectivity between the blocks and the auxiliary hypergraph,\nand that E = A0, V = A1.\nUnlike when performing full block reading, the code distance of the deformed code D•\nin partial block reading is not generically preserved, and can drop below d. Elementary\nexamples of this phenomenon include surface codes with defects and toric codes, which\nwe detail in Section 8.\nThis drop in distance can come from two causes. First, it is possible to multiply existing\nZ logicals in the codeblocks by new Z checks in order to lower the weights below d. Second,\nviewing the ancilla system defined by the mapping cone as a hypergraph H, there may\nbe cycles in H which are nontrivial X logicals, meaning that they do not correspond\nto stabilisers of the deformed code. When unmeasured, the cycles have symplectically\npaired Z logicals, which when multiplied by other logicals in the original code can result\nin weights lower than d [52].\nIn this section, we focus on partial block reading where the subcode has distance d, and\nthe measurements are assumed to be in the Z-basis. Dualising appropriately yields the\nsame results for the X-basis. For the first cause of the deformed code distance dropping,\nwe can thicken the hypergraph. This is equivalent to taking a tensor product of A• and\nthe dual of a repetition code P• before defining the mapping cone, as shown in Figure 2.\nWe can define chain maps from any copy of A• in the tensor product to Q, and the same\nfor Q′, Q′′..., and this choice does not need to be the same for each. An example is shown\n27\n\nX\nA1\n(a) A subcode A•,\nX\nX\n· · ·\nX\n(b) P•, the dual of a repetition code,\nX\nX\nZ\nX\nZ\n· · ·\n· · ·\nA1\nA1\nA1\nA⊺\n1\nA⊺\n1\nI\nI\nI\nI\nI\nI\nI\nI\n(c) The tensor product (P ⊗A)•,\nZ\nG\nE\nV\nZ\nG\nE\nV\nX\nZ\nG\nE\nV\nX\n· · ·\n· · ·\n(d) The thickened hypergraph H, which is (P ⊗A)• with degrees shifted by 1.\nFigure 2: Constructing a thickened hypergraph. Note that (b) is a conventional Tanner graph, not a\nscalable Tanner graph.\nin Figure 3\nLemma 4.2. Let (P ⊗A)• be the thickened subcode, and let f•, f′\n•, f′′\n• ... be chain maps\nfrom any copies of A• in (P ⊗A)• into Q, Q′, Q′′..., with h• = f• + f′\n• + f′′\n• · · · .\nThen if (P ⊗A)• has been thickened at least d times the deformed code D• has distance\nat least d.\nProof. Immediate from Corollary 7.10, Lemma 7.12 and Remark 7.11 further down, so we\ndefer the proof until then.\nThis is similar to devised sticking [59] and the original CKBB scheme [52] in that the\nauxiliary system is thickened d times, but in this case we are measuring many representa-\ntives of the same equivalence classes of operators simultaneously. As in the CKBB scheme,\nthickening d times is generally overkill [54], and distance preservation commonly requires\na smaller ancilla system than this upper bound. Indeed, in Section 6 we demonstrate\nthat soundness properties of the subcode can be used to reduce the required thickening\nwhile provably preserving the distance, and even that gives a sufficient but not gener-\nally necessary amount of thickening. If thickening d times were truly necessary then our\ntime overhead of d would essentially be replaced by a space overhead factor of d, at least\n28\n\nZ\nG\nE\nV\nZ\nG\nE\nV\nX\nZ\nG\nE\nV\nX\n· · ·\n· · ·\nX\nZ\nHX\nHZ\nQ\nX\nZ\nH′\nX\nH′\nZ\nQ′\nX\nZ\nH′′\nX\nH′′\nZ\nQ′′\nFigure 3: Example of partial block reading with a thickened hypergraph. The logical measurement is\nthe same as in Eq. 3, but the hypergraph has been thickened to prevent the weight of Z operators\ndropping below d.\nwhen A• is large, which is interesting but less useful than achieving a genuine reduction\nin spacetime overhead.\nFor the second cause of the deformed code distance dropping, that there can be cycles\nin H which correspond to X logicals, we use the following fact.\nAs the subcode has\ndistance d, the logical measurement protocol can also be performed in 1 round of error\ncorrection, similar to full block reading, by introducing redundant Z checks between each\nlevel of the hypergraph as shown in Figure 4.\nThis fact helps us to gauge-fix the cycles, because each data qubit in each cycle is\ninitialised in |+⟩immediately before and measured out in the X basis immediately after\nthe single round of error correction. The results of those X measurements can be used to\ninfer the outcome of X stabiliser measurements which would fix the gauges of the cycles,\nand so we do not need to explicitly add checks for those cycles [56].\nHowever, if the\nset of cycles does not have a basis of low weight cycles, this can result in detectors with\nhigh weight in the space direction, which impinge on the fault-tolerance against stochastic\nnoise.\nProposition 4.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d,\nand let A• be a subcode with distance d, with chain maps from A•, or a thickened version\nthereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code, which has code distance\nat least d.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before\nand after the partial block reading procedure. Then partial block reading performs logical\n29\n\nZ\nG\nE\nV\nZ\nG\nE\nV\nX\nZ\nG\nE\nV\nX\n· · ·\n· · ·\nX\nZ\nHX\nHZ\nQ\nX\nZ\nH′\nX\nH′\nZ\nQ′\nX\nZ\nH′′\nX\nH′′\nZ\nQ′′\nZ\nZ\nFigure 4: The same partial block reading as in Figure 3 but with additional redundant checks and\nmetachecks.\nmeasurement in O(1) rounds of syndrome measurements with fault-distance d.\nProof. See Appendix A.\nThe key here is that when thickening, each Z check error in level a is equivalent to a\nZ check error in level a −1 along with an X error on the data qubit between them. In\nthis way, we form a chain of metachecks up the thickened hypergraph. These allow all\nstabilisers in the subcode to be matched to sets of checks in each level of the hypergraph.\nEvidently, if the subcode A• has nA qubits, and the amount of thickening required is L,\nthen the space cost of a partial block reading is O(nAL).\nThe above proposition considers performing one partial block reading with d rounds\nof padding before and after the procedure. Once again, we would like to generalize this\nto performing t ≥d operations in O(t) time, thereby achieving constant time overhead in\namortisation. This introduces an extra complication. Instead of the deformed code, we\nconsider the compacted code, defined as follows.\nDefinition 4.4 (Compacted code). Let C• be a complex representing the memory CSS code\nblocks, and let A1,•, · · · , At,• be chain complexes each with a chain map fi,• : Ai,• →C•.\nThe compacted code is the code taken by applying all the mapping cones to the original\ncode, i.e.,\nCC• = cone(\nX\ni\n(fi,•).\nWhile this code is most likely not LDPC and potentially massive, it is just a proof\nconstruct and is never explicitly used in our surgery operations.\n30\n\nThere is an intuitive reason behind defining the compacted code. Ideally, we would like\nto construct partial block readings assuming just that every deformed code has distance d.\nThis assumption, however, may not be sufficient: if we perform two partial block readings\none after another with only O(1) syndrome rounds between them, we must carefully\nconsider how the weight of an unmeasured logical may be reduced by multiplication with\nthe new deformed code checks. While, by assumption, both the first and second partial\nblock readings would individually guarantee any weight-reduced operator remains weight\nd, there would be no such guarantee when both weight reductions occur together. This may\nseem irrelevant – after all, we are not performing the partial block readings simultaneously\n– but it happens that a sequence of check measurement errors in the O(1) syndrome\nrounds between the block readings can effectively reproduce this simultaneous scenario\nusing potentially just O(1) additional faults (since we are not assuming the original code\nis sound).12 Therefore, to ensure the overall surgery protocol is fault-tolerant, we impose\nthe condition that the compacted code, which can be seen as the memory blocks deformed\nby all the surgery operations applied simultaneously, has distance at least d.\nTheorem 4.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and\nlet Ξ be a set of Z-type partial block readings {Partial1, Partial2, Partial3, ..., Partialη}\nwhere each subcode has distance d and the compacted code has distance d.\nThen all η logical measurement rounds of partial block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the set of logical measurements.\nThe procedure has phenomenological\nfault-distance d, for any η ∈N.\nProof. See Appendix A.\nWhile it may seem daunting to study the compacted code and satisfy the distance con-\ndition, we show in Section 7 that standard techniques in code surgery, notably thickening\nand expansion, can be applied to satisfy this condition. We further remark that in the\ncase of full block reading with a full rank pattern matrix P, the resultant compacted code\nis a homological product code (C ⊗R)•, which has distance at least d.\nRemark 4.6. In Section 9 we study the relationship between surgery and homomorphic\nmeasurement [42]. For now, observe that given a subcode of distance d performing homo-\nmorphic measurement is straightforward, as one initialises an ancilla system that is a copy\nof that subcode and performs transversal CNOTs and single-qubit measurements. This ini-\ntialisation can take up to O(d) rounds to preserve fault-tolerance. Note that the method\nof algorithmic fault-tolerance [84] does not apply to generic homomorphic measurements.\nPerforming partial block reading with a distance d subcode requires additional conditions\non the distance of the deformed code to maintain fault-tolerance, which can be resolved at\nsome additional space cost by the thickening described above.\n5\nFast hypergraph surgery\nIn this section we generalise our partial block reading results to the case where A• is not\na subcode but some other complex with a suitable chain map into the original code.\n12We note that this simultaneous weight reduction issue, a scenario not considered in detail by prior code\nsurgery work, is also a concern when doing auxiliary graph surgeries [56, 58] separated by O(1) syndrome\nrounds if one does not assume expansion of the auxiliary graphs and instead just assumes large deformed\ncode distances. In practical constructions, e.g. Ref. [55], where one indeed tends not to guarantee expansion\nbut just deformed distance, this is an important issue to be aware of.\n31\n\nA2\nA1\nA0\nA−1\nC2\nC1\nC0\n0\nf2\nf1\nf0\n0\n0\nDefinition 5.1. We say that the chain map f• : A• →C• is non-overlapping on vertices\nif each row in the matrix f1 : A1 →C1 contains at most a single non-zero entry.\nThe degree 1 of A• is shifted by the mapping cone construction to be in cone(f)2, i.e.\nthe set of vertices in the hypergraph is the set of basis elements of A1.\nWe can directly transport Definition 4.1 to this more general setting.\nDefinition 5.2. Let A• be an auxiliary chain complex and let Q, Q′, Q′′... be CSS LDPC\ncodes, each of which have chain complexes C•, C′\n•, C′′\n• ..., such that there is a collection of\nsparse chain maps f•, f′\n•, f′′\n• ... of the forms\nf• : A• →C•;\nf′\n• : A• →C′\n•;\nf′′\n• : A• →C′′\n• ...\nwhich are non-overlapping on vertices.\nThen the hypergraph surgery of Z type defined by {f•, f′\n•, f′′\n• ...} is\nD• = cone(h•);\nh• = f• + f′\n• + f′′\n• + ...\nAs with partial block reading it is trivial to compute which logical operators are mea-\nsured using the Snake Lemma [90, Lem. 1.3.2].\nTheorem 5.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d,\nand let Ξ be a set of Z-type hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη}\nsuch that:\n• Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance d, with\nsparse cycle bases.\n• The compacted code has distance at least d.\nThen all η logical measurement rounds of generalised hypergraph reading can be per-\nformed sequentially using O(1) rounds of syndrome measurement each, and using d rounds\nof padding before and after the set of logical measurements. The procedure has phenomeno-\nlogical fault-distance d, for any η ∈N.\nProof. See Appendix A.\nIn the mapping cone that describes the surgery operation, the 1-cosystolic distance\ncaptures the metacheck distance. To satisfy this distance condition, it is often easier to\nstart with a homomorphic, high distance quantum code, as we did in full block reading.\nReturning to the second condition, in Section 7 we define a notion of hypergraph\nexpansion called modular expansion and show that it is a sufficient condition for the\ncompacted code to have distance d. We then show that a generic hypergraph can always\nbe thickened (by up to d layers) to a graph with the desired expansion properties. This\nis in analogy with prior surgery works studying graphs, where it is known that thickening\ncan be used to boost a property called relative expansion [57]. We remark that for both\nour work and prior works, expansion is a sufficient but not necessary condition, in practice\none can often find good constructions heuristically, such as adding random hyperedges.\n32\n\nRemark 5.4. In a generic hypergraph surgery, constant-weight local operators are mea-\nsured alongside high-weight global operators. If we require that the original codespace is\nrestored after the hypergraph surgery, the constant-weight local operators must all be in\nthe stabiliser group, and the high-weight operators must be logical operators. For a Z-\ntype measurement on a single code block, with an injective f1 function, this requires the\ngroup generated by F2-linear combinations of the hyperedges to include the truncation of\nthe X-type stabiliser group to the qubits addressed by the measurement. In the previous\nsections we have chosen the X-type checks of the subcode, but more generally we can pick\na different, potentially overcomplete, generating set.\n6\nIntermediate time overhead with locally testable codes\nWe can consider (a) block reading with a distance d subcode and (b) conventional LDPC\ncode surgery, which uses a distance 1 subcode, to each sit at opposite ends of an axis\ndetermining the time cost of logical measurements. In this section we consider interpolat-\ning along this axis, to find surgeries which use between O(1) and O(d) time each, while\nremaining fault-tolerant.\n6.1\nPartial block reading with a low distance subcode\nProposition 6.1. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d,\nand let A• be a subcode with distance dA = 1\nαd, with chain maps from A•, or a thickened\nversion thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code with code distance\nd and let H have a sparse cycle basis.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and\nafter the partial block reading procedure. First, ⌈α⌉rounds of syndrome measurements are\nnecessary (but not sufficient) to perform partial block reading with fault-distance d.\nThen if either: (a) α ≤2 or (b) the initial codeblocks are each locally testable codes with\nsoundness ρ ≥2n\nm , where n is the codeblock length and m the number of checks, then partial\nblock reading performs logical measurement in ⌈α⌉rounds of syndrome measurements with\nfault-distance d, maintaining the LDPC property throughout.\nProof. See Appendix A.\nRemark 6.2 (Cycle checks). When the block reading takes longer than O(1) round we\ngenuinely require a sparse cycle basis for H to maintain the LDPC property of the de-\nformed code, as the gauge checks at each timestep cannot easily be inferred from the state\npreparation and measurement of data qubits in the auxiliary system. In general, it is not\nguaranteed that H possesses a sparse cycle basis. In practice we expect to find sufficiently\nsparse cycle bases numerically for small codes.\nAdditionally, if a subcode has low weight X metachecks which have support on checks\nin the subcode, after degree shifting these metachecks become low weight cycle checks for a\nZ basis partial block reading.\nProposition 6.1 is interesting primarily because there are codes that are not expected\nto have distance d subcodes addressing targeted subsets of logical qubits. Indeed only\ntopological codes and certain product codes are known to allow for addressing any logical\nqubit in this manner [42, 43]. Instead, one can find subcodes with distance between 1 and\nd which still permit fast measurement, but not necessarily in constant time. The distance\nof A• is the minimum number of check errors on the auxiliary system which are required\nto cause an undetectable logical measurement error in a single round; this is not the same\n33\n\nas the single-shot distance of the entire deformed code, because there can be checks in the\nbulk which have far less protection against measurement errors.\nHowever, the errors which can occur in the bulk do not prevent us from performing\nmany of these partial block readings sequentially without d rounds of padding in between,\nimplying that interpolated partial block reading can indeed speed up computation sub-\nstantially.\nTheorem 6.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least\nd, and let Ξ be a set of partial block readings {Partial1, Partial2, Partial3, ..., Partialη}\nwhere each subcode has distance di ≥\n1\nαi d for i ∈[η] and the deformed codes each have code\ndistance d, with sparse cycle bases. Suppose the original codes have soundness ρ ≥2n\nm .\nThen all η logical measurement rounds of partial block reading can be performed sequen-\ntially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the logical measurements.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. See Appendix A.\nWe remark that when the base codes are locally testable codes with constant soundness,\nwe only require the weaker condition that each deformed code has distance d, instead of\nthe stronger condition that the compacted code has distance d. Intuitively, this is because\nwhen considering the potential weight reduction of a logical fault that is deformed into\ntwo different surgery ancilla systems, the support of that logical fault is necessarily shifted\nonto timelike check errors, which are proportional to the reduced space support as the\ncodes have constant soundness.\nRemark 6.4 (Interpolated surgery with lower soundness). Proposition 6.1 and Theo-\nrem 6.3 demand soundness ρ ≥2n\nm , which is asymptotically constant as m ∈Θ(n) for\nLDPC codes. If the soundness is lower than constant, which one may expect for practical\ncodes, it is sufficient to use asymptotically more rounds of syndrome measurement per\nlogical measurement, proportional to 1/ρ, to maintain phenomenological fault-distance d.\nFurthermore, one cannot guarantee that the surgeries can be performed within O(1) time\nof each other, without considering the compacted code (see Def. 4.4). Scaling the time\nbetween logical measurements to O(1/ρ) is then sufficient to preserve fault-distance.\nIf the soundness is vanishing then there is still a reduction in time overhead from\nmeasuring multiple representatives, but the reduction is additive rather than multiplicative,\nsee Proposition 9.3, and one is forced to consider the compacted code to ensure that the\nlogical measurements can be performed close together in time.\n6.2\nIntermediate time hypergraph surgery\nSimilar to the previous section on partial block reading, we can generalize Theorem 6.3 to\nthe case of hypergraph surgeries.\nTheorem 6.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d,\nand let Ξ be a set of hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη} such\nthat:\n• Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance di ≥\n1\nαi d for i ∈[η], with sparse cycle bases.\n• Each deformed code has distance at least d.\n34\n\n• The original codes have soundness ρ ≥2n\nm .\nThen all η logical measurement rounds of hypergraph surgery can be performed sequen-\ntially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the logical measurements.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. This follows by combining the proofs of Theorem 5.3 and Theorem 6.3.\nGauging logical measurement [56] can be seen as an instance of hypergraph surgery\nwhere the subcode A• has distance 1, and therefore requires O(d) rounds of syndrome\nmeasurement to maintain fault-distance d.\n7\nModular expansion\nIn previous sections, we studied hypergraph surgery operations where the compacted code\nor the deformed codes has distance d. This is critical for fault-tolerance. In this section,\nwe discuss sufficient expansion properties on the hypergraph for these conditions to hold.\nIn context of past surgery works, when A• contains only one logical operator (so α = d and\ndA = 1), partial block reading recovers the CKBB scheme [52]. It is known that there are\nsubstantially more efficient schemes for measuring individual logical operators in O(d) time\nby surgery [56, 58], using expander graphs. This section is essentially an extension of these\nresults to hypergraphs, which can measure multiple logical representatives simultaneously\nand quickly.\nDefinition 7.1. Let H(V, E) be a hypergraph, with edge-vertex incidence F2-matrix G.\nLet I be an index set of basis elements vi of a subspace W ⊂F2V. Then we can index\nelement in W as κλ for λ ∈2I.\nWe conflate κλ as a vector and as a set of vertices in V. To each basis element vi we\nassociate its set of vertices Vi = supp(vi), and define the vertex set U = S\ni\nVi.\nDefinition 7.2 (Modular expansion). Let H(V, E) be a hypergraph with incidence matrix\nG : F2E →F2V and specified subspace W ⊂F2V containing elements κλ. Let t ∈R+ be\na positive real number. The modular expansion Mt of H is the largest real number such\nthat, for all v ⊆V,\n|G⊺v| ≥Mt min(t, |κλ| −|v ∩κλ| + |v ∩U\\κλ| : κλ ∈W)\nAs this definition is quite complicated, we give some orienting remarks.\nModular\nexpansion is a generalisation of both relative expansion from Ref. [57] and the soundness\nof a classical locally testable code, and is therefore also as generalisation of the edge-\nexpansion of a graph. It is ‘modular’ in the sense that it contains distinguished subsets κλ\nsuch that, when they all exist in the same hypergraph, the notion of expansion for each\nof them combines to give a complete notion of expansion for the whole hypergraph.\nIn detail, recall that edge-expansion (henceforth called global expansion) can be defined\nfor hypergraphs as follows.\nDefinition 7.3 (Global expansion). Let H(V, E) be a hypergraph with incidence matrix\nG : F2E →F2V. The global expansion of H is the largest real number β such that\n|G⊺v| ≥β min(|v|, |V\\v|).\n35\n\nWhen G is a graph this reduces to traditional edge-expansion and β is the Cheeger\nconstant.\nRef. [57] defined relative expansion, a generalisation of global expansion.\nDefinition 7.4 (Relative expansion). Let H(V, E) be a hypergraph with incidence matrix\nG : F2E →F2V, distinguished subset U ⊂V and chosen parameter t ∈R+. The relative\nexpansion of H is the largest real number βt such that\n|G⊺v| ≥βt min(t, |v ∩U|, |U| −|v ∩U|).\nWhen t ≥|V| and U = V this reduces to the definition of global expansion. Similarly,\nwhen |I| = 1 modular expansion reduces to relative expansion, observing that there are\nonly two κλ vectors, one being the empty set and the other U.\nA different generalisation of global expansion is to the soundness of a locally testable\ncode. Recall Definition 2.10, with variables replaced to be suggestive of our hypergraph\nsetting.\nDefinition 7.5. A binary linear code C ⊂F2V is (ω, ρ)-locally testable if it has a parity-\ncheck matrix G⊺with rows of weight at most ω such that for any vector v ∈F2V,\n1\nm|G⊺v| ≥ρ\nnd(v, C)\nwhere m = |E|, n = |V| and d(v, C) = minx∈C(|v + x|). The values ω and ρ are the locality\nand soundness of the code respectively.\nIn particular, if we rearrange the equation defining soundness ρ then we get |G⊺v| ≥\nβd(v, C) for β = ρm\nn . We observe that in the case where the only codewords are 0 and\nV, β coincides with the definition of global expansion. For any sparse hypergraph H, if\nit has global expansion it is also a locally testable code, albeit a rather trivial one. In\nthis way the soundness of a code can be thought of as a generalisation of expansion to\nthe case where a hypergraph has ker G⊺with dimension greater than 1. As dim G⊺is\nequal to the number of logical representatives which are measured simultaneously by the\nauxiliary hypergraph, we require a more general definition than relative expansion in order\nto perform block reading.\nModular expansion coincides with soundness when U = V and t ≥|V| under the\nrearrangement Mt = ρm\nn , and when W = ker G⊺, so each element κλ is an element of the\nkernel. We thus have the informal commuting diagram,\nModular expansion\nRelative expansion\nk = 1\nSoundness of an LTC\nt ≥n\nGlobal expansion\nk = 1\nU = V\nt ≥n\nU = V\nwhere k = dim G⊺, n = |V|, and each κλ is assumed for simplicity of the diagram to\ncorrespond to an element of ker G⊺.\n36\n\nRemark 7.6. For a simply connected graph, the edge-expansion can be related to spec-\ntral properties of the adjacency matrix or Laplacian of the graph. We do not know of a\nsimilar relation between relative expansion, soundness or modular expansion and spectral\nproperties of the Laplacian of the hypergraph.\nTo explain our results on hypergraphs with modular expansion we use the terminology\nof port functions, similar to Ref. [57]. In short, given a CSS code Q and a hypergraph\nH(V, E), a port function fp is a map from a subset of data qubits in Q to a set of vertices\nin V. The deformed code is then defined as Q ↔H, with connections introduced between\nQ and H given by the port function fp. This definition of the deformed code is similar to\nthe definition of the mapping cone by a chain map, although the port function does not\nspecify the connectivity of deformed checks into the auxiliary hypergraph, and the port\nfunction is in the opposite direction to f1 in a chain map. Moreover, the port function\ndefinition can be applied to codes which are not CSS.\nTheorem 7.7 (Distance preservation). Let {Zi}i∈I be a set of logical representatives\n(trivial or nontrivial) on a distance d CSS code Q with supports ξi = supp(Zi).\nLet\nH(V, E) be a hypergraph with vertices in V being Z checks, hyperedges in E data qubits and\nX gauge checks forming a cycle basis, such that H measures all the logical representatives\n{Zi}i∈I. Let fp : S\ni ξi →V be an injective port function such that U = imfp ⊂V, and\nfp(ξi) = Vi ⊂V, ∀i ∈I. Let Md(H) ≥1, where W is generated by fp(ξi). Then the\ndistance of the deformed code Q ↔H is at least d.\nProof. See Appendix C.\nAs the above proof demands the port function fp from the original code to the hy-\npergraph be injective, it does not capture the most general form of hypergraph surgeries.\nIn that case, scaling the modular expansion proportional to the check incidence to the\noriginal code is sufficient to preserve the distance, see Remark 7.16.\nTheorem 7.8 (Thickening). Let H(V, E) be a hypergraph with modular expansion Mt(H).\nLet J L be the path graph with length L ≥\n1\nMt(H), i.e. L vertices and L −1 edges. Let\nHL := H□J L be the hypergraph of H thickened L times. Then HL has modular expansion\nMt(HL) ≥1 for Uℓ= S\ni\nV ℓ\ni at any level ℓ∈{1, 2, · · · , L}, where V ℓ\ni is the copy of Vi in\nthe ℓth level of the thickened hypergraph.\nProof. See Appendix C.\nLemma 7.9. Let H(V, E) be a hypergraph such that each element in ker(G⊺) has the\nsupporting set κλ for some λ ∈2I. Then Mt(H) ≥1\nt .\nProof. Consider an arbitrary vertex set v. We consider two cases: first, where |G⊺v| = 0,\nand then where |G⊺v| ≥1.\nIn the first case, we know that v ∈ker(G⊺) and therefore v = κm for some m ∈2I.\nTherefore |κm| −|v ∩κm| + |v ∩U\\κm| = 0, and so |G⊺v| ≥Mt(H) min(t, |κλ| −|v ∩κλ| +\n|v ∩U\\κλ| : λ ∈2I) for any Mt(H).\nIn the second case, |G⊺v| ≥1. We now consider the two values of the minimum. If\nmin(·) = t then |G⊺v| ≥1\nt t. If min(·) = |κλ| −|v ∩κλ| + |v ∩U\\κλ| for some λ ∈2I then\nt ≥|κλ| −|v ∩κλ| + |v ∩U\\κλ|, and so |G⊺v| ≥1 ≥1\nt (|κλ| −|v ∩κλ| + |v ∩U\\κλ|).\nCorollary 7.10. Let H(V, E) be any hypergraph such that each element in ker(G⊺) has\nthe supporting set κλ for some λ ∈2I. Let Ht := H□J t be the hypergraph of H thickened\nt times. Then Ht has modular expansion at least Mt(Ht) ≥1 for any level ℓ.\n37\n\nProof. As Mt(H) ≥1\nt and J L in Theorem 7.8 has L = t then L ≥\n1\nMt(H).\nCorollary 7.10 post-hoc justifies devised sticking from Ref. [59] by setting t = d, as\nthere the hypergraph is thickened d times to preserve fault-tolerance, and the CKBB\nscheme [52] is a special case of this.\nRemark 7.11. The expansion-based argument does not account for the cycles in the hy-\npergraph, however. For these see Ref. [52, Thm. 1], which shows that when the hypergraph\nis thickened d times the cycles can be considered gauge logicals and the deformed code still\nhas distance at least d as a subsystem code.\nTheorem 7.8 shows that it is not necessary to thicken d times, if the modular expansion\nbefore thickening is greater than 1\nd. This has previously been proved for the simpler case\nof measuring only one logical operator [55, 57].\nLemma 7.12 (Cross-block measurements). Let H be a hypergraph with modular expansion\nMd(H) ≥1\nL. Let H correspond to the degree-shifted subcode A• with distance at least d.\nVertices in H are Z checks and edges are data qubits. Let f•, f′\n•, f′′\n• ... be the inclusion\nchain maps of A• into C•, C′\n•, C′′\n• ..., each of which has distance at least d.\nLet HL be the the hypergraph thickened L times, with modular expansion Md(HL) ≥1\nat each level ℓ∈{1, 2, · · · , L}. Then the deformed code D• given by modifying each chain\nmap f•, f′\n•, f′′\n• ... to have their preimages on any arbitrary levels of the hypergraph and\ntaking the mapping cone has distance d if all cycles are gauge-fixed.\nProof. As all cycles are gauge-fixed, there are no new logical qubits, and no X logical can\nhave weight lower than d. We must therefore ensure that no Z logical which is unmeasured\ncan have its weight reduced lower than d by cleaning.\nWe prove the lemma by contradiction.\nIf any Z logical extending over all initial\ncodeblocks C•, C′\n•, C′′\n• ..., but not the hypergraph, could have its weight reduced below d\nby cleaning into the hypergraph, then a single logical in one of C•, C′\n•, C′′\n• ... must also have\nits weight reduced below d by cleaning. This fact is because removing the chain maps to\nall other codeblocks would leave the same set of Z checks which could clean ΛZ to the\nsame data qubits σ, with the exception of data qubits in the other codeblocks. This set\nof data qubits must satisfy |σ| < d for this lemma to be false. However, we know that\nMd(HL) ≥1 at each level of the hypergraph, therefore |σ| ≥d and the lemma holds.\nBy a combination of Corollary 7.10 and Lemma 7.12, observe that the earlier Lemma 4.2\nfrom Section 4 holds.\nTo summarise this section, if a subcode A• for partial block reading has sufficient\nsoundness then it must always also have modular expansion, which can then be boosted by\nthickening an appropriate amount to guarantee the deformed code’s distance is preserved.\nRecall the definition of soundness for quantum codes in Def. 2.11.\nCorollary 7.13. Let A• be the subcode used in a partial block reading.\nLet A• have\nsoundness ρA.\nThen, viewed as a hypergraph H, thickening\n2nA\nρAmA times preserves the\ndistance of the deformed code when all cycles are gauge-fixed.\nProof. By Lemma 2.12, H has soundness ρAmA\n2nA as a classical code with parity-check matrix\nG⊺. Setting W = ker G⊺, H also has modular expansion Md ≥ρAmA\n2nA . By Lemma 7.12, set\nL =\n2nA\nρAmA and the thickened hypergraph HL has modular expansion Md(HL) ≥1.\nRemark 7.14. As mA\n2nA ∈O(1) for LDPC codes, the asymptotic space cost of partial block\nreading when all cycles are gauge-fixed can be reduced to O(nA\nρA ).\n38\n\nAssuming the modular expansion Md(H) before thickening is greater than 1\nd, which\nis its minimum by Lemma 7.9, this thickening requires fewer than the d layers required\nby the CKBB scheme and similar [52, 59]. Note that as the soundness of a subcode is\nnot generally inherited from the soundness of the full code, finding subcodes with high\nsoundness is generally difficult.\nRegardless, having hypergraphs equipped with modular expansion is extremely useful\nto perform fast surgery.\nLemma 7.15. If each hypergraph Hi associated to a partial block reading Partiali for\ni ∈[n] has modular expansion Md(Hi) ≥1 then the compacted code CC• has distance d.\nProof. By Theorem 7.7, attaching H1 to the original code via the chain map (h•)1, to yield\ncone((h•)1), must always preserve the distance of the code. Applying the other mapping\ncones iteratively will then each preserve the distance, so CC• has distance d.\nAs a consequence, the conditions of Theorem 4.5 can be satisfied constructively, by\nchoosing subcodes of distance d and then thickening such that each hypergraph has mod-\nular expansion.\nRemark 7.16. A similar approach applies to Theorem 5.3, where instead it is sufficient\nfor each hypergraph to have modular expansion Md(Hi) ≥wi, where wi is the maximal\ncolumn weight of (f1)i, (f′\n1)i, (f′′\n1 )i, · · · .\n8\nExamples\nIn this section we illustrate block reading with a series of examples. We focus first on\n2D surface codes, upon which traditional lattice surgery is extremely well-understood\n[106, 110, 111].\n8.1\nTopological codes\n8.1.1\n2D surface codes\nWe demonstrate on unrotated surface codes, as the connection to block reading, and LDPC\ncode surgery in general, is more apparent in this case.\nWe start with two square surface code patches with distance d, with n = d2 + (d −1)2\ndata qubits each,\n39\n\nIn these diagrams qubits are edges, X and Z checks are vertices and faces respectively.\nThe deformed code for the full block reading performing a Z ⊗Z is then given by\nwhere the new edges are dashed grey lines. There are new vertically-oriented faces\nwhere there are new squares, and at the boundaries.\nEvidently the connectivity requirements are close to the requirements for transversal\nCNOTs.\n40\n\nThe only distance d subcode (as defined in Def. 2.5) of the square surface code is itself,\nand so the only partial block reading with a distance d subcode is full block reading. A\npartial block reading using a subcode A• with distance dA less than d can be constructed\nby choosing a strip of the surface codes and then connecting these together.\nIn the simple case of surface code patches as above the distance is always preserved by\npartial block reading, because the only logicals which can have reduced weight are those\non Z ⊗Z or X ⊗X, which are now stabilisers. If the surface code patches also have\ndefects, or are toric codes instead, partial block reading does not generically preserve code\ndistance and increasing modular expansion in the auxiliary hypergraph is necessary, by\nthickening or otherwise.\nRecall that surface codes have asymptotically vanishing soundness by [78, Cor. 1] so\nusing a partial block reading with dA < d on the surface code is not particularly useful for\nreducing the time overhead, unless dA ≥d\n2 by Proposition 6.1.\nIn the case where dA = 1 and the chosen subcode is along the boundary of the patches,\nwe recover traditional lattice surgery with unrotated surface codes:\n41\n\n8.1.2\n2D toric codes\nPartial block reading with toric codes is illustrative, as it showcases how the distance of\nthe deformed code can drop without thickening.\nFirst, consider the following d = 5 toric codeblocks, shown by tessellating their fun-\ndamental polygons. Blue and red edge pairs of the same lattice are identified to form\nloops.\n42\n\nA partial block reading between them can be performed with a subcode corresponding\nto a cylinder around the torus. Measuring only one representative from each torus recovers\ntraditional surgery with 2D toric codes,\n43\n\nwhich measures Z ⊗Z on one logical qubit from each codeblock, but not the other.\nGrowing the width of the cylinder such that the distance of A• is 4, we acquire the\ndeformed code:\n44\n\nDepending on the chosen subcode, the distance may drop in the deformed code, as\nthere can be a stringlike Z operator which extends from one torus to the other and\nthen back, skipping the part of the torus with the new connections and making a short\nnoncontractible closed curve. In our example the deformed code distance is still 5, as these\ncurves take the form shown by green edges,\n45\n\nwhich have length at least 6. However, were we to start with d = 7 toric codes, with\nthe width of the cylinder being 6, the distance would drop as a consequence of these\nsame curves. Increasing the length of this closed curve is then possible by thickening the\nsubcode, or boosting the modular expansion in some other manner. If we were to perform\na full block reading, these closed curves would become contractible due to the additional\nstabilisers, so the deformed code distance would be preserved regardless.\n8.1.3\n4D toric codes\n4D toric codes are known to be decodable in a single-shot manner, as they have some\nsoundness and metachecks of both X and Z-type [78, 82, 83, 107].\nA 4D toric code can be constructed by a 4-fold tensor product of the chain complex\nR• = R1\nR0\n∂1\nwhere R1 = R0 = Fm\n2 and the differential ∂1 is the incidence matrix of the cycle graph\nwith m vertices and edges, so dim ker(∂1) = 1 and dim Fm\n2 /im(∂1) = 1.\nThe 4-fold tensor product yields a chain complex with 5 nonzero components. Let the\n4D toric code be\nC• = (R ⊗R ⊗R ⊗R)• = C4\nC3\nC2\nC1\nC0\n46\n\nand the qubit component is fixed to be C2.13 More explicitly we have\nC2 =\nM\ni+j+k+l=2\nRi ⊗Rj ⊗Rk ⊗Rl,\nand the entire chain complex is given by\nR1R1R0R0\nR1R1R1R0\nR1R0R1R0\nR1R0R0R0\nR1R1R0R1\nR1R0R0R1\nR0R1R0R0\nR1R1R1R1\nR0R0R0R0\nR1R0R1R1\nR0R1R1R0\nR0R0R1R0\nR0R1R1R1\nR0R1R0R1\nR0R0R0R1\nR0R0R1R1\nwhere tensor products are suppressed for visibility and columns of terms are taken in a\ndirect sum.\nThe K¨unneth formula implies that dim H2(C) = 6. The distance of C• is d = m2, which\ncan be computed using Lemma 2.8. The intuition is that all nontrivial logical operators in\na 4D toric code form membranes through the complex, as opposed to strings in 2D toric\ncodes. For a given summand in C2, say R1 ⊗R1 ⊗R0 ⊗R0, one choice of corresponding\nnontrivial Z logical operator representative is then 1 ⊗1 ⊗(1, 0, · · · , 0) ⊗(1, 0, · · · , 0),\nwhere 1 = (1, 1, · · · , 1).\nTo give a treatment of partial block reading with 4D toric codes we specify a subcode\nA•. There are evidently many such choices. One choice is to measure just a single logical\nrepresentative, which by Thm. 1.6 must take Θ(d) syndrome measurement rounds to\nmaintain phenomenological fault distance.\nAnother choice of A• is to specify a set of Z-checks in C3, and to take the summands\nin C2, in the image of that set of Z-checks and so on. This is trivially guaranteed to be a\nsubcode. For example, set A• to be\nR1R1R0F2\nR1R0R0F2\nR1R1R1F2\nR1R0R1F2\nR0R1R0F2\nR0R0R0F2\nR0R1R1F2\nR0R0R1F2\n13This shift in qubit degree compared to convention is for convenience, so that there are no negative\nindices and basis elements in C2 can be identified with faces in a cell complex.\n47\n\nwhere the chain map component f3 maps R1R1R1F2 in A3 into R1R1R1R0 in C3 and so\non, choosing a consistent basis element of R0 to map F2 into. This choice measures 3 Z\nlogical operators, and many representatives of each. This is the choice taken in Ref. [107],\nand can be seen as setting A• to be a hyperplane of the lattice. While each Z logical\nrepresentative contained in A• must have weight at least m2, the X-distance of A• is only\nm, as this is a 3D toric code by identifying that R1R1R1F2 ∼= R1R1R1 etc, with the qubit\ncomponent at A2 rather than the customary A1.\nIn Ref. [60] it was observed that while one can perform single-shot lattice surgery using\na subcode given by a hyperplane with 4D surface and toric codes, the performance was\nreduced unless O(\n√\nd) = O(m) syndrome rounds were used. By Proposition 6.1 we in\nfact see that Ω(\n√\nd) rounds are required to maintain fault-distance, as α =\n√\nd in this\ncase. Thus one can perform single-shot lattice surgery, or lattice surgery using a constant\nnumber of syndrome rounds, with 4D toric codes, but with only O(\n√\nd) fault-distance,\ngiven that choice of subcode.\nRemark 8.1. Proposition 6.1 is not quite sufficient to justify Θ(\n√\nd) rounds of syndrome\nmeasurement to maintain fault-distance, only Ω(\n√\nd), as 4D toric codes do not have con-\nstant soundness [78, 82], see Remark 6.4.\nRemark 8.2. Lattice surgery with ‘geometrically enhanced’ 4D toric codes was studied in\nRef. [107, Sec. V]. There, the X-distance of A• was called the ‘boundary distance’, and\nsimilar arguments about the number of rounds required for fault tolerance were employed.\nThe lattice surgery there is performed in an ‘ancilla-free’ manner, where no new data\nqubits are used; only new checks.\nThis is a quotient rather than a mapping cone, see\nRefs. [53, 54, 64].\n8.2\nBivariate bicycle codes\nBivariate bicycle (BB) codes [86] are a class of Abelian 2-block group-algebra (2BGA)\ncodes [18, 112], and as such are lifted product and balanced product codes [16, 17, 113].\nThey are of substantial practical interest, as some bivariate bicycle codes are known to\nexhibit high pseudo-thresholds and low logical error rates under circuit noise and suitable\ndecoders, while having a higher rate than surface codes [63, 86], albeit requiring more\nconnectivity. BB codes can also be seen as toric codes boosted to have limited non-local\nconnectivity [114].\nIt is immediate that full block reading can be used to perform parallel Z ⊗Z or\nX ⊗X measurements between two BB codes using n + mZ = n + mX = 3\n2n new qubits,\nincluding check qubits, while maintaining the distance and LDPC property. For instance,\nthe J144, 12, 12K gross code, which uses 288 total qubits as a quantum memory when\nincluding check qubits, requires 216 additional qubits in total for a full block reading.\nThis is lower than for the same set of parallel measurements in Ref. [54, Sec. 3.3], which\nused 360 total new qubits (216 new data qubits and 144 new check qubits), increased\nthe check weight of the deformed code to 12, and required O(d) rounds of syndrome\nmeasurement. Thus even a naive application of block reading can reduce the spacetime\noverhead of logical measurements by an order of magnitude compared to conventional\nsurgery, where the connectivity allows for it.\nWe suspect that one can perform partial block readings using subcodes of the gross\ncode, and similar BB codes, in order to address desired logical qubits and reduce space\noverheads while maintaining fast and parallel logical measurement at high fault distance,\nand retaining a degree of addressability. As this requires substantial numerics we defer\nthis to future work.\n48\n\n9\nEquivalence between surgery and homomorphic measurement\nThe abstract relationship between (a) surgery by an auxiliary system and (b) homomorphic\nmeasurement [42] was folklore prior to this work, and indeed is implied by Ref. [58]. That\nis, given a chain map one can always construct a mapping cone and vice versa.\nThe\nrelationship we show in this section is more concrete: given the circuit for homomorphic\nmeasurement on a CSS code, it can always be rewritten into a logical measurement by\nsurgery using a mapping cone (called homological measurement in Ref. [58]) and vice\nversa.\nTheorem 9.1. Let C• be a CSS code with a homomorphic measurement procotol specified\nby a chain map f• : A• →C•.\nThen the circuit corresponding to the homomorphic\nmeasurement can be rewritten to a surgery specified by cone(f•).\nProof. See Appendix D.\nCorollary 9.2. Let C• be a CSS code with a surgery protocol specified by cone(f•) for a\nchain map f• : A• →C•. Then the circuit corresponding to the surgery protocol can be\nrewritten to a homomomorphic measurement specified by f•.\nProof. See Appendix D.\nThese proofs are performed using the ZX-calculus [115], a formal calculus for rewriting\ntensor networks on qubits.\nThis is a convenient formalism, as the rewriting requires\nconverting checks and data qubits in the homomorphic measurements into data qubits\nand checks respectively in the surgery procedure. In essence, the time and space axes of\nthe ancilla circuit are inverted.\nHowever, the rewriting does not generally preserve fault-distance, unlike the rewrites\nin Ref. [116, 117]. The reason why converting a surgery to a homomorphic measurement\ndoes not generally preserve fault-distance is because having a low timelike distance in a\nsurgery protocol can be resolved by measuring for multiple rounds. However, the equiv-\nalent homomorphic measurement has low space distance, which is fatal. We expect that\none can prepare many of those ancilla states and measure repeatedly, in the style of a Shor\nmeasurement [5]. However, this is no longer a homomorphic measurement in the sense\ndefined in Ref. [42].\nConversely, we do not know whether rewriting any homomorphic measurement to a\nhypergraph surgery preserves fault-distance.\nOur last results focus on general bounds on fast measurement by an auxiliary system.\nThe results are insensitive to the nature of the auxiliary system: whether it is surgery,\nhomomorphic measurement or any other scheme.\nProposition 9.3. Let Q be a quantum LDPC code. Let S be a set of logical operator\nrepresentatives measured at the same time by one or more auxiliary system for T syndrome\nrounds. Assume that the correct logical measurement outcomes of S are not known in\nadvance. Let f be the lowest weight data qubit fault in Q such that:\n• f flips at least one logical representative in S.\n• If f flips a logical representative in S then it flips all other representatives in the\nsame equivalence class in S.\nLet w be the weight of f, and s be the syndrome weight of f in Q. Then the phenomeno-\nlogical fault-distance of the measurement protocol is upper bounded by 2w + Ts.\n49\n\nProof. See Appendix E.\nWhen Q is CSS and the set S forms a Z-type subcode A•, w is the X-distance of A•.\nTheorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quan-\ntum LDPC code by an auxiliary system requires connections to more than one logical\nrepresentative to maintain phenomenological fault distance Ω(d), unless the correct logical\nmeasurement outcome is known in advance.\nProof. See Appendix E.\nImportantly, for fast surgery it is not enough for the deformed code in the idling\noperation to be a single-shot code. Because the correct outcomes of the new checks are\nnot generally known in advance, many connections between representatives of the original\ncode and the auxiliary hypergraph are essential to correct for measurement errors on these\nnew checks. Indeed, our earlier results on full and partial block reading with distance d\nsubcodes imply that this is the only thing that is required, assuming that the auxiliary\nhypergraph is sensibly constructed and that sufficent rounds of padding are used.\n10\nDiscussion\nIn this work we have introduced a variety of new, general LDPC code surgeries which\nreduce the time overhead of logical measurement below O(d) rounds and allow the mea-\nsurement of logical operators in parallel. We generalised prior work [55–58], which used\nexpansion properties of hypergraphs to efficiently measure individual logical operators,\nto use soundness instead, enabling measurement of many logical operators in parallel.\nWe elucidated the connection between homomorphic measurement and surgery, and gave\ngeneral bounds on the fault distance for any logical measurement by an auxiliary system.\nTo prove fault distances for fast surgery in Appendix A we developed an extension of\nthe fault complexes formalism [60] by taking mapping cones on idling spacetime volumes,\nwhich may be of independent interest.\nOur work raises many further questions. First, when is it possible to efficiently de-\nvise a sparse auxiliary hypergraph H with high modular expansion even if the original\nsubcodes have low soundness? In the case of individual logical operator representatives\nthis is straightforward by constructing an expander graph, but for more complicated par-\nallel measurements involving many representatives the numerical construction of suitable\nhypergraphs is challenging for large codes, and brute-force branching techniques [59, 61]\nhave formidable constant factor overheads.\nSimilarly, when H is a graph one can appeal to the Decongestion Lemma [118, App. A].\nWhen H is a small hypergraph one can typically devise a sparse cycle basis for H by\nnumerical brute-force [55], but for large codes such a method becomes impractical. Are\nthere families of LDPC codes for which such bases are efficient to construct, or unnecessary\nentirely?\nWe expect this to be true of any code that is geometrically local in a fixed\nspatial dimension. In fact, constant-time surgery operations raise the intriguing possibility\nof avoiding the need for a sparse cycle basis at all.\nThis is because cycle checks do\nnot need to be explicitly measured as they can be inferred from the initialization and\nreadout steps, forming detectors that have a constant extent in the time direction. These\ndetectors must provide fault-tolerance, but need not be strictly local. For instance, see the\nnonlocal hierarchical fault-tolerant detector complexes derived from local measurements\nin Ref. [109].\n50\n\nAn issue that arises when using hypergraphs is that the method of universal adapters [57]\nto connect heterogeneous subcodes between codeblocks does not generally apply. This is\nbecause the SkipTree algorithm assumes that the auxiliary system is a graph, rather than\na hypergraph. We do not know whether generalisations of the algorithm can be invented\nto connect heterogeneous hypergraphs for product measurements, either in full generality\nor for specific cases.\nHere, we have focused on hypergraph measurements that only measure local checks\nand global logicals, i.e. they do not locally deform the code after the measurement. It\nis possible to relax these requirements, which leads to a setting of dynamical local codes.\nCould this lead to more addressable fast surgery without sacrificing performance?\nOur proofs concerning interpolated partial block reading in Sec. 6 used a strict defini-\ntion of soundness, see Def. 2.11, which we believe can be loosened to a ‘small set soundness’,\nwith a meaning similar to small set expansion or modular expansion. Our proofs regarding\nfault-distance require d rounds of syndrome measurement (‘padding’) before and after the\nsurgery protocols in order to provably protect against timelike faults extending from one\ntimelike boundary to the other. We expect that these are typically unnecessary given sen-\nsible fault-tolerant operations performed before and after the surgery protocol. However,\nwe have not fully characterized when these rounds of padding can be reduced in number\nor removed.\nIt is known that high-distance subcodes which address specific logical qubits exist for\ntopological and tensor product codes [42, 43], but for other families of LDPC codes little\nis known.\nIt would be interesting to consider algebraic methods to find high-distance\nsubcodes for lifted or balanced product codes.\nAll our surgery methods used auxiliary hypergraphs, and there are other approaches\nto surgery which are ‘ancilla-free’, meaning that they use no new data qubits [83, 107].\nSuch approaches fall under the umbrella of quotients of codes, see Refs. [53, 64], rather\nthan mapping cones [58]. The arguments concerning time overhead appear to translate\nbetween the two approaches, but discerning a rigorous connection is left to future work.\nWe suggest that one could take a pushout or similar quotient directly on the spacetime\nvolumes described by fault complexes [60] in order to study such surgeries.\nWe do not know whether it is possible to take an arbitrary quantum LDPC code and\nprovably measure an arbitrary set of commuting logical Pauli products in constant time\noverhead and linear (up to polylog) space overhead while maintaining fault-distance and\nthe LDPC property. Such a proof is possible for the case of O(d) time overhead [61],\nbut the difficulties of finding appropriate subcodes and cycle bases require new methods.\nConversely, it may be possible to show that such a generic protocol is impossible, see\nRef. [119] for a related result in the classical setting.\nWe have developed a formalism for studying the phenomenological faults in generic\nCSS LDPC code surgery operations using fault complexes. We expect it is possible to\nprove thresholds of surgery on spacetime volumes in this formalism using combinatorial\nmethods on the relevant fault complexes [41].\nBlock reading entails metachecks, which means the addition of many more detectors\nto the decoding graph for a single round of measurement. We do not know whether there\nexist decoders which can efficiently decode such graphs quickly enough to be suitable for\nquantum hardware. In Ref. [83] a powerset decoder was used to decode the a geometrically\nenhanced 4D toric code memory, which also has metachecks. However, this was far too\nslow for realistic hardware, and we do not know how well the powerset decoder would\nhandle surgery operations. More realistic decoders for high-dimensional topological and\ntensor product codes were studied in Refs. [60, 120].\n51\n\nFinally, to bring the theory of block reading to application it will be necessary to\nperform many simulations and stability experiments [121], and numerically determine\npseudo-thresholds and logical error rates of the surgery operations.\n11\nAcknowledgements\nWe are grateful to Katie Chang and Louis Golowich for insightful discussions and for spot-\nting an error in an earlier version of this paper. A.C. is employed by Xanadu and would like\nto thank Ben Ide, Eric Sabo, Timo Hillmann, Michael Vasmer and Guillaume Dauphinais\nfor helpful discussions. Z.H. is supported by MIT Department of Mathematics and thanks\nHarry Zhou, Tomas Jochym-O’Connor and Guanyu Zhu for helpful discussions. D.J.W. is\nsupported by the Australian Research Council Discovery Early Career Research Award\n(DE220100625). 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URL\nhttps://arxiv.org/abs/2407.13826.\n[124] Andrew J. Landahl, Jonas T. Anderson, and Patrick R. Rice. Fault-tolerant quan-\ntum computing with color codes, 2011. URL https://arxiv.org/abs/1108.5738.\nhttps://doi.org/10.48550/arXiv.1108.5738.\n[125] John van de Wetering. Zx-calculus for the working quantum computer scientist,\n2020. URL https://arxiv.org/abs/2012.13966.\nA\nProofs concerning fault-distance\nIn this appendix we provide detailed proofs of the phenomenological fault-distance lower\nbounds described in the main text. The standard approach to proving phenomenological\nfault-distance is to construct a measurement circuit, where Pauli error faults can occur\nat integer timesteps and check faults can occur at half-integer timesteps, with detectors\n[92, 123], and then to show that any fault that does not trigger any detectors and is not\na product of spacetime stabilisers must have at least a given weight d.\nIt is difficult to prove the fault-distances which we require in that setting, and so we\nuse an alternative, almost equivalent one: that of fault complexes [60]. This enables us to\nuse homology to classify the logical faults and bound their corresponding weights. In this\nsetting, Pauli error faults of Z type and Z check errors occur on even timesteps, while\nPauli error faults of X type and X check errors occur on odd timesteps.\nA sketch of the proofs in this section works as follows: we construct fault complexes for\nthe idling operation, and then modify them using mapping cones to obtain fault complexes\nfor various surgery protocols. These result in spacetime volumes where the code is initially\nin the idling operation for d rounds, then undergoes a series of deformations to perform\nsurgery with no padding between surgery rounds, and finally all auxiliary systems are\nmeasured out and the code returns to the idling operation for d rounds. Because the\nspacetime volumes are defined in terms of chain complexes, and the deformations are\ndescribed in terms of mapping cones, we can construct long exact sequences which relate\nthe classes of logical faults in the spacetime volumes for surgery to the classes of logical\nfaults in the idling spacetime volume. We then organise these classes of logical faults and\nshow that they each must have weight at least d.\nOur techniques may be of independent interest to study other protocols in circuit-based\nor measurement-based quantum computing.\nBefore we move to discussing fault complexes, we give some elementary homological\nbackground which is required. The Snake Lemma [90, Lem. 1.3.2] and rank-nullity together\nimply that for a mapping cone cone(f•) defined by a chain map f• : A• →C•, we have\n|Hi(cone(f))| = |Hi(C)| + |Hi−1(A)| −|imHi(f)| −|imHi−1(f)|, where |V | is shorthand\nfor dim V when V is a vector space.\n60\n\nIn more detail, the Snake Lemma dictates that for the mapping cone cone(f•) we have\na long exact sequence 14:\n· · ·\nHi(A)\nHi(C)\nHi(cone(f))\nHi−1(A)\nHi−1(C)\n· · ·\nHi(f)\nι∗\nπ∗\nHi−1(f)\nIn this sequence the Hi(f) are the maps between homology spaces Hi(A) →Hi(C), which\nare well-defined by the commutative diagram in Eq. 1.\nι∗is a map Hi(C) →Hi(cone(f)) generated by the Snake Lemma. In homological\nmeasurement [58], this map at i = 1 takes logical operators in the original code to the\nlogical operators in the deformed code when performing surgery. In that case, the map ι∗\nwill be a surjection, as some of the logical qubits from the original code are measured out\nand no new logical qubits are added when the auxiliary hypergraph’s cycles are gauge-\nfixed.\nπ∗is a map Hi(cone(f)) →Hi−1(A). In homological measurement [58], this map at\ni = 1 picks out the new contributions to the logical space of the deformed code from\nHi−1(A); exactness at Hi(cone(f)) shows that the only new logical qubits which can arise\nare from cycles in the hypergraph.\nIn our case we will take mapping cones on spacetime volumes, rather than just codes,\nbut a similar intuition applies.\nFor any linear map T : V →W rank-nullity implies that |V | = | ker T| + |imT|, so we\ncan compute\n|Hi(cone(f))| = | ker π∗| + |imπ∗| = |imι∗| + |imπ∗|\n= |Hi(C)| −| ker ι∗| + | ker Hi−1(f)|\n= |Hi(C)| −|imHi(f)| + |Hi−1(A)| −|imHi−1(f)|.\n(4)\nThe premise of all of our proofs is that in a CSS-type code surgery, where the initial\ncode and deformed codes are all CSS codes, all logical faults must be either of Z type in\nspacetime, composed of Z Pauli errors and X check errors, or X type, composed of X\nPauli errors and Z check errors. As any logical fault composed of a product of these is\nat least as high weight as the corresponding logical fault of either Z or X type, we need\nonly show that the weights of all Z and X logical faults are at least d to prove that the\nfault-distance of a protocol is at least d.\nWe will frequently make use of the fact that when Hi(f) is injective |imHi(f)| =\n|Hi(A)|, by definition.\nA.1\nFault complexes\nA fault complex is defined as a chain complex with 4 terms,\nF• = F3\nF2\nF1\nF0\nwhere by convention the F3 and F0 components correspond to Z and X detectors respec-\ntively. In fault-tolerant measurement-based quantum computing (MBQC), F2 corresponds\nto a set of X fault locations, i.e. locations in spacetime which can experience only X-\ntype faults, while F1 corresponds to a set of Z-type fault locations. The fault complex is\ntherefore in bijection with a graph state.\nWhen the graph state corresponds to a foliated CSS code, this can be reinterpreted in\nthe circuit model, undergoing phenomenological noise. Now, F2 corresponds to a combined\n14This means that the image of each map coincides with the kernel of the next map.\n61\n\nset of spacetime locations at which data qubits can experience X Pauli faults and also Z\nchecks, which can experience measurement errors. F1 is the same but for Z Pauli faults\nand X checks. In this case, we can expand out a fault complex to have the following\ndiagram:\nF1,1\nF1,0\nF1,2\nF0,0\nF0,2\nF0,1\nwhere components in the overall fault complex are given by taking direct sums vertically\nin the diagram, so we have\nF1,2\nF1,1 ⊕F0,2\nF1,0 ⊕F0,1\nF0,0\nThroughout this section every diagram drawn in the above form is assumed to have direct\nsums vertically in the diagram. 15\nIn the phenomenological circuit model:\n• F0,0 corresponds to X detectors.\n• F0,1 corresponds to locations for Z Pauli faults on data qubits.\n• F1,0 corresponds to locations for X check faults.\n• F0,2 corresponds to locations for Z check faults.\n• F1,1 corresponds to locations for X Pauli faults on data qubits.\n• F1,2 corresponds to Z detectors.\nExpressed as a commutative diagram the fault complex then has the structure:\nX Pauli faults\nX check faults\nZ detectors\nX detectors\nZ check faults\nZ Pauli faults\nF1,1\n(X Pauli faults)\nF1,0\n(X check faults)\nF1,2\n(Z detectors)\nF0,0\n(X detectors)\nF0,2\n(Z check faults)\nF0,1\n(Z Pauli faults)\n15This is essentially passing to the “total complex” [90].\n62\n\nNote that as a fault complex describes a graph state, it does not have the ability to\ndescribe morphisms, i.e. MBQC-based or circuit-based protocols with input or output\nqubits.\nThe standard way to foliate a CSS code C• from the fault complex perspective is to\ntake (R⊗C)•, where R• : R1 →R0 has the differential R being either the full-rank parity-\ncheck matrix for the repetition code or its dual. In the former, the code is initialised in\nthe |0⟩state and measured out in the Z basis; in the latter, the code is initialised in the\n|+⟩state and measured out in the X basis. Hence the code ‘idles’ for l rounds, where l is\nthe blocklength of the repetition code.\nIf we do this, the fault complex has the explicit form:\nR1 ⊗C1\nR1 ⊗C0\nR1 ⊗C2\nR0 ⊗C0\nR0 ⊗C2\nR0 ⊗C1\n(5)\nIn the circuit model we have, in order,\n• R0 ⊗C0 corresponds to X detectors.\n• R0 ⊗C1 corresponds to locations for Z Pauli error faults.\n• R1 ⊗C0 corresponds to locations for X check error faults.\n• R0 ⊗C2 corresponds to locations for Z check error faults.\n• R1 ⊗C1 corresponds to locations for X Pauli error faults.\n• R1 ⊗C2 corresponds to Z detectors.\nNote that R1 ⊗C1 and R0 ⊗C1 correspond to the same data qubits but at different points\nin time; in the circuit model this is purely convention, and any given data qubit can still\nundergo either type of error, Z or X.\nIt is easy to compute the homologies using the K¨unneth formula. Then elements of\nH1(F) correspond to equivalence classes of logical Z faults in spacetime, composed of Z\nPauli errors and X check errors, and elements of H2(F) to logical X faults, composed of\nX Pauli errors and Z check errors.16\nFor completeness, we have,\n• H1(R ⊗C) = H1(R) ⊗H0(C) ⊕H0(R) ⊗H1(C).\n• H2(R ⊗C) = H1(R) ⊗H1(C) ⊕H0(R) ⊗H2(C).\nWhen R• is the repetition code, we have\nH1(R ⊗C) = H1(R) ⊗H0(C) = {0, 1} ⊗C0/imHX\nH2(R ⊗C) = H1(R) ⊗H1(C) = {0, (1, 0, 0, · · · , 0)} ⊗H1(C),\n16There are also the homology groups H1(F) and H4(F), corresponding to classes of detector triggers\nwhich are impossible.\n63\n\nFigure 5: A fault complex for a unit cell of the 2D surface code, starting and terminating at X-type\nboundaries. Time flows from bottom to top.\nwhere 0 and 1 are the all-zero and all-one vectors, and (1, 0, 0, · · · , 0) is a weight 1 vector\nrepresenting the nonzero equivalence class in H1(R); any vector with odd weight is in the\nsame class. When R• is dual to the repetition code we instead have\nH1(R ⊗C) = H0(R) ⊗H1(C) = {0, (1, 0, 0, · · · , 0)} ⊗H1(C)\nH2(R ⊗C) = H0(R) ⊗H2(C) = {0, 1} ⊗H2(C).\nIn Fig. 5 we show the fault complex for a small surface code as a Tanner graph, where\ncircles are fault locations and squares are detectors, and red, teal, yellow and blue indicate\nbasis elements of F0, F1, F2, F3 respectively.\nFault complexes are a useful algebraic framework for reasoning about fault distance\nof CSS codes in the phenomenological model: each location is a point in spacetime which\ncan experience a fault, which is also a basis vector in a space (either F1 or F 2) and so\nweights of faults can be related directly to weights of vectors in the fault complex.\nWe do not want to presume that the code is initialised and measured out in a given\nbasis, because we want to prove that a given protocol has sufficient fault-distance no\nmatter what other (assumed fault-tolerant) operations the logical circuit contains. One\napproach to this is to replace the parity-check matrix of R• with the parity-check matrix\nof a different repetition code, that of the cycle graph. This corresponds to ‘taking the\ntrace’ of the circuit operation being performed, and ensures that errors cannot vanish on\nthe initial and final boundaries. However, this leads to other, serious issues: a timelike\nfault which runs from the start to the finish of the spacetime volume is now a spacetime\nstabiliser, and there are other logical faults which we encounter that are equivalent to this\nfault, and so they will be unaccounted for.\nInstead, we modify the spacetime volume by adding checks to the start and end, which\nconstitute entry points for timelike errors to enter and exit the volume: depending on\nwhether R• is a full-rank repetition code or its dual, corresponding to initialising in the\n|0⟩or |+⟩state respectively, the new checks are Z checks or X checks.\nWe accomplish this using mapping cones.\nExplicitly, when R• is the dual of the\nrepetition code, so R• = Fl−1\n2\n→Fl\n2 for some l determining the number of rounds in the\n64\n\nspacetime volume, the code is initialised in |+⟩and measured out in the X-basis; we then\nintroduce new checks for timelike fault entrypoints by taking the mapping cone of the\nchain map g•,\n0\n0\n0\nC0 ⊕C0\nF3\nF2\nF1\nF0\ng0\nwhere g0 maps the first copy of C0 into the top layer of X-detectors in the fault complex,\nand the second copy of C0 into the bottom layer of X-detectors. Call the resultant fault\ncomplex F ′ = cone(g•). The unit cell from Fig. 5, once modified in this manner, is shown\nin Fig. 6.\nThe consequent classes of logical faults in cone(g•) can be computed using the Snake\nLemma, see Eq. 4. We have\n|H1(F ′)| = |H1(cone(g))| = |H1(F)| + |C0 ⊕C0| −|imH0(g)|\n= |H1(F)| + 2|C0| −|imH0(g)|.\nBy the K¨unneth formula, H0(F) = H0(R) ⊗H0(C) = {0, (1, 0, 0, · · · , 0)} ⊗H0(C), so\n|H0(F)| = |H0(C)|. By inspection, every element of one layer of X-detectors at the bottom\nof the fault complex is in the image of g•; those at the top are in the same equivalence\nclass. Hence |imH0(g)| = |H0(F)| = |H0(C)|, so\n|H1(F ′)| = |H1(cone(g))| = |H1(F)| + 2|C0| −|H0(C)|.\nWe therefore add 2mX −rX new basis elements to H1(F), where mX = |C0| is the\nnumber of X checks in C• and rX = |H0(C)| is the number of redundant X checks in\nC•. All new elements of H1(F ′) correspond to logical faults which enter from a timelike\nboundary on the top or bottom; mX −rX of these can be immediately annihilated by\nspacelike errors on the boundary, or by spacelike errors on further layers. We illustrate\nthis in Fig. 7, where incoming X check errors turn on X detectors, which can then be\nimmediately switched off again by Z data qubit Pauli errors.\nThe remaining mX are\nstrings of X check failures running from the bottom to the top boundary, illustrated in\nFig. 8. In both cases, all new logical faults must have timelike support on the new checks\nat the boundary, and these cannot be cleaned from the boundary into the bulk.\nWe can similarly calculate that\n|H0(F ′)| = |H0(F)| + 0 −|imH0(g)| −0\n= |H0(F)| −|H0(F)| = 0.\nFor the dual version, where R• = Fl\n2 →Fl−1\n2\n, we instead perform the mapping cone\non the cochain complex, and add 2mZ −rZ new elements to H2(F). Additionally, H3(F)\nbecomes 0.\nIn summary, when the initialisation and measurement is in the X basis the spacetime\nvolume for the idling operation now contains:\n• All Z timelike logical faults, extending through the volume.\n• Those X timelike logical faults with check errors in H2(C) which are not equivalent\nto spacelike faults.\n• All Z spacelike logical faults.\n65\n\nFigure 6: A fault complex for a unit cell of the 2D surface code, starting and terminating at X-type\nboundaries, but with new X checks to allow Z timelike errors to enter and exit the volume.\n66\n\nFigure 7: The fault complex from Fig. 6, where a Z timelike fault enters the fault complex and is\nimmediately annihilated by a spacelike fault. The highlighted fault locations are shown in green.\n67\n\nFigure 8: The fault complex from Fig. 6, where a Z timelike fault extends from one timelike boundary\nto the other. The highlighted fault locations are shown in green.\n68\n\n• No X spacelike logical faults.\nand when it is in the Z basis the volume is the same but with the X and Z logical\nfault characterisation inverted. Therefore if we use both of these fault complexes then,\ncombined, we account for all possible logical faults, as the code (and fault complex) is\nCSS.\nWe wish to describe complicated surgery protocols, rather than the idling state. Now,\na certain fault complex for the idling state is sufficient to study certain stability exper-\niments, but insufficient for the description of surgery protocols involving many logical\nmeasurements. For this the fault complex must describe the actual code deformation tak-\ning place. This can also be accomplished using mapping cones, as we shall demonstrate\npresently.\nA.2\nFull block reading\nWe are now ready to prove the fault-distance of full block reading. In this case, we start\nwith taking E• = Fc\n2 ⊗C• as our initial CSS code in the idling state, so the initial fault\ncomplex is F• = (R ⊗E)•. We then consider the two cases, one where the idling volume\nstarts and ends with Z basis measurements, and the other with X basis measurements,\nincluding the entrypoints for timelike errors, so that we acquire two different fault com-\nplexes. For convenience we label them both F ′\n•, and the difference will be made clear by\ndividing into the case for X-type boundaries and Z-type boundaries.\nIn each case, we are applying mapping cones to the idling volume. To perform a full\nblock reading in the Z basis, measuring Z logical operators, the mapping cone is on the\nchain complex, and in the X basis the mapping cone is on the cochain complex.\nIn the Z basis, the mapping cone on the fault complex is taken from the chain map f•\nwith components:\n0\nC2\nC1\nC0\nF ′\n3\nF ′\n2\nF ′\n1\nF ′\n0\nf3\nf2\nf1\nf0\nwhere the components map the auxiliary code C• to the codeblocks in E• to be measured,\nat a primal layer. In other words, in the case where we start and finish in the X basis we\n69\n\nnow have the fault complex\nE0\n(Bottom X\ncheck faults)\nE0\n(Top X check\nfaults)\nR1 ⊗E1\n(Old X Pauli\nfaults)\nR1 ⊗E0\n(Old X check\nfaults)\nR1 ⊗E2\n(Old Z\ndetectors)\nR0 ⊗E0\n(Old X\ndetectors)\nR0 ⊗E2\n(Old Z check\nfaults)\nR0 ⊗E1\n(Old Z Pauli\nfaults)\nC2\n(New Z\ndetectors)\nC1\n(New Z check\nfaults)\nC0\n(New Z Pauli\nfaults)\nwhere the copies of E0 at the top of the diagram are to add timelike entrypoints to the\nspacetime volume, and the C• at the bottom performs the full block reading.\nWe can now flip the boundary conditions, so that R• = Fl\n2 →Fl−1\n2\n, and flip the\nadditional timelike entrypoints accordingly, but still perform the same full block reading,\n70\n\nto acquire the fault complex\nE2\n(Bottom Z\ncheck faults)\nE2\n(Top Z check\nfaults)\nR1 ⊗E1\n(Old X Pauli\nfaults)\nR1 ⊗E0\n(Old X check\nfaults)\nR1 ⊗E2\n(Old Z\ndetectors)\nR0 ⊗E0\n(Old X\ndetectors)\nR0 ⊗E2\n(Old Z check\nfaults)\nR0 ⊗E1\n(Old Z Pauli\nfaults)\nC2\n(New Z\ndetectors)\nC1\n(New Z check\nfaults)\nC0\n(New Z Pauli\nfaults)\nDue to the change in boundary conditions, each of these fault complexes has different\nlogical faults, and we must account for them both.\nThere are some important consequences of using the fault complex formalism rather\nthan the phenomenological circuit-based model. It dictates that each fault location in the\nfault complex is a qubit initialised in the |+⟩state and then measured out in the X basis,\nwith the entangling CZ gates performing space- and time-like Hadamards to exchange\nerror types between layers. As a consequence, unlike in the circuit-based model, there\nare not separate timesteps on which the new ancilla data qubits are initialised in |+⟩and\nmeasured out in the X basis, as this is included in the fault complex layers. So, in the\ncircuit model, one would initialise the new data qubits in |+⟩, then measure all checks for\n1 round, then measure out the new data qubits in the X basis, which would generally be\nconsidered 3 timesteps: one each for the ‘merge’, measure, and ‘split’ steps. In the fault\ncomplex picture, this all happens on 1 timestep. This is why there is only one ‘copy’ of\nC• used for the auxiliary system in the mapping cone.\nAdditionally, the layers of Z and X checks are alternating, rather than occurring on\nthe same timestep as in the circuit-based model. When a surgery protocol only uses 1\nround of Z measurements, as in full block reading, the fault complex does not introduce\nX checks connections to the new data qubits at all. There are also no new fault locations\nfor X Pauli data qubit errors on the auxiliary system, as they would either act on data\nqubits in |+⟩or before the qubits are measured out in the X basis.\n71\n\nRemark A.1 (Multiple measurement rounds). In Section A.4 below we see cases in which\nthe number of measurement rounds required is greater than 1, in which case there are more\nthan one ‘copy’ of C• employed, and there are deformed X checks and new fault locations\nfor X Pauli data qubit errors.\nRemark A.2 (Cycle detectors). For all the full block reading procedures we do not require\nany cycle checks, or detectors related to those checks. In Thm. 3.12 we have imposed a\ncondition that no measured logical Paulis should be products of any others, i.e. the proof\nrequires that the set of measured logical Paulis is minimal for the group it generates. This\ncondition can be removed if one uses detectors for the cycles in the hypergraph: these\nare not cycle checks, merely detectors inferred from the initial state preparation and final\nmeasurement of the new data qubits, so the code will remain LDPC, but the detector matrix\nmay not unless the cycle basis is sparse.\nProposition 3.9. Let Q, Q′, Q′′,...\nbe a set of identical CSS LDPC codeblocks with\ndistance d, and let P be a pattern matrix determining a full block reading.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and\nafter the full block reading procedure. Then full block reading performs logical measurement\nin O(1) rounds of syndrome measurements with fault-distance d.\nProof. We treat the case of X-type boundary conditions first.\nX-type boundary conditions.\nBy the Snake Lemma, we can compute the following.\n|H1(cone(f))| = |H1(F ′)| −|imH1(f)| + |H0(C)| −|imH0(f)|\n= |H1(F ′)| −|H1(C)| + |H0(C)| −0\n= |H1(F ′)| −|H1(C)| + |H0(C)|.\nApplying the same argument to H2(cone(f)), we have:\n• |H1(cone(f))| = |H1(F ′)| −|H1(C)| + |H0(C)|.\n• |H2(cone(f))| = |H2(F ′)| −|H2(C)|.\nWe first discuss the negative contributions to the classes of logical faults.\nThe negative contribution to Z logical faults (faults that commute with all X detectors)\nappears because |H1(C)| logical measurements have occurred in the Z basis, and so there\nare |H1(C)| spacelike Z errors which no longer affect the logical state.\nThe negative contribution to X logical faults (faults that commute with all Z detectors)\nappears because the full block reading introduces Z detectors between codeblocks, which\nrequire timelike Z check errors to now be in ker HZ of the auxiliary code in order to be\nundetected.\nAs the space distance of the code and deformed code are each at least d, and there are\nno new equivalence classes of logical faults, we cannot clean any logical fault with these\nboundary conditions to have weight below d.\nZ-type boundary conditions. In this case the equivalence classes of logical faults\ncan be computed for H1(cone(f)) as\n|H1(cone(f))| = |H1(F ′)| −|imH1(f)| + |H0(C)| −|imH0(f)|\n= |H1(F ′)| −0 + |H0(C)| −0\n= |H1(F ′)| + |H0(C)|.\nApplying the same argument to H2(cone(f)) we have:\n72\n\n• |H1(cone(f))| = |H1(F ′)| + |H0(C)|.\n• |H2(cone(f))| = |H2(F ′)| + |H1(C)| −|H2(C)|.\nwhere H1(F ′) and H2(F ′) are equivalence classes already present before the full block\nreading, and H0(C) and H1(C) are new. Again we have a negative contribution of |H2(C)|\nto the classes of Z check faults.\nNote that some of the contributions to equivalence classes appear in both boundary\nconditions.\nFor the new X logical faults, it is straightforward that H1(C) are equivalence classes of\nundetectable Z check failures on the auxiliary code, which will flip the logical measurement\noutcomes of the full block reading.\nFor the new Z logical faults, the equivalence classes given by H0(C) are spacetime\nlogical faults which begin as Z Pauli errors on the auxiliary code, and then move into the\noriginal spacetime volume and become timelike errors to then exit from the start or end\nboundaries. Because the Z Pauli errors are not in im(HX), they cannot be cleaned into\nthe original spacetime volume and are genuinely new classes of logical faults. They are\nalso not products of space and time logical faults.\nTo prove the fault distance of a single full block reading, where there are at least d\nrounds of measurement before and after the block reading but only 1 round used for the\nblock reading itself, it is sufficient to show that every element of these equivalence classes,\nand their combinations, must have weight at least d.\nAny timelike logical fault in H1(F ′) or H2(F ′) must originate at the boundaries, and\ntherefore must have at least weight d to reach, and therefore affect, the protocol. They\ncannot be cleaned away from the boundary by spacetime stabilisers. Timelike errors which\noriginate at the boundary must extend for d rounds to affect the computation.\nLogical faults from the positive contribution of H0(C) must exit through one of the\nboundaries of the spacetime volume, and hence must have weights at least d.\nSpacelike logical faults in H1(F ′) and H2(F ′) must have weight at least d because the\noriginal code and the deformed code during block reading both have distance at least d.\nAdding spacetime stabilisers to these spacelike logical faults can only increase the weights,\nbecause whatever qubit faults are removed by cleaning are merely moved to another round\nin the volume due to the structure of the repetition code R•.\nLastly, logical faults in H1(C), which are new check errors on the auxiliary system,\nmust have weight at least d as timelike errors. Applying spacetime stabilisers to clean\nthese into the original code is possible, but the corresponding spacetime fault must act on\nat least d data qubits due to the 1-to-1 structure of the full block reading, corresponding\nto a spacelike logical fault, followed by some timelike errors, and then the same spacelike\nlogical fault, and so all such logical faults must have weight at least d.\nEvidently a dualised proof would apply were the full block reading to be applied in\nthe X basis instead.\nRemark A.3. As full block reading is only applied for 1 round of measurements, these\nnew classes of spacetime faults corresponding to H0(C) can be eliminated by adding an\nextra C−1 term to C•, which adds detectors reconstructed from the initialisation and mea-\nsurements of the ancilla system in the X-basis. Adding an extra C−1 term cannot make the\ndeformed code non-LDPC, but can make the detectors high weight. As we shall presently\nshow, these new classes of spacetime faults are not problematic anyway.\n73\n\nWe can then consider the case where multiple full block readings are performed in a row,\nbut with only O(1) rounds between them, and still leaving O(d) rounds of measurement\nbefore and after the procedure.\nIn this case all the previous calculations apply, by applying the mapping cone to\nthe chain or cochain complex versions of the fault complex.\nHowever, the analysis of\nlogical faults requires two more ingredients.\nFirst, in the case where only one block\nreading is performed in the spacetime volume, the elements of H0(C) must exit through\nthe boundaries of the volume. When there are multiple block readings performed in close\nproximity, however, these elements can connect and form noncontractible closed curves\nwithin the spacetime volume, with low fault weight. These noncontractible closed curves\nappear only when there is a full block reading whose logical measurements are a product\nof others performed in the same spacetime volume.\nSecond, we must consider the compacted code of the spacetime volume, see Defini-\ntion 4.4.\nLemma A.4. Let Ξ be a set of η full block reading procedures of Z-type such that no\nblock reading procedure in Ξ is a product of any others. Then the compacted code CC• has\ndistance d.\nProof. This is identical to the proof of code distance of the code (C ⊗R)•, where R• is a\nfull-rank matrix, see Lemma 3.8.\nTheorem 3.12. Let Ξ be a set of η full block reading procedures of Z-type such that no\nblock reading procedure in Ξ is a product of any others.\nThen all η logical measurement rounds of full block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the full set of procedures.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. Say that the η block readings being performed are each defined by a bitstring of\nlength c, and the set of bitstrings is divided into sets of size ηZ and ηX for Z and X type\nblock readings. Each bitstring contains a 1 in a location if that codeblock is involved in the\nblock reading specified by the bitstring, and a 0 otherwise. Whether the block readings\nare performed simultaneously or spread across multiple rounds, we can describe the block\nreadings being performed by a pair of pattern matrices PZ and PX over F2, with c columns\nand ηZ and ηX rows respectively. Note that, unlike when performing them simultaneously,\nthese matrices do not need to form a pattern complex.\nAssume that there is no block reading performed in the spacetime volume whose logical\nmeasurements are products of others. Then PZ and PX are each full-rank matrices.\nAny logical fault composed of Z Pauli errors in H0(C) on a Z-type block reading must\nextend to timelike X check faults on all blocks in that row of PZ. To prevent extending all\nthe way to the boundaries of the volume, such timelike faults must reach Z Pauli errors in\nH0(C) on a different row; these must subsequently extend onto all blocks in that row of\nPZ. Hence in order to form a noncontractible closed curve inside the volume the pattern\nmatrix PZ must not be full rank, and so there are some block readings which are products\nof previous ones.\nDualising the argument gives the same result for X Pauli errors in H2(C) on an X-type\nblock reading.\nAll other timelike logical faults follow the same arguments as before: they must either\nextend to a boundary or have distance d because of the new metachecks if they are timelike.\n74\n\nFor spacelike errors, we must be more careful, as spacelike errors in the different\ndeformed codes during the spacetime volume can be cleaned to have overlapping spacelike\nsupport. In particular, one can take products of spacelike logical faults in each deformed\ncode, or the undeformed code. Before applying spacetime stabilisers, such a product is a\nset of disjoint spacelike logical faults separated by some number of timesteps, which in the\nworst case is constant.\nUpon multiplying by spacetime stabilisers, such spacelike logical faults can have sup-\nport that is cleaned to the same timestep and hence cancel, introducing some check errors\nbetween the timesteps. Say that we have a set of ϑ spacelike Z logical faults Λ1, Λ2, · · · , Λϑ\nat different logical timesteps, acting on different logical qubits. These could each be sep-\narated by a constant number of timesteps, and induce a constant number of check errors\nwhen partially cleaned by a spacetime stabiliser S. Thus, if the spacelike component of\nthe product SΛ1Λ2 · · · Λϑ has weight below d the fault-distance may fall below d.\nWe now consider the compacted code CC• of the spacetime volume.\nEach logical\noperator Λi in each deformed code is guaranteed to be a logical operator of the CC•. Let\nΛ′\ni be the logical operator Λi viewed in CC•. We can hence consider the product | Q\ni Λ′\ni|,\nwhich is the minimal weight of the spacelike component of SΛ1Λ2 · · · Λϑ.\nAs CC• has\ndistance d, | Q\ni Λ′\ni| ≥d for any choice of logical operators Λi in any deformed codes.\nThen, | Q\ni SΛi| ≥d.\nFor spacelike X logical operators, the argument is similar.\nThe X-distance of the\ncompacted code CC• is at least d, and so cleaning X logical operators to have overlapping\nspacelike components can never reduce the fault weight below d.\nFor full block reading of CSS-type, we require an extension of the previous lemma\nconcerning the compacted code.\nLemma A.5. If the hypergraph surgeries are CSS-type full block readings such that no\nlogical measurement is the product of any other then the compacted code CC• has distance\nd.\nProof. This is identical to the proof of code distance of the code (C ⊗R)•, where each\npattern matrix in the complex R• is a full-rank matrix, see Lemma 3.8.\nLemma 3.16. Let Ξ be a set of η full block reading procedures of CSS-type such that no\nblock reading procedure in Ξ is a product of any others, and each of the measurements\ncommute.\nThen all η logical measurement rounds of full block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the full set of procedures.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. In this case we again take iterative mapping cones in the same way.\nIn the worst case, where there is short timelike length between full block readings, the\ncones are applied to only two layers in the fault complex: Z-type measurements to a layer\nwith X detectors (corresponding to a basis element of R0) and X-type measurements to\na close layer with Z detectors (a basis element of R1). By virtue of the mapping cone\nconstruction, the fault complex commutes as a chain complex.\nWe then use identical arguments to the proof of Z-type full block reading, where this\ntime the constant number of layers between measurements is reduced to 0. As the proof\nof fault-distance is independent of this number of rounds, and the compacted code has\n75\n\ndistance d, there are no spacelike logical faults which can be cleaned to have fault weight\nwith spacelike component lower than d.\nA.3\nPartial block reading with high-distance subcodes\nProposition 4.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d,\nand let A• be a subcode with distance d, with chain maps from A•, or a thickened version\nthereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code, which has code distance\nat least d.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before\nand after the partial block reading procedure. Then partial block reading performs logical\nmeasurement in O(1) rounds of syndrome measurements with fault-distance d.\nProof. We can rerun the arguments for Prop. 3.9 but where A• ̸= C•. A• is a subcode of\nC• or can be a subcode thickened up to d times (in the space direction). The thickening\ndoes not affect the homology of A•, or the maps on homology Hi(f) into C•.\nIn partial block reading with high-distance subcodes, unlike full block reading, we\ninclude additional X-detectors given by the initialisation and measurement of the new\nqubits in the X-basis. These give us detectors corresponding to a basis of cycles in the\nhypergraph; however, there is no guarantee that this basis is sparse. This does not affect\nthe LDPC property of the code, but does affect the size of the detectors when they are\nused for decoding. In other words, the deformed code is guaranteed to be LDPC, but the\nfault complex is not.\nNow, for a Z-type partial block reading the fault complex is given by cone(f•), where\nf• has components\n0\nA2\nA1\nA0\nA−1\nF ′\n3\nF ′\n2\nF ′\n1\nF ′\n0\n0\n0\nf2\nf1\nf0\n0\nwith im(f•) having support on a single primal layer, across potentially multiple codeblocks,\nand A−1 is a vector space with differential such that H0(A) = 0 to fix the cycles.\nFor the X-type boundary conditions we then have\n|H1(cone(f))| = |H1(F ′)| −|imH1(f)| + |H0(A)| −|imH0(f)|\n= |H1(F ′)| −|H1(A)| + 0 −0\n= |H1(F ′)| −|H1(A)|.\nAnd\n|H2(cone(f))| = |H2(F ′)| −|imH2(f)| + |H1(A)| −|imH1(f)|\n= |H2(F ′)| −|H2(A)| + |H1(A)| −|H1(A)|\n= |H2(F ′)| −|H2(A)|\n= |H2(cone(f)| = |H2(F ′)| −|H2(A)|.\n• |H1(cone(f))| = |H1(F ′)| −|H1(A)|.\n• |H2(cone(f))| = |H2(F ′)| −|H2(A)|.\nFor the Z-type boundary conditions, the calculation is the same except |imH1(f)| = 0, so:\n76\n\n• |H1(cone(f))| = |H1(F ′)|,\n• |H2(cone(f))| = |H2(F ′)| + |H1(A)| −|H2(A)|.\nHence we can categorise the logical faults in an identical way to previously and find that\nthey must all either extend from a boundary of the spacetime volume, or have spacelike\nweight at least d, or have timelike weight at least d, and that spacetime stabilisers cannot\nclean these faults to have weight lower than d by virtue of the repetition code structure\nof R•.\nBecause we added the X-detectors, which are present to allow the space distance of the\ndeformed code to be at least d, we also eliminate the spacetime faults which were present\nfor full-block reading and corresponded to contributions of H0(C). Now, H0(A) = 0 so\nthere are no such contributions, so we need not be concerned about noncontractible loops\nin the spacetime volume.\nTheorem 4.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and\nlet Ξ be a set of Z-type partial block readings {Partial1, Partial2, Partial3, ..., Partialη}\nwhere each subcode has distance d and the compacted code has distance d.\nThen all η logical measurement rounds of partial block reading can be performed sequen-\ntially using O(1) rounds of syndrome measurement each, and using d rounds of padding\nbefore and after the set of logical measurements.\nThe procedure has phenomenological\nfault-distance d, for any η ∈N.\nProof. All logical faults must either extend from a boundary or have distance d because\nof the new metachecks if they are timelike.\nFor the spacelike faults, we can apply the same argument as for full block reading.\nEach logical operator Λi in each deformed code is guaranteed to be a logical operator of\nthe compacted code CC•. Let Λ′\ni be the logical operator Λi viewed in CC•. We can hence\nconsider the product | Q\ni Λ′\ni|, which is the minimal weight of the spacelike component of\nSΛ1Λ2 · · · Λϑ. As CC• has distance d, | Q\ni Λ′\ni| ≥d for any choice of logical operators Λi in\nany deformed codes. Then, | Q\ni SΛi| ≥d.\nFor spacelike X logical operators, the argument is similar but more straightforward.\nThe X-distance of the compacted code CC• must always be at least d by Lemma 2.16,\nand so cleaning X logical operators to have overlapping spacelike components can never\nreduce the fault weight below d.\nA.4\nPartial block reading with low-distance subcodes\nConsider a partial block reading, where the subcode A• for the partial block reading has\ndistance less than d. For all our descriptions here we assume the block reading is measuring\nin the Z basis, and then one can dualise to acquire the corresponding construction to\nmeasure in the X basis.\nWe also presume that the auxiliary hypergraph for a Z-type partial block reading\nhas X-checks to gauge-fix cycles, and the same for X-type partial block readings with Z\nchecks. These are genuine checks, and not just detectors inferred from the initial and final\nmeasurements of the new qubits. In terms of the initial auxiliary system, this still appears\nas adding an A−1 term to A• such that H0(A) = 0,\nA• = A2\nA1\nA0\nA−1\nbut this is not yet the complex which we will perform the mapping cone with.\n77\n\nAs before, we ignore the thickening of A• in space, as one can compute that this\nleaves the homology and categorisation of logical fault equivalence classes unaffected, and\nis necessary only to preserve the spatial distance of the deformed code.\nHowever, as the distance of A• is now less than d, we must thicken A• in time to make\na fault complex K• corresponding to running the measurement procedure for more than\n1 round. Let dA be the distance of A• and let dA = 1\nαd for d the code distance of C•.\nThe fault complex K• corresponds to taking (P(α) ⊗A)•, where\nP(α)• = P(α)1\nP(α)0 = Fα−1\n2\nFα\n2\nis dual to the full-rank repetition code with length α, and has the differential matrix being\nthe incidence matrix of the path graph with α vertices. Explicitly K• is the chain complex:\nP(α)1 ⊗A1\nP(α)1 ⊗A0\nP(α)1 ⊗A−1\nP(α)1 ⊗A2\nP(α)0 ⊗A−1\nP(α)0 ⊗A2\nP(α)0 ⊗A1\nP(α)0 ⊗A0\nObserve that A• = K• when α = 1, recovering the case where we measure for only one\nround.\nNow, for a Z-type partial block reading the fault complex is given by cone(f•), where\nf• has the form:\nP(α)1 ⊗A1\nP(α)1 ⊗A0\nP(α)1 ⊗A−1\nP(α)1 ⊗A2\nP(α)0 ⊗A−1\nP(α)0 ⊗A2\nP(α)0 ⊗A1\nP(α)0 ⊗A0\nF ′\n3\nF ′\n2\nF ′\n1\nF ′\n0\n0\n0\n0\nf2\nf1\nf0\nThere are also maps from the top terms in the diagram – P(α)1 ⊗A1, P(α)1 ⊗A0 and\nP(α)1 ⊗A−1 – but these are zero maps, so we abuse notation slightly by labelling the\nmaps from the bottom layer f2, f1, f0. As before, the only nonzero maps in f• are into\nprimal layers.\nThe mapping cone then yields the fault complex cone(f•) for the measurement,\nK2\nK1\nK0\nK−1\nF ′\n3\nF ′\n2\nF ′\n1\nF ′\n0\nwhere the K3 term, equal to P(α)1 ⊗A2, is omitted as it has no physical meaning, being\na “detector on detectors”.\nBefore continuing further, let us make some orienting remarks regarding the terms and\nmaps in this mapping cone. Below we expand out to the full fault complex cone(f•) for\n78\n\nthe case where the spacetime volume is initialised and measured out in the X basis. As\nbefore, the P(α)1 ⊗A2 term is omitted. The diagram clearly commutes, by inheriting the\ncoherent monic of the subcode A• into the original codeblocks E•.\nP(α)1 ⊗A1\n(New Z\ndetectors)\nP(α)1 ⊗A0\n(New X Pauli\nfaults)\nP(α)1 ⊗A−1\n(New X check\nfaults)\nP(α)0 ⊗A−1\n(New X\ndetectors)\nP(α)0 ⊗A2\n(New Z\ndetectors)\nP(α)0 ⊗A1\n(New Z check\nfaults)\nP(α)0 ⊗A0\n(New Z Pauli\nfaults)\nR0 ⊗E2\n(Old Z check\nfaults)\nR0 ⊗E1\n(Old Z Pauli\nfaults)\nR1 ⊗E2\n(Old Z\ndetectors)\nR0 ⊗E0\n(Old X\ndetectors)\nR1 ⊗E1\n(Old X Pauli\nfaults)\nR1 ⊗E0\n(Old X check\nfaults)\nE0\n(Bottom X\nchecks)\nE0\n(Top X checks)\nf2\nf1\nf0\n• All terms in (R ⊗E)• are those from the fault complex of the idling operation.\n• The new E0 terms at the bottom are to add timelike entrypoints to the volume.\n• P(α)0 ⊗A−1 is a set of X detectors which arise from cycle checks throughout the\nsurgery procedure.\n• P(α)0 ⊗A0 is a set of fault locations for new qubits which can undergo Z Pauli\nerrors.\n79\n\n• P(α)1 ⊗A−1 is a set of new X checks corresponding to cycles in the hypergraph,\nwhich can undergo measurement errors.\n• P(α)0 ⊗A1 is a set of new Z checks, corresponding to vertices in the hypergraph,\nwhich can undergo measurement errors.\n• P(α)1 ⊗A0 is a set of fault locations for new qubits which can undergo X Pauli\nerrors. Note that when the block reading was performed for only 1 round these fault\nlocations were not present because of the immediately adjacent initialisations in |+⟩\nand measurements in the X-basis.\n• P(α)0 ⊗A2 is a set of Z detectors which are metachecks due to the block reading.\n• P(α)1 ⊗A1 is a set of Z detectors due to the vertex checks throughout the surgery\nprocedure.\nTo acquire the opposite spacetime volume, where it is initialised and measured in the\nZ basis, the R• is dualised, and F ′ has extra E2 components, rather than E0. This is\nidentical to the full block reading case.\nWe can hence rerun a similar proof as for the case of full block-reading, and acquire\nthe homology spaces from the Snake Lemma. We can then prove the following.\nProposition 6.1. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d,\nand let A• be a subcode with distance dA = 1\nαd, with chain maps from A•, or a thickened\nversion thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code with code distance\nd and let H have a sparse cycle basis.\nAssume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and\nafter the partial block reading procedure. First, ⌈α⌉rounds of syndrome measurements are\nnecessary (but not sufficient) to perform partial block reading with fault-distance d.\nThen if either: (a) α ≤2 or (b) the initial codeblocks are each locally testable codes with\nsoundness ρ ≥2n\nm , where n is the codeblock length and m the number of checks, then partial\nblock reading performs logical measurement in ⌈α⌉rounds of syndrome measurements with\nfault-distance d, maintaining the LDPC property throughout.\nProof. X-type boundary conditions.\nBy the Snake Lemma, we can compute the following.\n• |H1(cone(f))| = |H1(F ′)| −|H1(A)|.\n• |H2(cone(f))| = |H2(F ′)| −|H2(A)|.\nIn detail, using the K¨unneth formula and the fact that |H1(P(α)) = 0| and |H0(P(α))| = 1,\n|H1(cone(f))| = |H1(F ′)| + |H0(K)| −|imH1(f)| −|imH0(f)|\n= |H1(F ′)| + |H0(A)| −|imH1(f)| −|imH0(f)|\n= |H1(F ′)| + 0 −|H1(A)| −0\n= |H1(F ′)| −|H1(A)|.\nThe other computation works similarly.\nZ-type boundary conditions. In this case the equivalence classes of logical faults\ncan be computed as,\n• |H1(cone(f))| = |H1(F ′)|.\n80\n\n• |H2(cone(f))| = |H2(F ′)| + |H1(A)| −|H2(A)|.\nWe thus have a similar categorisation of all the equivalence classes of logical faults to\nthose in previous proofs: there are spacelike errors which must always have weight at least\nd by virtue of the deformed code having distance at least d, and there are timelike errors\nthat extend from a boundary and so have weight at least d. However, there are also the\ntimelike errors which correspond to the |H1(A)| term.\nAs check errors in the auxiliary code, these timelike logical faults must always have\nweight at least d if they extend for ⌈α⌉rounds, because they must have weight at least 1\nαd\nin each round. Should fewer than ⌈α⌉rounds be used then there always exists a logical\nfault with weight lower than d, proving the first part of the proposition.\nFor the second part, we must be careful about the cleaning with spacetime stabilisers.\nGiven such a timelike logical fault with weight at least d in the auxiliary code, at least dA\non each layer, cleaning it entirely into the bulk can leave only 2dA qubit Pauli X errors,\none set of size dA just before the surgery and one set of size dA just after, and a set of\ntimelike errors connecting them. We must therefore ensure that this set of timelike errors\nhas sufficent size to yield a high fault-distance. It is distributed over α layers, so we ensure\nthat each layer of check errors contains at least ν = (d−2v)\nα\nfaults, where v is the size of\nthe set of spacelike errors at the bottom (or equivalently the top), and v ≥dA. When\nv ≥d/2, ν < 0 so we are done; this is always the case if α ≤2, i.e. dA ≥d/2. The rest of\nthe proof therefore assumes dA ≤v < d/2. Note that ν < d\nα = dA.\nBecause the minimal size of the bottom set of Pauli errors is dA, there are at least\ndA check errors in the next round if |HZv| ≥dA, where v is the qubit fault vector for\nthe bottom set of Pauli errors. Now, assume that the HZ parity matrix of the original\ncode has soundness ρC ≥n\nm as a classical code. This is guaranteed if the original code is\nquantum locally testable with soundness ρ ≥2n\nm by Lemma 2.12.\nWe now show that d(v, C) ≥dA, where C = ker HZ. First, if |v + x| was lower than\ndA for any x ∈imH⊺\nX then the distance of the subcode would be lower than dA, as the\nsubcode has connections to all X checks incident to v, so multiplying v by x restricted to\nA1 would result in a nontrivial logical operator in A• with weight lower than dA. So we\nmust now consider |v+x| for any x ∈ker HZ\\imH⊺\nX. As v < d/2, the maximal overlap of v\nwith any other X logical in the original code is v. Hence |v + x| ≥|v| ∀x ∈ker HZ\\imH⊺\nX,\nand v ≥dA.\nAs a consequence of the soundness being ρC ≥n\nm, |HZv| ≥m\nn ρCd(v, C) ≥d(v, C) ≥dA,\nso each layer between the top and the bottom Pauli X errors must have at least dA check\nerrors.\nBy the same argument, applying cleaning to only a portion of the checks in the auxiliary\nsystem will distribute the faults such that the portion in the original code and the portion\non checks in the auxiliary system at each timestep must sum to dA, and so the fault\ndistance is at least d.\nWhen performing many partial block readings close together in the same spacetime\nvolume, we strengthen the condition to require soundness ρ ≥2n\nm regardless of whether α\nis above or below 2.\nTheorem 6.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least\nd, and let Ξ be a set of partial block readings {Partial1, Partial2, Partial3, ..., Partialη}\nwhere each subcode has distance di ≥\n1\nαi d for i ∈[η] and the deformed codes each have code\ndistance d, with sparse cycle bases. Suppose the original codes have soundness ρ ≥2n\nm .\nThen all η logical measurement rounds of partial block reading can be performed sequen-\ntially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding\n81\n\nbefore and after the logical measurements.\nThe procedure has phenomenological fault-\ndistance d, for any η ∈N.\nProof. By taking sequential mapping cones on the initial fault complex for the idling\noperation, one finds that the equivalence classes of logical faults are all either:\n• Timelike faults extending between boundaries.\n• Spacelike faults.\n• Timelike faults which act on the auxiliary systems, and cannot be cleaned to have\nweight below d in the original code.\nTimelike logical faults in the auxiliary system and between boundaries must still have\nweight at least d.\nCleaning spacelike logical faults Λ1, Λ2, · · · , Λϑ to have some shared spacelike support\nmust induce at least a proportional weight in check faults, as the initial code has sufficient\nsoundness, so the product SΛ1Λ2 · · · Λϑ cannot fall below weight d. Note that the condition\non the distance of the compacted code CC• is unnecessary in this case.\nTherefore the fault-distance of the procedure is at least d.\nA.5\nFast hypergraph surgery\nTheorem 5.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d,\nand let Ξ be a set of Z-type hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη}\nsuch that:\n• Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance d, with\nsparse cycle bases.\n• The compacted code has distance at least d.\nThen all η logical measurement rounds of generalised hypergraph reading can be per-\nformed sequentially using O(1) rounds of syndrome measurement each, and using d rounds\nof padding before and after the set of logical measurements. The procedure has phenomeno-\nlogical fault-distance d, for any η ∈N.\nProof. This proof proceeds in a largely similar manner to those of Appendix A.3. For\na single hypergraph surgery with an ancillary system A• we construct the same fault\ncomplex\n0\nA2\nA1\nA0\nA−1\nF ′\n3\nF ′\n2\nF ′\n1\nF ′\n0\n0\n0\nf2\nf1\nf0\n0\nbut where A• is now an arbitrary code such that f• is non-overlapping on vertices and\nH0(A) = 0.\nX-type boundary conditions.\n|H1(cone(f))| = |H1(F ′)| −|imH1(f)| + |H0(A)| −|imH0(f)|\n= |H1(F ′)| −|imH1(f)|,\n82\n\nand\n|H2(cone(f))| = |H2(F ′)| −|imH2(f)| + |H1(A)| −|imH1(f)|\n= |H2(F ′)| −|imH2(f)| + |H1(A)| −|H1(A)|\n= |H2(F ′)| −|imH2(f)|.\nIn summary we have:\n• |H1(cone(f))| = |H1(F ′)| −|imH1(f)|.\n• |H2(cone(f))| = |H2(F ′)| −|imH2(f)|.\nZ-type boundary conditions. Similar calculations apply but |imH2(f)| = |imH1(f)| =\n0.\n• |H1(cone(f))| = |H1(F ′)|,\n• |H2(cone(f))| = |H2(F ′)| + |H1(A)| −|imH2(f)|.\nWe can then apply such mapping cones sequentially, and so all logical faults are cate-\ngorised into equivalence classes with the following representatives:\n• Logical faults on checks in the auxiliary system.\n• Logical faults on data qubits in the original or deformed codes.\n• Logical faults on checks extending through the spacetime volume between bound-\naries.\nIn order,\n• Because each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic dis-\ntance d, the weight of a logical fault on checks in the auxiliary system must be at\nleast d.\n• Because the maps between the auxiliary systems and original codes are all non-\noverlapping on vertices, the weight of such a logical fault when cleaned into the\noriginal code must be at least d to flip d checks in the auxiliary system.\n• Because the compacted code has distance at least d, any spacelike fault must have\nweight at least d. This accounts for the fact that spacelike logical operators can\ncleaned by spacetime stabilisers to cancel their shared spacelike support.\n• All other timelike faults must extend from a timelike boundary, so to affect the\nprotocol must have weight at least d.\nRemark A.6. If the chain maps are overlapping on vertices (i.e. they do not satisfy\nDefinition 5.1) then the argument breaks down, as a timelike logical fault in the auxiliary\nsystem could be cleaned to a smaller logical fault in the original code, and so additional\nsyndrome rounds are required to increase the weights of these faults. In the extreme case,\nwhere multiple auxiliary systems measure a single logical representative, this coincides\nwith Theorem 1.6, which shows that such a measurement procedure will always require\nΩ(d) rounds to preserve the fault distance.\n83\n\nB\nNon-CSS measurements by full block reading\nIn this appendix we provide a more detailed explanation of how block reading extends to\nmore general logical Pauli measurements on codes that possess transversal gates.\nB.1\nY measurements\nWe can attach an ancillary system of the following form to any CSS code C:\nX\nY\nZ\nQ\nHZ\nHX\nI\nI\nI\nH⊺\nZ\nH⊺\nX\nwhere the data qubits Q are in the original code, and the r.h.s. scalable Tanner graph is\nthe ancillary system. The checks must always commute by inspection of the diagram.\nHowever, the properties of this combined code are complicated. Evidently, none of the\nZ or X logicals are measured out, as there are no new Z or X checks.\nDefinition B.1. Let C have a basis of representatives such that supp(Y i) = supp(Xi) =\nsupp(Zi), ∀i ∈[k]. Then we say C has matching representatives.\nLet C be self-dual, i.e.\nHZ = HX.\nIf C has matching representatives then Y i is\nmeasured out. As Y i ∈ker HZ = ker HX, cleaning by the new Y checks PY in bijection\nwith supp(Y i) gives PY Y i = 0 on both the new sets of data qubits. If the code does\nnot have matching representatives then the cleaning argument does not work, and so the\nancillary system may not measure the logical qubits.\nNote that to have supp(Xi) = supp(Zi), |supp(Y i)| = 1 mod 2, i.e. it must have odd\nsupport, in order for Xi to anticommute with Zi.\nIf C has matching representatives, all logical qubits will be measured out in the Y\nbasis. There are no new Z logicals, as any Z logical on the top right new data qubits\nmust be in ker(H⊺\nZ) so be products of old Z checks, and any Z operators on the bottom\nright new data qubits won’t commute with the old X checks. The inverse is true for new\nX checks.\nExample B.2. 2D colour codes are self-dual and have Y logicals composed of X and Z\nlogicals on the same support [124].\nRemark B.3. If C is self-dual but there exists a product of Pauli Y operators on data\nqubits, which is a composition of Zi and Xj with supp(Zi) = supp(Xj), but i ̸= j so the\nY operator is actually a representative of Zi ⊗Xj, then the Y ancillary system will still\nsend this Y operator to zero when cleaned, but the measurement is a Zi ⊗Xj measurement\ninstead.\nFor the measurement protocol, assume C is self-dual with matching representatives.\nFirst initialise new data qubits in the top(bottom) right set in the |0⟩(|+⟩) states respec-\ntively. Then measure all stabilisers. To commute with all Y logicals being measured, each\n84\n\nnew data qubit must be connected to an even number of new checks in bijection with each\nY logical, and so X(Z) errors on top right(bottom right) qubits will not affect the logical\nmeasurement outcome.\nAs the code is self-dual, products of Z and X checks with the same support are\nmapped to a subset of new Y checks, such that any nontrivial Y measurement error must\nbe in ker(HZ)\\im(H⊺\nX) = ker(HX)\\im(H⊺\nZ), and hence the logical measurement has some\nprotection against measurement faults.\nLemma B.4. Let C be a self-dual CSS code with matching representatives.\nThen C\nadmits a transversal S gate, acting as S⊗k on the logical space up to a Pauli correction.\nProof. From [34, Prop. 9], C admits a transversal S gate up to a Pauli correction if:\n• u · v = 0 mod 2 for all X stabilisers u, v.\n• For any X logical operator u and X stabiliser v, u · v = 0 mod 2.\nAs C is self-dual the first condition is satisfied. From the matching representatives\nproperty, any Xi logical operator u has the same support as its matching Zi operator,\nand Zi must commute with all stabilisers hence u · v = 0 mod 2.\nTherefore the procedure is not very useful, because the codeblock can have its mea-\nsurement basis permuted by transversal S and H and so Y measurements are unnecessary.\nThe same argument applies to performing block reading of multiple codeblocks in products\ncontaining Y terms. We do not know if partial block reading for non-CSS measurements,\nwhere the subcode is a stabiliser code but not CSS, can allow for fast measurements which\ncannot be performed using transversal gates to permute bases.\nB.2\nX ⊗Z measurements\nGiven a CSS code C, let D be the dual of that code, i.e. HD\nX = HC\nZ and HD\nZ = HC\nX.\nIntroduce the following ancillary system:\nX\nZ\nHX\nHZ\n[0, H⊺\nX]\nI\n[0, I]\nI\nHZ\nX\n[0, HX]\n[HZ, 0]\n[I, 0]\n[I, 0]\nI\nwhere C is on the l.h.s. and D on the r.h.s. The top right and middle checks are now of\nmixed-type, checking stabilisers with both X and Z terms. One can check by symplectic\nmultiplication that the above Tanner graph commutes. We now also have meta-checks\nwhich are of ‘mixed-type’, using both X and Z checks, indicated by the pink diamond.\nThis is a Z⊗Z measurement by full block reading, but conjugated by single-qubit Cliffords.\nHowever, because the codeblocks C and D are dual, this could equally be performed\nby using a modified version of homomorphic measurements but with transversal CX and\nCZ gates [24].\n85\n\nFigure 9: Venn diagram indicating the overlapping supports of a logical operator ΛZ and sets of checks\nκλ (magenta) and v ∩U (dashed).\nC\nProofs concerning modular expansion\nIn this appendix we provide technical proofs related to properties of modular expansion\nused in the main text.\nTheorem 7.7 (Distance preservation). Let {Zi}i∈I be a set of logical representatives\n(trivial or nontrivial) on a distance d CSS code Q with supports ξi = supp(Zi).\nLet\nH(V, E) be a hypergraph with vertices in V being Z checks, hyperedges in E data qubits and\nX gauge checks forming a cycle basis, such that H measures all the logical representatives\n{Zi}i∈I. Let fp : S\ni ξi →V be an injective port function such that U = imfp ⊂V, and\nfp(ξi) = Vi ⊂V, ∀i ∈I. Let Md(H) ≥1, where W is generated by fp(ξi). Then the\ndistance of the deformed code Q ↔H is at least d.\nProof. We show that for every nontrivial ΛZ logical operator in Q ↔H with some support\nin S\ni Vi, the weight of ΛZ cannot drop below d by multiplying with new Z checks (vertices)\nin H.\nLet v be an arbitrary subset of V, and define the stabiliser Z(G⊺v) to be the product\nof all checks in v. We must show that |ΛZ ◦Z(G⊺v)| ≥d.\nFirst, G⊺v ≥min(d, |κλ| −|v ∩κλ| + |v ∩U\\κλ| : λ ∈2I) by the definition of modular\nexpansion. If\nmin(d, |κλ| −|v ∩κλ| + |v ∩U\\κλ| : λ ∈2I) = d\nthen we are immediately done, as no matter which data qubits in Q are turned off by\nZ(G⊺v), there are at least d new data qubits with a Z Pauli in the ancillary hypergraph.\nThe other case is more difficult.\nIf |G⊺v| ≥|κλ|−|v∩κλ|+|v∩U\\κλ| for some λ ∈2I then we can construct the diagram\nin Fig. 9.\n86\n\nThe set in the dashed region is v ∩U, the top right circle is U, the bottom left circle\nis ΛZ and the magenta circle is κλ. We compute by inspection of the diagram that\n|ΛZ ◦Z(G⊺v)| ≥|τλ| + |σλ| + |Bλ| + |B′\nλ| + |Rλ| + |κλ| −|v ∩κλ| + |v ∩U\\κλ|.\nNext,\n|κλ| −|v ∩κλ| + |v ∩U\\κλ| = |φλ| + |σλ| + |B′\nλ| + |γλ|\nand\n|ΛZ ◦Zλ| = |Rλ| + |τλ| + |γλ| + |Bλ| + |φλ| ≥d\nwhere Zλ = Z(f−1κλ) acts as Z on the preimage of κλ in the original code. Zλ is a logical\noperator, but not necessarily a nontrivial one. As ΛZ is a nontrivial logical operator of\nthe deformed code,\n|ΛZ ◦Zλ| ≥d.\nPlugging these together we find that\n|ΛZ ◦Z(G⊺v)| ≥2(|σλ| + |B′\nλ|) + d ≥d.\nAs the cycles are gauge-fixed and no Z logical can be reduced below weight d, the\ndistance of Q ↔H is at least d.\nTheorem 7.8 (Thickening). Let H(V, E) be a hypergraph with modular expansion Mt(H).\nLet J L be the path graph with length L ≥\n1\nMt(H), i.e. L vertices and L −1 edges. Let\nHL := H□J L be the hypergraph of H thickened L times. Then HL has modular expansion\nMt(HL) ≥1 for Uℓ= S\ni\nV ℓ\ni at any level ℓ∈{1, 2, · · · , L}, where V ℓ\ni is the copy of Vi in\nthe ℓth level of the thickened hypergraph.\nProof. Let G : F2E →F2V be the incidence matrix of H, RL the incidence matrix of J L,\nand GL the incidence matrix of HL. Then\nGL =\n\u0010\nG ⊗IL\nIn ⊗RL\n\u0011\nwhere n = |V|.\nWe divide up the vertices of HL into sets Vℓfor ℓ∈[L], corresponding to each level of\nthe thickened graph. Similarly, for a given level ℓwe have V ℓ\ni and Uℓ= S\ni V ℓ\ni .\nLetting vℓ∈Vℓbe a set of vertices in level ℓ, we have the vector v = (v1 v2 · · · vL)⊺∈\nS\nℓVℓseen as a set of vertices in HL. Associated to this set we also have uℓ= vℓ∩Uℓ=\nv ∩Uℓ.\nNow, let r = arg minj min(t, |κj\nλ| −|uj ∩κj\nλ| + |uj\\κj\nλ| : λ ∈2I). Using the triangle\ninequality and the fact that LMt(H) ≥1,\n|G⊺\nLv| = |G⊺\nL(v1 v2 · · · vL)⊺| =\nL\nX\nj=2\n|vj−1 + vj| +\nL\nX\nj=1\n|G⊺vj|\n≥|vr + vℓ| + LMt(H) min(t, |κr\nλ| −|ur ∩κr\nλ| + |ur\\κr\nλ| : λ ∈2I)\n≥|ur + uℓ| + min(t, |κr\nλ| −|ur ∩κr\nλ| + |ur\\κr\nλ| : λ ∈2I).\nNow consider the different cases.\nIf min(·) = t then |ur + uℓ| + t ≥t and we are done.\n87\n\nIf min(·) = |κr\nλ| −|ur ∩κr\nλ| + |ur\\κr\nλ| for some λ ∈2I then\n|ur + uℓ| + |κr\nλ| −|ur ∩κr\nλ| + |ur\\κr\nλ|\n= |ur ∩κr\nλ + uℓ∩κr\nλ| + |ur\\κr\nλ + uℓ\\κℓ\nλ| + |κℓ\nλ| −|ur ∩κr\nλ| + |ur\\κr\nλ|\n≥|κℓ\nλ| −|uℓ∩κℓ\nλ| + |uℓ\\κℓ\nλ|,\nmaking use of the triangle inequality for the last line.\nTherefore |G⊺\nLv| ≥min(t, |κℓ\nλ| −|v ∩κℓ\nλ| + |v ∩Uℓ\\κℓ\nλ| : λ ∈2I) for any set of vertices\nv in HL, so HL has modular expansion Mt(HL) ≥1.\nD\nCircuit-equivalence of surgery and homomorphic measurement\nIn this appendix we derive a circuit equivalence between generalized surgery measurement\nand homomorphic measurement. The following proofs use the so-called “scalable” ZX-\ncalculus from Ref. [87]. This calculus has previously been used to describe LDPC surgeries\nin Ref. [64]. We assume familiarity with the ZX-calculus for this section; see Ref. [125] for\na comprehensive introduction.\nIn short, the scalable ZX-calculus (SZX-calculus) has thick wires indexed by a positive\ninteger n, denoting a register of n qubits. Green and red spiders then act on the n qubits,\nwith spider phases now represented by a vector v over C of length n, containing phases\nacting on each qubit. As in the conventional ZX-calculus, an all-zeros vector is omitted\nand the green or red spider left unlabelled. We also omit the label denoting the size of\nthe qubit register.\nThe SZX-calculus has other generators than green and red spiders. For example, we\nalso have the matrix arrows\nA\nand\nB\n(6)\nwhere A and B are matrices over F2. These generators can be defined by their action on the\nHilbert space. Suppose A ∈Fn×m\n2\nand B ∈Fp×q\n2\n. For a qubit state in the computational\nbasis |x⟩with x ∈Fm\n2 , the arrow pointing to the right in Eq. 6 corresponds to the linear\nmap RA : |x⟩7→|Ax⟩. Meanwhile, the arrow pointing to the left corresponds to the linear\nmap H⊗pRBT H⊗q = H⊗p \u0010\n|x⟩7→|BT x⟩\n\u0011\nH⊗q.\nStabiliser measurement on CSS codes can be neatly described using the SZX-calculus.\nGiven a CSS code with n qubits and the parity-check matrices HX, HZ, the measurements\ncan be described by\nH⊺\nX\nHZ\nsZπ\nsXπ\nwhere sZπ is the vector of phases representing syndrome outcomes sZ = {0, 1}mZ, and\nthe same for sXπ. In particular, when the input state is in the codespace, sZ = sX = 0\nand the phase vectors can be omitted. The same applies for the case where the the initial\nstate is not in the codespace but the measurements project it into the codespace.\nCommutation of stabilisers implies that\nH⊺\nX\nHZ\n=\nH⊺\nX\nHZ\n88\n\nand furthermore,\nHZ\nHZ\n=\nHZ\nwith a similar rewrite for X-checks. Obviously such rewrites do not generally preserve\nfault-tolerance of the circuit: we are taking multiple rounds of syndrome measurement\nand collapsing them into one round, which will generally reduce the fault-distance of the\nspacetime code.\nAlong with the merge and copy rules of the ZX-calculus we can now prove that surgery\nby an auxiliary system and homomorphic measurement using transversal gates are equiv-\nalent as circuits.\nTheorem 9.1. Let C• be a CSS code with a homomorphic measurement procotol specified\nby a chain map f• : A• →C•.\nThen the circuit corresponding to the homomorphic\nmeasurement can be rewritten to a surgery specified by cone(f•).\nProof. We describe an Z-basis logical measurement; the same result for X measurements\ncan be easily acquired similarly. The rewrites required are as follows.\nf⊺\n1\nαπ\nH′\nZ\n(H′\nX)⊺\nH⊺\nX\nHZ\n=\nf⊺\n1\nαπ\n(H′\nX)⊺\nH⊺\nX\nHZ\n=\nf⊺\n1\nαπ\n(H′\nX)⊺\nH⊺\nX\nHZ\n=\nf⊺\n1\nαπ\n(H′\nX)⊺\nH⊺\nX\nHZ\nf⊺\n2\nN⊺\nThe first SZX-diagram on the top left is depicting a homomorphic measurement. The\noriginal code is shown on the bottom wire, with parity-check matrices HZ, HX.\nAn\nancilla code is initialised in |0⟩⊗n′ then its stabiliser checks are measured. Then a set of\ntransversal CNOTs defined by a chain map f• are applied, with the controls on the initial\ncode and the targets on the ancilla code. Then the ancilla code is measured out in the Z\nbasis, with the vector απ where α ∈Fn′\n2 determines the logical measurement outcome.\nNext, the |0⟩initialisations are commuted through the Z-check measurements, and\nthen the merge rule is used to extend the check qubits of the X-check measurements to\ninstead become data qubits of the auxiliary system, initialised in |+⟩mX and measured\nin the X-basis. We now see that the data qubits of the homomorphic measurement have\nbecome check outcomes of Z measurements, which include both the new data qubits of\n89\n\nthe auxiliary system and data qubits of the original code. The outcomes απ of these Z\nmeasurements determine the logical measurement outcome.\nLast, we add X checks which gauge-fix the cycles (there are cases, such as in full-block\nreading and the CKBB scheme [52], where these are not necessary), with the check matrix\nN. We also deform the X checks of the original code to include qubits of the auxiliary\nsystem, such that these checks commute with the Z checks that determine the logical\nmeasurement outcome. The deformation of these checks is determined by the chain map\nf•.\nWe can then do the opposite rewriting to convert surgeries to homomorphic measure-\nments.\nCorollary 9.2. Let C• be a CSS code with a surgery protocol specified by cone(f•) for a\nchain map f• : A• →C•. Then the circuit corresponding to the surgery protocol can be\nrewritten to a homomomorphic measurement specified by f•.\nProof.\nF ⊺\nαπ\nG⊺\nH⊺\nX\nHZ\nM⊺\n=\nV\nE\nF ⊺\nαπ\nG⊺\nH⊺\nX\nHZ\nM⊺\nV\nE\n=\nF ⊺\nαπ\nG⊺\nH⊺\nX\nHZ\n=\nF ⊺\nαπ\nG⊺\nH⊺\nX\nHZ\nH′\nZ\nN⊺\nV\nV\nE\nE\nThus we see that the vertices V of the hypergraph in surgery, which are Z-checks, are\nrewritten to the data qubits in homomorphic measurement; the edges E of the hypergraph,\nwhich are data qubits, are rewritten to be X-check outcomes. The cycles vanish from the\ncircuit entirely, which is because they become X-type meta-checks on the ancilla code.\nThese relabellings correspond precisely to the degree-shifting of a mapping cone to recover\nthe original chain map f•.\n90\n\nE\nFast measurement requires multiple representatives\nIn this appendix we provide a proof of our results lower bounding the spacetime volume\nof a general procedure to measure logical representatives in a quantum code.\nProposition 9.3. Let Q be a quantum LDPC code. Let S be a set of logical operator\nrepresentatives measured at the same time by one or more auxiliary system for T syndrome\nrounds. Assume that the correct logical measurement outcomes of S are not known in\nadvance. Let f be the lowest weight data qubit fault in Q such that:\n• f flips at least one logical representative in S.\n• If f flips a logical representative in S then it flips all other representatives in the\nsame equivalence class in S.\nLet w be the weight of f, and s be the syndrome weight of f in Q. Then the phenomeno-\nlogical fault-distance of the measurement protocol is upper bounded by 2w + Ts.\nProof. The measurement scheme begins at time ti and ends at time to = ti + T, and\nmeasures all the logical operator representatives contained in S. Data qubit errors occur\nat integer timesteps and measurement errors occur at half-integer timesteps. To measure\nthis operator by an auxiliary system, entangling gates are applied from each qubit in S to\nsome other system Aux of checks and data qubits, such that the measurement outcomes of\nchecks in Aux infer the logical measurement outcomes. The entangling gates are applied\nwhile measuring checks in Aux for T rounds before ceasing the application of entangling\ngates.\nIn order to receive an incorrect logical measurement outcome without detecting an\nerror it is sufficient to flip any of those logical operators for the duration of the logi-\ncal measurement scheme, so long as there are no logical measurement outcomes between\nrepresentatives which are ‘mismatched’, meaning that measurements of different represen-\ntatives of the same logical operator yield different outcomes. Thus if any representative\nin S is flipped, all representatives in the same equivalence class in S must be flipped. Ob-\nserve that as none of the correct logical measurement outcomes of S are known in advance\n(which would occur if Q was initialised in a particular known state with deterministic\nlogical measurement outcomes) there are now no detectors which can detect the logical\nerror.\nTo flip some operators in the round at time ti, apply a Pauli operator fault f to Q\nwith weight w satisfying the conditions specified in the Proposition.\nThis set of faults is detectable by checks in the original code at timesteps ti + 1\n2, ti +\n3\n2, · · · , so in order to conceal the fault we add s check errors at each timestep for the\nduration of the logical measurement, where s is the number of checks which f flips. In\ntotal there are Ts of these check faults.\nLastly, apply an identical Pauli operator fault f after the logical measurement proce-\ndure concludes, at time to + 1. This is a logical fault of the measurement procedure with\nweight 2w + Ts.\nWere the code Q left in the idling operation instead, the above logical fault would be\na spacetime stabiliser of the code, and so this logical fault weight can be vastly lower than\nthe distance of Q depending on S, the time T and the syndrome weights s of operators.\nNote also that this is just an upper bound on the fault-distance: if the auxiliary system\nis poorly constructed, for example by connecting every qubit in S to a single new check\n91\n\nqubit, the fault-distance may be substantially lower due to check errors on the auxiliary\nsystem.\nTheorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quan-\ntum LDPC code by an auxiliary system requires connections to more than one logical\nrepresentative to maintain phenomenological fault distance Ω(d), unless the correct logical\nmeasurement outcome is known in advance.\nWe prove this by demonstrating that otherwise there exists a weight |f| logical fault,\nwhere |f| ∈o(d).\nProof. Assume w.l.o.g. that the measurement being performed is on a Z logical operator,\nas any Pauli measurement is locally equivalent to this measurement by conjugation of\nsingle-qubit Cliffords.\nIf the logical operator representative has support v, the measurement being performed\nis of the operator N\ni∈v\nZi. To measure this operator by an ancillary system, entangling gates\nare applied from each qubit i to some other system Aux of checks and data qubits, such\nthat the measurement outcomes of checks in Aux infer the logical measurement outcome\nof N\ni∈v\nZi.\nThe measurement scheme starts at time ti and concludes at time to, with o(d) rounds\nbetween. Data qubit errors occur at integer timesteps and measurement errors occur at\nhalf-integer timesteps.\nAssume that the entangling gates are applied while measuring\nchecks in Aux for o(d) rounds before ceasing the application of entangling gates. Regard-\nless of the system of checks and other structure of Aux, there exists a low weight fault\nwhich flips the logical measurement outcome.\nConsider an arbitrary qubit j ∈v. At any time t ∈[ti, to] the application of a Pauli\nerror Xt\nj, followed by measurement errors on all incident Y and Z checks at time t+ 1\n2, and\nthen a Pauli error Xt+1\nj\n, is a local spacetime stabiliser of the original code, and therefore\ndoes not violate any detectors. However, at time t + 1\n2 the outcome of measuring N\ni∈v\nZi is\nflipped, as Xj anticommutes with Zj. This is then corrected by the subsequent Pauli error\nXt+1\nj\n, and so at time t + 3\n2 the measurement of N\ni∈v\nZi would return the correct outcome\nagain.\nHence a logical fault f on the logical measurement can be constructed by applying a\nPauli error Xti\nj and measurement errors on all incident X and Y checks on qubit j from\ntime ti+ 1\n2 to time to+ 1\n2, and a final Pauli error Xto+1\nj\n. This is a spacetime stabiliser of the\noriginal code, as it is composed of local spacetime stabilisers at times t = ti, ti + 1, · · · , to,\nhence the fault does not violate any detectors. Each round of measurement of N\ni∈v\nZi will\nbe flipped by the original Xti\nj error, and the outcome of N\ni∈v\nZi is then flipped back after\nthe measurement procedure concludes. The fault f is then Xto+1\nj\nXti\nj\nS\nti≤t≤to M\nt+ 1\n2\nj\n, where\nM\nt+ 1\n2\nj\nis a measurement fault on each Y or Z check incident to qubit j at time t + 1\n2.\nBecause the code is LDPC, the fault at each round has weight bounded above by a con-\nstant, where the precise weight depends on the number of Y and Z checks incident to qubit\nj, but this cannot increase the weight asymptotically. Because the logical measurement is\nperformed for o(d) rounds, the weight |f| ∈o(d).\nNote that if the correct logical measurement outcome is known in advance, there\nis an additional detector at each round of N\ni∈v\nZi and so the above argument does not\n92\n\nhold. Similarly, the reason the proof no longer applies when measuring multiple logical\noperator representatives is because there are additional detectors which appear because\nthe measurements of each representative of the same logical operator must agree with\neach other to be correct; equivalently, these additional detectors correspond to sets of\nmeasurements whose correct outcomes are known in advance, as they are stabilisers of the\noriginal code.\nBoth of the above proofs are insensitive to the structure of Aux, which could be a\nmeasurement hypergraph in surgery, an ancilla block in homomomorphic measurement or\nany other logical measurement gadget.\n93\n"},"cleaned":{"kind":"string","value":"Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery Alexander Cowtan1, Zhiyang He2, Dominic J. Williamson3,4, and Theodore J. Yoder5 1 1 3QD, UK 2 02139, USA 3 2006, Australia 4IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA 5IBM Quantum, IBM T.J. Watson Research Center, Yorktown Heights, NY 10598, USA Quantum code surgery is a promising technique to perform fault-tolerant computation on quantum low-density parity-check codes. Recent developments have significantly reduced the space overhead of surgery. However, generic surgery operations still require O(d) rounds of repeated syndrome extraction to be made fault-tolerant. In this work, we focus on reducing the time overhead of surgery. We first present a general set of conditions that ensure fault-tolerant surgery operations can be performed with constant time overhead. This fast surgery necessarily makes use of an auxiliary complex described by a hypergraph rather than a graph. We then introduce a concrete scheme called block reading, which performs transversal surgery across multiple code blocks. We further investigate surgery operations with intermediate time overhead, between O(1) and O(d), which apply to quantum locally testable codes. Finally, we establish a circuit equivalence between homomorphic measurement and hypergraph surgery and derive bounds on the time overhead of generic logical measurement schemes. Overall, our results demonstrate that reducing the time cost of code surgery is not reliant on the quantum memory being single-shot. Instead it is chiefly the connectivity between a code and its measurement ancilla system that determines the achievable measurement time overhead. 1 Introduction Quantum error correction [1-5] is an essential building block in any fault-tolerant quantum computer that is capable of running large-scale applications. The surface code and other topological codes [6-10] have been extensively studied due to their rich mathematical structure and strong practical performance. Despite their many desirable properties, topological codes incur an appreciable space overhead [11] when performing fault-tolerant quantum computation (FTQC). This motivates the study of quantum low-density parity check (QLDPC) codes, which can achieve better parameters, notably constant encoding rate, in both the asymptotic and near-term regimes [12]. QLDPC codes possess the practically desirable LDPC property - each check (or detector [13]) involves a constant number Alexander Cowtan: 1 16 Oct 2025 of qubits and each qubit is in a constant number of checks. Following exciting advancements in both theoretical code design [14-18] and experimental qubit connectivity [1921], QLDPC codes are now competitive candidates to serve as low-overhead fault-tolerant quantum memories [22, 23]. To realize FTQC in low space overhead with QLDPC codes, we need to design schemes for performing logical computation on their encoded qubits. This is a flourishing area of research with many recent works. Popular techniques include constant-depth unitary circuits [23-36], resource state preparation and teleportation [37-46], and code switching [47-69]. Here, we focus on the technique of QLDPC code surgery, which has seen rapid recent developments [52-67] due to its general applicability. On a high level, code surgery is a technique for performing Pauli measurement on logical qubits encoded in a stabiliser code by introducing an ancilla system, consisting of new qubits and stabiliser generators, to deform the code in such a way that the chosen logical operators are decomposed into products of (newly introduced) stabilisers. The deformation procedure can be viewed as gauge fixing or weight reduction [70-74]. Code surgery can be made faulttolerant by repeatedly measuring all new stabiliser generators for a sufficient number of rounds (typically taken to be the code distance d) to prevent timelike errors. After obtaining the logical measurement outcome, the ancillary qubits are individually measured out to return to the original code. Unfortunately, measuring all stabiliser generators for d rounds incurs a substantial time overhead when compared to protocols that use constant-depth gates. For some computations this cost can be substantially ameliorated by using Pauli Based Computation [75] to reduce the number of gates that must be implemented [62, 76]. However, in the worst case a large time overhead remains. While a growing number of code families are known to be single-shot decodable [77-83], only high-dimensional topological codes are known to admit single-shot surgery [60]. Overall, prior to this work we lacked a rigorous understanding of the time overhead required for the fault-tolerance of surgery operations, except that d rounds is typically sufficient. In this work, we develop a theory of fast quantum code surgery, formulating the conditions needed for surgery operations to be fault-tolerant with only a constant number of rounds of error correction. Notably, the base memory code does not need to be singleshot decodable. As our primary example, we present a protocol called block reading which enables transversal surgery across identical CSS code blocks in constant space and time overhead. We discuss our main results in Section 1.2, and discuss how our work relates to prior work in the literature, including independent works which appeared shortly before ours [66-69], in Section 1.3. 1.1 Setting In this work, we aim to preserve the LDPC property of the codes and detectors throughout, and highlight where this is not guaranteed. However, we ignore any further specific connectivity requirements of potential physical implementations. We assume a phenomenological Pauli and measurement noise model in our notion of fault tolerance. We focus primarily on asymptotics rather than specific constant factors. We do not concern ourselves with decoders for block reading, which depend greatly on the initial CSS codeblocks. We leave this important question to future work. 2 1.2 Summary of main results To discuss our main results, we briefly review the usual setup of generalized code surgery, and refer readers to Section 2 for a more thorough introduction. One starts with, in general, a collection of memory code blocks with encoded logical qubits, on which the aim is to perform fault-tolerant logical Pauli measurements. One constructs an ancilla system of qubits and stabilisers, which can be specified by a hypergraph [56, 58]. One then connects this ancilla system to the memory blocks, in the process modifying the stabilisers of the ancilla system and the memory blocks according to the connections. The modified set of stabilisers is an abelian group, which specifies a stabiliser code (the deformed code). Certain logical operators from the memory blocks become products of stabilisers in this deformed code, which means they are measured when we measure these new stabilisers. By measuring the deformed code stabilisers for a sufficient number of rounds (typically, in prior work, d rounds, where d is the distance of the memory codes), one can deduce the logical measurement results fault-tolerantly. Finally, one measures out the ancilla system to return to the original code. The operators measured depend largely on the ancilla system (equivalently the hypergraph) and its connections to the memory. This formulation is general enough to capture all prior works on code surgery, see Section 3.2 of [62] for a brief review. In this work, we augment the above formulation by associating a 4-term chain complex with the hypergraph H = (V, E). Specifically, let C be a basis of cycles of H, where a cycle is a collections of hyperedges which include every vertex in V an even number of times. Let W be a set of O(1)-sized components of H, where a component is a collection of vertices which intersect every hyperedge on an even number of vertices.1 Then H• = FW 2 FV 2 FE 2 FC 2 δ2 δ1 δ0 is a 4-term cochain complex, where the coboundary maps δi are specified by the inclusion relations of W, V, E, C. Note that when H is a simple connected graph (i.e., every hyperedge is a regular edge), the only component is the set of all vertices V . Typically when treating cell complexes the above coboundary maps would instead be boundary maps: we flip this convention to suit our conventions in the main body. In prior works on surgery the focus was on the 3-term complex without W, which fully specifies the ancilla system. By including W, we can now consider the 1-cosystolic distance of H•, which is d• 1 = min{|u| : u ∈ker(δ2) (δ1)}. Our main result states that t ≥d hypergraph surgery operations, all satisfying certain distance conditions, can be performed sequentially with each operation taking a constant time, such that the overall protocol performs t distinct logical operations fault-tolerantly in O(t) time. Theorem 1.1 (Fast hypergraph surgery, informal statement of Theorem 5.3). Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks, each with distance at least d. Let t ≥d and H• 1, · · · , H• t be sparse chain complexes from hypergraphs which each define a Z-type surgery operation on the memory blocks, such that • All H• i have 1-cosystolic distances at least d, and 1Note here that W is a set of our choice and is often not a basis of all components of H. 3 • The compacted code, which is a CSS code defined by the t surgery operations, has distance at least d. Then the t surgery operations can be performed sequentially with O(1) time each. The phenomenological fault-distance of the protocol is at least d and the LDPC property is preserved throughout. We elaborate on the definition of the compacted code as well as other subtle conditions in the main text, but note here that there exist simpler sufficient conditions to require of each surgery operation individually or of the original code that together ensure the distance of the compacted code is large. Importantly, we consider the fault-tolerance of performing t ≥d surgery operations in time O(t), instead of performing one surgery operation in time O(1). This is because, strictly speaking, performing one surgery operation in time O(1) is not fault-tolerant. We did not assume that the memory code blocks are single-shot decodable, and the deformed codes are not necessarily single-shot decodable. Therefore, the syndrome information from O(1) rounds of noisy measurements is insufficient for fault-tolerance. In contrast, performing t operations in O(t) time generates enough syndrome information to ensure the whole protocol has phenomenological fault-distance at least d. In this work, we refer to this as amortised constant time. This theorem formulates general conditions on surgery operations which enable constanttime-overhead logical gates. Hypergraph surgeries that measure stabilisers and logical operators of a code are forced to incorporate its local structure, up to changing the choice of generators. In the remaining parts of this work, we study a concrete fast surgery operation which we call block reading. In its simplest form, given two identical code blocks of a CSS QLDPC code Q, block reading acts transversally on all pairs of data and check qubits of the two blocks and measure all pairs of logical qubits in the Z ⊗Z (or similarly X ⊗X) basis. It does so by taking the auxiliary complex (or hypergraph) to be the code Q itself, which by assumption has 1-cosystolic distance at least d. This example can be easily generalized to act on more than two code blocks and to measure more sophisticated Pauli product operators, including the checks of any CSS QLDPC code, transversally. We term this protocol full block reading. Theorem 1.2 (Full block reading, informal statement of Theorem 3.12). Let Q, Q′, Q′′... be a set of c identical Jn, k, dK CSS LDPC codeblocks. Then a set of sparse logical Z Pauli operators across Q, Q′, Q′′... can be measured in parallel using O(cn) additional qubits and amortised O(1) time overhead. Every measurement acts transversally on all k logical qubits in a given codeblock. The phenomenological fault-distance of the protocol is at least d, and the LDPC property is preserved throughout. Two important remarks are in order. First and foremost, parallels can be drawn between block reading and Steane-style error correction, or more generally logical measurements performed via ancillary code blocks and transversal CNOT gates. In terms of logical action, full block reading has the same effect as measurements via transversal gates, except with a code-deformation procedure in contrast to a unitary circuit. Notably, however, the latter protocol usually assumes the ancillary code block is fault-tolerantly prepared, which in general requires O(d) time overhead. The more subtle result of algorithmic fault-tolerance [84] showed that this need not be the case, and an ancillary code block prepared in O(1) time can still be used for measurements via transversal CNOT, such that the whole computation is fault-tolerant even though this individual operation is not. Our result on full block reading is similar to algorithmic fault-tolerance, in that 4 we prove d surgery operations can be performed fault-tolerantly in O(d) time despite the fact that a single operation performed in constant time is not necessarily fault-tolerant. Notably we introduce new surgery gadgets and develop a very different proof to Ref. [84]. This establishes a curious connection between transversal surgery and transversal gates. Secondly, it is instructive to consider the simplest case of full block reading, where there is a single memory block and we measure all logical qubits in the Z basis. This case mirrors Steane error correction, and showcases why a single fast hypergraph surgery operation is not necessarily fault-tolerant: if it were, we would have constant space-time overhead state preparation of arbitrary CSS QLDPC codes (see Remark 3.10)! Of course, full block reading with constant syndrome rounds applied to a non-single-shot code generates a frame error akin to that of Steane error correction with an ancillary code block (under-)prepared with constant syndrome rounds. This frame error is only eliminated in the subsequent O(d) time steps. Therefore, this is one of the reasons that performing either scheme in constant time is not strictly speaking fault-tolerant. With full block reading as a motivating example, we study more flexible fast hypergraph surgery operations by taking subcodes of the memory code and thickening them appropriately. We call this partial block reading, which acts transversally on all physical and logical qubits contained in the subcodes. Theorem 1.3 (Partial block reading, informal statement of Theorem 4.5 and Corollary 7.13). Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d. Let A be a subcode of each of the codeblocks Q, Q′, Q′′... with length nA and distance at least d. Then a suitable hypergraph H can be constructed using O(nAd) additional qubits. When A has soundness ρA this bound can be reduced to O( nA ρA ), assuming the existence of a sparse cycle basis. The precise notions of subcode, soundness and sparse cycle basis are defined in the main text. As usual in the study of code surgery, the factor O(d) space overhead is a worst case upper bound. When the subcode has suitable expansion properties, this space overhead can be significantly reduced, all the way to O(1) in some cases. We expect the space overhead of partial block reading to be highly optimizable in practice, a problem we leave to future work. Beyond constant time overhead surgery, we further study the case where the conditions of Theorem 1.1 are not satisfied, yet surgery can still be performed in lower than O(d) time. We show that when the memory codes have soundness properties, then partial block reading can be performed with less than O(d) syndrome rounds (in amortisation) when the subcode A has distance lower than d. Theorem 1.4 (Intermediate time hypergraph surgery, informal statement of Theorem 6.5). Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and Let t ≥d and H• 1, · · · , H• t be sparse chain complexes from hypergraphs which each define a surgery operation on the memory blocks, such that • Each auxiliary complex H• 1, · · · , H• t has 1-cosystolic distance di ≥ 1 αi d, • The deformed code each has distance at least d, and • The original codes have constant soundness. Then the t surgery operations can be performed sequentially with ⌈αi⌉time each. The phenomenological fault-distance of the protocol is at least d and the LDPC property is preserved throughout. 5 Our hypergraph surgery constructions rely on homomorphic chain maps between the auxiliary complex and the memory. This fact draws strong parallel to the technique of homomorphic measurement [42], which is a logical measurement scheme that employs transversal gates and generalizes Steane error correction. We formalize this connection by proving a general circuit equivalence between surgery and homomorphic measurement. Theorem 1.5 (Informal statement of results in Section 9). Let f define a homomorphic measurement protocol. Then the circuit for f can be rewritten via ZX calculus to an equivalent hypergraph surgery protocol and vice versa. Nevertheless, there are several subtleties, and this equivalence does not generally preserve the fault distance of the measurement protocols. At a high level, the data (check) qubits in an ancillary block for homomorphic measurement correspond to new check (data) qubits in the corresponding surgery protocol respectively. As a consequence, timelike errors in surgery protocols can be resolved by measuring stabilisers for multiple rounds. However, the corresponding spacelike errors on a homomorphic measurement ancilla code are more difficult to resolve in a similar manner. This is why surgery protocols tend to be more flexible in addressing logical qubits: one can create a highly targeted ancilla system with low measurement fault distance, and then boost that measurement fault distance by measuring for multiple rounds. Finally, we complement our constructions with a structural upper bound on the faultdistance of any logical measurement scheme, including surgery, homomorphic measurement and other similar protocols. The idea for this bound is based on quantifying the smallest undetectable fault that flips the logical representatives being measured while avoiding any ancilla system, see Proposition 9.3. This logical fault corresponds to a spacetime stabiliser in the fault-tolerant memory on the code without any logical measurement, that becomes a nontrivial logical fault when the code is measured. Here, we present a simplified statement of this general bound. Theorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quantum LDPC code by an auxiliary system requires connections to more than one logical representative to maintain phenomenological fault distance Ω(d), unless the correct logical measurement outcome is known in advance. In other words, reducing the time cost of code surgery is not reliant on the quantum memory being single-shot, nor does fast logical measurement require transversal gates between codeblocks. Instead it is chiefly the degree of connectivity between a code and its measurement ancilla system that determines the achievable measurement time overhead. 1.3 Related work Code surgery is an active topic of research, and our work has many connections to the literature. Notably, Ref. [67] appeared on arXiv shortly before this paper. Our works were developed independently, but share important results in common. We briefly discuss the areas of overlap and difference. Most importantly, Theorem 5 of Ref. [67] states that a surgery operation defined by a sparse chain complex and a sparse chain homomorphism can be performed in constant time overhead fault-tolerantly if the auxiliary complex has suitable expansion properties. This is similar to our Theorem 1.1 (formally Theorem 5.3), with a few points of distinction: 1. We study the fault-tolerance of performing t ≥d suitable surgery operations in O(t) time, and claim that the time overhead of each logical measurement is constant in amortisation. In contrast, Ref. [67] study the fault-tolerance of performing 6 one suitable surgery operation in constant time. As we argued, this operation is not fault-tolerant by itself, and Ref. [67] remedies this by assuming that the syndrome information on the memory is noiseless before and after the constant time surgery operation (see their Section IIIC). This is a strict assumption in the context of single-shot operations - for generic quantum codes, producing reliable syndrome information requires O(d) rounds of syndrome measurement. In this work, we consider a protocol where we start and end with O(d) rounds of syndrome measurement on the memory, and perform t ≥d hypergraph surgery operations in between. We demonstrate that each hypergraph surgery operation generates a round of syndrome information on the memory blocks, and accumulating these rounds guarantees faulttolerance of the full protocol. 2. Ref. [67] asks for the auxiliary complex to have suitable expansion properties, which is a sufficient but not necessary condition for the deformed code to have high distance. From this perspective our requirements are less stringent. Notably, full block reading for generic codes does not satisfy the expansion conditions in Ref. [67]. On the other hand, for partial block reading we prove that a similar expansion condition can satisfy the second condition of Theorem 1.1. Both Ref. [67] and our work show that the desired expansion can be achieved by taking a tensor product with a repetition code. The idea of using expansion to improve deformed code distance, and the idea of boosting expansion with a repetition code are both standard in the study of code surgery and weight reduction [52, 55, 56, 58, 72]. Our works are largely distinct and complimentary otherwise. As our primary examples we study full and partial block reading (Theorems 1.2 and 1.3), which perform transversal measurement of all logical operators contained in a (sub)code. Ref. [67] provides constructive, addressable surgery operations, capable of measuring one or more logical operators, using techniques including brute-force branching and auxiliary graph surgery [59, 61]. Ref. [67] considered concrete examples on abelian multi-cycle codes and demonstrated promising numerical simulation results. We further study the case of hypergraph surgery with intermediate time overhead (Theorem 1.4), where the conditions of Theorem 1.1 are not satisfied yet soundness of the memory codes enables fault-tolerant surgery in less than O(d) time.2 Our results on circuit-equivalence between surgery and homomorphic measurement, Theorem 1.5, and on the connectivity requirements for fast logical measurement, Theorem 1.6, have no counterparts in Ref. [67]. There are several other recent independent works which share some overlap with our results. Ref. [65] also considered the use of hypergraphs for code surgery. They explored randomized constructions demonstrating promising practical performance. While they did not explore the time overhead of these surgery operations, they stated a similar condition to boost the distance of the deformed code with expansion properties of the hypergraph (see their Lemma 2). In parallel, Ref. [68] and [69] developed single-shot dimension jumping techniques3 for three-dimensional homological product codes. Relatedly, Ref. [66] developed single-shot code switching techniques for more general homological product codes. In its general form, full block reading can be seen as measuring, for O(1) rounds, the stabilisers of a 2Ref. [67] has a brief remark with a similar observation. 3Dimension jump is a special form of code switching, where one switch from (multiple copies of) lower dimensional codes to (fewer copies) of higher dimensional codes and vice versa. 7 homological product code: that of the memory code and the pattern matrix.4 Our results show that this protocol can be combined with further hypergraph surgery protocols to be made fault-tolerant. In comparison, Refs. [66, 68, 69] all proved stronger fault-tolerance guarantees in their specific settings. They also developed techniques to switch from the resultant homological product code to other codes, which enable a range of interesting logical computation gadgets. We remark that the aforementioned works [65-69] all have other contributions not mentioned in our brief discussion above. Connecting to prior literature, while conventional quantum LDPC code surgery takes as input the support of a logical operator to be measured [52-56, 58], hypergraph surgery must incorporate the local structure of the code on the region being measured. We focus on the simplest choice of hypergraph corresponding to the subcode itself, namely block reading. This is similar to the CSS code surgery described in Ref. [64], as well as the finelydevised sticking in Ref. [59]. Our work is also related to Ref. [60], in which single-shot surgery with 3D and 4D toric codes was studied using the formalism of fault complexes. Our proofs in Appendix A use this formalism, and extend it to include deformations on spacetime volumes. We are aware of upcoming work in which the time overhead of lattice surgery with surface codes is reduced by using the complementary gap to estimate confidence in a surgery protocol and stop early if the confidence is high [85]. Our work is grounded in a different setting, concerned with phenomenological fault-distance and general CSS LDPC codes, and focuses on reducing the time overhead asymptotically. 1.4 Organization This work is lengthy and our results involve many subtleties. For the sake of the reader, we have deferred the majority of technical proofs to the appendices and aimed to build up complicated results from simpler ones. In Section 2, we review important background and definitions. In Section 3, we introduce full block reading as the motivating example, and demonstrate fault-tolerance guarantees with amortised constant time overhead surgery operations. In Section 4, we study partial block reading. In Section 5, we generalize this to hypergraph surgery. For readers familiar with the techniques of code surgery, we recommend skimming Sections 3, 4 and 5 on an initial reading. In Section 6, we study hypergraph surgery operations with intermediate time overheads, between O(1) and O(d). This type of surgery is enabled by quantum locally testable codes with constant soundness. In Section 7, we define a novel generalisation of the notions of both soundness and relative expansion from Ref. [57] which we call modular expansion. We show that hypergraphs with modular expansion can be used to perform hypergraph surgery in lower space overhead. In Section 8 we demonstrate our block reading schemes on a variety of examples. We show that our formalism captures fast surgery with topological codes such as 4D toric codes, and we describe block reading with 2D topological codes and bivariate bicycle codes [86]. Finally, in Section 9 we prove the circuit equivalence between homomorphic measurement and surgery, and prove an upper bound on the phenomenological fault distance for generic logical measurement protocol on quantum LDPC codes. 4More generally the pattern complex, if we use full block reading for both X and Z basis transversal logical measurements. 8 2 Preliminaries 2.1 Stabiliser codes and Tanner graphs Definition 2.1. Let Pn be the Pauli group over n qubits. A qubit stabiliser code Q is specified by an Abelian subgroup -1 /∈S ⊂Pn such that the codespace H is the mutual +1 eigenspace of S, i.e. U |ψ⟩= |ψ⟩ ∀U ∈S, |ψ⟩∈H. We say S is the stabiliser group for Q. A qubit stabiliser code Q can be specified by a stabiliser check matrix H = [HX|HZ] ∈Fr×2n 2 , such that H 0 1 1 0 ! H⊺= 0, where a row [u|v] corresponds to a generator of the stabiliser group, and therefore check on Q, iu·vX(u)Z(v), for u, v ∈Fn 2. Definition 2.2. A qubit CSS code Q is a qubit stabiliser code where the generators of S can be split into two sets SX and SZ. SX contains Pauli products with terms drawn from {X, I} and SZ terms drawn from {Z, I}. Thus there is a stabiliser check matrix H for Q such that H = HX 0 0 HZ ! , HXH⊺ Z = 0. Definition 2.3. A CSS low-density parity check (LDPC) code family is a family of CSS codes such that the column and row weights of HX and HZ are bounded above by some constant σ. Logical operators in a stabiliser code are Pauli products that commute with all the checks of the code. As the stabiliser group is Abelian, all stabilisers are equivalent to the trivial logical operator. The code distance d of Q is the minimum Pauli weight of all the nontrivial logical operator representatives. Definition 2.4. A Tanner graph is a bipartite graph with a vertex for each qubit and each check. A qubit is connected to each check in which it participates with an edge labeled [1|0], [0|1], or [1|1] depending on whether the check acts on the qubit as X, Z, or Y . In our convention, data qubits are black circles and checks are boxes. If a check only measures Z or X terms then we omit the edge label, and instead label the box with a Z or X. Therefore for a CSS code we have no edge labels, and all boxes contain either Z or X labels. In this case, the condition that all stabilisers commute is the same as saying that the number of paths between every Z and X check is even. Depicting large Tanner graphs directly becomes unwieldy, so we use scalable notation5 to make them more compact. Qubits are gathered into disjoint named sets Q0, Q1, Q2, ..., and the same for checks C0, C1, C2, ... . An edge between Qi and Cj is then labeled with the stabiliser check matrix of Q restricted to Qi and lifted to Cj, so we have [CX|CZ] ∈F|Cj|×2|Qi| 2 . 5Terminology borrowed from Ref. [87]. 9 [HX|HZ] (a) A generic stabiliser code, X Z HX HZ (b) A CSS code, X Z X L ∂1 ∂T 1 ∂1 I I I I (c) A hypergraph product code. Figure 1: Examples of scalable Tanner graphs. If that check matrix is all-zeros then omit the edge entirely. If the checks are all of the same type then we omit the all-zeros half of the check matrix; for example if Cj contains only Z checks then we label the check box Z and the edge [CZ] ∈F|Cj|×|Qi| 2 . For example, in Figure 1 we show some basic scalable Tanner graphs with qubit sets drawn as circles and check sets drawn as boxes. We have (a) a generic stabiliser code, (b) a CSS code, and (c) the hypergraph product [14] of a classical code C, with bits L and check matrix ∂1, by a repetition code R with blocksize 2 and check matrix 1 1 . We use scalable Tanner graphs to describe our procedures. 2.2 CSS codes and chain complexes The well-known CSS code-homology correspondence relates quantum CSS codes to chain complexes, allowing for the study of various properties using the techniques of homological algebra [17, 29, 53, 58, 64, 88, 89]. As we employ this formalism to study block reading, we give a brief summary of this correspondence here. A based chain complex over F2, C• = · · · Cl Cl-1 Cl-2 · · · ∂l+1 ∂l ∂l-1 ∂l-2 is a Z-graded vector space L i∈Z Ci over F2 equipped with a basis at each degree i, and boundary operators ∂i : Ci →Ci-1 such that ∂i-1 ◦∂i = 0, ∀i ∈Z. Equivalently, im∂i+1 ⊆ker ∂i. The homology space at degree i is Hi(C•) = ker ∂i/im∂i+1. A CSS code must satisfy the property that HXH⊺ Z = 0, and therefore we can view an Jn, k, dK CSS code as a chain complex with 3 nonzero terms: C• = C2 C1 C0 ∂2 ∂1 where explicitly at each degree we have C• = FmZ 2 Fn 2 FmX 2 H⊺ Z HX An undetectable Z operator must have support on a vector v ∈ker HX, and so every such operator is of the form Z(v). Z stabilisers are elements of imH⊺ Z. Thus equivalence classes 10 of Z logical operators are elements of H1(C•) = ker HX/imH⊺ Z, the homology space of C• at degree 1. Hence k = dim H1(C•), as the number of logical qubits must be equal to the number of independent equivalence classes of Z logical operators. The Z-distance dZ of a CSS code is the weight of the smallest nontrivial Z operator, so dZ = min v∈ker HX ⊺ Z |v|. Dual to the notion of chain complex is a cochain complex, C• = · · · Cl Cl+1 Cl+2 · · · δl-1 δl δl+1 δl+2 which is defined similarly to a chain complex but with boundary operators transposed to become coboundary operators δi, over F2. The cohomology space is Hi(C•) = ker δi/imδi-1, and enjoys the isomorphism Hi(C•) ∼= Hi(C•). As HXH⊺ Z = 0, we also have HZH⊺ X = 0, hence we can equally view a CSS code as a cochain complex with 3 terms: C• = FmX 2 Fn 2 FmZ 2 H⊺ X HZ This view prioritises X operators, rather than Z operators. We then have k = dim H1(C•) and dX = min u∈ker HZ ⊺ X |u|. Furthermore, d = min(dX, dZ) as the error types are detected independently. A chain map, or homomorphism between chain complexes, is a linear map: f• : C• →D• which has a component fi : Ci →Di at each degree i, such that the following diagram commutes, · · · Cl+1 Cl Cl-1 · · · · · · Dl+1 Dl Dl-1 · · · ∂C l+1 fl+1 ∂C l fl fl-1 ∂D l+1 ∂D l (1) i.e. ∂D i+1fi+1 = fi∂C i+1 ∀i ∈Z. A dual definition applies to a cochain map f•, which can be found by taking the transpose of each map in the diagram. Definition 2.5 (Subcode). Let a subcode of the CSS code defined by a chain complex C• be a code defined by the chain complex A• with an injective chain map f• : A• →C•. We stipulate further that f• must take each data qubit in A• to a single data qubit in C•, and the same for any Z or X checks in A•.6 The additional stipulation is made to maintain control over weights, as this is essential for our analysis; if C• is LDPC, so is A•. Each matrix in the chain map f• is therefore a permutation matrix, up to some all-zero rows. We can equally take the subcode in a dual manner, viewing the original CSS code via its cochain complex C• and stipulating that there is a similar cochain map g• : A• →C•. In this work, it is always made clear from context whether the subcode is taken in the chain (Z) or cochain (X) sense. 6In Ref. [53] this property of f• is called basis-preservation. 11 2.2.1 Tensor products The tensor product of (co)chain complexes is inherited from the tensor product of Z-graded vector spaces. Namely, Definition 2.6. [90, Sec. 2.7] Let C•, D• be chain complexes over F2. Define (C ⊗D)• with components (C ⊗D)l = M i+j=l Ci ⊗Dj where the latter tensor product is the normal tensor product of vector spaces. Differentials between components are given for ∂(C ⊗D) l by matrices ∂C i ⊗I I ⊗∂D j ! for a given i, j, then stacked horizontally for each term i, j | i + j = l, and vertically for each i′, j′ | i′ + j′ = l -1. One can check that ∂(C ⊗D) l-1 ◦∂(C ⊗D) l = 0 (mod 2), as desired. Lemma 2.7 (K ̈unneth formula). [90, Thm 3.6.3] Hl(C ⊗D) ∼= M i+j=l Hi(C) ⊗Hj(D) That is, the homology spaces factor through the tensor product conveniently. Both the tensor product and K ̈unneth formula can be applied dually to cochain complexes. By virtue of the fact that we are working with finite-dimensional vector spaces with explicit bases, we can strengthen the K ̈unneth formula: Lemma 2.8. [91, Prop. 1.13] Hl(C ⊗D) = M i+j=l Hi(C) ⊗Hj(D)7 Proof. We first observe that any basis element of Hi(C) ⊗Hj(D) for i + j = l, which is an equivalence class of elements in Ci ⊗Dj, is an element of Hl(C ⊗D), and that linearly independent elements remain independent. Then by a counting argument, Hl(C ⊗D) = L i+j=l Hi(C) ⊗Hj(D). As CSS codes can be viewed as (co)chain complexes, the tensor product can be used to construct new codes [14, 91]. 2.2.2 Mapping cones Given a chain map f• : A• →C• over F2, the mapping cone of f• is defined to be the chain complex with spaces cone(f)i = Ci ⊕Ai-1 and maps ∂cone(f) i : cone(f)i →cone(f)i-1 given by Ai-1 Ai-2 Ci Ci-1 fi-1 ∂C i ∂A i-1 ⊕ ⊕ 7Strictly speaking this is still an isomorphism, but the isomorphism is canonical. 12 ∂cone(f) i = ∂C i fi-1 0 ∂A i-1 ! . The homology of mapping cones can be studied using exact sequences and the Snake Lemma [90, Lem. 1.3.2], see App. A. Using chain maps and mapping cones, one can construct new quantum codes. In particular, Z logical measurement of a CSS code C• can be seen as constructing a particular auxiliary system A• and then using the chain map f• to build the deformed code cone(f•) [58]; X logical measurements are then mapping cones from cochain maps. This perspective originates from the quantum weight reduction of Hastings [71, 72]. 2.3 Metachecks and soundness Typically, quantum codes require O(d) rounds of error correction for every logical timestep in order to maintain fault-tolerance. This is because in addition to errors occurring on data qubits, measurement errors can occur on checks. As checks are not typically protected against measurement errors, one must measure repeatedly in order to diagnose such errors successfully. This can be seen as constructing a spacetime code whose timelike component is a repetition code [60]. There are, however, quantum codes that do have protection against measurement errors [77-82]. Although there are multiple forms of such protection, here we focus on the approach that uses metachecks [78]. A metacheck is a set of stabilisers that has a product of outcomes known to be +1 in advance, even in the presence of faults on data qubits. Hence a metacheck is a detector [92] on the spacetime code, which detects solely measurement errors. It is possible to construct codes with sufficient metacheck structures, such that the minimum number of faulty measurements in a single error-correction round required to cause a logical fault is the same as the code distance d. Such codes can be single-shot codes, which in principle require only a single error-correction round against adversarial noise [78]. Single-shot LDPC codes with metachecks require redundancy in their stabiliser generators and so can be expensive to construct, as having enough redundant checks to yield sufficient detectors for protection against up to d measurement errors can entail a larger space overhead. Given a scalable Tanner graph for a CSS code, we can incorporate metachecks with new sets of vertices and edges: X Z HX HZ MZ MX where blue (red) diamonds represent metachecks on Z (X) checks respectively, and MZ (MX) is the F2-linear map taking a syndrome outcome to its corresponding set of metachecks. Equivalently, metachecks can be viewed as additional terms in the (co)chain complex of a code: C• = C3 C2 C1 C0 C-1 ∂3 ∂2 ∂1 ∂0 C• = C3 C2 C1 C0 C-1 δ2 δ1 δ0 δ-1 where C3 = C3 represents metachecks on Z checks and C-1 = C-1 represents metachecks on X checks. 13 In scalable Tanner graph notation, maps extend outwards from data qubits, to checks and then metachecks. In (co)chain complexes, maps begin at the lowest or highest degree and point towards the higher or lower degree respectively. Definition 2.9. [78] Let Q be a CSS code. The Z-single shot distance is dss Z = min v∈ker MZ |v| = min v∈ker δ2 δ1 |v| and is the weight of the smallest Z-measurement fault that is undetectable by Zmetachecks and is not equivalent to an X fault on data qubits. Similarly, dss X = min u∈ker MX |u| = min u∈ker ∂0 ∂1 |u| The single-shot distance of Q is therefore dss = min(dss X, dss Z ), as the Z and X checks are independent. If ker ∂0 ∂1 = ∅then by convention we say that dss X = ∞, and similar for dss Z . Like logical operators, measurement faults can be partitioned into equivalence classes, in H2(C) and H0(C) for Z measurement errors and X measurement errors respectively. Even if the single-shot distance is at least the distance of the code, i.e. dss ≥d, it is still possible for a low-weight measurement fault to prevent fault-tolerant single-shot error correction. This is because there could be a low-weight measurement fault which is equivalent to a fault on data qubits, but that fault is very large. Hence the measurement fault may be interpreted by the decoder as a large fault on data qubits, and a large, erroneous recovery operator applied. As a consequence, the next logical cycle would require only a small fault on data qubits to cause a logical error. This motivates the study of LDPC codes with good soundness [78, 93, 94], where every valid small set of syndrome outcomes has a sufficiently small set of data qubit errors which produces them; thus a minimum distance decoder will not misinterpret a small measurement fault as a large data qubit fault. There are several different definitions of soundness for both classical and quantum codes. For classical codes, we adopt a simple combinatorial definition of local-testability from Ref. [95, Def. 11]. Definition 2.10 (Local testability). A binary linear code C is (ω, ρ)-locally testable if it has a parity-check matrix H : Fn 2 →Fm 2 with rows of weight at most ω such that for any vector v ∈Fn 2, 1 m|Hv| ≥ρ nd(v, C) where d(v, C) = minx∈C(|v + x|) and | · | is the Hamming weight. The values ω and ρ are the locality and soundness of the code respectively. For quantum stabiliser codes, we follow Ref. [96]. For an n-qubit quantum code Q, let Qt = Span{(A1 ⊗· · · ⊗An) |ψ⟩: |ψ⟩∈Q, #{i ∈[n], Ai ̸= I} ≤t} be the t-fattening of Q, which is the set of states with distance at most t from the codespace. Then define DQ = X t≥1 t(ΠQt -ΠQt-1). where ΠQt is the projector onto the space Qt. 14 Definition 2.11. A qubit stabiliser code Q with generators S1, · · · , Sm is locally testable with locality ω and soundness ρ if all generators have weight at most ω and 1 m m X i=1 1 2(I -Si) ⪰ρ nDQ where A ⪰B means the operator A -B is positive semidefinite. Lemma 2.12. [96, Fact 17] A quantum CSS code CSS(HX, HZ) with parity-check matrices HX, HZ is quantum locally testable with soundness ρ if ker HX, ker HZ are classical locally testable codes with soundness ρ. Conversely, if CSS(HX, HZ) has soundness ρ then ker HX, ker HZ are classical locally testable codes with soundness at least ρ/2. Remark 2.13. Note that Def. 2.11 is subtly different from the version of soundness given in Ref. [78], and is closer to the definition of confinement in Ref. [82]. Intuitively, the version in Ref. [78] stipulates that every valid small set of syndrome outcomes has a sufficiently small set of data qubit errors which produces them. Def. 2.11 and confinement instead stipulate that every small set of data qubit errors produces a sufficiently large set of syndrome outcomes. See Ref. [82] for more discussion about the distinction between these definitions. Def. 2.11 aligns more closely with the definition of soundness for classical codes, so is preferable for our purposes. It is well-known that a quantum code with both sufficiently high single-shot distance and sufficient soundness, of a certain type, can be decoded in a single-shot manner against adversarial noise using a minimum-weight decoder [78], and similar results apply too to confinement [82]. We remark that the development of quantum locally testable codes with good parameters is an ongoing line of research [97-100]. In summary, a metachecked code being single-shot is reliant on the decoder and noise model, in addition to other code properties than just the single-shot distance. For phenomenological adversarial noise, a minimum weight decoder is sufficient when the codes have sufficient properties related to soundness [78, 82]. In this work we do not require the quantum memory to be single-shot, or even have high single-shot distance, but we make use of the general framework of metachecks, soundness and their relation to homology. 2.4 Quantum code surgery Generalised lattice surgery, or quantum code surgery, is a method of performing computation by code deformation, see Section 3.2 of Ref. [62] for a brief review. In this work, we presume that the codes are all stabiliser codes, although one can perform surgery with non-Abelian codes [101-105]. We also focus on logical measurements by the introduction of an auxiliary system, while surgery can be performed in other ways to implement a larger class of operations [53, 64, 106, 107]. Those exceptions aside, all prior works in quantum LDPC code surgery are known to be unified by the construction of a measurement hypergraph [56, 58, 62]. In brief, for a given initial quantum memory Q and logical measurement to be performed, we construct a hypergraph H(V, E), consisting of vertices V, edges E and cycles.8 Edges are associated to new data qubits, and there are two types of new stabiliser checks: vertex checks, whose measurement outcomes are used to infer the logical measurement outcome, and cycle checks, which are present to gauge-fix and thus prevent any new logical qubits 8A cycle in a hypergraph is a collection of edges that contains every vertex an even number of times. 15 from appearing in the deformed code.9 The measurement hypergraph is connected to the initial code Q. Specifically, a subset of vertices (new checks) are connected to qubits in Q in order to perform the logical measurement, while a subset of edges (new data qubits) are connected to checks in Q such that the deformed code commutes as a stabiliser code. Definition 2.14. Let H be a hypergraph with edge-vertex incidence matrix G : F2E →F2V. A cycle basis is a full-rank matrix N : F2F →F2E such that imN = ker G. F is identified as a set of faces, or cycles, in H. We say that the cycle basis is sparse if N is a sparse matrix. Given a cycle basis of a measurement hypergraph, each cycle in the basis can be associated to a new check, which fixes the gauge of the logical which would otherwise have support on the cycle. If the cycle basis is sparse then this can be done while maintaining the LDPC property of the deformed code. There are several methods of constructing such measurement hypergraphs [52-59, 61, 62], which are described in more detail in Ref. [62, Sec. 3]. The most asymptotically efficient method currently known to measure a generic set of Pauli product operators on disjoint logical qubits is by brute-force branching and gauging logical measurements with universal adapters [61]. For an arbitrary collection of logically disjoint Pauli product measurements supported on t logical qubits, this scheme uses O tω(log t + log3 ω) ancilla qubits, where ω ≥d is the maximum weight of the single logical Pauli representatives involved in the measurements, and d is the code distance. This is all done in time O(d) independent of t, while preserving the phenomenological fault distance and LDPC property. This is a vast improvement to the original work on surgery with quantum LDPC codes [52], which required O(dω) ancilla qubits to measure a single weight ω logical operator representative10 and could not generally perform parallel measurements without losing the LDPC property. However, despite these advances, all contemporary methods require O(d) rounds of error correction per logical cycle for generic codes in order to maintain fault-tolerance11. It also means that the study of LDPC code surgery does not, prior to this work, cover the well-known cases of single-shot surgery with 3D and 4D topological codes [60]. The primary reason for this stark contrast is that, in LDPC code surgery, the measurement outcomes of new checks on vertices in the measurement hypergraph H are used to infer logical measurement outcomes; and in prior works on LDPC code surgery these new checks did not have any protection against measurement errors. When measuring a single logical operator using gauging logical measurements or homological measurements [56, 58], for example, the product of vertex check outcomes is interpreted as the logical measurement outcome. Hence, were they measured for only one round, just a single measurement error would flip the product, and constitute a logical error. As such, the checks must be measured for O(d) rounds in order to infer the correct logical measurement outcome. In this work, we focus on reducing the number of rounds required by introducing metachecks which protect against measurement errors in H. All of our protocols yield CSS-type surgery. 9These new logical qubits could have low weight logical operators, and lower the dressed weights of other operators acting on logical qubits in the code [52], which would lower the fault-distance. 10The representative was also required to be irreducible, meaning that it did not contain any other logical representatives, a problem which was fully resolved in Refs [56, 58]. 11This assumes a single representative is measured; Ref. [56] described a method to decrease the time overhead to O( d m) via parallel surgery measurement of 2m -1 equivalent logical operators. 16 Definition 2.15. A CSS-type code surgery is a surgery protocol which begins with a CSS code memory, then performs code deformation such that the code remains CSS throughout. CSS-type code surgery is a limited subset of more general code surgery, as one cannot measure e.g. Y terms or products of the form X ⊗Z while preserving the CSS property. Thus all product measurements of CSS-type each contain terms drawn from either {Z, I} or {X, I}. We remark that CSS surgery is easily extended to measure more general logical Pauli operators on codes that are self-dual. Lemma 2.16. Given any CSS code Q, measurement of Z logicals by a measurement hypergraph results in a deformed code with X-distance at least as high as the X-distance of Q, so long as all cycles are gauge-fixed or otherwise stabilisers. This is an immediate corollary of Ref. [58, Thm. 1]. A dual result follows for the Z-distance when performing measurement of X logicals. 3 Full block reading To understand and reduce the time overhead of surgery operations, we first consider full block reading, which is a surgery operation that acts transversally on all the physical qubits, and thereby logical qubits, of identical code blocks. The fact that one can perform block reading with surface codes is a widely known folklore result, as these are in a sense 'transversal' measurements between patches. Our results extend substantially beyond this simple case. We start with the simple example of two blocks. 3.1 Full block reading on two blocks Given two identical Jn, k, dK quantum memories, which are CSS LDPC codeblocks labelled Q and Q′, X Z HX HZ Q X Z HX HZ Q′ we can initialise a hypergraph H(V, E) between the two X Z HX HZ Q X Z HX HZ Q′ Z I I I I G E V (2) 17 such that new data qubits are edges, new Z checks are vertices and H has the edge-vertex incidence matrix G ∈FV×E 2 . H has 1-to-1 maps from edges to X checks in Q and to X checks in Q′, and similar for vertices. Thus in order for the checks of the deformed code to commute, we have G = H⊺ X. Let D be this deformed code. Lemma 3.1. Measuring the checks of D performs a Z ⊗Z measurement on each pair of logical qubits (qi, q′ i) for i ∈[k], where qi is a logical qubit in Q and q′ i the same logical qubit in Q′. Proof. Consider an arbitrary Z logical operator representative acting on qi, and call it Zi = Z(vi). Let Z′ i be the identical copy of Zi in Q′. By definition, vi ∈ker(HX). Multiplying Zi by the bijective Z checks P in V does not leave any support on E, as Pvi ∈ker(G⊺) = ker(HX), but does clean Zi into Q′. The operator which it is cleaned to is precisely Z′ i, hence the two logicals are now stabiliser equivalent and so we have performed a measurement of Zi ⊗Z′ i. Now because ker(G⊺) = ker(HX), the only operators which can be cleaned are of this form and so we have performed a Z⊗Z measurement on each pair of logical qubits between the two codes. Specifically, each set of Z checks in V in bijection with a logical Z operator in Q and Q′ measures the product of those operators. The logical measurement outcome is inferred as the product of the syndrome outcomes in that set. Lemma 3.2. D has no additional logical qubits. Proof. New logical qubits can only appear when performing measurement hypergraph surgery if there are cycles in the hypergraph [56, 58], which do not have corresponding faces, i.e. checks, to gauge-fix. In general if ker(G) ̸= 0 then there can exist a new X logical operator with support on u ∈ker(G). This hypergraph may have cycles, but all logical operators on these cycles are also products of X stabilisers from Q (or, alternatively and symmetrically, from Q′). In particular, given any element u ∈ker(G), multiply X(u) by the bijective X checks S in Q. Because u ∈ker(G), Su ∈ker(H⊺ X) and so the cleaned operator maps to 0 on data qubits in Q, hence is a stabiliser. We shall presently prove that the code distance is preserved by the deformation, but in order to do so it is easier to use the language of cohomology. Lemma 3.3. Let C• be the initial codeblock Q viewed as a cochain complex. Then the deformed code D has the cochain complex D• = (C ⊗R)• where R• = R0 R1 P , R0 = F2 2, R1 = F2, P = 1 1 . Proof. (C⊗R)• = C0 ⊗F2 2 C1 ⊗F2 2 ⊕C0 ⊗F2 C2 ⊗F2 2 ⊕C1 ⊗F2 C2 ⊗F2 In words, 18 • at the level of X checks we have two copies of C0, the X checks of Q, • at the level of data qubits we have two copies of C1 the data qubits of Q, and an additional set of data qubits in bijection with the X checks of Q, • at the level of Z checks we have two copies of C2 and an additional set of Z checks in bijection with the data qubits of Q, • there is another term, which we ignore for now and consider only the first 3 terms. We now check that the coboundary maps are correct. δ0 C⊗R = H⊺ X ⊗I I ⊗P ! , δ1 C⊗R = HZ ⊗I 0 I ⊗P H⊺ X ⊗I ! which we recognise as the check matrices, with appropriate transposes, from Eq. 2. Using the perspective of cochains it is easy to verify the logical dimension using the K ̈unneth formula. Moreover, by Lemma 2.8 we have the equation: H1(C ⊗R) = H1(C) ⊗H0(R) ⊕H0(C) ⊗H1(R) hence the set of X logical operators in D is given by representatives of H1(C) ⊗H0(R) ⊕H0(C) ⊗H1(R) which explicitly is ker HZ/imH⊺ X ⊗ker P ⊕C2/imHZ ⊗0 = ker HZ/imH⊺ X ⊗ker P so every X logical operator belongs to a stabiliser equivalence class [ui, u′ i] of the identical logical operators ui, u′ i in Q and Q′, or a nontrivial sum thereof. Lemma 3.4. The distance of D is at least d. Proof. As there are no new logical qubits, to prove the distance bound it is sufficient to ensure that cleaning Z logicals cannot reduce their weight below d. The X-distance is guaranteed by Lemma 2.16. For Z logicals, observe that multiplying any Z logical in Q by Z checks in V adds support to the corresponding data qubits in Q′ by a 1-to-1 map, and potentially adds support on E as well. Any Z logical which has support on both Q and Q′, say v and v′, can in the worst case be cleaned to have weight |v + v′| on Q and Q′, considering both v and v′ to be in Fn 2, and v + v′ is a valid logical of Q so must have weight at least d. As a consequence of Lemmas 3.2 and 3.4 it is guaranteed by the arguments in Ref. [56] that the block reading can be performed with fault distance d in d rounds of error correction, as the procedure can be viewed as a parallel gauging logical measurement. All new data qubits are initialised in |+⟩, and all checks are measured for d rounds, before measuring out the new data qubits in the X basis. We reduce this from d to O(1) in Section 3.3. 19 3.2 Full block reading of Z type We can extend block reading straightforwardly to acting on many codeblocks simultaneously. We start with a set of c identical Jn, k, dK LDPC codeblocks. Definition 3.5 (Pattern Matrix). Let P : Fc 2 →Fφ 2 be a full-rank matrix, which defines the pattern of logical measurements to be performed between codeblocks by deforming to a new code D. We call P the pattern matrix. Each row r of P corresponds to a set of parallel measurements to be performed. Each entry of 1 in r denotes that a codeblock is involved in that set of measurements, so if rj = 1 for column j then codeblock j is involved, and if rj = 0 then it is not. Then r specifies a measurement of P 1 i ⊗P 2 i ⊗P 3 i · · · for every logical qubit i ∈[k] in codeblocks Q1, Q2, Q3... in the basis P j ∈{Z, I} depending on whether rj = 1 or 0. If P is not full rank, and has some linearly dependent rows, then those rows correspond to logical measurements which are already performed by other rows and so are redundant. Additionally, if any row has weight 1, with the 1 in column j, that measurement is equivalent to measuring out codeblock j in the Z basis. Hence for simplicity we assume that P is full rank, with no linearly dependent rows, and that there are no rows with weight 1. Definition 3.6 (Z-type full block reading). A full block reading of Z type takes as input c CSS LDPC codes, each with cochain complex C•, and a pattern matrix P : Fc 2 →Fφ 2. The cochain complex for the deformed code D of the full block reading denoted by P is given by (C ⊗R)• where R• = Fc 2 Fφ 2 P , First we check that the definition above is consistent with the definition of a pattern matrix. We have (C⊗R)• = C0 ⊗Fc 2 C1 ⊗Fc 2 ⊕C0 ⊗Fφ 2 C2 ⊗Fc 2 ⊕C1 ⊗Fφ 2 C2 ⊗Fφ 2 where we currently ignore the last C2 ⊗Fφ 2 term. We can pick out C0 ⊗Fc 2 →C1 ⊗Fc 2 →C2 ⊗Fc 2 as the c original codeblocks. Then, δ0 C⊗R = H⊺ X ⊗I I ⊗P ! , δ1 C⊗R = HZ ⊗I 0 I ⊗P H⊺ X ⊗I ! as for the example in Eq. 2 but where P is now a more general matrix. The Z logical operators in D are then H1(C⊗R) = H1(C)⊗H0(R)⊕H0(C)⊗H1(R) = ker HX/imH⊺ Z⊗Fc 2/imP⊺⊕C0/imHX⊗ker P⊺ where ker P⊺= 0 as P has no surplus rows, so H1(C ⊗R) = ker HX/imH⊺ Z ⊗Fc 2/imP⊺. Given that our initial code had Z logical operators given by ker HX/imH⊺ Z ⊗Fc 2, we see that those in imP⊺are now stabilisers, and so block reading has measured precisely the rows of P on each logical qubit. Furthermore, no new equivalence classes are present, so there are no new logical qubits. 20 Remark 3.7. Let P be LDPC and each identical codeblock be LDPC. Then the deformed code (C ⊗R)• during measurement is LDPC, and the procedure uses O(cn) ancilla qubits. This is immediate from Definition 3.6. Lemma 3.8. The deformed code has code distance at least d. Proof. The X-distance follows from Lemma 2.16. The Z logicals are given by H1(C ⊗R) = ker HX/imH⊺ Z ⊗Fc 2/imP⊺. We start with a Z logical ΛZ composed of different Z logicals v, v′, v′′, ... in each codeblock, at least one of which must have weight at least d and be a nontrivial Z logical. Applying a new Z check in V only cleans support from the codeblocks if the incident data qubits all have support in ΛZ; otherwise the check applies at least 1 new Z to a data qubit in a codeblock, and the weight of ΛZ cannot be reduced below d in this way. If we apply new Z checks in V to clean support from ccl codeblocks, in the worst case we have support |v| + |v′| + |v′′| + · · · -ccl|v ∩v′ ∩v′′ ∩· · · | left in the codeblocks. But |v| + |v′| + |v′′| + · · · -ccl|v ∩v′ ∩v′′ ∩· · · | = |v ∪v′ ∪v′′ ∪· · · | -|v ∩v′ ∩v′′ ∩· · · | ≥|v∆v′∆v′′∆· · · | = |v + v′ + v′′ + · · · | ≥d where ∆is the symmetric difference of sets. Should the row weights of P be too high, meaning that the weights of these logical measurements are high and so the degree of the Tanner graph is high, then one can introduce ancilla blocks to break those measurements up, making P sparse. This corresponds to adding new bits, i.e., columns to the check matrix P, such that a row can be split up into multiple rows. The ancilla codeblocks used to reduce its weight can be initialised at the same time as the code deformation procedure begins, and can be considered part of the measurement hypergraph, where data qubits in ancilla blocks are edges, Z checks are vertices and X checks are cycles, as normal. Consequently, the data qubits are initialised in |+⟩. This can be thought of as thickening the hypergraph, see Figure 4 later on. 3.3 Time overhead of full block reading We now return to our original example, Eq. 2 from Section 3.1. When computing the cochain complex in Lemma 3.3 we had an additional term (C ⊗R)3 = C2 ⊗F2 = C2, which corresponds to a collection of metachecks on the Z-checks of the deformed code. Thus the full scalable Tanner graph for the logical measurement is X Z HX HZ Q X Z HX HZ Q′ Z I I I I G E I I G V 21 where G = HZ and MZ = δ2 C⊗R = I ⊗P HZ ⊗I = I ⊗P HZ . Consider first the case where measurement faults can occur only on checks in V. As G = HZ, if a measurement fault v /∈ker(HZ) then it is detected by the metachecks. If v ∈ker(HZ) then either (a) it is equivalent to an X data qubit error on E or (b) it has weight on checks at least d. (b) follows from the fact that G = H⊺ X, and the lowest weight vector in ker HZ ⊺ X has by definition weight at least d. For (a), as all the data qubits in E are initialised in the |+⟩basis, X errors leave the state invariant so the logical measurement outcome is unaffected. Therefore to cause an incorrect logical measurement outcome, the measurement fault weight on V must be at least d, in the case where measurement faults cannot happen elsewhere. Note further that any vector v ∈ker(G) representing an undetectable measurement fault on E, regardless of weight, is equivalent to an undetectable X data qubit fault on Q (or Q′), as multiplying by X faults on bijective qubits leaves 0 on the Z checks in Q (or Q′). To study the more general case it is fruitful to use the cohomology of (C ⊗R)•. H2(C ⊗R) = H2(C) ⊗H0(R) ⊕H1(C) ⊗H1(R), which is explicitly C2/imHZ ⊗ker P ⊕ker HZ/imH⊺ X ⊗0 = C2/imHZ ⊗ker P. Thus any Z measurement fault which is neither detected by metachecks nor equivalent to a data qubit fault must be a linear combination of pairs of identical Z check faults in Q and Q′, which are not equivalent to X data qubit faults, and Z check faults which are equivalent to X data qubit faults. Note that the Z single shot distance dss Z of (C ⊗R)• can be very low even in the more general case, as C2/imHZ can have low weight representatives. Nevertheless, any measurement fault which includes Z checks in V must be able to be cleaned, by applying some subset of X errors to Q and Q′, so that it has support only on the Z checks of Q and Q′. Only if the set of X errors which it is cleaned to is large does the measurement fault cause a logical measurement fault. In our analysis of the time overhead, we have first a weak result: Proposition 3.9. Let Q, Q′, Q′′,... be a set of identical CSS LDPC codeblocks with distance d, and let P be a pattern matrix determining a full block reading. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the full block reading procedure. Then full block reading performs logical measurement in O(1) rounds of syndrome measurements with fault-distance d. Proof. See Appendix A. The intuition is that in the timestep where the full block reading is performed, any measurement fault on the check qubits in V with weight < d which is undetectable by metachecks is either (a) in imG and so does not affect the logical measurement outcome or (b) has a paired measurement fault on Z checks outside of V, in the original code, such that the measurement faults on the original code must extend to future and previous timesteps to avoid detection. In order to maintain the fault-distance of the entire procedure we must assume d rounds of syndrome measurements on the original code before and after block reading. This is because there may be low weight faults on data qubits, Z checks and X checks outside of H which are not detected within 1 timestep, and so could lead to logical errors by connecting to timelike errors extending from the top or bottom boundaries. This is equivalent to d measurement rounds of 'padding' before and after block reading [108]. 22 Remark 3.10 (State preparation). Proposition 3.9 does not allow one to generically use full block reading on a single codeblock to prepare a logical |+⟩k or |0⟩k state in constant time, because of the padding required to provably preserve fault-distance. Remark 3.11 (Frame change). At the end of the block reading protocol, the ancilla data qubits are measured out in the X-basis, and the X checks in the original code are undeformed back to their original incidence. These measurements do not commute with the preceding Z checks, and so have random outcomes, although products of these measurement outcomes which correspond to cycles in the hypergraph must be +1 in the absence of errors, as these are precisely the sets of X measurements which do commute with the preceding Z checks. Because each new data qubit is in 1-to-1 correspondence with an X check in each incident codeblock, the anticipated outcomes of X-checks in the original codeblocks are updated after the block reading protocol by taking a product with the outcomes of those data qubit X measurements, inducing a frame change. When the X measurements on ancilla data qubits undergo faults, the frame is updated incorrectly. These measurement errors are therefore in bijection with X check errors on the original code, detected by preceding and subsequent rounds of measurement. The reason why preceding rounds of measurement can diagnose such frame errors is because in the measurement round where ancilla data qubits are measured out the product of such a measurement and its corresponding X-check in an original codeblock commutes with the Z checks in the deformed code, and so must yield a +1 outcome in the absence of errors. Thus having a consistent history of checks before the frame change can detect frame errors. Proposition 3.9 is at first not particularly helpful, because the overall time cost of the procedure is still O(d). However, the proof extends to performing full block readings sequentially without padding between logical measurement rounds, with each logical measurement taking O(1) rounds. Theorem 3.12. Let Ξ be a set of η full block reading procedures of Z-type such that no block reading procedure in Ξ is a product of any others. Then all η logical measurement rounds of full block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the full set of procedures. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. See Appendix A. Remark 3.13. In the above statement, the assumption that the pattern matrix is full rank is for technical reasons related to our proof technique. We expect that this requirement can be relaxed. 3.4 Full block reading of CSS-type Full block reading can be generalised to measure sets of commuting operators of the form Z ⊗Z ⊗· · · ⊗Z and X ⊗X ⊗· · · ⊗X simultaneously. This allows for fast code switching into a transversally concatenated code, similar to a procedure in Ref. [66] and the basic operation in the protocol of Ref. [109]. To see this, let PZ be the pattern matrix for Z-type logical measurements. Recall that the rows of PZ each specify a set of parallel Z-type measurements as per Definition 3.5. In order for us to perform X-type block reading at the same time, the X-type logical operators to measure must commute with the Z-type logical operators. Thus each row s 23 of a PX pattern matrix, specifying a set of parallel X-type logical measurements, must satisfy r · s for each row r of PZ, where · is the usual dot product on vectors over F2. Therefore, PZP⊺ X = 0. Our pattern matrix is now upgraded to a pattern complex. Definition 3.14 (CSS-type full block reading). A full block reading of CSS-type takes as input c CSS LDPC codes, each with cochain complex C•, and a pattern cochain complex R•. The cochain complex for the deformed code of the full block reading denoted by P is given by (C ⊗R)• where R• = Fθ 2 Fc 2 Fφ 2 P⊺ X PZ with R-1 = Fθ 2, R0 = Fc 2 and R1 = Fφ 2, and we assume that both PZ and PX are full rank with no redundant rows, and that all rows have weight greater than 1. As the tensor product of cochain complexes is also a cochain complex, the checks in the deformed code commute. We have (C ⊗R)-1 = C0 ⊗R-1 (C ⊗R)0 = C1 ⊗R-1 ⊕C0 ⊗R0 (C ⊗R)1 = C2 ⊗R-1 ⊕C1 ⊗R0 ⊕C0 ⊗R1 (C ⊗R)2 = C2 ⊗R0 ⊕C1 ⊗R1 (C ⊗R)3 = C2 ⊗R1 and H1(C ⊗R) = C2/imHZ ⊗ker P⊺ X ⊕ker HZ/imH⊺ X ⊗ker PZ/imP⊺ X ⊕C0 ⊗R1/imPZ which given that ker P⊺ X = 0 and R1/imPZ = 0 reduces to H1(C ⊗R) = ker HZ/imH⊺ X ⊗ker PZ/imP⊺ X. A similar calculation gives H1(C ⊗R) = ker HX/imH⊺ Z ⊗ker PX/imP⊺ Z, so we see that logical Z operators in imP⊺ Z, i.e. the row space of PZ, are now stabilisers, and the same for logical X operators in the row space of PX, for every logical qubit in the codeblocks. That the distance of the CSS-type block reading deformed code is at least d is immediate by applying the arguments of Lemma 3.8 twice, first for the Z-type auxiliary system and then for the X-type. In the CSS-type full block reading measurement protocol, all new data qubits included in the Z-type (X-type) measurements are initialised in |+⟩(|0⟩) states respectively. All syndromes are measured for O(1) rounds, and then all new data qubits are measured out in the X (Z) bases respectively. Remark 3.15. Let R• be LDPC and each identical codeblock Q, Q′, Q\"... be LDPC. Then the deformed code (C ⊗R)• is LDPC. 24 Lemma 3.16. Let Ξ be a set of η full block reading procedures of CSS-type such that no block reading procedure in Ξ is a product of any others, and each of the measurements commute. Then all η logical measurement rounds of full block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the full set of procedures. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. See Appendix A. In Appendix B we also show that full block reading can be extended to perform nonCSS measurements, namely those with X ⊗Z and Y terms, on CSS codes, but that our constructions for these require specific structure on the codes, such as transversal S gates, so they add no extra computational power to full block reading with CSS-type measurements. Full block reading gives no addressability on logical qubits within a codeblock, as the measurements are on all logical qubits, so the utility of full block reading is limited in the same manner as 1-to-1 transversal CNOT gates between CSS codes. The logical computational power of a full block reading is identical to initialising an ancilla block, then applying transversal CNOTs, then measuring out the ancilla block. Consequently, it is unclear how useful full block reading would be for practical faulttolerant computation. Compared to measurement using transversal gates and ancilla codeblocks, full block reading requires fewer qubits overall, but full block reading also requires more connections between the initial codeblocks and the auxiliary system. Nevertheless, our analysis of full block reading sets the stage for the more sophisticated operations in the next sections. We first consider the case of reading a subcode, then present our general result on fast hypergraph surgery. 4 Partial block reading Returning again to the example in Eq. 2, consider a different generalisation. We start with two codeblocks Q and Q′, which may be different, X Z HX HZ Q X Z H′ X H′ Z Q′ 25 and construct the measurement hypergraph H to connect only to a subcode of each, but where the subcodes are identical. X Z HX HZ Q X Z H′ X H′ Z Q′ Z Ml Mr Fl Fr G E V That is, up to permutation and the addition of all-zero rows, the matrices satisfy Ml = Mr; Fl = Fr. Given an appropriate choice of subcode and measurement hypergraph, the measurements are only between those logical qubits whose logical representatives have connections to the hypergraph, and so are contained in the subcode. This is called a partial block reading. The construction no longer yields a tensor product of cochain complexes, so we instead use mapping cones from Section 2.2.2 to describe partial block reading. As the constructions become quite complicated, we also refrain from studying multiple partial block readings occurring simultaneously, unlike in the full block reading case. All partial block readings are assumed to be performed sequentially, albeit with only a constant number of syndrome rounds between them. Definition 4.1. Let A be a subcode of Q, Q′, Q′′..., each of which have chain complexes C•, C′ •, C′′ • ..., such that there is a collection of injective chain maps f•, f′ •, f′′ • ... of the forms f• : A• →C•; f′ • : A• →C′ •; f′′ • : A• →C′′ • ... Then the partial block reading of Z type defined by {f•, f′ •, f′′ • ...} is D• = cone(h•); h• = f• + f′ • + f′′ • + ... Explicitly, D• is the chain complex A1 A0 0 C2 C1 C0 C′ 2 C′ 1 C′ 0 C′′ 2 C′′ 1 C′′ 0 ... ... ... ∂A 1 0 ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ ⊕ 26 where labels for the differentials in each codeblock, and labels for the chain maps, have been suppressed for visibility. Note that also A• generally possesses a nonzero A2 component. This component corresponds to metachecks, so we ignore that term for now. It is trivial to compute, using the Snake Lemma [90, Lem. 1.3.2], that the measured logical qubits are those in each code which have Z logical representatives contained in the subcode A•. As a Tanner graph, we have X Z HX HZ Q X Z H′ X H′ Z Q′ Z f0 f⊺ 1 G X Z H′′ X H′′ Z Q′′ f′ 0 f′ 1 ⊺ f′′ 0 f′′ 1 ⊺ E V (3) for a partial block reading between three codeblocks. Observe that the chain maps f•, f′ •, f′′ • determine the connectivity between the blocks and the auxiliary hypergraph, and that E = A0, V = A1. Unlike when performing full block reading, the code distance of the deformed code D• in partial block reading is not generically preserved, and can drop below d. Elementary examples of this phenomenon include surface codes with defects and toric codes, which we detail in Section 8. This drop in distance can come from two causes. First, it is possible to multiply existing Z logicals in the codeblocks by new Z checks in order to lower the weights below d. Second, viewing the ancilla system defined by the mapping cone as a hypergraph H, there may be cycles in H which are nontrivial X logicals, meaning that they do not correspond to stabilisers of the deformed code. When unmeasured, the cycles have symplectically paired Z logicals, which when multiplied by other logicals in the original code can result in weights lower than d [52]. In this section, we focus on partial block reading where the subcode has distance d, and the measurements are assumed to be in the Z-basis. Dualising appropriately yields the same results for the X-basis. For the first cause of the deformed code distance dropping, we can thicken the hypergraph. This is equivalent to taking a tensor product of A• and the dual of a repetition code P• before defining the mapping cone, as shown in Figure 2. We can define chain maps from any copy of A• in the tensor product to Q, and the same for Q′, Q′′..., and this choice does not need to be the same for each. An example is shown 27 X A1 (a) A subcode A•, X X · · · X (b) P•, the dual of a repetition code, X X Z X Z · · · · · · A1 A1 A1 A⊺ 1 A⊺ 1 I I I I I I I I (c) The tensor product (P ⊗A)•, Z G E V Z G E V X Z G E V X · · · · · · (d) The thickened hypergraph H, which is (P ⊗A)• with degrees shifted by 1. Figure 2: Constructing a thickened hypergraph. Note that (b) is a conventional Tanner graph, not a scalable Tanner graph. in Figure 3 Lemma 4.2. Let (P ⊗A)• be the thickened subcode, and let f•, f′ •, f′′ • ... be chain maps from any copies of A• in (P ⊗A)• into Q, Q′, Q′′..., with h• = f• + f′ • + f′′ • · · · . Then if (P ⊗A)• has been thickened at least d times the deformed code D• has distance at least d. Proof. Immediate from Corollary 7.10, Lemma 7.12 and Remark 7.11 further down, so we defer the proof until then. This is similar to devised sticking [59] and the original CKBB scheme [52] in that the auxiliary system is thickened d times, but in this case we are measuring many representatives of the same equivalence classes of operators simultaneously. As in the CKBB scheme, thickening d times is generally overkill [54], and distance preservation commonly requires a smaller ancilla system than this upper bound. Indeed, in Section 6 we demonstrate that soundness properties of the subcode can be used to reduce the required thickening while provably preserving the distance, and even that gives a sufficient but not generally necessary amount of thickening. If thickening d times were truly necessary then our time overhead of d would essentially be replaced by a space overhead factor of d, at least 28 Z G E V Z G E V X Z G E V X · · · · · · X Z HX HZ Q X Z H′ X H′ Z Q′ X Z H′′ X H′′ Z Q′′ Figure 3: Example of partial block reading with a thickened hypergraph. The logical measurement is the same as in Eq. 3, but the hypergraph has been thickened to prevent the weight of Z operators dropping below d. when A• is large, which is interesting but less useful than achieving a genuine reduction in spacetime overhead. For the second cause of the deformed code distance dropping, that there can be cycles in H which correspond to X logicals, we use the following fact. As the subcode has distance d, the logical measurement protocol can also be performed in 1 round of error correction, similar to full block reading, by introducing redundant Z checks between each level of the hypergraph as shown in Figure 4. This fact helps us to gauge-fix the cycles, because each data qubit in each cycle is initialised in |+⟩immediately before and measured out in the X basis immediately after the single round of error correction. The results of those X measurements can be used to infer the outcome of X stabiliser measurements which would fix the gauges of the cycles, and so we do not need to explicitly add checks for those cycles [56]. However, if the set of cycles does not have a basis of low weight cycles, this can result in detectors with high weight in the space direction, which impinge on the fault-tolerance against stochastic noise. Proposition 4.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let A• be a subcode with distance d, with chain maps from A•, or a thickened version thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code, which has code distance at least d. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the partial block reading procedure. Then partial block reading performs logical 29 Z G E V Z G E V X Z G E V X · · · · · · X Z HX HZ Q X Z H′ X H′ Z Q′ X Z H′′ X H′′ Z Q′′ Z Z Figure 4: The same partial block reading as in Figure 3 but with additional redundant checks and metachecks. measurement in O(1) rounds of syndrome measurements with fault-distance d. Proof. See Appendix A. The key here is that when thickening, each Z check error in level a is equivalent to a Z check error in level a -1 along with an X error on the data qubit between them. In this way, we form a chain of metachecks up the thickened hypergraph. These allow all stabilisers in the subcode to be matched to sets of checks in each level of the hypergraph. Evidently, if the subcode A• has nA qubits, and the amount of thickening required is L, then the space cost of a partial block reading is O(nAL). The above proposition considers performing one partial block reading with d rounds of padding before and after the procedure. Once again, we would like to generalize this to performing t ≥d operations in O(t) time, thereby achieving constant time overhead in amortisation. This introduces an extra complication. Instead of the deformed code, we consider the compacted code, defined as follows. Definition 4.4 (Compacted code). Let C• be a complex representing the memory CSS code blocks, and let A1,•, · · · , At,• be chain complexes each with a chain map fi,• : Ai,• →C•. The compacted code is the code taken by applying all the mapping cones to the original code, i.e., CC• = cone( X i (fi,•). While this code is most likely not LDPC and potentially massive, it is just a proof construct and is never explicitly used in our surgery operations. 30 There is an intuitive reason behind defining the compacted code. Ideally, we would like to construct partial block readings assuming just that every deformed code has distance d. This assumption, however, may not be sufficient: if we perform two partial block readings one after another with only O(1) syndrome rounds between them, we must carefully consider how the weight of an unmeasured logical may be reduced by multiplication with the new deformed code checks. While, by assumption, both the first and second partial block readings would individually guarantee any weight-reduced operator remains weight d, there would be no such guarantee when both weight reductions occur together. This may seem irrelevant - after all, we are not performing the partial block readings simultaneously - but it happens that a sequence of check measurement errors in the O(1) syndrome rounds between the block readings can effectively reproduce this simultaneous scenario using potentially just O(1) additional faults (since we are not assuming the original code is sound).12 Therefore, to ensure the overall surgery protocol is fault-tolerant, we impose the condition that the compacted code, which can be seen as the memory blocks deformed by all the surgery operations applied simultaneously, has distance at least d. Theorem 4.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let Ξ be a set of Z-type partial block readings {Partial1, Partial2, Partial3, ..., Partialη} where each subcode has distance d and the compacted code has distance d. Then all η logical measurement rounds of partial block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the set of logical measurements. The procedure has phenomenological fault-distance d, for any η ∈N. Proof. See Appendix A. While it may seem daunting to study the compacted code and satisfy the distance condition, we show in Section 7 that standard techniques in code surgery, notably thickening and expansion, can be applied to satisfy this condition. We further remark that in the case of full block reading with a full rank pattern matrix P, the resultant compacted code is a homological product code (C ⊗R)•, which has distance at least d. Remark 4.6. In Section 9 we study the relationship between surgery and homomorphic measurement [42]. For now, observe that given a subcode of distance d performing homomorphic measurement is straightforward, as one initialises an ancilla system that is a copy of that subcode and performs transversal CNOTs and single-qubit measurements. This initialisation can take up to O(d) rounds to preserve fault-tolerance. Note that the method of algorithmic fault-tolerance [84] does not apply to generic homomorphic measurements. Performing partial block reading with a distance d subcode requires additional conditions on the distance of the deformed code to maintain fault-tolerance, which can be resolved at some additional space cost by the thickening described above. 5 Fast hypergraph surgery In this section we generalise our partial block reading results to the case where A• is not a subcode but some other complex with a suitable chain map into the original code. 12We note that this simultaneous weight reduction issue, a scenario not considered in detail by prior code surgery work, is also a concern when doing auxiliary graph surgeries [56, 58] separated by O(1) syndrome rounds if one does not assume expansion of the auxiliary graphs and instead just assumes large deformed code distances. In practical constructions, e.g. Ref. [55], where one indeed tends not to guarantee expansion but just deformed distance, this is an important issue to be aware of. 31 A2 A1 A0 A-1 C2 C1 C0 0 f2 f1 f0 0 0 Definition 5.1. We say that the chain map f• : A• →C• is non-overlapping on vertices if each row in the matrix f1 : A1 →C1 contains at most a single non-zero entry. The degree 1 of A• is shifted by the mapping cone construction to be in cone(f)2, i.e. the set of vertices in the hypergraph is the set of basis elements of A1. We can directly transport Definition 4.1 to this more general setting. Definition 5.2. Let A• be an auxiliary chain complex and let Q, Q′, Q′′... be CSS LDPC codes, each of which have chain complexes C•, C′ •, C′′ • ..., such that there is a collection of sparse chain maps f•, f′ •, f′′ • ... of the forms f• : A• →C•; f′ • : A• →C′ •; f′′ • : A• →C′′ • ... which are non-overlapping on vertices. Then the hypergraph surgery of Z type defined by {f•, f′ •, f′′ • ...} is D• = cone(h•); h• = f• + f′ • + f′′ • + ... As with partial block reading it is trivial to compute which logical operators are measured using the Snake Lemma [90, Lem. 1.3.2]. Theorem 5.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and let Ξ be a set of Z-type hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη} such that: • Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance d, with sparse cycle bases. • The compacted code has distance at least d. Then all η logical measurement rounds of generalised hypergraph reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the set of logical measurements. The procedure has phenomenological fault-distance d, for any η ∈N. Proof. See Appendix A. In the mapping cone that describes the surgery operation, the 1-cosystolic distance captures the metacheck distance. To satisfy this distance condition, it is often easier to start with a homomorphic, high distance quantum code, as we did in full block reading. Returning to the second condition, in Section 7 we define a notion of hypergraph expansion called modular expansion and show that it is a sufficient condition for the compacted code to have distance d. We then show that a generic hypergraph can always be thickened (by up to d layers) to a graph with the desired expansion properties. This is in analogy with prior surgery works studying graphs, where it is known that thickening can be used to boost a property called relative expansion [57]. We remark that for both our work and prior works, expansion is a sufficient but not necessary condition, in practice one can often find good constructions heuristically, such as adding random hyperedges. 32 Remark 5.4. In a generic hypergraph surgery, constant-weight local operators are measured alongside high-weight global operators. If we require that the original codespace is restored after the hypergraph surgery, the constant-weight local operators must all be in the stabiliser group, and the high-weight operators must be logical operators. For a Ztype measurement on a single code block, with an injective f1 function, this requires the group generated by F2-linear combinations of the hyperedges to include the truncation of the X-type stabiliser group to the qubits addressed by the measurement. In the previous sections we have chosen the X-type checks of the subcode, but more generally we can pick a different, potentially overcomplete, generating set. 6 Intermediate time overhead with locally testable codes We can consider (a) block reading with a distance d subcode and (b) conventional LDPC code surgery, which uses a distance 1 subcode, to each sit at opposite ends of an axis determining the time cost of logical measurements. In this section we consider interpolating along this axis, to find surgeries which use between O(1) and O(d) time each, while remaining fault-tolerant. 6.1 Partial block reading with a low distance subcode Proposition 6.1. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let A• be a subcode with distance dA = 1 αd, with chain maps from A•, or a thickened version thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code with code distance d and let H have a sparse cycle basis. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the partial block reading procedure. First, ⌈α⌉rounds of syndrome measurements are necessary (but not sufficient) to perform partial block reading with fault-distance d. Then if either: (a) α ≤2 or (b) the initial codeblocks are each locally testable codes with soundness ρ ≥2n m , where n is the codeblock length and m the number of checks, then partial block reading performs logical measurement in ⌈α⌉rounds of syndrome measurements with fault-distance d, maintaining the LDPC property throughout. Proof. See Appendix A. Remark 6.2 (Cycle checks). When the block reading takes longer than O(1) round we genuinely require a sparse cycle basis for H to maintain the LDPC property of the deformed code, as the gauge checks at each timestep cannot easily be inferred from the state preparation and measurement of data qubits in the auxiliary system. In general, it is not guaranteed that H possesses a sparse cycle basis. In practice we expect to find sufficiently sparse cycle bases numerically for small codes. Additionally, if a subcode has low weight X metachecks which have support on checks in the subcode, after degree shifting these metachecks become low weight cycle checks for a Z basis partial block reading. Proposition 6.1 is interesting primarily because there are codes that are not expected to have distance d subcodes addressing targeted subsets of logical qubits. Indeed only topological codes and certain product codes are known to allow for addressing any logical qubit in this manner [42, 43]. Instead, one can find subcodes with distance between 1 and d which still permit fast measurement, but not necessarily in constant time. The distance of A• is the minimum number of check errors on the auxiliary system which are required to cause an undetectable logical measurement error in a single round; this is not the same 33 as the single-shot distance of the entire deformed code, because there can be checks in the bulk which have far less protection against measurement errors. However, the errors which can occur in the bulk do not prevent us from performing many of these partial block readings sequentially without d rounds of padding in between, implying that interpolated partial block reading can indeed speed up computation substantially. Theorem 6.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and let Ξ be a set of partial block readings {Partial1, Partial2, Partial3, ..., Partialη} where each subcode has distance di ≥ 1 αi d for i ∈[η] and the deformed codes each have code distance d, with sparse cycle bases. Suppose the original codes have soundness ρ ≥2n m . Then all η logical measurement rounds of partial block reading can be performed sequentially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding before and after the logical measurements. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. See Appendix A. We remark that when the base codes are locally testable codes with constant soundness, we only require the weaker condition that each deformed code has distance d, instead of the stronger condition that the compacted code has distance d. Intuitively, this is because when considering the potential weight reduction of a logical fault that is deformed into two different surgery ancilla systems, the support of that logical fault is necessarily shifted onto timelike check errors, which are proportional to the reduced space support as the codes have constant soundness. Remark 6.4 (Interpolated surgery with lower soundness). Proposition 6.1 and Theorem 6.3 demand soundness ρ ≥2n m , which is asymptotically constant as m ∈Θ(n) for LDPC codes. If the soundness is lower than constant, which one may expect for practical codes, it is sufficient to use asymptotically more rounds of syndrome measurement per logical measurement, proportional to 1/ρ, to maintain phenomenological fault-distance d. Furthermore, one cannot guarantee that the surgeries can be performed within O(1) time of each other, without considering the compacted code (see Def. 4.4). Scaling the time between logical measurements to O(1/ρ) is then sufficient to preserve fault-distance. If the soundness is vanishing then there is still a reduction in time overhead from measuring multiple representatives, but the reduction is additive rather than multiplicative, see Proposition 9.3, and one is forced to consider the compacted code to ensure that the logical measurements can be performed close together in time. 6.2 Intermediate time hypergraph surgery Similar to the previous section on partial block reading, we can generalize Theorem 6.3 to the case of hypergraph surgeries. Theorem 6.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and let Ξ be a set of hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη} such that: • Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance di ≥ 1 αi d for i ∈[η], with sparse cycle bases. • Each deformed code has distance at least d. 34 • The original codes have soundness ρ ≥2n m . Then all η logical measurement rounds of hypergraph surgery can be performed sequentially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding before and after the logical measurements. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. This follows by combining the proofs of Theorem 5.3 and Theorem 6.3. Gauging logical measurement [56] can be seen as an instance of hypergraph surgery where the subcode A• has distance 1, and therefore requires O(d) rounds of syndrome measurement to maintain fault-distance d. 7 Modular expansion In previous sections, we studied hypergraph surgery operations where the compacted code or the deformed codes has distance d. This is critical for fault-tolerance. In this section, we discuss sufficient expansion properties on the hypergraph for these conditions to hold. In context of past surgery works, when A• contains only one logical operator (so α = d and dA = 1), partial block reading recovers the CKBB scheme [52]. It is known that there are substantially more efficient schemes for measuring individual logical operators in O(d) time by surgery [56, 58], using expander graphs. This section is essentially an extension of these results to hypergraphs, which can measure multiple logical representatives simultaneously and quickly. Definition 7.1. Let H(V, E) be a hypergraph, with edge-vertex incidence F2-matrix G. Let I be an index set of basis elements vi of a subspace W ⊂F2V. Then we can index element in W as κλ for λ ∈2I. We conflate κλ as a vector and as a set of vertices in V. To each basis element vi we associate its set of vertices Vi = supp(vi), and define the vertex set U = S i Vi. Definition 7.2 (Modular expansion). Let H(V, E) be a hypergraph with incidence matrix G : F2E →F2V and specified subspace W ⊂F2V containing elements κλ. Let t ∈R+ be a positive real number. The modular expansion Mt of H is the largest real number such that, for all v ⊆V, |G⊺v| ≥Mt min(t, |κλ| -|v ∩κλ| + |v ∩U\\κλ| : κλ ∈W) As this definition is quite complicated, we give some orienting remarks. Modular expansion is a generalisation of both relative expansion from Ref. [57] and the soundness of a classical locally testable code, and is therefore also as generalisation of the edgeexpansion of a graph. It is 'modular' in the sense that it contains distinguished subsets κλ such that, when they all exist in the same hypergraph, the notion of expansion for each of them combines to give a complete notion of expansion for the whole hypergraph. In detail, recall that edge-expansion (henceforth called global expansion) can be defined for hypergraphs as follows. Definition 7.3 (Global expansion). Let H(V, E) be a hypergraph with incidence matrix G : F2E →F2V. The global expansion of H is the largest real number β such that |G⊺v| ≥β min(|v|, |V |). 35 When G is a graph this reduces to traditional edge-expansion and β is the Cheeger constant. Ref. [57] defined relative expansion, a generalisation of global expansion. Definition 7.4 (Relative expansion). Let H(V, E) be a hypergraph with incidence matrix G : F2E →F2V, distinguished subset U ⊂V and chosen parameter t ∈R+. The relative expansion of H is the largest real number βt such that |G⊺v| ≥βt min(t, |v ∩U|, |U| -|v ∩U|). When t ≥|V| and U = V this reduces to the definition of global expansion. Similarly, when |I| = 1 modular expansion reduces to relative expansion, observing that there are only two κλ vectors, one being the empty set and the other U. A different generalisation of global expansion is to the soundness of a locally testable code. Recall Definition 2.10, with variables replaced to be suggestive of our hypergraph setting. Definition 7.5. A binary linear code C ⊂F2V is (ω, ρ)-locally testable if it has a paritycheck matrix G⊺with rows of weight at most ω such that for any vector v ∈F2V, 1 m|G⊺v| ≥ρ nd(v, C) where m = |E|, n = |V| and d(v, C) = minx∈C(|v + x|). The values ω and ρ are the locality and soundness of the code respectively. In particular, if we rearrange the equation defining soundness ρ then we get |G⊺v| ≥ βd(v, C) for β = ρm n . We observe that in the case where the only codewords are 0 and V, β coincides with the definition of global expansion. For any sparse hypergraph H, if it has global expansion it is also a locally testable code, albeit a rather trivial one. In this way the soundness of a code can be thought of as a generalisation of expansion to the case where a hypergraph has ker G⊺with dimension greater than 1. As dim G⊺is equal to the number of logical representatives which are measured simultaneously by the auxiliary hypergraph, we require a more general definition than relative expansion in order to perform block reading. Modular expansion coincides with soundness when U = V and t ≥|V| under the rearrangement Mt = ρm n , and when W = ker G⊺, so each element κλ is an element of the kernel. We thus have the informal commuting diagram, Modular expansion Relative expansion k = 1 Soundness of an LTC t ≥n Global expansion k = 1 U = V t ≥n U = V where k = dim G⊺, n = |V|, and each κλ is assumed for simplicity of the diagram to correspond to an element of ker G⊺. 36 Remark 7.6. For a simply connected graph, the edge-expansion can be related to spectral properties of the adjacency matrix or Laplacian of the graph. We do not know of a similar relation between relative expansion, soundness or modular expansion and spectral properties of the Laplacian of the hypergraph. To explain our results on hypergraphs with modular expansion we use the terminology of port functions, similar to Ref. [57]. In short, given a CSS code Q and a hypergraph H(V, E), a port function fp is a map from a subset of data qubits in Q to a set of vertices in V. The deformed code is then defined as Q ↔H, with connections introduced between Q and H given by the port function fp. This definition of the deformed code is similar to the definition of the mapping cone by a chain map, although the port function does not specify the connectivity of deformed checks into the auxiliary hypergraph, and the port function is in the opposite direction to f1 in a chain map. Moreover, the port function definition can be applied to codes which are not CSS. Theorem 7.7 (Distance preservation). Let {Zi}i∈I be a set of logical representatives (trivial or nontrivial) on a distance d CSS code Q with supports ξi = supp(Zi). Let H(V, E) be a hypergraph with vertices in V being Z checks, hyperedges in E data qubits and X gauge checks forming a cycle basis, such that H measures all the logical representatives {Zi}i∈I. Let fp : S i ξi →V be an injective port function such that U = imfp ⊂V, and fp(ξi) = Vi ⊂V, ∀i ∈I. Let Md(H) ≥1, where W is generated by fp(ξi). Then the distance of the deformed code Q ↔H is at least d. Proof. See Appendix C. As the above proof demands the port function fp from the original code to the hypergraph be injective, it does not capture the most general form of hypergraph surgeries. In that case, scaling the modular expansion proportional to the check incidence to the original code is sufficient to preserve the distance, see Remark 7.16. Theorem 7.8 (Thickening). Let H(V, E) be a hypergraph with modular expansion Mt(H). Let J L be the path graph with length L ≥ 1 Mt(H), i.e. L vertices and L -1 edges. Let HL := H□J L be the hypergraph of H thickened L times. Then HL has modular expansion Mt(HL) ≥1 for Ul= S i V l i at any level l∈{1, 2, · · · , L}, where V l i is the copy of Vi in the lth level of the thickened hypergraph. Proof. See Appendix C. Lemma 7.9. Let H(V, E) be a hypergraph such that each element in ker(G⊺) has the supporting set κλ for some λ ∈2I. Then Mt(H) ≥1 t . Proof. Consider an arbitrary vertex set v. We consider two cases: first, where |G⊺v| = 0, and then where |G⊺v| ≥1. In the first case, we know that v ∈ker(G⊺) and therefore v = κm for some m ∈2I. Therefore |κm| -|v ∩κm| + |v ∩U\\κm| = 0, and so |G⊺v| ≥Mt(H) min(t, |κλ| -|v ∩κλ| + |v ∩U\\κλ| : λ ∈2I) for any Mt(H). In the second case, |G⊺v| ≥1. We now consider the two values of the minimum. If min(·) = t then |G⊺v| ≥1 t t. If min(·) = |κλ| -|v ∩κλ| + |v ∩U\\κλ| for some λ ∈2I then t ≥|κλ| -|v ∩κλ| + |v ∩U\\κλ|, and so |G⊺v| ≥1 ≥1 t (|κλ| -|v ∩κλ| + |v ∩U\\κλ|). Corollary 7.10. Let H(V, E) be any hypergraph such that each element in ker(G⊺) has the supporting set κλ for some λ ∈2I. Let Ht := H□J t be the hypergraph of H thickened t times. Then Ht has modular expansion at least Mt(Ht) ≥1 for any level l. 37 Proof. As Mt(H) ≥1 t and J L in Theorem 7.8 has L = t then L ≥ 1 Mt(H). Corollary 7.10 post-hoc justifies devised sticking from Ref. [59] by setting t = d, as there the hypergraph is thickened d times to preserve fault-tolerance, and the CKBB scheme [52] is a special case of this. Remark 7.11. The expansion-based argument does not account for the cycles in the hypergraph, however. For these see Ref. [52, Thm. 1], which shows that when the hypergraph is thickened d times the cycles can be considered gauge logicals and the deformed code still has distance at least d as a subsystem code. Theorem 7.8 shows that it is not necessary to thicken d times, if the modular expansion before thickening is greater than 1 d. This has previously been proved for the simpler case of measuring only one logical operator [55, 57]. Lemma 7.12 (Cross-block measurements). Let H be a hypergraph with modular expansion Md(H) ≥1 L. Let H correspond to the degree-shifted subcode A• with distance at least d. Vertices in H are Z checks and edges are data qubits. Let f•, f′ •, f′′ • ... be the inclusion chain maps of A• into C•, C′ •, C′′ • ..., each of which has distance at least d. Let HL be the the hypergraph thickened L times, with modular expansion Md(HL) ≥1 at each level l∈{1, 2, · · · , L}. Then the deformed code D• given by modifying each chain map f•, f′ •, f′′ • ... to have their preimages on any arbitrary levels of the hypergraph and taking the mapping cone has distance d if all cycles are gauge-fixed. Proof. As all cycles are gauge-fixed, there are no new logical qubits, and no X logical can have weight lower than d. We must therefore ensure that no Z logical which is unmeasured can have its weight reduced lower than d by cleaning. We prove the lemma by contradiction. If any Z logical extending over all initial codeblocks C•, C′ •, C′′ • ..., but not the hypergraph, could have its weight reduced below d by cleaning into the hypergraph, then a single logical in one of C•, C′ •, C′′ • ... must also have its weight reduced below d by cleaning. This fact is because removing the chain maps to all other codeblocks would leave the same set of Z checks which could clean ΛZ to the same data qubits σ, with the exception of data qubits in the other codeblocks. This set of data qubits must satisfy |σ| < d for this lemma to be false. However, we know that Md(HL) ≥1 at each level of the hypergraph, therefore |σ| ≥d and the lemma holds. By a combination of Corollary 7.10 and Lemma 7.12, observe that the earlier Lemma 4.2 from Section 4 holds. To summarise this section, if a subcode A• for partial block reading has sufficient soundness then it must always also have modular expansion, which can then be boosted by thickening an appropriate amount to guarantee the deformed code's distance is preserved. Recall the definition of soundness for quantum codes in Def. 2.11. Corollary 7.13. Let A• be the subcode used in a partial block reading. Let A• have soundness ρA. Then, viewed as a hypergraph H, thickening 2nA ρAmA times preserves the distance of the deformed code when all cycles are gauge-fixed. Proof. By Lemma 2.12, H has soundness ρAmA 2nA as a classical code with parity-check matrix G⊺. Setting W = ker G⊺, H also has modular expansion Md ≥ρAmA 2nA . By Lemma 7.12, set L = 2nA ρAmA and the thickened hypergraph HL has modular expansion Md(HL) ≥1. Remark 7.14. As mA 2nA ∈O(1) for LDPC codes, the asymptotic space cost of partial block reading when all cycles are gauge-fixed can be reduced to O(nA ρA ). 38 Assuming the modular expansion Md(H) before thickening is greater than 1 d, which is its minimum by Lemma 7.9, this thickening requires fewer than the d layers required by the CKBB scheme and similar [52, 59]. Note that as the soundness of a subcode is not generally inherited from the soundness of the full code, finding subcodes with high soundness is generally difficult. Regardless, having hypergraphs equipped with modular expansion is extremely useful to perform fast surgery. Lemma 7.15. If each hypergraph Hi associated to a partial block reading Partiali for i ∈[n] has modular expansion Md(Hi) ≥1 then the compacted code CC• has distance d. Proof. By Theorem 7.7, attaching H1 to the original code via the chain map (h•)1, to yield cone((h•)1), must always preserve the distance of the code. Applying the other mapping cones iteratively will then each preserve the distance, so CC• has distance d. As a consequence, the conditions of Theorem 4.5 can be satisfied constructively, by choosing subcodes of distance d and then thickening such that each hypergraph has modular expansion. Remark 7.16. A similar approach applies to Theorem 5.3, where instead it is sufficient for each hypergraph to have modular expansion Md(Hi) ≥wi, where wi is the maximal column weight of (f1)i, (f′ 1)i, (f′′ 1 )i, · · · . 8 Examples In this section we illustrate block reading with a series of examples. We focus first on 2D surface codes, upon which traditional lattice surgery is extremely well-understood [106, 110, 111]. 8.1 Topological codes 8.1.1 2D surface codes We demonstrate on unrotated surface codes, as the connection to block reading, and LDPC code surgery in general, is more apparent in this case. We start with two square surface code patches with distance d, with n = d2 + (d -1)2 data qubits each, 39 In these diagrams qubits are edges, X and Z checks are vertices and faces respectively. The deformed code for the full block reading performing a Z ⊗Z is then given by where the new edges are dashed grey lines. There are new vertically-oriented faces where there are new squares, and at the boundaries. Evidently the connectivity requirements are close to the requirements for transversal CNOTs. 40 The only distance d subcode (as defined in Def. 2.5) of the square surface code is itself, and so the only partial block reading with a distance d subcode is full block reading. A partial block reading using a subcode A• with distance dA less than d can be constructed by choosing a strip of the surface codes and then connecting these together. In the simple case of surface code patches as above the distance is always preserved by partial block reading, because the only logicals which can have reduced weight are those on Z ⊗Z or X ⊗X, which are now stabilisers. If the surface code patches also have defects, or are toric codes instead, partial block reading does not generically preserve code distance and increasing modular expansion in the auxiliary hypergraph is necessary, by thickening or otherwise. Recall that surface codes have asymptotically vanishing soundness by [78, Cor. 1] so using a partial block reading with dA < d on the surface code is not particularly useful for reducing the time overhead, unless dA ≥d 2 by Proposition 6.1. In the case where dA = 1 and the chosen subcode is along the boundary of the patches, we recover traditional lattice surgery with unrotated surface codes: 41 8.1.2 2D toric codes Partial block reading with toric codes is illustrative, as it showcases how the distance of the deformed code can drop without thickening. First, consider the following d = 5 toric codeblocks, shown by tessellating their fundamental polygons. Blue and red edge pairs of the same lattice are identified to form loops. 42 A partial block reading between them can be performed with a subcode corresponding to a cylinder around the torus. Measuring only one representative from each torus recovers traditional surgery with 2D toric codes, 43 which measures Z ⊗Z on one logical qubit from each codeblock, but not the other. Growing the width of the cylinder such that the distance of A• is 4, we acquire the deformed code: 44 Depending on the chosen subcode, the distance may drop in the deformed code, as there can be a stringlike Z operator which extends from one torus to the other and then back, skipping the part of the torus with the new connections and making a short noncontractible closed curve. In our example the deformed code distance is still 5, as these curves take the form shown by green edges, 45 which have length at least 6. However, were we to start with d = 7 toric codes, with the width of the cylinder being 6, the distance would drop as a consequence of these same curves. Increasing the length of this closed curve is then possible by thickening the subcode, or boosting the modular expansion in some other manner. If we were to perform a full block reading, these closed curves would become contractible due to the additional stabilisers, so the deformed code distance would be preserved regardless. 8.1.3 4D toric codes 4D toric codes are known to be decodable in a single-shot manner, as they have some soundness and metachecks of both X and Z-type [78, 82, 83, 107]. A 4D toric code can be constructed by a 4-fold tensor product of the chain complex R• = R1 R0 ∂1 where R1 = R0 = Fm 2 and the differential ∂1 is the incidence matrix of the cycle graph with m vertices and edges, so dim ker(∂1) = 1 and dim Fm 2 /im(∂1) = 1. The 4-fold tensor product yields a chain complex with 5 nonzero components. Let the 4D toric code be C• = (R ⊗R ⊗R ⊗R)• = C4 C3 C2 C1 C0 46 and the qubit component is fixed to be C2.13 More explicitly we have C2 = M i+j+k+l=2 Ri ⊗Rj ⊗Rk ⊗Rl, and the entire chain complex is given by R1R1R0R0 R1R1R1R0 R1R0R1R0 R1R0R0R0 R1R1R0R1 R1R0R0R1 R0R1R0R0 R1R1R1R1 R0R0R0R0 R1R0R1R1 R0R1R1R0 R0R0R1R0 R0R1R1R1 R0R1R0R1 R0R0R0R1 R0R0R1R1 where tensor products are suppressed for visibility and columns of terms are taken in a direct sum. The K ̈unneth formula implies that dim H2(C) = 6. The distance of C• is d = m2, which can be computed using Lemma 2.8. The intuition is that all nontrivial logical operators in a 4D toric code form membranes through the complex, as opposed to strings in 2D toric codes. For a given summand in C2, say R1 ⊗R1 ⊗R0 ⊗R0, one choice of corresponding nontrivial Z logical operator representative is then 1 ⊗1 ⊗(1, 0, · · · , 0) ⊗(1, 0, · · · , 0), where 1 = (1, 1, · · · , 1). To give a treatment of partial block reading with 4D toric codes we specify a subcode A•. There are evidently many such choices. One choice is to measure just a single logical representative, which by Thm. 1.6 must take Θ(d) syndrome measurement rounds to maintain phenomenological fault distance. Another choice of A• is to specify a set of Z-checks in C3, and to take the summands in C2, in the image of that set of Z-checks and so on. This is trivially guaranteed to be a subcode. For example, set A• to be R1R1R0F2 R1R0R0F2 R1R1R1F2 R1R0R1F2 R0R1R0F2 R0R0R0F2 R0R1R1F2 R0R0R1F2 13This shift in qubit degree compared to convention is for convenience, so that there are no negative indices and basis elements in C2 can be identified with faces in a cell complex. 47 where the chain map component f3 maps R1R1R1F2 in A3 into R1R1R1R0 in C3 and so on, choosing a consistent basis element of R0 to map F2 into. This choice measures 3 Z logical operators, and many representatives of each. This is the choice taken in Ref. [107], and can be seen as setting A• to be a hyperplane of the lattice. While each Z logical representative contained in A• must have weight at least m2, the X-distance of A• is only m, as this is a 3D toric code by identifying that R1R1R1F2 ∼= R1R1R1 etc, with the qubit component at A2 rather than the customary A1. In Ref. [60] it was observed that while one can perform single-shot lattice surgery using a subcode given by a hyperplane with 4D surface and toric codes, the performance was reduced unless O( √ d) = O(m) syndrome rounds were used. By Proposition 6.1 we in fact see that Ω( √ d) rounds are required to maintain fault-distance, as α = √ d in this case. Thus one can perform single-shot lattice surgery, or lattice surgery using a constant number of syndrome rounds, with 4D toric codes, but with only O( √ d) fault-distance, given that choice of subcode. Remark 8.1. Proposition 6.1 is not quite sufficient to justify Θ( √ d) rounds of syndrome measurement to maintain fault-distance, only Ω( √ d), as 4D toric codes do not have constant soundness [78, 82], see Remark 6.4. Remark 8.2. Lattice surgery with 'geometrically enhanced' 4D toric codes was studied in Ref. [107, Sec. V]. There, the X-distance of A• was called the 'boundary distance', and similar arguments about the number of rounds required for fault tolerance were employed. The lattice surgery there is performed in an 'ancilla-free' manner, where no new data qubits are used; only new checks. This is a quotient rather than a mapping cone, see Refs. [53, 54, 64]. 8.2 Bivariate bicycle codes Bivariate bicycle (BB) codes [86] are a class of Abelian 2-block group-algebra (2BGA) codes [18, 112], and as such are lifted product and balanced product codes [16, 17, 113]. They are of substantial practical interest, as some bivariate bicycle codes are known to exhibit high pseudo-thresholds and low logical error rates under circuit noise and suitable decoders, while having a higher rate than surface codes [63, 86], albeit requiring more connectivity. BB codes can also be seen as toric codes boosted to have limited non-local connectivity [114]. It is immediate that full block reading can be used to perform parallel Z ⊗Z or X ⊗X measurements between two BB codes using n + mZ = n + mX = 3 2n new qubits, including check qubits, while maintaining the distance and LDPC property. For instance, the J144, 12, 12K gross code, which uses 288 total qubits as a quantum memory when including check qubits, requires 216 additional qubits in total for a full block reading. This is lower than for the same set of parallel measurements in Ref. [54, Sec. 3.3], which used 360 total new qubits (216 new data qubits and 144 new check qubits), increased the check weight of the deformed code to 12, and required O(d) rounds of syndrome measurement. Thus even a naive application of block reading can reduce the spacetime overhead of logical measurements by an order of magnitude compared to conventional surgery, where the connectivity allows for it. We suspect that one can perform partial block readings using subcodes of the gross code, and similar BB codes, in order to address desired logical qubits and reduce space overheads while maintaining fast and parallel logical measurement at high fault distance, and retaining a degree of addressability. As this requires substantial numerics we defer this to future work. 48 9 Equivalence between surgery and homomorphic measurement The abstract relationship between (a) surgery by an auxiliary system and (b) homomorphic measurement [42] was folklore prior to this work, and indeed is implied by Ref. [58]. That is, given a chain map one can always construct a mapping cone and vice versa. The relationship we show in this section is more concrete: given the circuit for homomorphic measurement on a CSS code, it can always be rewritten into a logical measurement by surgery using a mapping cone (called homological measurement in Ref. [58]) and vice versa. Theorem 9.1. Let C• be a CSS code with a homomorphic measurement procotol specified by a chain map f• : A• →C•. Then the circuit corresponding to the homomorphic measurement can be rewritten to a surgery specified by cone(f•). Proof. See Appendix D. Corollary 9.2. Let C• be a CSS code with a surgery protocol specified by cone(f•) for a chain map f• : A• →C•. Then the circuit corresponding to the surgery protocol can be rewritten to a homomomorphic measurement specified by f•. Proof. See Appendix D. These proofs are performed using the ZX-calculus [115], a formal calculus for rewriting tensor networks on qubits. This is a convenient formalism, as the rewriting requires converting checks and data qubits in the homomorphic measurements into data qubits and checks respectively in the surgery procedure. In essence, the time and space axes of the ancilla circuit are inverted. However, the rewriting does not generally preserve fault-distance, unlike the rewrites in Ref. [116, 117]. The reason why converting a surgery to a homomorphic measurement does not generally preserve fault-distance is because having a low timelike distance in a surgery protocol can be resolved by measuring for multiple rounds. However, the equivalent homomorphic measurement has low space distance, which is fatal. We expect that one can prepare many of those ancilla states and measure repeatedly, in the style of a Shor measurement [5]. However, this is no longer a homomorphic measurement in the sense defined in Ref. [42]. Conversely, we do not know whether rewriting any homomorphic measurement to a hypergraph surgery preserves fault-distance. Our last results focus on general bounds on fast measurement by an auxiliary system. The results are insensitive to the nature of the auxiliary system: whether it is surgery, homomorphic measurement or any other scheme. Proposition 9.3. Let Q be a quantum LDPC code. Let S be a set of logical operator representatives measured at the same time by one or more auxiliary system for T syndrome rounds. Assume that the correct logical measurement outcomes of S are not known in advance. Let f be the lowest weight data qubit fault in Q such that: • f flips at least one logical representative in S. • If f flips a logical representative in S then it flips all other representatives in the same equivalence class in S. Let w be the weight of f, and s be the syndrome weight of f in Q. Then the phenomenological fault-distance of the measurement protocol is upper bounded by 2w + Ts. 49 Proof. See Appendix E. When Q is CSS and the set S forms a Z-type subcode A•, w is the X-distance of A•. Theorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quantum LDPC code by an auxiliary system requires connections to more than one logical representative to maintain phenomenological fault distance Ω(d), unless the correct logical measurement outcome is known in advance. Proof. See Appendix E. Importantly, for fast surgery it is not enough for the deformed code in the idling operation to be a single-shot code. Because the correct outcomes of the new checks are not generally known in advance, many connections between representatives of the original code and the auxiliary hypergraph are essential to correct for measurement errors on these new checks. Indeed, our earlier results on full and partial block reading with distance d subcodes imply that this is the only thing that is required, assuming that the auxiliary hypergraph is sensibly constructed and that sufficent rounds of padding are used. 10 Discussion In this work we have introduced a variety of new, general LDPC code surgeries which reduce the time overhead of logical measurement below O(d) rounds and allow the measurement of logical operators in parallel. We generalised prior work [55-58], which used expansion properties of hypergraphs to efficiently measure individual logical operators, to use soundness instead, enabling measurement of many logical operators in parallel. We elucidated the connection between homomorphic measurement and surgery, and gave general bounds on the fault distance for any logical measurement by an auxiliary system. To prove fault distances for fast surgery in Appendix A we developed an extension of the fault complexes formalism [60] by taking mapping cones on idling spacetime volumes, which may be of independent interest. Our work raises many further questions. First, when is it possible to efficiently devise a sparse auxiliary hypergraph H with high modular expansion even if the original subcodes have low soundness? In the case of individual logical operator representatives this is straightforward by constructing an expander graph, but for more complicated parallel measurements involving many representatives the numerical construction of suitable hypergraphs is challenging for large codes, and brute-force branching techniques [59, 61] have formidable constant factor overheads. Similarly, when H is a graph one can appeal to the Decongestion Lemma [118, App. A]. When H is a small hypergraph one can typically devise a sparse cycle basis for H by numerical brute-force [55], but for large codes such a method becomes impractical. Are there families of LDPC codes for which such bases are efficient to construct, or unnecessary entirely? We expect this to be true of any code that is geometrically local in a fixed spatial dimension. In fact, constant-time surgery operations raise the intriguing possibility of avoiding the need for a sparse cycle basis at all. This is because cycle checks do not need to be explicitly measured as they can be inferred from the initialization and readout steps, forming detectors that have a constant extent in the time direction. These detectors must provide fault-tolerance, but need not be strictly local. For instance, see the nonlocal hierarchical fault-tolerant detector complexes derived from local measurements in Ref. [109]. 50 An issue that arises when using hypergraphs is that the method of universal adapters [57] to connect heterogeneous subcodes between codeblocks does not generally apply. This is because the SkipTree algorithm assumes that the auxiliary system is a graph, rather than a hypergraph. We do not know whether generalisations of the algorithm can be invented to connect heterogeneous hypergraphs for product measurements, either in full generality or for specific cases. Here, we have focused on hypergraph measurements that only measure local checks and global logicals, i.e. they do not locally deform the code after the measurement. It is possible to relax these requirements, which leads to a setting of dynamical local codes. Could this lead to more addressable fast surgery without sacrificing performance? Our proofs concerning interpolated partial block reading in Sec. 6 used a strict definition of soundness, see Def. 2.11, which we believe can be loosened to a 'small set soundness', with a meaning similar to small set expansion or modular expansion. Our proofs regarding fault-distance require d rounds of syndrome measurement ('padding') before and after the surgery protocols in order to provably protect against timelike faults extending from one timelike boundary to the other. We expect that these are typically unnecessary given sensible fault-tolerant operations performed before and after the surgery protocol. However, we have not fully characterized when these rounds of padding can be reduced in number or removed. It is known that high-distance subcodes which address specific logical qubits exist for topological and tensor product codes [42, 43], but for other families of LDPC codes little is known. It would be interesting to consider algebraic methods to find high-distance subcodes for lifted or balanced product codes. All our surgery methods used auxiliary hypergraphs, and there are other approaches to surgery which are 'ancilla-free', meaning that they use no new data qubits [83, 107]. Such approaches fall under the umbrella of quotients of codes, see Refs. [53, 64], rather than mapping cones [58]. The arguments concerning time overhead appear to translate between the two approaches, but discerning a rigorous connection is left to future work. We suggest that one could take a pushout or similar quotient directly on the spacetime volumes described by fault complexes [60] in order to study such surgeries. We do not know whether it is possible to take an arbitrary quantum LDPC code and provably measure an arbitrary set of commuting logical Pauli products in constant time overhead and linear (up to polylog) space overhead while maintaining fault-distance and the LDPC property. Such a proof is possible for the case of O(d) time overhead [61], but the difficulties of finding appropriate subcodes and cycle bases require new methods. Conversely, it may be possible to show that such a generic protocol is impossible, see Ref. [119] for a related result in the classical setting. We have developed a formalism for studying the phenomenological faults in generic CSS LDPC code surgery operations using fault complexes. We expect it is possible to prove thresholds of surgery on spacetime volumes in this formalism using combinatorial methods on the relevant fault complexes [41]. Block reading entails metachecks, which means the addition of many more detectors to the decoding graph for a single round of measurement. We do not know whether there exist decoders which can efficiently decode such graphs quickly enough to be suitable for quantum hardware. In Ref. [83] a powerset decoder was used to decode the a geometrically enhanced 4D toric code memory, which also has metachecks. However, this was far too slow for realistic hardware, and we do not know how well the powerset decoder would handle surgery operations. More realistic decoders for high-dimensional topological and tensor product codes were studied in Refs. [60, 120]. 51 Finally, to bring the theory of block reading to application it will be necessary to perform many simulations and stability experiments [121], and numerically determine pseudo-thresholds and logical error rates of the surgery operations. 11 Acknowledgements We are grateful to Katie Chang and Louis Golowich for insightful discussions and for spotting an error in an earlier version of this paper. A.C. is employed by Xanadu and would like to thank Ben Ide, Eric Sabo, Timo Hillmann, Michael Vasmer and Guillaume Dauphinais for helpful discussions. Z.H. is supported by MIT -O'Connor and Guanyu Zhu for helpful discussions. 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A Proofs concerning fault-distance In this appendix we provide detailed proofs of the phenomenological fault-distance lower bounds described in the main text. The standard approach to proving phenomenological fault-distance is to construct a measurement circuit, where Pauli error faults can occur at integer timesteps and check faults can occur at half-integer timesteps, with detectors [92, 123], and then to show that any fault that does not trigger any detectors and is not a product of spacetime stabilisers must have at least a given weight d. It is difficult to prove the fault-distances which we require in that setting, and so we use an alternative, almost equivalent one: that of fault complexes [60]. This enables us to use homology to classify the logical faults and bound their corresponding weights. In this setting, Pauli error faults of Z type and Z check errors occur on even timesteps, while Pauli error faults of X type and X check errors occur on odd timesteps. A sketch of the proofs in this section works as follows: we construct fault complexes for the idling operation, and then modify them using mapping cones to obtain fault complexes for various surgery protocols. These result in spacetime volumes where the code is initially in the idling operation for d rounds, then undergoes a series of deformations to perform surgery with no padding between surgery rounds, and finally all auxiliary systems are measured out and the code returns to the idling operation for d rounds. Because the spacetime volumes are defined in terms of chain complexes, and the deformations are described in terms of mapping cones, we can construct long exact sequences which relate the classes of logical faults in the spacetime volumes for surgery to the classes of logical faults in the idling spacetime volume. We then organise these classes of logical faults and show that they each must have weight at least d. Our techniques may be of independent interest to study other protocols in circuit-based or measurement-based quantum computing. Before we move to discussing fault complexes, we give some elementary homological background which is required. The Snake Lemma [90, Lem. 1.3.2] and rank-nullity together imply that for a mapping cone cone(f•) defined by a chain map f• : A• →C•, we have |Hi(cone(f))| = |Hi(C)| + |Hi-1(A)| -|imHi(f)| -|imHi-1(f)|, where |V | is shorthand for dim V when V is a vector space. 60 In more detail, the Snake Lemma dictates that for the mapping cone cone(f•) we have a long exact sequence 14: · · · Hi(A) Hi(C) Hi(cone(f)) Hi-1(A) Hi-1(C) · · · Hi(f) ι∗ π∗ Hi-1(f) In this sequence the Hi(f) are the maps between homology spaces Hi(A) →Hi(C), which are well-defined by the commutative diagram in Eq. 1. ι∗is a map Hi(C) →Hi(cone(f)) generated by the Snake Lemma. In homological measurement [58], this map at i = 1 takes logical operators in the original code to the logical operators in the deformed code when performing surgery. In that case, the map ι∗ will be a surjection, as some of the logical qubits from the original code are measured out and no new logical qubits are added when the auxiliary hypergraph's cycles are gaugefixed. π∗is a map Hi(cone(f)) →Hi-1(A). In homological measurement [58], this map at i = 1 picks out the new contributions to the logical space of the deformed code from Hi-1(A); exactness at Hi(cone(f)) shows that the only new logical qubits which can arise are from cycles in the hypergraph. In our case we will take mapping cones on spacetime volumes, rather than just codes, but a similar intuition applies. For any linear map T : V →W rank-nullity implies that |V | = | ker T| + |imT|, so we can compute |Hi(cone(f))| = | ker π∗| + |imπ∗| = |imι∗| + |imπ∗| = |Hi(C)| -| ker ι∗| + | ker Hi-1(f)| = |Hi(C)| -|imHi(f)| + |Hi-1(A)| -|imHi-1(f)|. (4) The premise of all of our proofs is that in a CSS-type code surgery, where the initial code and deformed codes are all CSS codes, all logical faults must be either of Z type in spacetime, composed of Z Pauli errors and X check errors, or X type, composed of X Pauli errors and Z check errors. As any logical fault composed of a product of these is at least as high weight as the corresponding logical fault of either Z or X type, we need only show that the weights of all Z and X logical faults are at least d to prove that the fault-distance of a protocol is at least d. We will frequently make use of the fact that when Hi(f) is injective |imHi(f)| = |Hi(A)|, by definition. A.1 Fault complexes A fault complex is defined as a chain complex with 4 terms, F• = F3 F2 F1 F0 where by convention the F3 and F0 components correspond to Z and X detectors respectively. In fault-tolerant measurement-based quantum computing (MBQC), F2 corresponds to a set of X fault locations, i.e. locations in spacetime which can experience only Xtype faults, while F1 corresponds to a set of Z-type fault locations. The fault complex is therefore in bijection with a graph state. When the graph state corresponds to a foliated CSS code, this can be reinterpreted in the circuit model, undergoing phenomenological noise. Now, F2 corresponds to a combined 14This means that the image of each map coincides with the kernel of the next map. 61 set of spacetime locations at which data qubits can experience X Pauli faults and also Z checks, which can experience measurement errors. F1 is the same but for Z Pauli faults and X checks. In this case, we can expand out a fault complex to have the following diagram: F1,1 F1,0 F1,2 F0,0 F0,2 F0,1 where components in the overall fault complex are given by taking direct sums vertically in the diagram, so we have F1,2 F1,1 ⊕F0,2 F1,0 ⊕F0,1 F0,0 Throughout this section every diagram drawn in the above form is assumed to have direct sums vertically in the diagram. 15 In the phenomenological circuit model: • F0,0 corresponds to X detectors. • F0,1 corresponds to locations for Z Pauli faults on data qubits. • F1,0 corresponds to locations for X check faults. • F0,2 corresponds to locations for Z check faults. • F1,1 corresponds to locations for X Pauli faults on data qubits. • F1,2 corresponds to Z detectors. Expressed as a commutative diagram the fault complex then has the structure: X Pauli faults X check faults Z detectors X detectors Z check faults Z Pauli faults F1,1 (X Pauli faults) F1,0 (X check faults) F1,2 (Z detectors) F0,0 (X detectors) F0,2 (Z check faults) F0,1 (Z Pauli faults) 15This is essentially passing to the \"total complex\" [90]. 62 Note that as a fault complex describes a graph state, it does not have the ability to describe morphisms, i.e. MBQC-based or circuit-based protocols with input or output qubits. The standard way to foliate a CSS code C• from the fault complex perspective is to take (R⊗C)•, where R• : R1 →R0 has the differential R being either the full-rank paritycheck matrix for the repetition code or its dual. In the former, the code is initialised in the |0⟩state and measured out in the Z basis; in the latter, the code is initialised in the |+⟩state and measured out in the X basis. Hence the code 'idles' for l rounds, where l is the blocklength of the repetition code. If we do this, the fault complex has the explicit form: R1 ⊗C1 R1 ⊗C0 R1 ⊗C2 R0 ⊗C0 R0 ⊗C2 R0 ⊗C1 (5) In the circuit model we have, in order, • R0 ⊗C0 corresponds to X detectors. • R0 ⊗C1 corresponds to locations for Z Pauli error faults. • R1 ⊗C0 corresponds to locations for X check error faults. • R0 ⊗C2 corresponds to locations for Z check error faults. • R1 ⊗C1 corresponds to locations for X Pauli error faults. • R1 ⊗C2 corresponds to Z detectors. Note that R1 ⊗C1 and R0 ⊗C1 correspond to the same data qubits but at different points in time; in the circuit model this is purely convention, and any given data qubit can still undergo either type of error, Z or X. It is easy to compute the homologies using the K ̈unneth formula. Then elements of H1(F) correspond to equivalence classes of logical Z faults in spacetime, composed of Z Pauli errors and X check errors, and elements of H2(F) to logical X faults, composed of X Pauli errors and Z check errors.16 For completeness, we have, • H1(R ⊗C) = H1(R) ⊗H0(C) ⊕H0(R) ⊗H1(C). • H2(R ⊗C) = H1(R) ⊗H1(C) ⊕H0(R) ⊗H2(C). When R• is the repetition code, we have H1(R ⊗C) = H1(R) ⊗H0(C) = {0, 1} ⊗C0/imHX H2(R ⊗C) = H1(R) ⊗H1(C) = {0, (1, 0, 0, · · · , 0)} ⊗H1(C), 16There are also the homology groups H1(F) and H4(F), corresponding to classes of detector triggers which are impossible. 63 Figure 5: A fault complex for a unit cell of the 2D surface code, starting and terminating at X-type boundaries. Time flows from bottom to top. where 0 and 1 are the all-zero and all-one vectors, and (1, 0, 0, · · · , 0) is a weight 1 vector representing the nonzero equivalence class in H1(R); any vector with odd weight is in the same class. When R• is dual to the repetition code we instead have H1(R ⊗C) = H0(R) ⊗H1(C) = {0, (1, 0, 0, · · · , 0)} ⊗H1(C) H2(R ⊗C) = H0(R) ⊗H2(C) = {0, 1} ⊗H2(C). In Fig. 5 we show the fault complex for a small surface code as a Tanner graph, where circles are fault locations and squares are detectors, and red, teal, yellow and blue indicate basis elements of F0, F1, F2, F3 respectively. Fault complexes are a useful algebraic framework for reasoning about fault distance of CSS codes in the phenomenological model: each location is a point in spacetime which can experience a fault, which is also a basis vector in a space (either F1 or F 2) and so weights of faults can be related directly to weights of vectors in the fault complex. We do not want to presume that the code is initialised and measured out in a given basis, because we want to prove that a given protocol has sufficient fault-distance no matter what other (assumed fault-tolerant) operations the logical circuit contains. One approach to this is to replace the parity-check matrix of R• with the parity-check matrix of a different repetition code, that of the cycle graph. This corresponds to 'taking the trace' of the circuit operation being performed, and ensures that errors cannot vanish on the initial and final boundaries. However, this leads to other, serious issues: a timelike fault which runs from the start to the finish of the spacetime volume is now a spacetime stabiliser, and there are other logical faults which we encounter that are equivalent to this fault, and so they will be unaccounted for. Instead, we modify the spacetime volume by adding checks to the start and end, which constitute entry points for timelike errors to enter and exit the volume: depending on whether R• is a full-rank repetition code or its dual, corresponding to initialising in the |0⟩or |+⟩state respectively, the new checks are Z checks or X checks. We accomplish this using mapping cones. Explicitly, when R• is the dual of the repetition code, so R• = Fl-1 2 →Fl 2 for some l determining the number of rounds in the 64 spacetime volume, the code is initialised in |+⟩and measured out in the X-basis; we then introduce new checks for timelike fault entrypoints by taking the mapping cone of the chain map g•, 0 0 0 C0 ⊕C0 F3 F2 F1 F0 g0 where g0 maps the first copy of C0 into the top layer of X-detectors in the fault complex, and the second copy of C0 into the bottom layer of X-detectors. Call the resultant fault complex F ′ = cone(g•). The unit cell from Fig. 5, once modified in this manner, is shown in Fig. 6. The consequent classes of logical faults in cone(g•) can be computed using the Snake Lemma, see Eq. 4. We have |H1(F ′)| = |H1(cone(g))| = |H1(F)| + |C0 ⊕C0| -|imH0(g)| = |H1(F)| + 2|C0| -|imH0(g)|. By the K ̈unneth formula, H0(F) = H0(R) ⊗H0(C) = {0, (1, 0, 0, · · · , 0)} ⊗H0(C), so |H0(F)| = |H0(C)|. By inspection, every element of one layer of X-detectors at the bottom of the fault complex is in the image of g•; those at the top are in the same equivalence class. Hence |imH0(g)| = |H0(F)| = |H0(C)|, so |H1(F ′)| = |H1(cone(g))| = |H1(F)| + 2|C0| -|H0(C)|. We therefore add 2mX -rX new basis elements to H1(F), where mX = |C0| is the number of X checks in C• and rX = |H0(C)| is the number of redundant X checks in C•. All new elements of H1(F ′) correspond to logical faults which enter from a timelike boundary on the top or bottom; mX -rX of these can be immediately annihilated by spacelike errors on the boundary, or by spacelike errors on further layers. We illustrate this in Fig. 7, where incoming X check errors turn on X detectors, which can then be immediately switched off again by Z data qubit Pauli errors. The remaining mX are strings of X check failures running from the bottom to the top boundary, illustrated in Fig. 8. In both cases, all new logical faults must have timelike support on the new checks at the boundary, and these cannot be cleaned from the boundary into the bulk. We can similarly calculate that |H0(F ′)| = |H0(F)| + 0 -|imH0(g)| -0 = |H0(F)| -|H0(F)| = 0. For the dual version, where R• = Fl 2 →Fl-1 2 , we instead perform the mapping cone on the cochain complex, and add 2mZ -rZ new elements to H2(F). Additionally, H3(F) becomes 0. In summary, when the initialisation and measurement is in the X basis the spacetime volume for the idling operation now contains: • All Z timelike logical faults, extending through the volume. • Those X timelike logical faults with check errors in H2(C) which are not equivalent to spacelike faults. • All Z spacelike logical faults. 65 Figure 6: A fault complex for a unit cell of the 2D surface code, starting and terminating at X-type boundaries, but with new X checks to allow Z timelike errors to enter and exit the volume. 66 Figure 7: The fault complex from Fig. 6, where a Z timelike fault enters the fault complex and is immediately annihilated by a spacelike fault. The highlighted fault locations are shown in green. 67 Figure 8: The fault complex from Fig. 6, where a Z timelike fault extends from one timelike boundary to the other. The highlighted fault locations are shown in green. 68 • No X spacelike logical faults. and when it is in the Z basis the volume is the same but with the X and Z logical fault characterisation inverted. Therefore if we use both of these fault complexes then, combined, we account for all possible logical faults, as the code (and fault complex) is CSS. We wish to describe complicated surgery protocols, rather than the idling state. Now, a certain fault complex for the idling state is sufficient to study certain stability experiments, but insufficient for the description of surgery protocols involving many logical measurements. For this the fault complex must describe the actual code deformation taking place. This can also be accomplished using mapping cones, as we shall demonstrate presently. A.2 Full block reading We are now ready to prove the fault-distance of full block reading. In this case, we start with taking E• = Fc 2 ⊗C• as our initial CSS code in the idling state, so the initial fault complex is F• = (R ⊗E)•. We then consider the two cases, one where the idling volume starts and ends with Z basis measurements, and the other with X basis measurements, including the entrypoints for timelike errors, so that we acquire two different fault complexes. For convenience we label them both F ′ •, and the difference will be made clear by dividing into the case for X-type boundaries and Z-type boundaries. In each case, we are applying mapping cones to the idling volume. To perform a full block reading in the Z basis, measuring Z logical operators, the mapping cone is on the chain complex, and in the X basis the mapping cone is on the cochain complex. In the Z basis, the mapping cone on the fault complex is taken from the chain map f• with components: 0 C2 C1 C0 F ′ 3 F ′ 2 F ′ 1 F ′ 0 f3 f2 f1 f0 where the components map the auxiliary code C• to the codeblocks in E• to be measured, at a primal layer. In other words, in the case where we start and finish in the X basis we 69 now have the fault complex E0 (Bottom X check faults) E0 (Top X check faults) R1 ⊗E1 (Old X Pauli faults) R1 ⊗E0 (Old X check faults) R1 ⊗E2 (Old Z detectors) R0 ⊗E0 (Old X detectors) R0 ⊗E2 (Old Z check faults) R0 ⊗E1 (Old Z Pauli faults) C2 (New Z detectors) C1 (New Z check faults) C0 (New Z Pauli faults) where the copies of E0 at the top of the diagram are to add timelike entrypoints to the spacetime volume, and the C• at the bottom performs the full block reading. We can now flip the boundary conditions, so that R• = Fl 2 →Fl-1 2 , and flip the additional timelike entrypoints accordingly, but still perform the same full block reading, 70 to acquire the fault complex E2 (Bottom Z check faults) E2 (Top Z check faults) R1 ⊗E1 (Old X Pauli faults) R1 ⊗E0 (Old X check faults) R1 ⊗E2 (Old Z detectors) R0 ⊗E0 (Old X detectors) R0 ⊗E2 (Old Z check faults) R0 ⊗E1 (Old Z Pauli faults) C2 (New Z detectors) C1 (New Z check faults) C0 (New Z Pauli faults) Due to the change in boundary conditions, each of these fault complexes has different logical faults, and we must account for them both. There are some important consequences of using the fault complex formalism rather than the phenomenological circuit-based model. It dictates that each fault location in the fault complex is a qubit initialised in the |+⟩state and then measured out in the X basis, with the entangling CZ gates performing space- and time-like Hadamards to exchange error types between layers. As a consequence, unlike in the circuit-based model, there are not separate timesteps on which the new ancilla data qubits are initialised in |+⟩and measured out in the X basis, as this is included in the fault complex layers. So, in the circuit model, one would initialise the new data qubits in |+⟩, then measure all checks for 1 round, then measure out the new data qubits in the X basis, which would generally be considered 3 timesteps: one each for the 'merge', measure, and 'split' steps. In the fault complex picture, this all happens on 1 timestep. This is why there is only one 'copy' of C• used for the auxiliary system in the mapping cone. Additionally, the layers of Z and X checks are alternating, rather than occurring on the same timestep as in the circuit-based model. When a surgery protocol only uses 1 round of Z measurements, as in full block reading, the fault complex does not introduce X checks connections to the new data qubits at all. There are also no new fault locations for X Pauli data qubit errors on the auxiliary system, as they would either act on data qubits in |+⟩or before the qubits are measured out in the X basis. 71 Remark A.1 (Multiple measurement rounds). In Section A.4 below we see cases in which the number of measurement rounds required is greater than 1, in which case there are more than one 'copy' of C• employed, and there are deformed X checks and new fault locations for X Pauli data qubit errors. Remark A.2 (Cycle detectors). For all the full block reading procedures we do not require any cycle checks, or detectors related to those checks. In Thm. 3.12 we have imposed a condition that no measured logical Paulis should be products of any others, i.e. the proof requires that the set of measured logical Paulis is minimal for the group it generates. This condition can be removed if one uses detectors for the cycles in the hypergraph: these are not cycle checks, merely detectors inferred from the initial state preparation and final measurement of the new data qubits, so the code will remain LDPC, but the detector matrix may not unless the cycle basis is sparse. Proposition 3.9. Let Q, Q′, Q′′,... be a set of identical CSS LDPC codeblocks with distance d, and let P be a pattern matrix determining a full block reading. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the full block reading procedure. Then full block reading performs logical measurement in O(1) rounds of syndrome measurements with fault-distance d. Proof. We treat the case of X-type boundary conditions first. X-type boundary conditions. By the Snake Lemma, we can compute the following. |H1(cone(f))| = |H1(F ′)| -|imH1(f)| + |H0(C)| -|imH0(f)| = |H1(F ′)| -|H1(C)| + |H0(C)| -0 = |H1(F ′)| -|H1(C)| + |H0(C)|. Applying the same argument to H2(cone(f)), we have: • |H1(cone(f))| = |H1(F ′)| -|H1(C)| + |H0(C)|. • |H2(cone(f))| = |H2(F ′)| -|H2(C)|. We first discuss the negative contributions to the classes of logical faults. The negative contribution to Z logical faults (faults that commute with all X detectors) appears because |H1(C)| logical measurements have occurred in the Z basis, and so there are |H1(C)| spacelike Z errors which no longer affect the logical state. The negative contribution to X logical faults (faults that commute with all Z detectors) appears because the full block reading introduces Z detectors between codeblocks, which require timelike Z check errors to now be in ker HZ of the auxiliary code in order to be undetected. As the space distance of the code and deformed code are each at least d, and there are no new equivalence classes of logical faults, we cannot clean any logical fault with these boundary conditions to have weight below d. Z-type boundary conditions. In this case the equivalence classes of logical faults can be computed for H1(cone(f)) as |H1(cone(f))| = |H1(F ′)| -|imH1(f)| + |H0(C)| -|imH0(f)| = |H1(F ′)| -0 + |H0(C)| -0 = |H1(F ′)| + |H0(C)|. Applying the same argument to H2(cone(f)) we have: 72 • |H1(cone(f))| = |H1(F ′)| + |H0(C)|. • |H2(cone(f))| = |H2(F ′)| + |H1(C)| -|H2(C)|. where H1(F ′) and H2(F ′) are equivalence classes already present before the full block reading, and H0(C) and H1(C) are new. Again we have a negative contribution of |H2(C)| to the classes of Z check faults. Note that some of the contributions to equivalence classes appear in both boundary conditions. For the new X logical faults, it is straightforward that H1(C) are equivalence classes of undetectable Z check failures on the auxiliary code, which will flip the logical measurement outcomes of the full block reading. For the new Z logical faults, the equivalence classes given by H0(C) are spacetime logical faults which begin as Z Pauli errors on the auxiliary code, and then move into the original spacetime volume and become timelike errors to then exit from the start or end boundaries. Because the Z Pauli errors are not in im(HX), they cannot be cleaned into the original spacetime volume and are genuinely new classes of logical faults. They are also not products of space and time logical faults. To prove the fault distance of a single full block reading, where there are at least d rounds of measurement before and after the block reading but only 1 round used for the block reading itself, it is sufficient to show that every element of these equivalence classes, and their combinations, must have weight at least d. Any timelike logical fault in H1(F ′) or H2(F ′) must originate at the boundaries, and therefore must have at least weight d to reach, and therefore affect, the protocol. They cannot be cleaned away from the boundary by spacetime stabilisers. Timelike errors which originate at the boundary must extend for d rounds to affect the computation. Logical faults from the positive contribution of H0(C) must exit through one of the boundaries of the spacetime volume, and hence must have weights at least d. Spacelike logical faults in H1(F ′) and H2(F ′) must have weight at least d because the original code and the deformed code during block reading both have distance at least d. Adding spacetime stabilisers to these spacelike logical faults can only increase the weights, because whatever qubit faults are removed by cleaning are merely moved to another round in the volume due to the structure of the repetition code R•. Lastly, logical faults in H1(C), which are new check errors on the auxiliary system, must have weight at least d as timelike errors. Applying spacetime stabilisers to clean these into the original code is possible, but the corresponding spacetime fault must act on at least d data qubits due to the 1-to-1 structure of the full block reading, corresponding to a spacelike logical fault, followed by some timelike errors, and then the same spacelike logical fault, and so all such logical faults must have weight at least d. Evidently a dualised proof would apply were the full block reading to be applied in the X basis instead. Remark A.3. As full block reading is only applied for 1 round of measurements, these new classes of spacetime faults corresponding to H0(C) can be eliminated by adding an extra C-1 term to C•, which adds detectors reconstructed from the initialisation and measurements of the ancilla system in the X-basis. Adding an extra C-1 term cannot make the deformed code non-LDPC, but can make the detectors high weight. As we shall presently show, these new classes of spacetime faults are not problematic anyway. 73 We can then consider the case where multiple full block readings are performed in a row, but with only O(1) rounds between them, and still leaving O(d) rounds of measurement before and after the procedure. In this case all the previous calculations apply, by applying the mapping cone to the chain or cochain complex versions of the fault complex. However, the analysis of logical faults requires two more ingredients. First, in the case where only one block reading is performed in the spacetime volume, the elements of H0(C) must exit through the boundaries of the volume. When there are multiple block readings performed in close proximity, however, these elements can connect and form noncontractible closed curves within the spacetime volume, with low fault weight. These noncontractible closed curves appear only when there is a full block reading whose logical measurements are a product of others performed in the same spacetime volume. Second, we must consider the compacted code of the spacetime volume, see Definition 4.4. Lemma A.4. Let Ξ be a set of η full block reading procedures of Z-type such that no block reading procedure in Ξ is a product of any others. Then the compacted code CC• has distance d. Proof. This is identical to the proof of code distance of the code (C ⊗R)•, where R• is a full-rank matrix, see Lemma 3.8. Theorem 3.12. Let Ξ be a set of η full block reading procedures of Z-type such that no block reading procedure in Ξ is a product of any others. Then all η logical measurement rounds of full block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the full set of procedures. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. Say that the η block readings being performed are each defined by a bitstring of length c, and the set of bitstrings is divided into sets of size ηZ and ηX for Z and X type block readings. Each bitstring contains a 1 in a location if that codeblock is involved in the block reading specified by the bitstring, and a 0 otherwise. Whether the block readings are performed simultaneously or spread across multiple rounds, we can describe the block readings being performed by a pair of pattern matrices PZ and PX over F2, with c columns and ηZ and ηX rows respectively. Note that, unlike when performing them simultaneously, these matrices do not need to form a pattern complex. Assume that there is no block reading performed in the spacetime volume whose logical measurements are products of others. Then PZ and PX are each full-rank matrices. Any logical fault composed of Z Pauli errors in H0(C) on a Z-type block reading must extend to timelike X check faults on all blocks in that row of PZ. To prevent extending all the way to the boundaries of the volume, such timelike faults must reach Z Pauli errors in H0(C) on a different row; these must subsequently extend onto all blocks in that row of PZ. Hence in order to form a noncontractible closed curve inside the volume the pattern matrix PZ must not be full rank, and so there are some block readings which are products of previous ones. Dualising the argument gives the same result for X Pauli errors in H2(C) on an X-type block reading. All other timelike logical faults follow the same arguments as before: they must either extend to a boundary or have distance d because of the new metachecks if they are timelike. 74 For spacelike errors, we must be more careful, as spacelike errors in the different deformed codes during the spacetime volume can be cleaned to have overlapping spacelike support. In particular, one can take products of spacelike logical faults in each deformed code, or the undeformed code. Before applying spacetime stabilisers, such a product is a set of disjoint spacelike logical faults separated by some number of timesteps, which in the worst case is constant. Upon multiplying by spacetime stabilisers, such spacelike logical faults can have support that is cleaned to the same timestep and hence cancel, introducing some check errors between the timesteps. Say that we have a set of θ spacelike Z logical faults Λ1, Λ2, · · · , Λθ at different logical timesteps, acting on different logical qubits. These could each be separated by a constant number of timesteps, and induce a constant number of check errors when partially cleaned by a spacetime stabiliser S. Thus, if the spacelike component of the product SΛ1Λ2 · · · Λθ has weight below d the fault-distance may fall below d. We now consider the compacted code CC• of the spacetime volume. Each logical operator Λi in each deformed code is guaranteed to be a logical operator of the CC•. Let Λ′ i be the logical operator Λi viewed in CC•. We can hence consider the product | Q i Λ′ i|, which is the minimal weight of the spacelike component of SΛ1Λ2 · · · Λθ. As CC• has distance d, | Q i Λ′ i| ≥d for any choice of logical operators Λi in any deformed codes. Then, | Q i SΛi| ≥d. For spacelike X logical operators, the argument is similar. The X-distance of the compacted code CC• is at least d, and so cleaning X logical operators to have overlapping spacelike components can never reduce the fault weight below d. For full block reading of CSS-type, we require an extension of the previous lemma concerning the compacted code. Lemma A.5. If the hypergraph surgeries are CSS-type full block readings such that no logical measurement is the product of any other then the compacted code CC• has distance d. Proof. This is identical to the proof of code distance of the code (C ⊗R)•, where each pattern matrix in the complex R• is a full-rank matrix, see Lemma 3.8. Lemma 3.16. Let Ξ be a set of η full block reading procedures of CSS-type such that no block reading procedure in Ξ is a product of any others, and each of the measurements commute. Then all η logical measurement rounds of full block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the full set of procedures. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. In this case we again take iterative mapping cones in the same way. In the worst case, where there is short timelike length between full block readings, the cones are applied to only two layers in the fault complex: Z-type measurements to a layer with X detectors (corresponding to a basis element of R0) and X-type measurements to a close layer with Z detectors (a basis element of R1). By virtue of the mapping cone construction, the fault complex commutes as a chain complex. We then use identical arguments to the proof of Z-type full block reading, where this time the constant number of layers between measurements is reduced to 0. As the proof of fault-distance is independent of this number of rounds, and the compacted code has 75 distance d, there are no spacelike logical faults which can be cleaned to have fault weight with spacelike component lower than d. A.3 Partial block reading with high-distance subcodes Proposition 4.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let A• be a subcode with distance d, with chain maps from A•, or a thickened version thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code, which has code distance at least d. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the partial block reading procedure. Then partial block reading performs logical measurement in O(1) rounds of syndrome measurements with fault-distance d. Proof. We can rerun the arguments for Prop. 3.9 but where A• ̸= C•. A• is a subcode of C• or can be a subcode thickened up to d times (in the space direction). The thickening does not affect the homology of A•, or the maps on homology Hi(f) into C•. In partial block reading with high-distance subcodes, unlike full block reading, we include additional X-detectors given by the initialisation and measurement of the new qubits in the X-basis. These give us detectors corresponding to a basis of cycles in the hypergraph; however, there is no guarantee that this basis is sparse. This does not affect the LDPC property of the code, but does affect the size of the detectors when they are used for decoding. In other words, the deformed code is guaranteed to be LDPC, but the fault complex is not. Now, for a Z-type partial block reading the fault complex is given by cone(f•), where f• has components 0 A2 A1 A0 A-1 F ′ 3 F ′ 2 F ′ 1 F ′ 0 0 0 f2 f1 f0 0 with im(f•) having support on a single primal layer, across potentially multiple codeblocks, and A-1 is a vector space with differential such that H0(A) = 0 to fix the cycles. For the X-type boundary conditions we then have |H1(cone(f))| = |H1(F ′)| -|imH1(f)| + |H0(A)| -|imH0(f)| = |H1(F ′)| -|H1(A)| + 0 -0 = |H1(F ′)| -|H1(A)|. And |H2(cone(f))| = |H2(F ′)| -|imH2(f)| + |H1(A)| -|imH1(f)| = |H2(F ′)| -|H2(A)| + |H1(A)| -|H1(A)| = |H2(F ′)| -|H2(A)| = |H2(cone(f)| = |H2(F ′)| -|H2(A)|. • |H1(cone(f))| = |H1(F ′)| -|H1(A)|. • |H2(cone(f))| = |H2(F ′)| -|H2(A)|. For the Z-type boundary conditions, the calculation is the same except |imH1(f)| = 0, so: 76 • |H1(cone(f))| = |H1(F ′)|, • |H2(cone(f))| = |H2(F ′)| + |H1(A)| -|H2(A)|. Hence we can categorise the logical faults in an identical way to previously and find that they must all either extend from a boundary of the spacetime volume, or have spacelike weight at least d, or have timelike weight at least d, and that spacetime stabilisers cannot clean these faults to have weight lower than d by virtue of the repetition code structure of R•. Because we added the X-detectors, which are present to allow the space distance of the deformed code to be at least d, we also eliminate the spacetime faults which were present for full-block reading and corresponded to contributions of H0(C). Now, H0(A) = 0 so there are no such contributions, so we need not be concerned about noncontractible loops in the spacetime volume. Theorem 4.5. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let Ξ be a set of Z-type partial block readings {Partial1, Partial2, Partial3, ..., Partialη} where each subcode has distance d and the compacted code has distance d. Then all η logical measurement rounds of partial block reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the set of logical measurements. The procedure has phenomenological fault-distance d, for any η ∈N. Proof. All logical faults must either extend from a boundary or have distance d because of the new metachecks if they are timelike. For the spacelike faults, we can apply the same argument as for full block reading. Each logical operator Λi in each deformed code is guaranteed to be a logical operator of the compacted code CC•. Let Λ′ i be the logical operator Λi viewed in CC•. We can hence consider the product | Q i Λ′ i|, which is the minimal weight of the spacelike component of SΛ1Λ2 · · · Λθ. As CC• has distance d, | Q i Λ′ i| ≥d for any choice of logical operators Λi in any deformed codes. Then, | Q i SΛi| ≥d. For spacelike X logical operators, the argument is similar but more straightforward. The X-distance of the compacted code CC• must always be at least d by Lemma 2.16, and so cleaning X logical operators to have overlapping spacelike components can never reduce the fault weight below d. A.4 Partial block reading with low-distance subcodes Consider a partial block reading, where the subcode A• for the partial block reading has distance less than d. For all our descriptions here we assume the block reading is measuring in the Z basis, and then one can dualise to acquire the corresponding construction to measure in the X basis. We also presume that the auxiliary hypergraph for a Z-type partial block reading has X-checks to gauge-fix cycles, and the same for X-type partial block readings with Z checks. These are genuine checks, and not just detectors inferred from the initial and final measurements of the new qubits. In terms of the initial auxiliary system, this still appears as adding an A-1 term to A• such that H0(A) = 0, A• = A2 A1 A0 A-1 but this is not yet the complex which we will perform the mapping cone with. 77 As before, we ignore the thickening of A• in space, as one can compute that this leaves the homology and categorisation of logical fault equivalence classes unaffected, and is necessary only to preserve the spatial distance of the deformed code. However, as the distance of A• is now less than d, we must thicken A• in time to make a fault complex K• corresponding to running the measurement procedure for more than 1 round. Let dA be the distance of A• and let dA = 1 αd for d the code distance of C•. The fault complex K• corresponds to taking (P(α) ⊗A)•, where P(α)• = P(α)1 P(α)0 = Fα-1 2 Fα 2 is dual to the full-rank repetition code with length α, and has the differential matrix being the incidence matrix of the path graph with α vertices. Explicitly K• is the chain complex: P(α)1 ⊗A1 P(α)1 ⊗A0 P(α)1 ⊗A-1 P(α)1 ⊗A2 P(α)0 ⊗A-1 P(α)0 ⊗A2 P(α)0 ⊗A1 P(α)0 ⊗A0 Observe that A• = K• when α = 1, recovering the case where we measure for only one round. Now, for a Z-type partial block reading the fault complex is given by cone(f•), where f• has the form: P(α)1 ⊗A1 P(α)1 ⊗A0 P(α)1 ⊗A-1 P(α)1 ⊗A2 P(α)0 ⊗A-1 P(α)0 ⊗A2 P(α)0 ⊗A1 P(α)0 ⊗A0 F ′ 3 F ′ 2 F ′ 1 F ′ 0 0 0 0 f2 f1 f0 There are also maps from the top terms in the diagram - P(α)1 ⊗A1, P(α)1 ⊗A0 and P(α)1 ⊗A-1 - but these are zero maps, so we abuse notation slightly by labelling the maps from the bottom layer f2, f1, f0. As before, the only nonzero maps in f• are into primal layers. The mapping cone then yields the fault complex cone(f•) for the measurement, K2 K1 K0 K-1 F ′ 3 F ′ 2 F ′ 1 F ′ 0 where the K3 term, equal to P(α)1 ⊗A2, is omitted as it has no physical meaning, being a \"detector on detectors\". Before continuing further, let us make some orienting remarks regarding the terms and maps in this mapping cone. Below we expand out to the full fault complex cone(f•) for 78 the case where the spacetime volume is initialised and measured out in the X basis. As before, the P(α)1 ⊗A2 term is omitted. The diagram clearly commutes, by inheriting the coherent monic of the subcode A• into the original codeblocks E•. P(α)1 ⊗A1 (New Z detectors) P(α)1 ⊗A0 (New X Pauli faults) P(α)1 ⊗A-1 (New X check faults) P(α)0 ⊗A-1 (New X detectors) P(α)0 ⊗A2 (New Z detectors) P(α)0 ⊗A1 (New Z check faults) P(α)0 ⊗A0 (New Z Pauli faults) R0 ⊗E2 (Old Z check faults) R0 ⊗E1 (Old Z Pauli faults) R1 ⊗E2 (Old Z detectors) R0 ⊗E0 (Old X detectors) R1 ⊗E1 (Old X Pauli faults) R1 ⊗E0 (Old X check faults) E0 (Bottom X checks) E0 (Top X checks) f2 f1 f0 • All terms in (R ⊗E)• are those from the fault complex of the idling operation. • The new E0 terms at the bottom are to add timelike entrypoints to the volume. • P(α)0 ⊗A-1 is a set of X detectors which arise from cycle checks throughout the surgery procedure. • P(α)0 ⊗A0 is a set of fault locations for new qubits which can undergo Z Pauli errors. 79 • P(α)1 ⊗A-1 is a set of new X checks corresponding to cycles in the hypergraph, which can undergo measurement errors. • P(α)0 ⊗A1 is a set of new Z checks, corresponding to vertices in the hypergraph, which can undergo measurement errors. • P(α)1 ⊗A0 is a set of fault locations for new qubits which can undergo X Pauli errors. Note that when the block reading was performed for only 1 round these fault locations were not present because of the immediately adjacent initialisations in |+⟩ and measurements in the X-basis. • P(α)0 ⊗A2 is a set of Z detectors which are metachecks due to the block reading. • P(α)1 ⊗A1 is a set of Z detectors due to the vertex checks throughout the surgery procedure. To acquire the opposite spacetime volume, where it is initialised and measured in the Z basis, the R• is dualised, and F ′ has extra E2 components, rather than E0. This is identical to the full block reading case. We can hence rerun a similar proof as for the case of full block-reading, and acquire the homology spaces from the Snake Lemma. We can then prove the following. Proposition 6.1. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance d, and let A• be a subcode with distance dA = 1 αd, with chain maps from A•, or a thickened version thereof, to Q, Q′, Q′′.... Let D• = cone(h•) be the deformed code with code distance d and let H have a sparse cycle basis. Assume that there are d rounds of syndrome measurements on Q, Q′, Q′′... before and after the partial block reading procedure. First, ⌈α⌉rounds of syndrome measurements are necessary (but not sufficient) to perform partial block reading with fault-distance d. Then if either: (a) α ≤2 or (b) the initial codeblocks are each locally testable codes with soundness ρ ≥2n m , where n is the codeblock length and m the number of checks, then partial block reading performs logical measurement in ⌈α⌉rounds of syndrome measurements with fault-distance d, maintaining the LDPC property throughout. Proof. X-type boundary conditions. By the Snake Lemma, we can compute the following. • |H1(cone(f))| = |H1(F ′)| -|H1(A)|. • |H2(cone(f))| = |H2(F ′)| -|H2(A)|. In detail, using the K ̈unneth formula and the fact that |H1(P(α)) = 0| and |H0(P(α))| = 1, |H1(cone(f))| = |H1(F ′)| + |H0(K)| -|imH1(f)| -|imH0(f)| = |H1(F ′)| + |H0(A)| -|imH1(f)| -|imH0(f)| = |H1(F ′)| + 0 -|H1(A)| -0 = |H1(F ′)| -|H1(A)|. The other computation works similarly. Z-type boundary conditions. In this case the equivalence classes of logical faults can be computed as, • |H1(cone(f))| = |H1(F ′)|. 80 • |H2(cone(f))| = |H2(F ′)| + |H1(A)| -|H2(A)|. We thus have a similar categorisation of all the equivalence classes of logical faults to those in previous proofs: there are spacelike errors which must always have weight at least d by virtue of the deformed code having distance at least d, and there are timelike errors that extend from a boundary and so have weight at least d. However, there are also the timelike errors which correspond to the |H1(A)| term. As check errors in the auxiliary code, these timelike logical faults must always have weight at least d if they extend for ⌈α⌉rounds, because they must have weight at least 1 αd in each round. Should fewer than ⌈α⌉rounds be used then there always exists a logical fault with weight lower than d, proving the first part of the proposition. For the second part, we must be careful about the cleaning with spacetime stabilisers. Given such a timelike logical fault with weight at least d in the auxiliary code, at least dA on each layer, cleaning it entirely into the bulk can leave only 2dA qubit Pauli X errors, one set of size dA just before the surgery and one set of size dA just after, and a set of timelike errors connecting them. We must therefore ensure that this set of timelike errors has sufficent size to yield a high fault-distance. It is distributed over α layers, so we ensure that each layer of check errors contains at least ν = (d-2v) α faults, where v is the size of the set of spacelike errors at the bottom (or equivalently the top), and v ≥dA. When v ≥d/2, ν < 0 so we are done; this is always the case if α ≤2, i.e. dA ≥d/2. The rest of the proof therefore assumes dA ≤v < d/2. Note that ν < d α = dA. Because the minimal size of the bottom set of Pauli errors is dA, there are at least dA check errors in the next round if |HZv| ≥dA, where v is the qubit fault vector for the bottom set of Pauli errors. Now, assume that the HZ parity matrix of the original code has soundness ρC ≥n m as a classical code. This is guaranteed if the original code is quantum locally testable with soundness ρ ≥2n m by Lemma 2.12. We now show that d(v, C) ≥dA, where C = ker HZ. First, if |v + x| was lower than dA for any x ∈imH⊺ X then the distance of the subcode would be lower than dA, as the subcode has connections to all X checks incident to v, so multiplying v by x restricted to A1 would result in a nontrivial logical operator in A• with weight lower than dA. So we must now consider |v+x| for any x ∈ker HZ ⊺ X. As v < d/2, the maximal overlap of v with any other X logical in the original code is v. Hence |v + x| ≥|v| ∀x ∈ker HZ ⊺ X, and v ≥dA. As a consequence of the soundness being ρC ≥n m, |HZv| ≥m n ρCd(v, C) ≥d(v, C) ≥dA, so each layer between the top and the bottom Pauli X errors must have at least dA check errors. By the same argument, applying cleaning to only a portion of the checks in the auxiliary system will distribute the faults such that the portion in the original code and the portion on checks in the auxiliary system at each timestep must sum to dA, and so the fault distance is at least d. When performing many partial block readings close together in the same spacetime volume, we strengthen the condition to require soundness ρ ≥2n m regardless of whether α is above or below 2. Theorem 6.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and let Ξ be a set of partial block readings {Partial1, Partial2, Partial3, ..., Partialη} where each subcode has distance di ≥ 1 αi d for i ∈[η] and the deformed codes each have code distance d, with sparse cycle bases. Suppose the original codes have soundness ρ ≥2n m . Then all η logical measurement rounds of partial block reading can be performed sequentially using ⌈αi⌉rounds of syndrome measurement each, and using d rounds of padding 81 before and after the logical measurements. The procedure has phenomenological faultdistance d, for any η ∈N. Proof. By taking sequential mapping cones on the initial fault complex for the idling operation, one finds that the equivalence classes of logical faults are all either: • Timelike faults extending between boundaries. • Spacelike faults. • Timelike faults which act on the auxiliary systems, and cannot be cleaned to have weight below d in the original code. Timelike logical faults in the auxiliary system and between boundaries must still have weight at least d. Cleaning spacelike logical faults Λ1, Λ2, · · · , Λθ to have some shared spacelike support must induce at least a proportional weight in check faults, as the initial code has sufficient soundness, so the product SΛ1Λ2 · · · Λθ cannot fall below weight d. Note that the condition on the distance of the compacted code CC• is unnecessary in this case. Therefore the fault-distance of the procedure is at least d. A.5 Fast hypergraph surgery Theorem 5.3. Let Q, Q′, Q′′... be a set of CSS LDPC codeblocks with distance at least d, and let Ξ be a set of Z-type hypergraph surgeries {Hyper1, Hyper2, Hyper3, ..., Hyperη} such that: • Each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance d, with sparse cycle bases. • The compacted code has distance at least d. Then all η logical measurement rounds of generalised hypergraph reading can be performed sequentially using O(1) rounds of syndrome measurement each, and using d rounds of padding before and after the set of logical measurements. The procedure has phenomenological fault-distance d, for any η ∈N. Proof. This proof proceeds in a largely similar manner to those of Appendix A.3. For a single hypergraph surgery with an ancillary system A• we construct the same fault complex 0 A2 A1 A0 A-1 F ′ 3 F ′ 2 F ′ 1 F ′ 0 0 0 f2 f1 f0 0 but where A• is now an arbitrary code such that f• is non-overlapping on vertices and H0(A) = 0. X-type boundary conditions. |H1(cone(f))| = |H1(F ′)| -|imH1(f)| + |H0(A)| -|imH0(f)| = |H1(F ′)| -|imH1(f)|, 82 and |H2(cone(f))| = |H2(F ′)| -|imH2(f)| + |H1(A)| -|imH1(f)| = |H2(F ′)| -|imH2(f)| + |H1(A)| -|H1(A)| = |H2(F ′)| -|imH2(f)|. In summary we have: • |H1(cone(f))| = |H1(F ′)| -|imH1(f)|. • |H2(cone(f))| = |H2(F ′)| -|imH2(f)|. Z-type boundary conditions. Similar calculations apply but |imH2(f)| = |imH1(f)| = 0. • |H1(cone(f))| = |H1(F ′)|, • |H2(cone(f))| = |H2(F ′)| + |H1(A)| -|imH2(f)|. We can then apply such mapping cones sequentially, and so all logical faults are categorised into equivalence classes with the following representatives: • Logical faults on checks in the auxiliary system. • Logical faults on data qubits in the original or deformed codes. • Logical faults on checks extending through the spacetime volume between boundaries. In order, • Because each auxiliary complex (A•)1, (A•)2, (A•)3, ..., (A•)η has 1-cosystolic distance d, the weight of a logical fault on checks in the auxiliary system must be at least d. • Because the maps between the auxiliary systems and original codes are all nonoverlapping on vertices, the weight of such a logical fault when cleaned into the original code must be at least d to flip d checks in the auxiliary system. • Because the compacted code has distance at least d, any spacelike fault must have weight at least d. This accounts for the fact that spacelike logical operators can cleaned by spacetime stabilisers to cancel their shared spacelike support. • All other timelike faults must extend from a timelike boundary, so to affect the protocol must have weight at least d. Remark A.6. If the chain maps are overlapping on vertices (i.e. they do not satisfy Definition 5.1) then the argument breaks down, as a timelike logical fault in the auxiliary system could be cleaned to a smaller logical fault in the original code, and so additional syndrome rounds are required to increase the weights of these faults. In the extreme case, where multiple auxiliary systems measure a single logical representative, this coincides with Theorem 1.6, which shows that such a measurement procedure will always require Ω(d) rounds to preserve the fault distance. 83 B Non-CSS measurements by full block reading In this appendix we provide a more detailed explanation of how block reading extends to more general logical Pauli measurements on codes that possess transversal gates. B.1 Y measurements We can attach an ancillary system of the following form to any CSS code C: X Y Z Q HZ HX I I I H⊺ Z H⊺ X where the data qubits Q are in the original code, and the r.h.s. scalable Tanner graph is the ancillary system. The checks must always commute by inspection of the diagram. However, the properties of this combined code are complicated. Evidently, none of the Z or X logicals are measured out, as there are no new Z or X checks. Definition B.1. Let C have a basis of representatives such that supp(Y i) = supp(Xi) = supp(Zi), ∀i ∈[k]. Then we say C has matching representatives. Let C be self-dual, i.e. HZ = HX. If C has matching representatives then Y i is measured out. As Y i ∈ker HZ = ker HX, cleaning by the new Y checks PY in bijection with supp(Y i) gives PY Y i = 0 on both the new sets of data qubits. If the code does not have matching representatives then the cleaning argument does not work, and so the ancillary system may not measure the logical qubits. Note that to have supp(Xi) = supp(Zi), |supp(Y i)| = 1 mod 2, i.e. it must have odd support, in order for Xi to anticommute with Zi. If C has matching representatives, all logical qubits will be measured out in the Y basis. There are no new Z logicals, as any Z logical on the top right new data qubits must be in ker(H⊺ Z) so be products of old Z checks, and any Z operators on the bottom right new data qubits won't commute with the old X checks. The inverse is true for new X checks. Example B.2. 2D colour codes are self-dual and have Y logicals composed of X and Z logicals on the same support [124]. Remark B.3. If C is self-dual but there exists a product of Pauli Y operators on data qubits, which is a composition of Zi and Xj with supp(Zi) = supp(Xj), but i ̸= j so the Y operator is actually a representative of Zi ⊗Xj, then the Y ancillary system will still send this Y operator to zero when cleaned, but the measurement is a Zi ⊗Xj measurement instead. For the measurement protocol, assume C is self-dual with matching representatives. First initialise new data qubits in the top(bottom) right set in the |0⟩(|+⟩) states respectively. Then measure all stabilisers. To commute with all Y logicals being measured, each 84 new data qubit must be connected to an even number of new checks in bijection with each Y logical, and so X(Z) errors on top right(bottom right) qubits will not affect the logical measurement outcome. As the code is self-dual, products of Z and X checks with the same support are mapped to a subset of new Y checks, such that any nontrivial Y measurement error must be in ker(HZ) (H⊺ X) = ker(HX) (H⊺ Z), and hence the logical measurement has some protection against measurement faults. Lemma B.4. Let C be a self-dual CSS code with matching representatives. Then C admits a transversal S gate, acting as S⊗k on the logical space up to a Pauli correction. Proof. From [34, Prop. 9], C admits a transversal S gate up to a Pauli correction if: • u · v = 0 mod 2 for all X stabilisers u, v. • For any X logical operator u and X stabiliser v, u · v = 0 mod 2. As C is self-dual the first condition is satisfied. From the matching representatives property, any Xi logical operator u has the same support as its matching Zi operator, and Zi must commute with all stabilisers hence u · v = 0 mod 2. Therefore the procedure is not very useful, because the codeblock can have its measurement basis permuted by transversal S and H and so Y measurements are unnecessary. The same argument applies to performing block reading of multiple codeblocks in products containing Y terms. We do not know if partial block reading for non-CSS measurements, where the subcode is a stabiliser code but not CSS, can allow for fast measurements which cannot be performed using transversal gates to permute bases. B.2 X ⊗Z measurements Given a CSS code C, let D be the dual of that code, i.e. HD X = HC Z and HD Z = HC X. Introduce the following ancillary system: X Z HX HZ [0, H⊺ X] I [0, I] I HZ X [0, HX] [HZ, 0] [I, 0] [I, 0] I where C is on the l.h.s. and D on the r.h.s. The top right and middle checks are now of mixed-type, checking stabilisers with both X and Z terms. One can check by symplectic multiplication that the above Tanner graph commutes. We now also have meta-checks which are of 'mixed-type', using both X and Z checks, indicated by the pink diamond. This is a Z⊗Z measurement by full block reading, but conjugated by single-qubit Cliffords. However, because the codeblocks C and D are dual, this could equally be performed by using a modified version of homomorphic measurements but with transversal CX and CZ gates [24]. 85 Figure 9: Venn diagram indicating the overlapping supports of a logical operator ΛZ and sets of checks κλ (magenta) and v ∩U (dashed). C Proofs concerning modular expansion In this appendix we provide technical proofs related to properties of modular expansion used in the main text. Theorem 7.7 (Distance preservation). Let {Zi}i∈I be a set of logical representatives (trivial or nontrivial) on a distance d CSS code Q with supports ξi = supp(Zi). Let H(V, E) be a hypergraph with vertices in V being Z checks, hyperedges in E data qubits and X gauge checks forming a cycle basis, such that H measures all the logical representatives {Zi}i∈I. Let fp : S i ξi →V be an injective port function such that U = imfp ⊂V, and fp(ξi) = Vi ⊂V, ∀i ∈I. Let Md(H) ≥1, where W is generated by fp(ξi). Then the distance of the deformed code Q ↔H is at least d. Proof. We show that for every nontrivial ΛZ logical operator in Q ↔H with some support in S i Vi, the weight of ΛZ cannot drop below d by multiplying with new Z checks (vertices) in H. Let v be an arbitrary subset of V, and define the stabiliser Z(G⊺v) to be the product of all checks in v. We must show that |ΛZ ◦Z(G⊺v)| ≥d. First, G⊺v ≥min(d, |κλ| -|v ∩κλ| + |v ∩U\\κλ| : λ ∈2I) by the definition of modular expansion. If min(d, |κλ| -|v ∩κλ| + |v ∩U\\κλ| : λ ∈2I) = d then we are immediately done, as no matter which data qubits in Q are turned off by Z(G⊺v), there are at least d new data qubits with a Z Pauli in the ancillary hypergraph. The other case is more difficult. If |G⊺v| ≥|κλ|-|v∩κλ|+|v∩U\\κλ| for some λ ∈2I then we can construct the diagram in Fig. 9. 86 The set in the dashed region is v ∩U, the top right circle is U, the bottom left circle is ΛZ and the magenta circle is κλ. We compute by inspection of the diagram that |ΛZ ◦Z(G⊺v)| ≥|τλ| + |σλ| + |Bλ| + |B′ λ| + |Rλ| + |κλ| -|v ∩κλ| + |v ∩U\\κλ|. Next, |κλ| -|v ∩κλ| + |v ∩U\\κλ| = |φλ| + |σλ| + |B′ λ| + |γλ| and |ΛZ ◦Zλ| = |Rλ| + |τλ| + |γλ| + |Bλ| + |φλ| ≥d where Zλ = Z(f-1κλ) acts as Z on the preimage of κλ in the original code. Zλ is a logical operator, but not necessarily a nontrivial one. As ΛZ is a nontrivial logical operator of the deformed code, |ΛZ ◦Zλ| ≥d. Plugging these together we find that |ΛZ ◦Z(G⊺v)| ≥2(|σλ| + |B′ λ|) + d ≥d. As the cycles are gauge-fixed and no Z logical can be reduced below weight d, the distance of Q ↔H is at least d. Theorem 7.8 (Thickening). Let H(V, E) be a hypergraph with modular expansion Mt(H). Let J L be the path graph with length L ≥ 1 Mt(H), i.e. L vertices and L -1 edges. Let HL := H□J L be the hypergraph of H thickened L times. Then HL has modular expansion Mt(HL) ≥1 for Ul= S i V l i at any level l∈{1, 2, · · · , L}, where V l i is the copy of Vi in the lth level of the thickened hypergraph. Proof. Let G : F2E →F2V be the incidence matrix of H, RL the incidence matrix of J L, and GL the incidence matrix of HL. Then GL = G ⊗IL In ⊗RL where n = |V|. We divide up the vertices of HL into sets Vlfor l∈[L], corresponding to each level of the thickened graph. Similarly, for a given level lwe have V l i and Ul= S i V l i . Letting vl∈Vlbe a set of vertices in level l, we have the vector v = (v1 v2 · · · vL)⊺∈ S lVlseen as a set of vertices in HL. Associated to this set we also have ul= vl∩Ul= v ∩Ul. Now, let r = arg minj min(t, |κj λ| -|uj ∩κj λ| + |uj\\κj λ| : λ ∈2I). Using the triangle inequality and the fact that LMt(H) ≥1, |G⊺ Lv| = |G⊺ L(v1 v2 · · · vL)⊺| = L X j=2 |vj-1 + vj| + L X j=1 |G⊺vj| ≥|vr + vl| + LMt(H) min(t, |κr λ| -|ur ∩κr λ| + |ur\\κr λ| : λ ∈2I) ≥|ur + ul| + min(t, |κr λ| -|ur ∩κr λ| + |ur\\κr λ| : λ ∈2I). Now consider the different cases. If min(·) = t then |ur + ul| + t ≥t and we are done. 87 If min(·) = |κr λ| -|ur ∩κr λ| + |ur\\κr λ| for some λ ∈2I then |ur + ul| + |κr λ| -|ur ∩κr λ| + |ur\\κr λ| = |ur ∩κr λ + ul∩κr λ| + |ur\\κr λ + ul\\κl λ| + |κl λ| -|ur ∩κr λ| + |ur\\κr λ| ≥|κl λ| -|ul∩κl λ| + |ul\\κl λ|, making use of the triangle inequality for the last line. Therefore |G⊺ Lv| ≥min(t, |κl λ| -|v ∩κl λ| + |v ∩Ul\\κl λ| : λ ∈2I) for any set of vertices v in HL, so HL has modular expansion Mt(HL) ≥1. D Circuit-equivalence of surgery and homomorphic measurement In this appendix we derive a circuit equivalence between generalized surgery measurement and homomorphic measurement. The following proofs use the so-called \"scalable\" ZXcalculus from Ref. [87]. This calculus has previously been used to describe LDPC surgeries in Ref. [64]. We assume familiarity with the ZX-calculus for this section; see Ref. [125] for a comprehensive introduction. In short, the scalable ZX-calculus (SZX-calculus) has thick wires indexed by a positive integer n, denoting a register of n qubits. Green and red spiders then act on the n qubits, with spider phases now represented by a vector v over C of length n, containing phases acting on each qubit. As in the conventional ZX-calculus, an all-zeros vector is omitted and the green or red spider left unlabelled. We also omit the label denoting the size of the qubit register. The SZX-calculus has other generators than green and red spiders. For example, we also have the matrix arrows A and B (6) where A and B are matrices over F2. These generators can be defined by their action on the Hilbert space. Suppose A ∈Fn×m 2 and B ∈Fp×q 2 . For a qubit state in the computational basis |x⟩with x ∈Fm 2 , the arrow pointing to the right in Eq. 6 corresponds to the linear map RA : |x⟩7→|Ax⟩. Meanwhile, the arrow pointing to the left corresponds to the linear map H⊗pRBT H⊗q = H⊗p |x⟩7→|BT x⟩ H⊗q. Stabiliser measurement on CSS codes can be neatly described using the SZX-calculus. Given a CSS code with n qubits and the parity-check matrices HX, HZ, the measurements can be described by H⊺ X HZ sZπ sXπ where sZπ is the vector of phases representing syndrome outcomes sZ = {0, 1}mZ, and the same for sXπ. In particular, when the input state is in the codespace, sZ = sX = 0 and the phase vectors can be omitted. The same applies for the case where the the initial state is not in the codespace but the measurements project it into the codespace. Commutation of stabilisers implies that H⊺ X HZ = H⊺ X HZ 88 and furthermore, HZ HZ = HZ with a similar rewrite for X-checks. Obviously such rewrites do not generally preserve fault-tolerance of the circuit: we are taking multiple rounds of syndrome measurement and collapsing them into one round, which will generally reduce the fault-distance of the spacetime code. Along with the merge and copy rules of the ZX-calculus we can now prove that surgery by an auxiliary system and homomorphic measurement using transversal gates are equivalent as circuits. Theorem 9.1. Let C• be a CSS code with a homomorphic measurement procotol specified by a chain map f• : A• →C•. Then the circuit corresponding to the homomorphic measurement can be rewritten to a surgery specified by cone(f•). Proof. We describe an Z-basis logical measurement; the same result for X measurements can be easily acquired similarly. The rewrites required are as follows. f⊺ 1 απ H′ Z (H′ X)⊺ H⊺ X HZ = f⊺ 1 απ (H′ X)⊺ H⊺ X HZ = f⊺ 1 απ (H′ X)⊺ H⊺ X HZ = f⊺ 1 απ (H′ X)⊺ H⊺ X HZ f⊺ 2 N⊺ The first SZX-diagram on the top left is depicting a homomorphic measurement. The original code is shown on the bottom wire, with parity-check matrices HZ, HX. An ancilla code is initialised in |0⟩⊗n′ then its stabiliser checks are measured. Then a set of transversal CNOTs defined by a chain map f• are applied, with the controls on the initial code and the targets on the ancilla code. Then the ancilla code is measured out in the Z basis, with the vector απ where α ∈Fn′ 2 determines the logical measurement outcome. Next, the |0⟩initialisations are commuted through the Z-check measurements, and then the merge rule is used to extend the check qubits of the X-check measurements to instead become data qubits of the auxiliary system, initialised in |+⟩mX and measured in the X-basis. We now see that the data qubits of the homomorphic measurement have become check outcomes of Z measurements, which include both the new data qubits of 89 the auxiliary system and data qubits of the original code. The outcomes απ of these Z measurements determine the logical measurement outcome. Last, we add X checks which gauge-fix the cycles (there are cases, such as in full-block reading and the CKBB scheme [52], where these are not necessary), with the check matrix N. We also deform the X checks of the original code to include qubits of the auxiliary system, such that these checks commute with the Z checks that determine the logical measurement outcome. The deformation of these checks is determined by the chain map f•. We can then do the opposite rewriting to convert surgeries to homomorphic measurements. Corollary 9.2. Let C• be a CSS code with a surgery protocol specified by cone(f•) for a chain map f• : A• →C•. Then the circuit corresponding to the surgery protocol can be rewritten to a homomomorphic measurement specified by f•. Proof. F ⊺ απ G⊺ H⊺ X HZ M⊺ = V E F ⊺ απ G⊺ H⊺ X HZ M⊺ V E = F ⊺ απ G⊺ H⊺ X HZ = F ⊺ απ G⊺ H⊺ X HZ H′ Z N⊺ V V E E Thus we see that the vertices V of the hypergraph in surgery, which are Z-checks, are rewritten to the data qubits in homomorphic measurement; the edges E of the hypergraph, which are data qubits, are rewritten to be X-check outcomes. The cycles vanish from the circuit entirely, which is because they become X-type meta-checks on the ancilla code. These relabellings correspond precisely to the degree-shifting of a mapping cone to recover the original chain map f•. 90 E Fast measurement requires multiple representatives In this appendix we provide a proof of our results lower bounding the spacetime volume of a general procedure to measure logical representatives in a quantum code. Proposition 9.3. Let Q be a quantum LDPC code. Let S be a set of logical operator representatives measured at the same time by one or more auxiliary system for T syndrome rounds. Assume that the correct logical measurement outcomes of S are not known in advance. Let f be the lowest weight data qubit fault in Q such that: • f flips at least one logical representative in S. • If f flips a logical representative in S then it flips all other representatives in the same equivalence class in S. Let w be the weight of f, and s be the syndrome weight of f in Q. Then the phenomenological fault-distance of the measurement protocol is upper bounded by 2w + Ts. Proof. The measurement scheme begins at time ti and ends at time to = ti + T, and measures all the logical operator representatives contained in S. Data qubit errors occur at integer timesteps and measurement errors occur at half-integer timesteps. To measure this operator by an auxiliary system, entangling gates are applied from each qubit in S to some other system Aux of checks and data qubits, such that the measurement outcomes of checks in Aux infer the logical measurement outcomes. The entangling gates are applied while measuring checks in Aux for T rounds before ceasing the application of entangling gates. In order to receive an incorrect logical measurement outcome without detecting an error it is sufficient to flip any of those logical operators for the duration of the logical measurement scheme, so long as there are no logical measurement outcomes between representatives which are 'mismatched', meaning that measurements of different representatives of the same logical operator yield different outcomes. Thus if any representative in S is flipped, all representatives in the same equivalence class in S must be flipped. Observe that as none of the correct logical measurement outcomes of S are known in advance (which would occur if Q was initialised in a particular known state with deterministic logical measurement outcomes) there are now no detectors which can detect the logical error. To flip some operators in the round at time ti, apply a Pauli operator fault f to Q with weight w satisfying the conditions specified in the Proposition. This set of faults is detectable by checks in the original code at timesteps ti + 1 2, ti + 3 2, · · · , so in order to conceal the fault we add s check errors at each timestep for the duration of the logical measurement, where s is the number of checks which f flips. In total there are Ts of these check faults. Lastly, apply an identical Pauli operator fault f after the logical measurement procedure concludes, at time to + 1. This is a logical fault of the measurement procedure with weight 2w + Ts. Were the code Q left in the idling operation instead, the above logical fault would be a spacetime stabiliser of the code, and so this logical fault weight can be vastly lower than the distance of Q depending on S, the time T and the syndrome weights s of operators. Note also that this is just an upper bound on the fault-distance: if the auxiliary system is poorly constructed, for example by connecting every qubit in S to a single new check 91 qubit, the fault-distance may be substantially lower due to check errors on the auxiliary system. Theorem 1.6. Any logical measurement protocol performed in o(d) rounds on a quantum LDPC code by an auxiliary system requires connections to more than one logical representative to maintain phenomenological fault distance Ω(d), unless the correct logical measurement outcome is known in advance. We prove this by demonstrating that otherwise there exists a weight |f| logical fault, where |f| ∈o(d). Proof. Assume w.l.o.g. that the measurement being performed is on a Z logical operator, as any Pauli measurement is locally equivalent to this measurement by conjugation of single-qubit Cliffords. If the logical operator representative has support v, the measurement being performed is of the operator N i∈v Zi. To measure this operator by an ancillary system, entangling gates are applied from each qubit i to some other system Aux of checks and data qubits, such that the measurement outcomes of checks in Aux infer the logical measurement outcome of N i∈v Zi. The measurement scheme starts at time ti and concludes at time to, with o(d) rounds between. Data qubit errors occur at integer timesteps and measurement errors occur at half-integer timesteps. Assume that the entangling gates are applied while measuring checks in Aux for o(d) rounds before ceasing the application of entangling gates. Regardless of the system of checks and other structure of Aux, there exists a low weight fault which flips the logical measurement outcome. Consider an arbitrary qubit j ∈v. At any time t ∈[ti, to] the application of a Pauli error Xt j, followed by measurement errors on all incident Y and Z checks at time t+ 1 2, and then a Pauli error Xt+1 j , is a local spacetime stabiliser of the original code, and therefore does not violate any detectors. However, at time t + 1 2 the outcome of measuring N i∈v Zi is flipped, as Xj anticommutes with Zj. This is then corrected by the subsequent Pauli error Xt+1 j , and so at time t + 3 2 the measurement of N i∈v Zi would return the correct outcome again. Hence a logical fault f on the logical measurement can be constructed by applying a Pauli error Xti j and measurement errors on all incident X and Y checks on qubit j from time ti+ 1 2 to time to+ 1 2, and a final Pauli error Xto+1 j . This is a spacetime stabiliser of the original code, as it is composed of local spacetime stabilisers at times t = ti, ti + 1, · · · , to, hence the fault does not violate any detectors. Each round of measurement of N i∈v Zi will be flipped by the original Xti j error, and the outcome of N i∈v Zi is then flipped back after the measurement procedure concludes. The fault f is then Xto+1 j Xti j S ti≤t≤to M t+ 1 2 j , where M t+ 1 2 j is a measurement fault on each Y or Z check incident to qubit j at time t + 1 2. Because the code is LDPC, the fault at each round has weight bounded above by a constant, where the precise weight depends on the number of Y and Z checks incident to qubit j, but this cannot increase the weight asymptotically. Because the logical measurement is performed for o(d) rounds, the weight |f| ∈o(d). Note that if the correct logical measurement outcome is known in advance, there is an additional detector at each round of N i∈v Zi and so the above argument does not 92 hold. Similarly, the reason the proof no longer applies when measuring multiple logical operator representatives is because there are additional detectors which appear because the measurements of each representative of the same logical operator must agree with each other to be correct; equivalently, these additional detectors correspond to sets of measurements whose correct outcomes are known in advance, as they are stabilisers of the original code. Both of the above proofs are insensitive to the structure of Aux, which could be a measurement hypergraph in surgery, an ancilla block in homomomorphic measurement or any other logical measurement gadget. 93"}}},{"rowIdx":90,"cells":{"id":{"kind":"string","value":"2510.14891"},"raw":{"kind":"string","value":"A Performance Portable Matrix Free Dense\nMTTKRP in GenTen*\nGabriel Kosmacher1, Eric T. Phipps2, Sivasankaran Rajamanickam2\nAbstract—We extend the GenTen tensor decomposi-\ntion package by introducing an accelerated dense ma-\ntricized tensor times Khatri-Rao product (MTTKRP),\nthe workhorse kernel for canonical polyadic (CP) tensor\ndecompositions, that is portable and performant on\nmodern CPU and GPU architectures. In contrast to\nthe state-of-the-art matrix multiply based MTTKRP\nkernels used by Tensor Toolbox, TensorLy, etc., that ex-\nplicitly form Khatri-Rao matrices, we develop a matrix-\nfree element-wise parallelization approach whose mem-\nory cost grows with the rank R like the sum of the\ntensor shape O(R(n+m+k)), compared to matrix-based\nmethods whose memory cost grows like the product of\nthe tensor shape O(R(mnk)). For the largest problem\nwe study, a rank 2000 MTTKRP, the smaller growth\nrate yields a matrix-free memory cost of just 2% of the\nmatrix-based methods, a 50x improvement. In practice,\nthe reduced memory impact means our matrix-free\nMTTKRP can compute a rank 2000 tensor decompo-\nsition on a single NVIDIA H100 instead of six H100s\nusing a matrix-based MTTKRP. We also compare our\noptimized matrix-free MTTKRP to baseline matrix-\nfree implementations on different devices, showing a 3x\nsingle-device speedup on an Intel 8480+ CPU and an\n11x speedup on a H100 GPU. In addition to numerical\nresults, we provide fine grained performance models\nfor an ideal multi-level cache machine, compare analyt-\nical performance predictions to empirical results, and\nprovide a motivated heuristic selection for selecting an\nalgorithmic hyperparameter.\nI. Introduction\nDense tensors are relevant in the fields of numerical\nsimulations [1], medical imaging [2], signal processing [3],\nand color perception [4], among others. Low-rank approx-\nimations of dense tensors are a powerful tool used to\ncompress such datasets and reveal relationships between\nmodes. A popular decomposition for computing such ap-\nproximations is the CANDECOMP/PARAFAC (CP), also\ncalled canonical polyadic, decomposition [5], taking a user\n*Sandia National Laboratories is a multimission laboratory man-\naged and operated by National Technology & Engineering Solutions\nof Sandia, LLC, a wholly owned subsidiary of Honeywell Interna-\ntional Inc., for the U.S. Department of Energy’s National Nuclear\nSecurity Administration under contract DE-NA0003525. This paper\ndescribes objective technical results and analysis. Any subjective\nviews or opinions that might be expressed in the paper do not\nnecessarily represent the views of the U.S. Department of Energy\nor the United States Government. SAND2025-13211O.\n1The University of Texas at Austin\n2Sandia National Laboratories\nCorresponding author: Gabriel Kosmacher\nEmail: gkosmacher@utexas.edu\nsupplied rank R and outputting a weighted sum of R rank-\n1 tensors. The bottleneck kernel for CP algorithms [6] is\nthe matrix action of a mode-k unfolding of a tensor with\nthe Khatri-Rao product of factor matrices and is called\nthe matricized tensor times tensor Khatri-Rao product\n(MTTKRP).\nMost open-source tensor decomposition packages (see\nSection I-A) compute the MTTKRP by explicitly forming\nthe Khatri-Rao product matrix, allowing for invocations to\nBLAS-3 algorithms but compromising on memory usage.\nIn contrast, we seek to compute the MTTKRP without\nexplicitly forming the Khatri-Rao matrix and instead take\nadvantage of the problem structure to develop matrix-free,\nalso called element-based, algorithms. Our work extends\nthe open-source GenTen1 tensor decomposition package\nwith performance portable MTTKRPs and, as far as we\nare aware, introduces the first GPU algorithm for a matrix-\nfree MTTKRP with dense tensors.\nContributions: Our work offers the following method-\nological contributions and experimental results:\n• The design and open-source implementation of per-\nformance portable, matrix-free algorithms for dense\nMTTKRPs (see Section III). For a tensor Y of shape\nI1, . . . , Id and CP-rank R, our matrix-free algorithms\nfor a mode-k MTTKRP have O(R P\nn In) memory\nusage while standard matrix-based approaches have\na O(R Q\nn̸=k In) memory cost.\n• Multiple analytic memory and computational models\nfor our algorithmic variants based on an ideal multi-\nlevel cache machine with different caching assump-\ntions (see Section IV-B).\n• A heuristic model for an algorithmic hyper-parameter\nthat aligns with measured times on a CPU and GPU\n(see Section IV-C1).\n• CPU and GPU algorithmic evaluations (see Sec-\ntions IV-D1 and IV-D2). In particular, our matrix-free\nMTTKRP can perform a tensor decomposition on 1\nGPU that requires 6 GPUs for a matrix-based MT-\nTKRP. We also compare our optimized, matrix-free\nMTTKRP to baseline matrix-free implementations,\nshowing a 3x speedup on an Intel 8480+ CPU and an\n11x speedup on an NVIDIA H100 GPU.\n1https://github.com/sandialabs/GenTen\n1\narXiv:2510.14891v1 [cs.MS] 16 Oct 2025\n\nA. Related work\nThere are a variety of open-source packages offering\ndense CP decompositions. N-Way Toolbox [7], TensorLy\n[8], and rTensor [9] compute MTTKRPs by explicitly\nunfolding the tensor and forming the Khatri-Rao product.\nThe MATLAB Tensor Toolbox [6], [10], PyTTB [11] and\nGenTen [12] use and algorithm introduced by Phan et.\nal. [13], described in Section III-A, that avoids expensive\ntensor unfolding and reduces memory impact by forming\npartial Khatri-Rao matrices. Vannieuwenhoven et. al. [14]\nbuild upon [13] by constructing an algorithm, implemented\nusing the Eigen3 C++ library [15], that sequentially stores\nblocks of the Khatri-Rao matrix in constant memory,\nhence achieving the desired O(R P\nn In) storage complex-\nity. However, the blocking algorithm is only applicable\nto the special case where all k-mode MTTKRPs are\ncomputed at once, and can not generalize to compute\nindividual mode-k MTTKRPs needed by, e.g., the CP-\nALS algorithm. Table I gives an overview of features\nfor different dense mode-K MTTKRP algorithms and\ncodebases.\nThere has been considerable more work for optimizing\nMTTKRPs in the sparse tensor case. DeFacTo [16] stores\ntensors as collections of sparse matrices and relies on opti-\nmized sparse matrix-vector multiply routines to compute\nthe MTTKRP. SPLATT [17] offers a variety of compressed\ntensor formats and a cache-blocking scheme to speedup\nsparse MTTKRPs by parallelizing over rows of the output\nmatrix. HiCOO [18] proposes a new storage format that\npartitions the sparse tensor into sparse blocks to increase\nmemory bandwidth and develops an algorithm that paral-\nlelizes over groups of sparse blocks. Both SPLATT and\nHiCOO were developed primarily for low-thread-count\nCPUs and do not consider portability to high-thread GPU\narchitectures. Laukemann et. al. [19] introduce another\nsparse tensor format, ALTO, that linearizes the tensors\nmulti-index set such that tensor entries that are close\nin space are close in memory. Tensor entries are then\ndecomposed into line segments that encode subspaces, and\na parallel algorithm is introduced that assigns threads\nto chunks of line segments. Though developed for CPUs,\nincreasing the number of chunks per line segment should\nallow for straightforward portability to GPUs. GenTen [20]\noffers an performance portable approach that parallelizes\nover tensor nonzeros and permutes the nonzeros to avoid\natomic contention. This algorithm is similar to ours (in\nthe dense case) as it avoids forming the Khatri-Rao prod-\nuct explicitly. However, permuting the tensor nonzeros\nrequires an additional storage cost of O(dN), where d\nis the dimension of the tensor and N is the number of\nnonzeros. As we show in IV-B, explicitly permuting tensor\nentries in such a fashion necessary for an efficient dense\nMTTKRP.\nII. Background\nA. Notation\nA tensor X ∈RI1×···×Id is d-way array of size N =\nI1 × · · · × Id typically stored in a flattened row-major or\ncolumn-major format. It is standard practice [5] to denote\ntensors as bold uppercase letters in Euler calligraphic font\n(e.g., X), matrices by bold uppercase letters (e.g., A),\nvectors by bold lowercase letters (e.g., a), and scalars by\nlowercase letters (e.g., a). We adopt Matlab notation for\narray indexing2.\nThe Khatri-Rao product ⊙of two matrices A ∈Rm×p\nand B ∈Rn×p is the column-wise Kronecker product of\nA and B defined by\nA ⊙B = [a1 ⊗b1| . . . |ap ⊗bp] ∈Rmn×p,\n(1)\nwhere ⊗is the Kronecker product and a(k)\nj\n≡Ak(:, j) is a\ncolumn vector.\nTensors are indexed via multi-indexed arrays i\n=\n(i1, . . . , id) where in ∈[In]. The set notation [a] is shortand\nfor {z : z ∈Z, 1 ≤z ≤a}. We call i ∈[N] the linear index\nof a tensor and (i1, . . . , id) the multi-index, we assume\na bijective mapping between the two index sets, and we\nrefer to a tensor element as xi ≡X(i) ≡X(i1, . . . , id).\nThe forward map i 7→(i1, . . . , id) is given by the operator\nind2sub(X, i) and is defined like the Matlab function of\nthe same name3, while the backward map (i1, . . . id) 7→i\nis given by the operator sub2ind(X, (i1, . . . , id)), again\ndefined like the corresponding Matlab function4. The nth\nmode-k slice of a tensor S(k)\nn\nis a d −1 subtensor obtained\nby fixing the kth value of the multi-index to n and allowing\nthe other values to range, i.e., S(1)\nn\n= X(n, :, . . . , :). The\nmode-k matricization of a tensor X rearranges the tensor\nelements into a matrix X(k) ∈RIk×N/Ik such that element\ni maps to (ik, i′\nk) by\ni′\nk = 1 +\nk−1\nX\nℓ=1\n(iℓ−1)Iℓ+\nd\nX\nℓ=k+1\n(iℓ−1)Iℓ/Ik.\n(2)\nThe reshape operator reshape(X, [s1, . . . , sm]) does not\npermute the underlying tensor elements (unlike tensor\nmatricization) and is defined like the Matlab function of\nthe same name5.\nThe rank-R canonical polyadic (CP) decomposition [21],\n[22] of a tensor X is a rank-R tensor M of the form\nX ≈M =\nR\nX\nj=1\nλja(1)\nj\n◦· · · ◦a(d)\nj ,\n(3)\n2https://www.mathworks.com/help/matlab/math/array-\nindexing.html\n3https://www.mathworks.com/help/matlab/ref/ind2sub.html\n4https://www.mathworks.com/help/matlab/ref/sub2ind.html\n5https://www.mathworks.com/help/matlab/ref/double.reshape.\nhtml\n2\n\nTABLE I\nMethods and codebases for dense mode-k MTTKRPs. Given a tensor Y of shape I1, . . . , Id with factor matrices A1, . . . , Ad,\nlet the Khatri-Rao matrix be Z = Ad ⊙· · · ⊙Ak+1 ⊙Ak−1 ⊙· · · ⊙A1 (see Section II-B for details). We list memory usage and\nopen-source codes for each method, and we list the parallelization method on CPUs and GPUs for each code base. All\nmethods have the same computational cost of O(Rd(Q\nn In)).\nmethods\nmemory\ncode\nCPU\nGPU\nfull Z [6]\nO\n\u0012\nR Q\nn̸=k\nIn\n\u0013\nN-way Toolbox [7]\nBLAS-3\nTensorLy [8]\nBLAS-3\nBLAS-3\nrTensor [9]\nBLAS-3\npartial Z [13]\nO\n\u0012\nR\n\u0012k−1\nQ\nn=1\nIn +\ndQ\nk+1\nIn\n\u0013\u0013\nTTB [6]\nBLAS-3\nPyTTB [11]\nBLAS-3\nGenTen [12]\nBLAS-3\nBLAS-3\nno Z (our method)\nO\n\u0012\nR\ndP\nn=1\nIn\n\u0013\nGenTen\nSIMD\nSIMT\nwhere λ ∈RR is a weight vector and ◦is a d-way outer\nproduct. We call Ak = [a(1)\n1 , . . . , a(k)\nd ] ∈RIk×R the mode-\nk factor matrix of a tensor X. The form M = JA1, . . . , AdK\nis referred to as a Kruskal tensor.\nB. The MTTKRP kernel\nAlgorithms to compute CP decompositions generally fall\ninto two categories: all-at-once [23] or alternating [22],\n[24]. In both cases, the workhorse kernel is the mode-k\nmatricized tensor times Khatri-Rao product (MTTKRP)\ndefined as\nG(k) = Y(k) diag(λ)ZT\n(4)\nwhere Z = Ad ⊙· · · ⊙Ak+1 ⊙Ak−1 ⊙· · · ⊙A1 for some\na Kruskal tensor M = JA1, . . . , AdK, d-way tensor Y, and\nweight vector λ ∈RR\n+. We call algorithms that explicitly\nconstruct Z matrix-based MTTKRPs.\nMatrix-based MTTKRPs have two distinct downsides:\nthe explicit formation the Khatri-Rao matrix Z and ma-\ntricization Y(k). For a mode-k MTTKRP with CP-rank\nR, the Khatri-Rao matrix Z is size Q\nn̸=k In × R, which\ncan dwarf the cost R P\nn̸=k In of simply storing the fac-\ntor matrices An individually, leading to O(R Q\nn̸=k In)\nmemory usage. As for the matricization Y(k), tensors\nare typically stored in memory as a flat array, and in\nsuch cases, the mode-k (for 1 < k < d) matricization\nrequires reordering the non-contiguous tensor entries into\ncontiguous chunks. These reorderings have a memory-\nbound cost of O((d −1)N) and prove significant costs to\nan otherwise computationally bound kernel.\nThese drawbacks motivate a matrix-free, or element-\nwise, definition of the mode-k MTTKRP (common in the\nsparse case [20]), requiring only the explicit storage of Y,\nλ and {Am}d\nm=1, given by\nG(k)(n, j) = λj\nN\nX\ni=1\nik=n\nY(i)\nd\nY\nm=1\nm̸=k\nAm(im, j),\n(5)\nfor n ∈[Ik] and j ∈[R]. As the CP-rank R grows,\nthe dominant memory cost of this approach becomes the\nstorage of the factor and output matrices, yielding an\noverall O(R P\nn In) memory burden.\nC. Kokkos programming framework\nKokkos [25] is a C++ framework providing abstractions\nto write one-off functions for single-instruction multiple-\ndata (SIMD) parallelism on CPUs and single-instruction\nmultiple-threads (SIMT) parallelism on GPUs. It is the\nKokkos convention to discuss levels of parallelism mirror-\ning that of the OpenMP 4.0 specification [26], i.e., leagues\nof teams comprising team-threads which may execute\nvector-thread instructions in parallel. Leagues are virtual\nand correspond to the highest level of parallel space (e.g.,\na Cuda grid or a collection of OpenMP threads), while a\nteam within a league corresponds to hyperthreads on a\nCPU and y-dimension threads within a block on a GPU\nand vector parallelism corresponds to vector instructions\non CPUs and x-dimension threads (i.e., threads within a\nwarp) on GPUs. We use LeagueRange, TeamRange,\nand VectorRange to denote league, team, and vector\nparallel ranges, respectively. We use bx to denote team\nsize and by to denote vector size, and we use px do denote\nthreads within a team and py to denote vector-threads\nwithin a thread.\n1) SIMD arrays: Compile-time polymorphic SIMD ar-\nrays were introduced in [20] as an extension of Kokkos that\nallow for the allocation of small arrays as register arrays\non GPUs or thread-private stack arrays on CPUs. We\nutilize these arrays to allow for a single vector-thread py to\ncalculate many (e.g., 4) updates to the output matrix G(k)\nper vector loop, increasing memory-bandwidth efficiency.\nWe denote the SIMD vector size as F and SIMD vectors as\ngreek leters with the vector arrow, e.g., ⃗π, ⃗φ. SIMD vectors\nare in-effect for loops that the compiler is forced to unroll\nand vectorize.\nIII. Parallel MTTKRP algorithms\nThe state of the art dense MTTKRP was introduced in\n[13] and takes the viewpoint of Eq. (4). The algorithm,\ncalled MTTKRP-GEMM, uses smart partitioning of the\n3\n\nKhatri-Rao product matrix ZT to avoid costly reorders\nof Y, utilizes temporary memory allocations for a (some-\ntimes) lower storage cost, and gets its name from its use\nof the BLAS-3 general matrix-matrix multiplies (GEMMs)\nfor highly-optimized parallelization.\nWe introduce element based algorithms all taking the\nviewpoint of Eq. (5): MTTKRP-ELEM, MTTKRP-\nSLICE, and MTTKRP-TILE. Each of our algorithms\nutilize the same vector parallelization over columns of\n{Ak} and differ in how they assign tensor elements Y(i)\n(or groups of Y(i)’s) to Kokkos leagues and teams. The\nvector parallelism scheme is the same as used in [20]\nto introduce parallelism over the columns of the factor\nmatrices {Ak} by assigning multiple column indices j in\nEq. (5) to a thread vector py. Along with the enforcement\nof a row-wise layout, this allows for coalesced reads of the\nfactor matrices, allowing the compute device to achieve a\nhigher percentage of memory bandwidth. For simplicity,\nwe assume that the CP-rank R divides the product of the\nvector size and the SIMD array length (by × F) for the\nremainder of this discussion.\nA. MTTKRP-GEMM\nThe key insight of [13] is to avoid the costly matriciza-\ntions Y(k) by using the reshape operator and breaking\nup the full Khatri-Rao product ZT into left and right\ncomponents ZT = [ZT\nR ∈RR×IR|ZT\nL ∈RR×IL], where\nIL = Qk−1\nm=1 Im and IR = Qd\nm=k+1 Im.\nWe briefly summarize the algorithm given in [13]. For\nmore detail, please refer to the paper. For mode-1, re-\nplace Y(k) with reshape(Y, [I1, IR]) in Eq. (4) to avoid\npermutations. Similarly, for mode d, replace Y(k) with\nreshape(Y, [IL, Id])T in Eq. (4) to avoid permutations.\nThe steps for modes 1 < k < d are given in Algorithm 1.\nNote that although MTTKRP-GEMM has a storage\nAlgorithm 1 MTTKRP-GEMM\nInput: Y, {A(m)}d\nm=1, R, k\nOutput: G(k)\n1: ZT\nR = diag(λ)(Ad ⊙· · · ⊙Ak+1)T\n2: C = reshape(Y, [IL · Ik, IR])ZR\n// GEMM\n3: C = reshape(C, [IL, Ik, R])\n// tensorize the matrix\n4: ZT\nL = diag(λ)(Ak−1 ⊙· · · ⊙A1)T\n5: G(k)(ℓ, j) = PIL\nq=1 C(q, ℓ, j)ZL(q, j)\n// parallelize\nwith einsum, GEMM, etc.\ncomplexity across modes-k of O(R(IL + IR)) as the left\nand right factor matrices can be stored in the same\ntemporary memory space, the storage cost for modes-1, d\nis O(R(Qd\nm=1 In/ min{I1, Id})), the same cost as forming\nZT explicitly in 4.\nB. MTTKRP-ELEM\nThe simplest attempt to parallelize Eq. (5) is to assign\nevery thread within a team px to a tensor element Y(i)\nwithout enforcing any particular league-wise structure.\nThis algorithm, MTTKRP-ELEM, is described in line 2.\nTensor indices i are given by the league offset plus team\nrank in Algorithm 2, hence the tensor reads in line 4\nare packed on CPUs and coalesced on GPUs. Reading\n{Am}d\nm=1 and writing to G(k) require a conversion of the\nlinear index i to a multi-index (i1, . . . , id) in line 5 with the\nfunction ind2sub(Y, i) which uses costly integer divisions.\nLines 6-20 are the vector parallelism over factor matrices\n{Am}d\nm=1 columns. Line 8 ensures that each vector-thread\npy process at least F columns, and the while loop on\nlines 9-19 increases the number of columns processed by\neach thread to R/(by×F). Line 18 ensures that the column\nstrides are packed/coalesced for a fixed px, but this is\nnegated by line 14 as the multi-index entry im is not\ncoalesced for neighboring px. These pseudo-random reads\nof the factor matrices have a significant negative effect on\nmemory bandwidth as R(d−1) factor matrix columns are\nread per tensor element Y(i). Finally, G(k) is atomically\nupdated for each N tensor element R times on line 17.\nAlgorithm 2 MTTKRP-ELEM\nInput: Y, {Am}d\nm=1, R, k, F\nOutput: G(k)\n1: LeagueRange l ∈[N/bx]\n2:\nTeamRange px ∈bx\n3:\ni ←l × N/bx + px\n4:\nyi ←Y[i]\n// packed/coalesced reads\n5:\n(i1, . . . , id) ←ind2sub(Y, i)\n6:\nVecRange py ∈[by]\n7:\njj ←0\n8:\n⃗φ ←0\n9:\nwhile jj < R do\n// read column chunks\n10:\nj ←jj + F × by + py\n11:\n⃗φ ←xi × λ[j : j + F]\n12:\nfor m = 1, . . . , d do\n13:\nif m ̸= k then\n14:\n⃗φ ←⃗φ × Am[im, j : j + F]\n//\npseudo-random reads\n15:\nend if\n16:\nend for\n17:\nAtomicAdd(V[ik, j : j + F], ⃗φ)\n//\nN × R atomic updates\n18:\njj ←jj + F × by\n19:\nend while\n20:\nEnd VecRange\n21:\nEnd TeamRange\n22: End LeagueRange\nC. MTTKRP-SLICE\nWe would like our element-based parallelization scheme\nto avoid the atomic updates and pseudo-random read-\ns/writes of the MTTKRP-ELEM algorithm. To avoid\natomic operations, Eq. (5) informs us to processes each\nmode-k slice (see Fig. 1) of the tensor S(k)\nn\nserially.\n4\n\nFig. 1. Slicing of a 3-way tensor Y ∈RI1×I2×I3 along each mode.\nWe thus construct an algorithm MTTKRP-SLICE\nthat assigns league and team parallelism to each slice S(k)\nn\nof Y. The MTTKRP-SLICE algorithm is described by\nAlgorithm 3 with Nt = N/Ik (i.e., the number of elements\nin a given slice S(k)\nn ) and modifying the atomic add on\nline 25 to be a conflict-free global update. The algorithm\nbegins by assigning each team to a slice S(k)\nn\nin line 4.\nLine 6 then computes an anchor multi-index a for the\nslice, in this case given by am = 0 for m ̸= k and ak = n.\nLines 8-28 are the vector parallelism over factor matrices\n{Am}d\nm=1 columns and differs from the vector parallelism\nin Algorithm 2 as each vector-thread py now processes Nt\ntensor elements yi (lines 12-24). Line 13 computes the\ntensor multi-index (i1, . . . id) as the sum of the anchor\nmulti-index (a1, . . . , ad) and the ind2sub(S(k)\nn , ii). Note\nthat the resulting multi-index i is length d (i.e., the\nsame length of a multi-index for Y), and the kth entry\nof the multi-index ik = n. For different slices S(k)\nn\nand\nS(k)\nm , the multi-indices differ only in their kth element,\nhence in practice we implement a special ind2sub(S(k)\nn , ii)\nfunction that uses the same precomputed offsets for each\nslice, avoiding the need to explicitly form the slice and\ncompute costly inner loop integer divisions used in the\nclassic implementation. Line 14 takes the resulting multi-\nindex and calls a function sub2ind(Y, (i1, . . . , id)) built\non cheap integer additions and multiplications to get the\ntensor linear index i. The inner loop over tensor elements\nin a slice allows the factor matrices columns to be cached\nand reused (line 20) in the computation of the element\nHadamard product ⃗φ = yi\nQ\nm̸=k Am(im, j). The element\nHadamard product ⃗φ is then added to the slice Hadamard\nproduct ⃗π on line 23. After the inner slice loop terminates\non line 28, the slice Hadamard products ⃗π are written to\nG(k)(ik, j : j + F) without atomics.\nD. MTTKRP-TILE\nThe MTTKRP-SLICE algorithm has the advantage\nover MTTKRP-ELEM in that it avoids atomic conflicts\nand has beneficial cache blocking for columns of the fac-\ntor matrices {Am}d\nm=1. However, MTTKRP-SLICE has\nNs× more serial operations, which can lead to a significant\nloss of parallelization for MTTKRPs on modes with large\nslices. We attempt to remedy this loss of parallelization by\nassigning leagues to slices and teams to tiles within a slice\n(see Fig. 2). Given a slice S(n)\nk , a tile partition is defined by\na tile volume Nt that divides the slice volume Ns = N/Ik\nsuch that S(k)\nn\n= {S(k)\nn ((t−1)NT : tNT )}NS/NT\nt=1\nand whose\nset elements S(k)\nn ((t −1)NT : tNT ) are called tiles.\nWe call the resulting algorithm that parallelizes over\ntiles MTTKRP-TILE and describe it in Algorithm 3.\nThe algorithm is controlled by the tile volume hyperpa-\nrameter NT , which controls both the length of the serial\naccumulation of the tile Hadamard product in lines 12-24\nand the number of atomic operations in line 25, which is\ngiven by In(NS/NT ) × R. Note that setting a tile size of\nNT = NS and removing the atomic stipulation on line 25\nreverts the algorithm to MTTKRP-SLICE, while setting\na tile size of NT = 1 reverts the algorithm to MTTKRP-\nELEM, albeit with more structure in the league and team\nassignments. We refer to the discussion in Section III-C for\na detailed discussion of the pseudo-code in Algorithm 3.\nAlgorithm 3 MTTKRP-TILE\nInput: Y, {Am}d\nm=1, R, k, F\nOutput: G(k)\n1: NS ←N/Ik\n// mode-n slice volume\n2: LeagueRange l ∈[N/(NS/NT )]\n// league & team\nparallelism over the tiles\n3:\nTeamRange px ∈[bx]\n4:\nn ←intdiv(l × bx + px, NS/NT )\n5:\nt ←intmod(l × bx + px, NS/NT )\n6:\n(a1, . . . , am) ←ind2sub(S(k)\nn [(t −1)NT ], 0)\n7:\njj ←0\n8:\nVecRange py ∈[by]\n9:\n⃗π ←0\n// init. tile Hadamard product\n10:\n⃗φ ←0 // init. element Hadamard product\n11:\nwhile jj < R do\n12:\nfor ii ∈[NT ] do\n13:\n(i1, . . . , id)\n←\n(a1, . . . , ad)\n+\nind2sub(S(k)\nn [(t −1)NT ], ii)\n14:\ni ←sub2ind(Y, (i1, . . . , id))\n15:\nyi ←Y[i]\n// re-read tensor element\n16:\nj ←jj + F × by + py\n17:\n⃗φ ←yi × λ[j : j + F]\n18:\nfor m = 1, . . . , d do\n19:\nif m ̸= k then\n20:\n⃗φ ←⃗φ × Am[im, j : j + F] //\ncolumns are reused\n21:\nend if\n22:\nend for\n23:\n⃗π ←⃗π + ⃗φ\n// update tile product\n24:\nend for\n25:\nAtomicAdd(G(k)[n, j : j + F],⃗π)\n//\nIn(NS/NT ) × R atomic updates\n26:\njj ←jj + F × by\n27:\nend while\n28:\nEnd VecRange\n29:\nEnd TeamRange\n30: End LeagueRange\n5\n\nFig. 2. A tensor Y ∈RI1×I2×I3 decomposed into mode-2 slices S(2)\nn . Each slice in then assigned to a Kokkos league of two teams p1, p2, where\neach team computes a tile Hadamard product and atomically reduces their local contributions to G(k)(n, j). In this instructive example we\nhave one league per slice, but in practice we allow for more than one league per slice. However, the map between teams and tiles is always\nbijective.\nIV. Numerical results\nWe study the performance and portability of the dif-\nferent MTTKRP algorithms described in Section III. Our\nnumerical experiments are performed on two tensors with\nrandomly generated values based of relevant scientific\nsimulations [1]: the 2D compressible tearing mode tensor\n[27], [28] A ∈R401×201×12×501, about 3.7 GB in dou-\nble precision, and the 3D island coalescence tensor [29]\nB ∈R129×129×129×12×39, about 7.5 GB. Our experiments\nuse CP-ranks R = 10, 500, 1000, 2000.\nA. Experimental setup\nEach algorithmic variant is implemented in the pub-\nlicly available GenTen library6 built atop Kokkos for\nperformance portability. Our numerical experiments are\nperformed on cluster nodes the Hops machine at Sandia\nNational Labs. Each node has two 3.8 GHz Intel Xeon\nPlatinum 8480+ CPUs and 4 Nvidia H100 80GB 700W\nSXM5 GPUs. For the CPUs, we set bx = by = 1 (i.e., we\nonly use league parallelism), use GenTen heuristics (see\n[20] Tables 3,4) for selecting F based on R, and compile\nwith the Intel MKL LAPACK and BLAS and architec-\nture specific vectorization flags set by Kokkos . We also\nset the environment variables OMP\nNUM\nTHREADS = 56,\nOMP\nPROC\nBIND = close, and OMP\nPLACES = threads.\nOn GPUs, we set bx = 128/by and choose by and F based\non R using the aforementioned GenTen heuristics.\nB. Performance analysis\nWe use a model f = NRd to measure the flop count of\nthe MTTKRP as for each N tensor elements, we do R(d−\n1) multiplications and R additions. Memory bandwidth is\nmeasured for each element-based algorithmic variant using\na 0-cache model m0, an infinite-cache model m∞, and a\n0, LM-cache model, where LM is the middle level of cache\non a device, i.e., L1 cache on a GPU and L2 cache on a\nCPU. The infinite-cache model is given by m∞= sf(N +\nR P\nn In), were sf is the size in bytes of the floating-point\ntype, N is the size of the tensor, R P\nn̸=k In is the size of\nreading all but the kth factor matrix, and RIk is the size\n6https://github.com/sandialabs/GenTen\nof the output matrix. The 0-cache model is given by m0 =\nsf(N+NR(d−1)+(NS/NT )IkR) as the whole tensor must\nbe read in, every tensor element N must read each factor\nmatrix column R for d −1 factor matrices, and each R\ncolumns of the output matrix are updated NS/NT times\nfor each Ik slice. The 0, LM model assumes that the tensor\nand output matrices cannot be cached as they are not read\nrepeatedly, but the factor matrices are cached, yielding the\nmodel m0,LM = sf(N + NS/NT (IkR)) + 1\nl sf(NR(d −1)),\nwhere l is the ratio between LM bandwidth and device\nmemory bandwidth.\nWe use these models to predict the total time T0 =\nf/τf + m0τm, T∞= f/τf + m∞/τm, and T0,LM = f/τf +\nm0,LM/τm, where τf is the peak machine measured in\nflops and τm is the peak machine bandwidth measured\nin read/write operations per second. We define arithmetic\nintensity to be I = f/m∞and we say that a model\nis compute bound if I > τf/τm. Given a wall-time t,\nthe throughput of the model is measured as gflops =\nf/t/10243 and the memory bandwidths for each model are\ngiven by mops0 = m0/t and mops∞= m∞/t.\nPeak\nthroughput\nτf,\nbandwidth\nτm,\nand\nLM\nbandwidth\nratio\nl\nfor\neach\ndevice\nis\nreported\ninTable II. For the H100, throughput and bandwidth\nare\ntaken\nfrom\nNVIDIA’s\nwhitepaper7\nand\nwe\ncalculate\nthe\npeak\nL1\ncache\nbandwidth\nas\n#SM’s x L1 transfer bytes/cycle x clock rate,\nwhich is 123 × 128.98/103\n≈33TB/s, where the L1\ntransfer bytes per clock cycle is taken from NVIDIAs\nhopper tuning guide8. Hence, for the H100, we set\nl = 10. Throughput for the 8480+ is taken from Intel’s\nAPP metrics sheet9 while memory bandwidth and L2\ncache bandwidth are measured with the STREAM triad\nbenchmark [30], where we use a 2GB array to measure\nmemory bandwidth and a 0.6MB array to measure the\nL2 cache bandwidth.\nWe analyze the efficacy of our performance models in\n7https://www.nvidia.com/en-us/data-center/h100/\n8https://docs.nvidia.com/cuda/hopper-tuning-guide/index.html\n9https://www.intel.com/content/www/us/en/support/articles/\n000005755/processors.html\n6\n\nTABLE II\nPeak compute throughput τf , memory bandwidth τm, and\ncache bandwidth ratio l for each device.\nDevice\n1012τf\n10244τm\nl\n8480+\n2.28\n0.12\n8\nH100\n34\n3.35\n10\nTable III. Our results show that the cache effects play\na significant role for each algorithmic variant: the T∞\nprediction is much too optimistic for each variant. The\nMTTKRP-ELEM algorithm fails to surpass even the 0-\ncache predicted time T0 for either device. As discussed in\nSection III-B, this can be attributed to the uncoalesced\nrepeated reads of the factor-matrices. The MTTKRP-\nSLICE algorithm also fails to surpass T0 on the GPU,\nthough it does so on the CPU. We hypothesize that this\ndiscrepancy is due to the much larger cache size of the\nCPU compared to the GPU. For MTTKRP-SLICE on\nthe CPU and MTTKRP-TILE on both devices, the 0-\ncache model is far to pessimistic while the ∞-cache model\nis too optimistic, indicating that the algorithms achieve (to\nsome extent) the desired caching effects on the repeated\nreads of the factor matrices, Our empirical results for\nthese methods align most with the 0, LM-cache model,\nachieving at least 90% of the predicted time on the CPU\nand 40% of the predicted time on the GPU. We surmise\nthat the higher accuracy of our models on the CPU is\ndue to the fact that we use empirically measured peak\nbandwidth, while on the GPU we use the vendor reported\nideal world numbers.\nC. Experiments\n1) Selecting the tile size: Our first goal is to select a\nvalue for the heuristic tile volume parameter NT in the\nMTTKRP-TILE algorithm. Table IV presents the nu-\nmerical performance of a sweep over tile-widths\nd−1√NT ∈\n{2, 4, 6, 8, 10, 12, 14} for the tearing-mode and island coa-\nlescence tensors for a fixed R = 32. On the GPU, we find\nthat for the 4-way tensor A, a tile size of 216 is optimal,\nwhile for the 5-way tensor B, a tile size of 256 is optimal.\nOn an H100, our maximum occupancy with our stan-\ndard block configuration of (4, 32) is 64 warps per SM.\nNSIGHT-compute reports our achieved occupancy to be\n60%, but even if we assume a more conservative occupancy\nof 50%, we have 32 tiles per SM, with each tile rereading\n8 × 256 bytes of factor matrices entries, or 64KB of\ninformation per SM, 25% of the total L1 cache, a good\ntarget given that the L1 cache handles read, write, and\natomic instructions on a GPU.\nOn the 8480+, we find that a tile width of 12, i.e., the\nsize of the minimum dimension, is optimal. Our parallel\nCPU implementation assigns one tile per core, and given\nthat each core has 2MB of L2 cache, this means that the\nfactor matrices take up at most ∼0.5% of the L1 cache.\nHowever, our implementation assumes that the tile size\nis regular, i.e., that\nd−1√NT ∈Z. As such, performance\ndecreases when\nd−1√NT > minn=1,...,d In.\nTo impose tile regularity and cache awareness, we use\nthe simple heuristic\nNT = min\n\n\n\n \nd−1\ns\nsLM/4\nsf(c/2)\n!d−1\n,\nmin\nn=1,...,d{In}d−1\n\n\n, (6)\nwhere sLM is the size in bytes of the middle level of cache\n(i.e., L1 cache (per SM) on a GPU, L2 cache (per core) on\na CPU), and c is the maximum number of tiles per cache\nunit (i.e., SM/core). We find that this heuristic is valid for\nmost R.\nD. Memory impact of MTTKRP variants\nAs discussed in Section III-A, forming the left and right\nKhatri-Rao matrices ZT\nL and ZT\nR can incur a large storage\ncost. This cost dominates the total memory required for\nthe mode-k MTTKRP-GEMM algorithm, which is given\nby mGEMM\n∞\n= N + R(IL + IR + Ik). It is important to\nnote the storage cost of the MTTKRP-GEMM algorithm\nrelies heavily on the initial shape of a given tensor. For\nexample, the maximum memory usage over all modes k\nfor the island coalescence tensor B for a R = 2000 is\n391GB, but if the initial tensor shape was permuted to be\n129 × 12 × 129 × 39 × 129 (best case), the memory usage\nwould be 123GB, while if the tensor was permuted to be\n129 × 129 × 129 × 39 × 12 (worst case), the memory usage\nwould be 1255GB. However, especially in the context in\nsitu decomposition of scientific simulation data [1], [31],\nreshaping tensors can eliminate multidimensional relation-\nships between tensor entries and can be expensive and im-\npractical. Fig. 3 shows the maximum memory usage over\nall modes k for the MTTKRP-GEMM and MTTKRP-\nTILE algorithms for the tearing mode tensor A and the\nisland coalescence tensor B. The y axis of the plot denotes\nthe minimum number of 80GB H100 GPUs needed for\nstorage, however, the distribution protocol in IV-D2 does\nnot evenly assign memory to MPI ranks, hence in practice\nmore GPUs may be required for a given problem size.\n1) Single node performance: The linear-in-rank memory\nscaling of the MTTKRP-GEMM algorithm described\nin Section IV-D motivates the use of the tearing-mode\ntensor A to study single-node performance of the different\nMTTKRP variants. Fig. 4 reports the average MTTKRP\nperformance, measured in gflops, over all modes, for the\nOpenMP and Cuda execution spaces. We compare the\nSLICE, TILE, and GEMM variants on the OpenMP space\nand the ELEM, TILE, and GEMM variants on the Cuda\nspace, where the tile size NT is chosen by Eq. (6). We omit\nthe ELEM variant for OpenMP and the SLICE variant for\nCuda given their poor performance alluded to in Table III.\nFor R > 10, MTTKRP-TILE achieves at least 20%\nof MTTKRP-GEMM’s performance, an efficient result\ngiven MTTKRP-TILE’s memory footprint. Additionally,\nfor R > 10, MTTKRP-TILE performs at least 2× faster\n7\n\nTABLE III\nExecution time T in seconds measured against analytical performance model times T0, T∞for the tensors A, B on the\nOpenMP and Cuda execution spaces. The tile size NT for the MTTKRP-TILE algorithm is selected via the heuristic in\nEq. (6). The CP-rank is fixed to R = 32.\nOpenMP\nCuda\nT\nT0\nT0,LM\nT∞\nT\nT0\nT0,LM\nT∞\nA\nELEM\n10.394\n3.817\n1.349\n0.056\n1.823\n0.138\n0.047\n0.003\nSLICE\n0.543\n2.877\n0.409\n0.056\n8.528\n0.104\n0.013\n0.003\nTILE\n0.526\n2.955\n0.487\n0.056\n0.043\n0.110\n0.019\n0.003\nB\nELEM\n25.167\n9.876\n3.054\n0.130\n3.924\n0.356\n0.105\n0.007\nSLICE\n1.137\n7.927\n1.105\n0.130\n23.032\n0.286\n0.035\n0.007\nTILE\n1.517\n8.414\n1.592\n0.130\n0.120\n0.304\n0.052\n0.007\nTABLE IV\nSweep over tile widths\nd−1√NT for the MTTKRP-TILE\nalgorithmic variant with R = 32 for each tensor A, B on\nboth execution spaces, with performance in gflops (higher is\nbetter) averaged over all modes. These results motivate a\nheuristic to choose NT given by Eq. (6).\nMTTKRP-TILE gflops\nd−1√NT\nTearing mode tensor A\nIsland coalescence tensor B\nOpenMP\nCuda\nOpenMP\nCuda\n2\n32.9\n253.3\n51.4\n585.1\n4\n89.9\n1313.6\n99.0\n1232.2\n6\n98.9\n1328.3\n106.8\n1173.4\n8\n98.1\n1103.7\n104.9\n944.1\n10\n96.5\n947.3\n103.3\n884.4\n12\n99.6\n1230.0\n110.8\n948.8\n14\n97.4\n1159.2\n101.4\n843.9\n10\n500 1000\n2000\nCP-ranks (R)\n0\n100\n200\n300\n400\nMemory Usage (GB)\n1 H100\n2 H100s\n3 H100s\n4 H100s\n5 H100sTearing mode\nGEMM\nTILE\n10\n500 1000\n2000\nCP-ranks (R)\n1 H100\n2 H100s\n3 H100s\n4 H100s\n5 H100s\nIsland coalescence\nGEMM\nTILE\nFig. 3.\nWorst case memory usage for the GEMM and TILE\nMTTKRP variants for the tearing mode tensor A and the island\ncoalescence tensor B for different CP-ranks. The plot is annotated\nwith lines showing the minimum number of NVIDIA H100 GPUs\nrequired incur the storage costs.\nthan the SLICE variant on the OpenMP space and at least\n3× faster than ELEM on the Cuda space, with a peak\nspeedup of 11× for R = 500. These speedups reinforce the\nnecessity for data-reuse and reduced atomic contention for\nelement based parallelization approaches.\n10\n500\n1000\n2000\nCP-ranks (R)\n0\n2000\n4000\n6000\ngflops\n29\n87\n88\n101\n31\n192\n204\n213\n220\n630\n807\n923\nOpenMP\nSLICE\nTILE\nGEMM\n10\n500\n1000\n2000\nCP-ranks (R)\n12\n122\n238\n445\n522\n1385\n1386\n1401\n665\n6888\n6830\n7035\nCuda\nELEM\nTILE\nGEMM\nFig. 4. Tearing mode tensor A average (over all modes) MTTKRP\nperformance (higher is better) for different algorithmic variants on\ndifferent execution spaces.\n2) Comparison against distributed MTTKRP-GEMM:\nThe disparities of memory costs between the MTTKRP-\nTILE and MTTKRP-GEMM algorithms for the island\ncoalescence tensor B (see Fig. 3) motivate us to compare\nrunning the MTTKRP-TILE algorithm on a single GPU\nagainst running MTTKRP-GEMM algorithm on (the\nrequired) multiple GPUs. Distributed MTTKRPs use the\ncommutation pattern introduced in [32] whose communi-\ncation overhead consists of an initial tensor redistribution\nand all-reduce communication to combine contributions\nacross processors. Fig. 5 compares the execution time\nof the CP-ALS algorithmon B with tolerance 10−4 over\nranks R = 10, 500, 1000, 2000 on a single device when\nusing the MTTKRP-TILE algorithm and on multiple\ndevices when using the MTTKRP-GEMM algorithm.\nWhile tensor and factor matrices fit snugly into memory\nfor the TILE variant, MTTKRP-GEMM requires at least\n6 H100s for rank 2000, 3 H100s for rank 1000, and 2\nH100s for rank 500 (findings in line with our lower bound\nmemory model). Note that the hops machine requires two\nnodes to distribute across 6 GPUs, with the multi-node\ncommunication bandwidth drastically increasing tensor\nredistribution and MTTKRP communication time. When\nR = 2000, MTTKRP-TILE based CP-ALS on a sin-\ngle GPU is able to achieve 67% of the performance of\nMTTKRP-GEMM based CP-ALS run on the minumum\n8\n\n6 H100s. For R = 1000, 4 H100s are required to beat\nthe MTTKRP-TILE based CP-ALS for and at least 5 are\nrequired to beat the MTTKRP-TILE based CP-ALs for\nR = 500. For all tested ranks, tensor redistribution is the\nmost expensive overhead incurred in the distributed CP-\nALS, dwarfing the all-reduce communication time.\n10\n500\n1000\n2000\nCP-ranks (R)\n0\n25\n50\n75\n100\n125\n150\n175\nTime (s)\nTILE\nGEMM-nGPUs=6\nGEMM-nGPUs=4\nGEMM-nGPUs=3\nGEMM-nGPUs=2\nother CP\nredist\ncomm\nFig. 5. Time (lower is better) for CP-ALS on the island coalecence\ntensor B with MTTKRP-TILE algorithm on a single H100 compared\nwith MTTKRP-GEMM on one H100 per process. MTTKRP time is\ndenoted with hatches, other CP-ALS operations (i.e., inner-product,\nCholesky solve, etc.) by a grey fill, tensor redistribution time with a\npurple fill, and MTTKRP communication time with an orange fill.\nV. Conclusion\nWe introduce fast and memory efficient algorithms for\nmatrix free dense MTTKRPs together with detailed per-\nformance analysis and evaluations on CPUs and GPUs.\nWe extend the open-source GenTen package with our\nmethods and demonstrate that state-of-the-art matrix\nbased MTTKRPs need many GPUs to compute the same\ntensor decomposition that, when replaced with our MT-\nTKRP, can be computed on a single device. Our methods\nhave limitations. Fig. 4 shows that we have yet to achieve\nthe efficiency of the matrix based MTTKRP-GEMM algo-\nrithm when the Khatri-Rao product matrix fits in device\nmemory. Table III motivates us to further improve our\ncaching strategies to achieve a higher percentage of the\npredicted infinite-cache time T∞. Such strategies include\nGEMM-like tilings of the output matrix and a Hilbert\ncurve or ALTO-like [19] ordering of the input tensor to\nincrease cache locality.\nReferences\n[1] D. M. Dunlavy, E. T. Phipps, H. Kolla, J. N. Shadid, and\nE. 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In contrast to the state-of-the-art matrix multiply based MTTKRP kernels used by Tensor Toolbox, TensorLy, etc., that explicitly form Khatri-Rao matrices, we develop a matrixfree element-wise parallelization approach whose memory cost grows with the rank R like the sum of the tensor shape O(R(n+m+k)), compared to matrix-based methods whose memory cost grows like the product of the tensor shape O(R(mnk)). For the largest problem we study, a rank 2000 MTTKRP, the smaller growth rate yields a matrix-free memory cost of just 2% of the matrix-based methods, a 50x improvement. In practice, the reduced memory impact means our matrix-free MTTKRP can compute a rank 2000 tensor decomposition on a single NVIDIA H100 instead of six H100s using a matrix-based MTTKRP. We also compare our optimized matrix-free MTTKRP to baseline matrixfree implementations on different devices, showing a 3x single-device speedup on an Intel 8480+ CPU and an 11x speedup on a H100 GPU. In addition to numerical results, we provide fine grained performance models for an ideal multi-level cache machine, compare analytical performance predictions to empirical results, and provide a motivated heuristic selection for selecting an algorithmic hyperparameter. I. Introduction Dense tensors are relevant in the fields of numerical simulations [1], medical imaging [2], signal processing [3], and color perception [4], among others. Low-rank approximations of dense tensors are a powerful tool used to compress such datasets and reveal relationships between modes. A popular decomposition for computing such approximations is the CANDECOMP/PARAFAC (CP), also called canonical polyadic, decomposition [5], taking a user *Sandia National Laboratories is a multimission laboratory managed and operated by National Technology & Engineering Solutions of Sandia, LLC, a wholly owned subsidiary of Honeywell International Inc., for the U.S. 's National Nuclear Security Administration under contract DE-NA0003525. This paper describes objective technical results and analysis. Any subjective views or opinions that might be expressed in the paper do not necessarily represent the views of the U.S. . SAND2025-13211O. 1The 2Sandia National Laboratories Corresponding author: Gabriel Kosmacher Email: supplied rank R and outputting a weighted sum of R rank1 tensors. The bottleneck kernel for CP algorithms [6] is the matrix action of a mode-k unfolding of a tensor with the Khatri-Rao product of factor matrices and is called the matricized tensor times tensor Khatri-Rao product (MTTKRP). Most open-source tensor decomposition packages (see Section I-A) compute the MTTKRP by explicitly forming the Khatri-Rao product matrix, allowing for invocations to BLAS-3 algorithms but compromising on memory usage. In contrast, we seek to compute the MTTKRP without explicitly forming the Khatri-Rao matrix and instead take advantage of the problem structure to develop matrix-free, also called element-based, algorithms. Our work extends the open-source GenTen1 tensor decomposition package with performance portable MTTKRPs and, as far as we are aware, introduces the first GPU algorithm for a matrixfree MTTKRP with dense tensors. Contributions: Our work offers the following methodological contributions and experimental results: • The design and open-source implementation of performance portable, matrix-free algorithms for dense MTTKRPs (see Section III). For a tensor Y of shape I1, . . . , Id and CP-rank R, our matrix-free algorithms for a mode-k MTTKRP have O(R P n In) memory usage while standard matrix-based approaches have a O(R Q n̸=k In) memory cost. • Multiple analytic memory and computational models for our algorithmic variants based on an ideal multilevel cache machine with different caching assumptions (see Section IV-B). • A heuristic model for an algorithmic hyper-parameter that aligns with measured times on a CPU and GPU (see Section IV-C1). • CPU and GPU algorithmic evaluations (see Sections IV-D1 and IV-D2). In particular, our matrix-free MTTKRP can perform a tensor decomposition on 1 GPU that requires 6 GPUs for a matrix-based MTTKRP. We also compare our optimized, matrix-free MTTKRP to baseline matrix-free implementations, showing a 3x speedup on an Intel 8480+ CPU and an 11x speedup on an NVIDIA H100 GPU. 1https://github.com/sandialabs/GenTen 1 16 Oct 2025 A. Related work There are a variety of open-source packages offering dense CP decompositions. N-Way Toolbox [7], TensorLy [8], and rTensor [9] compute MTTKRPs by explicitly unfolding the tensor and forming the Khatri-Rao product. The MATLAB Tensor Toolbox [6], [10], PyTTB [11] and GenTen [12] use and algorithm introduced by Phan et. al. [13], described in Section III-A, that avoids expensive tensor unfolding and reduces memory impact by forming partial Khatri-Rao matrices. Vannieuwenhoven et. al. [14] build upon [13] by constructing an algorithm, implemented using the Eigen3 C++ library [15], that sequentially stores blocks of the Khatri-Rao matrix in constant memory, hence achieving the desired O(R P n In) storage complexity. However, the blocking algorithm is only applicable to the special case where all k-mode MTTKRPs are computed at once, and can not generalize to compute individual mode-k MTTKRPs needed by, e.g., the CPALS algorithm. Table I gives an overview of features for different dense mode-K MTTKRP algorithms and codebases. There has been considerable more work for optimizing MTTKRPs in the sparse tensor case. DeFacTo [16] stores tensors as collections of sparse matrices and relies on optimized sparse matrix-vector multiply routines to compute the MTTKRP. SPLATT [17] offers a variety of compressed tensor formats and a cache-blocking scheme to speedup sparse MTTKRPs by parallelizing over rows of the output matrix. HiCOO [18] proposes a new storage format that partitions the sparse tensor into sparse blocks to increase memory bandwidth and develops an algorithm that parallelizes over groups of sparse blocks. Both SPLATT and HiCOO were developed primarily for low-thread-count CPUs and do not consider portability to high-thread GPU architectures. Laukemann et. al. [19] introduce another sparse tensor format, ALTO, that linearizes the tensors multi-index set such that tensor entries that are close in space are close in memory. Tensor entries are then decomposed into line segments that encode subspaces, and a parallel algorithm is introduced that assigns threads to chunks of line segments. Though developed for CPUs, increasing the number of chunks per line segment should allow for straightforward portability to GPUs. GenTen [20] offers an performance portable approach that parallelizes over tensor nonzeros and permutes the nonzeros to avoid atomic contention. This algorithm is similar to ours (in the dense case) as it avoids forming the Khatri-Rao product explicitly. However, permuting the tensor nonzeros requires an additional storage cost of O(dN), where d is the dimension of the tensor and N is the number of nonzeros. As we show in IV-B, explicitly permuting tensor entries in such a fashion necessary for an efficient dense MTTKRP. II. Background A. Notation A tensor X ∈RI1×···×Id is d-way array of size N = I1 × · · · × Id typically stored in a flattened row-major or column-major format. It is standard practice [5] to denote tensors as bold uppercase letters in Euler calligraphic font (e.g., X), matrices by bold uppercase letters (e.g., A), vectors by bold lowercase letters (e.g., a), and scalars by lowercase letters (e.g., a). We adopt Matlab notation for array indexing2. The Khatri-Rao product ⊙of two matrices A ∈Rm×p and B ∈Rn×p is the column-wise Kronecker product of A and B defined by A ⊙B = [a1 ⊗b1| . . . |ap ⊗bp] ∈Rmn×p, (1) where ⊗is the Kronecker product and a(k) j ≡Ak(:, j) is a column vector. Tensors are indexed via multi-indexed arrays i = (i1, . . . , id) where in ∈[In]. The set notation [a] is shortand for {z : z ∈Z, 1 ≤z ≤a}. We call i ∈[N] the linear index of a tensor and (i1, . . . , id) the multi-index, we assume a bijective mapping between the two index sets, and we refer to a tensor element as xi ≡X(i) ≡X(i1, . . . , id). The forward map i 7→(i1, . . . , id) is given by the operator ind2sub(X, i) and is defined like the Matlab function of the same name3, while the backward map (i1, . . . id) 7→i is given by the operator sub2ind(X, (i1, . . . , id)), again defined like the corresponding Matlab function4. The nth mode-k slice of a tensor S(k) n is a d -1 subtensor obtained by fixing the kth value of the multi-index to n and allowing the other values to range, i.e., S(1) n = X(n, :, . . . , :). The mode-k matricization of a tensor X rearranges the tensor elements into a matrix X(k) ∈RIk×N/Ik such that element i maps to (ik, i′ k) by i′ k = 1 + k-1 X l=1 (il-1)Il+ d X l=k+1 (il-1)Il/Ik. (2) The reshape operator reshape(X, [s1, . . . , sm]) does not permute the underlying tensor elements (unlike tensor matricization) and is defined like the Matlab function of the same name5. The rank-R canonical polyadic (CP) decomposition [21], [22] of a tensor X is a rank-R tensor M of the form X ≈M = R X j=1 λja(1) j ◦· · · ◦a(d) j , (3) 2https://www.mathworks.com/help/matlab/math/arrayindexing.html 3https://www.mathworks.com/help/matlab/ref/ind2sub.html 4https://www.mathworks.com/help/matlab/ref/sub2ind.html 5https://www.mathworks.com/help/matlab/ref/double.reshape. html 2 TABLE I Methods and codebases for dense mode-k MTTKRPs. Given a tensor Y of shape I1, . . . , Id with factor matrices A1, . . . , Ad, let the Khatri-Rao matrix be Z = Ad ⊙· · · ⊙Ak+1 ⊙Ak-1 ⊙· · · ⊙A1 (see Section II-B for details). We list memory usage and open-source codes for each method, and we list the parallelization method on CPUs and GPUs for each code base. All methods have the same computational cost of O(Rd(Q n In)). methods memory code CPU GPU full Z [6] O R Q n̸=k In N-way Toolbox [7] BLAS-3 TensorLy [8] BLAS-3 BLAS-3 rTensor [9] BLAS-3 partial Z [13] O R k-1 Q n=1 In + dQ k+1 In TTB [6] BLAS-3 PyTTB [11] BLAS-3 GenTen [12] BLAS-3 BLAS-3 no Z (our method) O R dP n=1 In GenTen SIMD SIMT where λ ∈RR is a weight vector and ◦is a d-way outer product. We call Ak = [a(1) 1 , . . . , a(k) d ] ∈RIk×R the modek factor matrix of a tensor X. The form M = JA1, . . . , AdK is referred to as a Kruskal tensor. B. The MTTKRP kernel Algorithms to compute CP decompositions generally fall into two categories: all-at-once [23] or alternating [22], [24]. In both cases, the workhorse kernel is the mode-k matricized tensor times Khatri-Rao product (MTTKRP) defined as G(k) = Y(k) diag(λ)ZT (4) where Z = Ad ⊙· · · ⊙Ak+1 ⊙Ak-1 ⊙· · · ⊙A1 for some a Kruskal tensor M = JA1, . . . , AdK, d-way tensor Y, and weight vector λ ∈RR +. We call algorithms that explicitly construct Z matrix-based MTTKRPs. Matrix-based MTTKRPs have two distinct downsides: the explicit formation the Khatri-Rao matrix Z and matricization Y(k). For a mode-k MTTKRP with CP-rank R, the Khatri-Rao matrix Z is size Q n̸=k In × R, which can dwarf the cost R P n̸=k In of simply storing the factor matrices An individually, leading to O(R Q n̸=k In) memory usage. As for the matricization Y(k), tensors are typically stored in memory as a flat array, and in such cases, the mode-k (for 1 τf/τm. Given a wall-time t, the throughput of the model is measured as gflops = f/t/10243 and the memory bandwidths for each model are given by mops0 = m0/t and mops∞= m∞/t. Peak throughput τf, bandwidth τm, and LM bandwidth ratio l for each device is reported inTable II. For the H100, throughput and bandwidth are taken from NVIDIA's whitepaper7 and we calculate the peak L1 cache bandwidth as #SM's x L1 transfer bytes/cycle x clock rate, which is 123 × 128.98/103 ≈33TB/s, where the L1 transfer bytes per clock cycle is taken from NVIDIAs hopper tuning guide8. Hence, for the H100, we set l = 10. Throughput for the 8480+ is taken from Intel's APP metrics sheet9 while memory bandwidth and L2 cache bandwidth are measured with the STREAM triad benchmark [30], where we use a 2GB array to measure memory bandwidth and a 0.6MB array to measure the L2 cache bandwidth. We analyze the efficacy of our performance models in 7https://www.nvidia.com/en-us/data-center/h100/ 8https://docs.nvidia.com/cuda/hopper-tuning-guide/index.html 9https://www.intel.com/content/www/us/en/support/articles/ 000005755/processors.html 6 TABLE II Peak compute throughput τf , memory bandwidth τm, and cache bandwidth ratio l for each device. Device 1012τf 10244τm l 8480+ 2.28 0.12 8 H100 34 3.35 10 Table III. Our results show that the cache effects play a significant role for each algorithmic variant: the T∞ prediction is much too optimistic for each variant. The MTTKRP-ELEM algorithm fails to surpass even the 0cache predicted time T0 for either device. As discussed in Section III-B, this can be attributed to the uncoalesced repeated reads of the factor-matrices. The MTTKRPSLICE algorithm also fails to surpass T0 on the GPU, though it does so on the CPU. We hypothesize that this discrepancy is due to the much larger cache size of the CPU compared to the GPU. For MTTKRP-SLICE on the CPU and MTTKRP-TILE on both devices, the 0cache model is far to pessimistic while the ∞-cache model is too optimistic, indicating that the algorithms achieve (to some extent) the desired caching effects on the repeated reads of the factor matrices, Our empirical results for these methods align most with the 0, LM-cache model, achieving at least 90% of the predicted time on the CPU and 40% of the predicted time on the GPU. We surmise that the higher accuracy of our models on the CPU is due to the fact that we use empirically measured peak bandwidth, while on the GPU we use the vendor reported ideal world numbers. C. Experiments 1) Selecting the tile size: Our first goal is to select a value for the heuristic tile volume parameter NT in the MTTKRP-TILE algorithm. Table IV presents the numerical performance of a sweep over tile-widths d-1√NT ∈ {2, 4, 6, 8, 10, 12, 14} for the tearing-mode and island coalescence tensors for a fixed R = 32. On the GPU, we find that for the 4-way tensor A, a tile size of 216 is optimal, while for the 5-way tensor B, a tile size of 256 is optimal. On an H100, our maximum occupancy with our standard block configuration of (4, 32) is 64 warps per SM. NSIGHT-compute reports our achieved occupancy to be 60%, but even if we assume a more conservative occupancy of 50%, we have 32 tiles per SM, with each tile rereading 8 × 256 bytes of factor matrices entries, or 64KB of information per SM, 25% of the total L1 cache, a good target given that the L1 cache handles read, write, and atomic instructions on a GPU. On the 8480+, we find that a tile width of 12, i.e., the size of the minimum dimension, is optimal. Our parallel CPU implementation assigns one tile per core, and given that each core has 2MB of L2 cache, this means that the factor matrices take up at most ∼0.5% of the L1 cache. However, our implementation assumes that the tile size is regular, i.e., that d-1√NT ∈Z. As such, performance decreases when d-1√NT > minn=1,...,d In. To impose tile regularity and cache awareness, we use the simple heuristic NT = min d-1 s sLM/4 sf(c/2) !d-1 , min n=1,...,d{In}d-1 , (6) where sLM is the size in bytes of the middle level of cache (i.e., L1 cache (per SM) on a GPU, L2 cache (per core) on a CPU), and c is the maximum number of tiles per cache unit (i.e., SM/core). We find that this heuristic is valid for most R. D. Memory impact of MTTKRP variants As discussed in Section III-A, forming the left and right Khatri-Rao matrices ZT L and ZT R can incur a large storage cost. This cost dominates the total memory required for the mode-k MTTKRP-GEMM algorithm, which is given by mGEMM ∞ = N + R(IL + IR + Ik). It is important to note the storage cost of the MTTKRP-GEMM algorithm relies heavily on the initial shape of a given tensor. For example, the maximum memory usage over all modes k for the island coalescence tensor B for a R = 2000 is 391GB, but if the initial tensor shape was permuted to be 129 × 12 × 129 × 39 × 129 (best case), the memory usage would be 123GB, while if the tensor was permuted to be 129 × 129 × 129 × 39 × 12 (worst case), the memory usage would be 1255GB. However, especially in the context in situ decomposition of scientific simulation data [1], [31], reshaping tensors can eliminate multidimensional relationships between tensor entries and can be expensive and impractical. Fig. 3 shows the maximum memory usage over all modes k for the MTTKRP-GEMM and MTTKRPTILE algorithms for the tearing mode tensor A and the island coalescence tensor B. The y axis of the plot denotes the minimum number of 80GB H100 GPUs needed for storage, however, the distribution protocol in IV-D2 does not evenly assign memory to MPI ranks, hence in practice more GPUs may be required for a given problem size. 1) Single node performance: The linear-in-rank memory scaling of the MTTKRP-GEMM algorithm described in Section IV-D motivates the use of the tearing-mode tensor A to study single-node performance of the different MTTKRP variants. Fig. 4 reports the average MTTKRP performance, measured in gflops, over all modes, for the OpenMP and Cuda execution spaces. We compare the SLICE, TILE, and GEMM variants on the OpenMP space and the ELEM, TILE, and GEMM variants on the Cuda space, where the tile size NT is chosen by Eq. (6). We omit the ELEM variant for OpenMP and the SLICE variant for Cuda given their poor performance alluded to in Table III. For R > 10, MTTKRP-TILE achieves at least 20% of MTTKRP-GEMM's performance, an efficient result given MTTKRP-TILE's memory footprint. Additionally, for R > 10, MTTKRP-TILE performs at least 2× faster 7 TABLE III Execution time T in seconds measured against analytical performance model times T0, T∞for the tensors A, B on the OpenMP and Cuda execution spaces. The tile size NT for the MTTKRP-TILE algorithm is selected via the heuristic in Eq. (6). The CP-rank is fixed to R = 32. OpenMP Cuda T T0 T0,LM T∞ T T0 T0,LM T∞ A ELEM 10.394 3.817 1.349 0.056 1.823 0.138 0.047 0.003 SLICE 0.543 2.877 0.409 0.056 8.528 0.104 0.013 0.003 TILE 0.526 2.955 0.487 0.056 0.043 0.110 0.019 0.003 B ELEM 25.167 9.876 3.054 0.130 3.924 0.356 0.105 0.007 SLICE 1.137 7.927 1.105 0.130 23.032 0.286 0.035 0.007 TILE 1.517 8.414 1.592 0.130 0.120 0.304 0.052 0.007 TABLE IV Sweep over tile widths d-1√NT for the MTTKRP-TILE algorithmic variant with R = 32 for each tensor A, B on both execution spaces, with performance in gflops (higher is better) averaged over all modes. These results motivate a heuristic to choose NT given by Eq. (6). MTTKRP-TILE gflops d-1√NT Tearing mode tensor A Island coalescence tensor B OpenMP Cuda OpenMP Cuda 2 32.9 253.3 51.4 585.1 4 89.9 1313.6 99.0 1232.2 6 98.9 1328.3 106.8 1173.4 8 98.1 1103.7 104.9 944.1 10 96.5 947.3 103.3 884.4 12 99.6 1230.0 110.8 948.8 14 97.4 1159.2 101.4 843.9 10 500 1000 2000 CP-ranks (R) 0 100 200 300 400 Memory Usage (GB) 1 H100 2 H100s 3 H100s 4 H100s 5 H100sTearing mode GEMM TILE 10 500 1000 2000 CP-ranks (R) 1 H100 2 H100s 3 H100s 4 H100s 5 H100s Island coalescence GEMM TILE Fig. 3. Worst case memory usage for the GEMM and TILE MTTKRP variants for the tearing mode tensor A and the island coalescence tensor B for different CP-ranks. The plot is annotated with lines showing the minimum number of NVIDIA H100 GPUs required incur the storage costs. than the SLICE variant on the OpenMP space and at least 3× faster than ELEM on the Cuda space, with a peak speedup of 11× for R = 500. These speedups reinforce the necessity for data-reuse and reduced atomic contention for element based parallelization approaches. 10 500 1000 2000 CP-ranks (R) 0 2000 4000 6000 gflops 29 87 88 101 31 192 204 213 220 630 807 923 OpenMP SLICE TILE GEMM 10 500 1000 2000 CP-ranks (R) 12 122 238 445 522 1385 1386 1401 665 6888 6830 7035 Cuda ELEM TILE GEMM Fig. 4. Tearing mode tensor A average (over all modes) MTTKRP performance (higher is better) for different algorithmic variants on different execution spaces. 2) Comparison against distributed MTTKRP-GEMM: The disparities of memory costs between the MTTKRPTILE and MTTKRP-GEMM algorithms for the island coalescence tensor B (see Fig. 3) motivate us to compare running the MTTKRP-TILE algorithm on a single GPU against running MTTKRP-GEMM algorithm on (the required) multiple GPUs. Distributed MTTKRPs use the commutation pattern introduced in [32] whose communication overhead consists of an initial tensor redistribution and all-reduce communication to combine contributions across processors. Fig. 5 compares the execution time of the CP-ALS algorithmon B with tolerance 10-4 over ranks R = 10, 500, 1000, 2000 on a single device when using the MTTKRP-TILE algorithm and on multiple devices when using the MTTKRP-GEMM algorithm. While tensor and factor matrices fit snugly into memory for the TILE variant, MTTKRP-GEMM requires at least 6 H100s for rank 2000, 3 H100s for rank 1000, and 2 H100s for rank 500 (findings in line with our lower bound memory model). Note that the hops machine requires two nodes to distribute across 6 GPUs, with the multi-node communication bandwidth drastically increasing tensor redistribution and MTTKRP communication time. When R = 2000, MTTKRP-TILE based CP-ALS on a single GPU is able to achieve 67% of the performance of MTTKRP-GEMM based CP-ALS run on the minumum 8 6 H100s. For R = 1000, 4 H100s are required to beat the MTTKRP-TILE based CP-ALS for and at least 5 are required to beat the MTTKRP-TILE based CP-ALs for R = 500. For all tested ranks, tensor redistribution is the most expensive overhead incurred in the distributed CPALS, dwarfing the all-reduce communication time. 10 500 1000 2000 CP-ranks (R) 0 25 50 75 100 125 150 175 Time (s) TILE GEMM-nGPUs=6 GEMM-nGPUs=4 GEMM-nGPUs=3 GEMM-nGPUs=2 other CP redist comm Fig. 5. Time (lower is better) for CP-ALS on the island coalecence tensor B with MTTKRP-TILE algorithm on a single H100 compared with MTTKRP-GEMM on one H100 per process. MTTKRP time is denoted with hatches, other CP-ALS operations (i.e., inner-product, Cholesky solve, etc.) by a grey fill, tensor redistribution time with a purple fill, and MTTKRP communication time with an orange fill. V. Conclusion We introduce fast and memory efficient algorithms for matrix free dense MTTKRPs together with detailed performance analysis and evaluations on CPUs and GPUs. We extend the open-source GenTen package with our methods and demonstrate that state-of-the-art matrix based MTTKRPs need many GPUs to compute the same tensor decomposition that, when replaced with our MTTKRP, can be computed on a single device. Our methods have limitations. Fig. 4 shows that we have yet to achieve the efficiency of the matrix based MTTKRP-GEMM algorithm when the Khatri-Rao product matrix fits in device memory. Table III motivates us to further improve our caching strategies to achieve a higher percentage of the predicted infinite-cache time T∞. Such strategies include GEMM-like tilings of the output matrix and a Hilbert curve or ALTO-like [19] ordering of the input tensor to increase cache locality. References [1] D. M. Dunlavy, E. T. Phipps, H. Kolla, J. N. Shadid, and E. 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Nevertheless, here we find that the nontrivial quantum metric of the occupied Bloch\nstates allows for an electric field induced Berry curvature dipole, which generates a strong and\ntunable second-order Hall current, enabling it to be switched on or off by simply adjusting the\nrelative orientation between the symmetry-reducing dc field and the ac probe field. Specifically,\nwe investigate the electric field induced second-order anomalous Hall response in a two-dimensional\nRashba-coupled hybrid altermagnet that interpolates between dx2−y2 (B1g) and dxy (B2g) altermag-\nnet symmetry, motivated by recent proposals for mixed-symmetry states. Crucially, the nonlinear\nsignal is highly sensitive to the underlying symmetry of the altermagnetic order at specific doping\nlevels, offering a purely electrical method to distinguish distinct altermagnetic orders. Our results\nposition hybrid altermagnets as a promising platform for controllable nonlinear transport and spin-\ntronic applications.\nThe discovery of altermagnets (AMs) has revealed a new\nclass of magnetic order that bridges ferromagnets (FMs)\nand antiferromagnets (AFMs), exhibiting momentum-\ndependent spin splitting but zero net magnetization [1–\n7]. Unlike conventional (N´eel) AFMs that preserve time-\nreversal symmetry (T ) through spin-compensated sublat-\ntices related by translation or inversion (P), AM breaks\nglobal T while their sublattices are instead connected\nby crystalline point-group operations, such as rotations\n(Cn) or mirrors (M) [7, 8].\nConsequently, AMs have\nbeen shown to host exotic electronic features, such as\nsymmetry-protected nodal lines, Weyl planes [9], and\nmomentum-locked spin textures [10], all while suppress-\ning stray magnetic fields thus, making AMs promising\ncandidates for advanced spintronic and topological de-\nvices [11, 12].\nIn AMs, the Berry curvature (BC) exhibits quadrupo-\nlar or higher-order multipolar symmetry in momentum\nspace, resulting in a vanishing momentum-space aver-\nage and, consequently, the absence of any BC-driven\nlinear anomalous Hall response [13, 14], despite broken\nT . Consequently, the leading anomalous Hall response\nmay only emerge at second order. Recently, such second-\norder anomalous Hall effect (SAHE) has attracted con-\nsiderable interest due to its fundamental connection to\nthe quantum geometry of Bloch states and its poten-\ntial for promising technological applications [15–22]. The\nprimary driver for this SAHE is the dipole moment of\nthe asymmetric distribution of the BC over the occupied\nstate, namely the Berry curvature dipole (BCD) [15, 22].\nIn addition to requiring broken P, SAHE is subject to\nstrict crystallographic symmetry constraints [15]. In d-\nwave AMs, both T and four-fold rotational (C4) symme-\ntries are broken individually, but the combined C4T is\n∗ankita.bhattacharya@physics.uu.se\npreserved. As a consequence, recent works have shown\nthat the second-order anomalous Hall response is forbid-\nden in AMs and subsequently, only nonlinear Hall re-\nsponses at the third order have so far been considered\n[13, 23–26].\nThrough a parallel development, recent theoretical [27]\nand experimental [28, 29] works have demonstrated that\nthe SAHE can also occur in systems where the inherent\nBCD is symmetry forbidden, if instead mediated by the\nBerry connection polarizability (BCP) or, equivalently,\nthe band-normalized quantum metric (QM), i.e., the real\npart of the quantum geometric tensor. [18, 30, 31]. An\nexternal dc electric field can induce a finite field-induced\nBCD through the nontrivial QM of the occupied Bloch\nbands, making the SAHE present even in systems where\nthe inherent BCD-induced SAHE is absent and, impor-\ntantly, also provide greater external tunability. Notably,\nthis electric-field driven SAHE is distinct from the QM-\ndipole-induced intrinsic (independent of relaxation time)\nSAHE, reported in PT -symmetric AFM [18].\nIn this work we consider two-dimensional (2D) hybrid\nAMs. While prototypical AMs such as RuO2 [32–34] and\nMnTe [35–37] realize distinct dx2−y2 (B1g) and dxy (B2g)\nspin-splitting, respectively [7, 32, 38], theoretical classi-\nfications have emphasized that these two order param-\neters, belonging to different irreducible representations\nof the tetragonal point group D4h, can in principle co-\nexist once the crystal symmetry is reduced to one of its\nlower-symmetry subgroups [39, 40]. Moreover, lowering\nthe crystal symmetry, through strain [41–44] or surface\ntermination [40, 45], may induce mixing between other-\nwise pure altermagnetic orders. Experimental techniques\nsuch as nanoscale domain engineering [36, 37] also al-\nlow for controlled tuning between these symmetry limits.\nMotivated by these developments, we consider a hybrid\naltermagnetic order that continuously interpolates be-\ntween the B1g and B2g irreducible representations, cap-\narXiv:2510.14899v1 [cond-mat.mes-hall] 16 Oct 2025\n\n2\nturing the possibility of mixed-symmetry states in real-\nistic materials.\nWe also consider AMs that inherently\nhost Rashba spin-orbit coupling (RSOC) due to common\nstructural asymmetry [46, 47], which ensures the broken\nP required for any finite SAHE [15].\nUsing hybrid AMs with RSOC as realistic platform for\nAMs, we theoretically establish the existence of an elec-\ntric field induced BCD-assisted SAHE in AMs. In par-\nticular, this exploits the BCP or equivalently, the QM, to\novercome the C4T constraints that otherwise prevents an\ninherent BCD-induced SAHE. Quite remarkably, we also\nfind that the electric field induced SAHE varies substan-\ntially in magnitude between the dx2−y2 and dxy limits in\ncertain doping regimes, thus enabling a purely electrical\nroute to distinguish the two altermagnetic orders. Our\nfindings establish AMs as a powerful platform to realize\nhighly tunable nonlinear Hall transport already at second\norder, driven by quantum geometric effects.\nModel hybrid altermagnet (AM).— To investigate the\nSAHE in hybrid AMs, we introduce a minimal two-band\nHamiltonian defined on a 2D square lattice.\nIn this\nmodel, we phenomenologically mix the B1g and B2g com-\nponents of the altermagnetic orderings to capture the es-\nsential physics of a hybrid AM. The tight-binding Hamil-\ntonian is given by [48–50]\nH(k) = ϵ(k) σ0 + h(k) · σ,\n(1)\nwhere σν denote the Pauli matrices for ν = 1, 2, 3 and\nthe 2 × 2 identity matrix for ν = 0 acting in the spin-\nsublattice subspace. The components of the Hamiltonian\nare\nϵ(k) = −2t (cos kx + cos ky) −µ,\n(2)\nh(k) = {−λ sin ky, λ sin kx, tam α (cos kx −cos ky)\n+ tam (1 −α) sin kx sin ky}\n(3)\nwhere t denotes the nearest-neighbor hopping amplitude,\nand µ is the chemical potential, λ denotes the RSOC\nstrength, and tam is the strength of the altermagnetic\nspin splitting . We parameterize the relative weight of\nthe two altermagnetic orders by 0 ≤α ≤1.\nIn two\ndimensions, AMs inherently exhibit RSOC due to struc-\ntural asymmetry [46, 47]. The case α = 1 corresponds\nto a pure dx2−y2 AM order arising from nearest-neighbor\nspin-dependent hopping, while α = 0 yields pure dxy AM\norder governed by next-nearest-neighbor spin-dependent\nhopping. Intermediate values of α thus describe a hybrid\nAM that seamlessly interpolates between the two irre-\nducible representations B1g and B2g of the point group\nD4h. Such hybridization significantly enriches the spin\nsplitting .\nThe square lattice Hamiltonian in Eq. (1) may be rele-\nvant for tetragonal AMs such as RuO2 [32–34] and MnTe\n[35–37], and is directly relevant to quasi-2D layered com-\npounds like Ca3Ru2O7 that host both altermagnetism\nand RSOC [41]. Additionally, it naturally accommodates\nthe dx2−y2 (B1g) and dxy (B2g) [2, 3] orders whose hy-\nbridization we study.\nSecond-order anomalous Hall effect (SAHE).— Al-\nthough the Hamiltonian in Eq. (1) breaks T , it exhibits\nno anomalous Hall response at linear order in an ap-\nplied ac electric field Eω with frequency ω, owing to the\npresence of combined C4T symmetry. Consequently, the\nleading anomalous Hall transport may at most arise at\nsecond order in Eω.\nWithin semi-classical Boltzmann\ntheory, the second-order anomalous Hall current can be\nexpressed as jAH\ni\n= χAH\nijk Eω\nj Eω\nk , with the second-order\ntransverse conductivity tensor χAH\nijk [15]\nχAH\nijk = ϵilk\ne3τ\n2ℏ2(1 + iωτ)\nX\nn\nZ\nddk\n(2π)d f0 (∂kjΩn\nl ),\n(4)\nwhere ϵilk is the Levi-Civita symbol and f0 is the equilib-\nrium Fermi-Dirac distribution function and Ωn\nl is the l-th\ncomponent of BC of the nth band. Here, τ is the relax-\nation time that makes the SAHE an extrinsic effect. The\nintegral is the first moment of the BC over the occupied\nstates, known as the Berry curvature dipole (BCD) [15]\nDjl =\nX\nn\nZ\nddk\n(2π)d f0 (∂kjΩn\nl ).\n(5)\nFor the BCD, or equivalently χAH\nijk , to be nonzero, in-\nversion symmetry P must be broken, otherwise, the mo-\nmentum derivative ∂kjΩn\nl (k) transforms as an odd func-\ntion of momentum, causing its Brillouin zone integral to\nvanish. In Eq. (1), the Rashba term λ explicitly breaks\nP. Additionally, the BCD obeys a severe symmetry con-\nstraint, namely the presence of two or more mirror lines\nin a crystal forces the SAHE to still vanish, while a sin-\ngle mirror line forces it to be orthogonal to that mirror\nplane [15, 51, 52]. Thus, mirror symmetries Mx,y, when\npresent, constrain BCD and thus, the SAHE.\nPure dx2−y2 altermagnetic order, i.e., when α = 1 in\nEq. (1), breaks both Mx and My mirror symmetries,\nwhile purely dxy, i.e., α = 0, preserves both Mx,y. The\npresence of both Mx,y thus directly prohibits any BCD-\ninduced SAHE for α = 0. In contrast, the absence of both\nmirror planes for α = 1, in principle, should allow SAHE.\nHowever, the presence of combined C4T symmetry in\nAMs has been shown to completely forbid BCD-driven\nSAHE for any values of α, thus forcing the leading order\nHall transport to be third order [13, 14, 53, 54]. Next\nwe will show that the nontrivial QM of the Bloch states\nactually still enables an extrinsic second-order transverse\nresponse in the presence of a dc electric field in AMs.\nElectric field induced SAHE.— In the presence of a\nsymmetry-reducing dc electromagnetic field Edc, the\nBC of the Bloch electrons is modified due to the po-\nsitional shift of Bloch electrons arising from interband\nmixing [17]. This results in a field-induced Berry con-\nnection that is expressed as AE =\n↔\nG Edc, where\n↔\nG is a\nsecond-rank tensor, called the BCP. The component Gn\nab\nof the BCP tensor is defined as [17, 55, 56]\nGn\nab(k) = 2 Re\nX\nm̸=n\nAnm\na\n(k) Amn\nb\n(k)\nϵn(k) −ϵm(k) .\n(6)\n\n3\n−π\n2\n0\nπ\n2\nky\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−20\n0\n20\n−50\n0\n50\nFIG. 1. Berry connection polarizability (BCP) tensor com-\nponents for one of the band: (a) G1\nxx, (b) G1\nxy, and (c) G1\nyy\nin the first Brillouin zone. Field-induced Berry curvature for\nthe same band , ΩE(k), for dc-field orientations (d) θ = 0, (e)\nπ/4, and (f) π/2, with range arrows indicating electric field\ndirections. Calculations performed with t = 1.0, tam = 0.5t,\nα = 0.5, µ = 0.3t, and λ = 0.08t.\nHere, the interband Berry connection is Anm\na\n(k) =\n⟨un|i∂a|um⟩, where ∂a ≡∂ka and |un⟩is the periodic part\nof the n-th unperturbed Bloch state, with ϵn the corre-\nsponding band energy. Notably, Gn\nab(k) is closely linked\nwith the QM, gn\nab = 2 P\nm̸=n Re[Anm\na\n(k) Amn\nb\n(k)], the\nreal part of the quantum geometric tensor [30]. A non-\ntrivial BCP, or equivalently QM, generates in turn a field-\ninduced BC ΩE\nn = ∇k×AE\nn , which, similar to the original\nBC of the Bloch band, acts like a pseudo-magnetic field\nin k-space which yields a SAHE, just as the original BC\nin Eq. (4). The field-induced BC is especially important\nin those systems where either the intrinsic BC vanishes\nor the original BCD-induced SAHE is forbidden due to\nthe symmetry constraints, as discussed above for AMs.\nIn typical Hall transport measurements, the applied\nelectric field and the measured current are both restricted\nto the x−y plane. Then, an external electric field applied\nat an angle θ relative to the x-axis, such that Edc =\nEdc(cos θ, sin θ), induces a field-induced BC of the form\n[27, 55, 57],\nΩE\nnz = Edc[(∂kxGn\nyx −∂kyGn\nxx) cos θ\n(7)\n+ (∂kxGn\nyy −∂kyGn\nxy) sin θ],\nwhich is dependent on both the magnitude and direction\nof Edc. This field-induced BC can lead to a finite field-\ninduced BCD following Eq. (5) with an angular depen-\ndence DE(θ) =\n\u0000DE\nxz(θ), DE\nyz(θ)\n\u0001\n.\nDue to the lowered\nsymmetry in the presence of Edc, a field-induced BCD\nthen gives rise to the SAHE.\nFinally, plugging the angular dependence of the BCD\ninto Eq. (4), and recasting it through χAH = j2ω/(Eω)2,\nwe arrive at the second-harmonic current j2ω [27, 28]\nj2ω = −\ne3τ\n2(1 + iωτ)ℏ2 (ˆz × Eω) [DE(θ) · Eω],\n(8)\nwhere Eω is the in-plane ac Hall probe field that makes an\nangle ϕ with the x-axis and satisfies the condition Eω ≪\nEdc. As seen from Eq. (8), the maximum Hall response\nis obtained when Eω ∥DE(θ), while it vanishes for Eω ⊥\nDE(θ). Overall, the angular dependence j2ω is intricate\nas it not only varies with θ through DE(θ) but also with\nϕ, thus providing a high degree of external tunability of\nthe SAHE. In 2D, only the two components χyxx and\nχxyy of the tensor χAH\nijk are finite and independent. Thus\nχAH simplifies to [18, 27]\nχAH(θ, ϕ) = χyxx(θ) cos ϕ −χxyy(θ) sin ϕ.\n(9)\nSAHE in altermagnets.— To illustrate how the electric\nfield induced SAHE emerges in AMs described by Eq. (1),\nwe start by showcasing the k-resolved distribution of the\ncomponents of the BCP tensor, or equivalently, band-\nnormalized QM for one of the two bands, setting α = 0.5\nin Fig. 1 (a-c). For the other band, the contributions are\nexactly equal in magnitude but opposite in sign. For pure\naltermagnetic orders, i.e., for α = 0 or α = 1, the com-\nponents of BCP tensors are shown in the Supplementary\nMaterial (SM) [58]. The corresponding band structures\nfor these three α values are shown in the SM [58]. Consis-\ntent with expectations, the BCP values exhibit maxima\nat the band-touching points, due to the denominator in\nEq. (6). The presence of a finite BCP leads to a finite\nBC, in accordance with Eq. (7), when an external dc\nelectric field Edc is applied. For Edc oriented along the\n−5.0\n−2.5\n0.0\n2.5\n5.0\n−π\n2\n0\nπ\n2\nky\n(a)\n(b)\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\n(c)\n−π\n2\n0\nπ\n2\nkx\n(d)\nFIG. 2.\nMomentum distribution of the Fermi-function-\nweighted derivatives of the field-induced Berry curvature in\nlog-scale for the two bands, (a) f0 ∂kxΩE\n1 , (b) f0 ∂kyΩE\n1 , and\nfor band 2, (c) f0 ∂kxΩE\n2 , (d) f0 ∂kyΩE\n2 , evaluated at θ = π/4.\nParameters are the same as in Fig. 1.\n\n4\n0\nπ\n2\nπ\n3π\n2\n2π\nφ\n−0.02\n0.00\n0.02\nχAH\nθ = 0\nθ = π\n6\nθ = π\n4\nθ = π\n2\nθ = 3π\n4\nFIG. 3. Second-order Hall conductivity χAH as a function of\nangle ϕ, set by the ac driving field Eω, for several different\nangles θ, set by the symmetry-breaking static field Edc, with\nthe x-axis. χAH is scaled by the prefactor A =\ne3τ\n2(1+iωτ)ℏ2 and\nthe parameters are the same as in Fig. 1.\nx (θ = 0), x = y (θ = π/4), and y(θ = π/2), the field-\ninduced BC ΩE are shown in Fig. 1 (d)-(f), respectively.\nThe k-resolved distribution of ΩE shows a clear dipolar\nstructure. Not only the magnitude, but also the direc-\ntion of the ΩE and, thus, also the resulting BCD, can be\nexternally tuned by changing the direction of Edc [27].\nOnce a finite and field-tunable BCD is confirmed, it\nis straightforward to express χAH in terms of Dxz and\nDyz following Eq. (4-5). Figure 2 shows the momentum-\nresolved components of the Fermi-weighted Berry curva-\nture dipole, i.e., the integrand of Eq. (5) for both bands\nagain for α = 0.5, with Edc oriented along the x = y\ndirection. These are then integrated over the Brillouin\nzone and summed over bands to yield the finite field-\ninduced BCD. Finally, following Eq. (4), with it angular\ndependence in Eq. (9), we find an overall nonzero second-\norder Hall response. The complex angular variation of\nthe resulting second-order Hall conductivity χAH(θ, ϕ) in\na hybrid AM is depicted in Fig. 3. It offers significant\nexperimental control over the second-order Hall current,\nenabling it to be switched on or off simply by adjusting\nthe relative orientation of Edc and Eω. These behaviors\nare similarly observed for the pure altermagnetic order,\ni.e., for α = 1 and α = 0, see the SM [58].\nRemarkably, we find that the electric field induced\nSAHE exhibits a strong sensitivity in magnitude between\ndifferent altermagnetic orders, at least for a range of dop-\ning levels, set by µ. The response is maximized at α = 0,\ncorresponding to a pure dxy-type altermagnetic order\nwhen Edc ⊥Eω, but as α increases, the response χAH\ndecreases and becomes negligible for α →1, i.e., pure\ndx2−y2 order, see Fig. 4(a). In contrast, when Edc ∥Eω,\nχAH is found to be vanishingly small for both α = 0\nand α = 1 as shown in Fig. 4(b). Thus, simply tuning\nthe direction of either Edc or Eω, provides a straight-\nforward way to conclusively distinguish between pure or-\nders dxy and dx2−y2. For the parameter choices used in\nthe presented results, this doping range is found to be\n0.0\n0.5\n1.0\nα\n0.0\n0.5\n1.0\n1.5\nχAH\n(a) θ = 0, φ = π\n2\nλ = 0.08\nλ = 0.1\nλ = 0.15\n0.0\n0.5\n1.0\nα\n−0.8\n−0.6\n−0.4\n−0.2\n0.0\n(b) θ = 0, φ = π\nλ = 0.08\nλ = 0.1\nλ = 0.15\nFIG. 4.\nSecond-order Hall conductivity χAH as a function\nof α for different RSOC λ values and field orientations: (a)\nθ = 0, ϕ = π/2 and (b) θ = 0, ϕ = π with t = 1, tam = 0.5t,\nand µ = 0.3t (same as Fig. 1).\n−t < µ < t. We have also tested our results for smaller\nand larger tam, which shifts the corresponding doping\nrange, although it is still clearly present. In this doping\nregime, the system exhibits two hole-like Fermi pockets\nfor α = 0, whereas for α = 0.5 and α = 1, it features\none electron and one hole Fermi pocket, see SM [58].\nWhen the Fermi surfaces (FSs) have the opposite curva-\nture, their contributions to Hall transport tend to cancel\neach other, resulting in a nearly vanishing response. In\ncontrast, when both FSs have the same curvature, their\ncontributions add constructively, leading to a finite over-\nall response.\nFor example, in the doping regime with\ntwo electron pockets (regardless of α), the field-induced\nSAHE remains nonzero for both pure and hybrid alter-\nmagnetic symmetries, see the SM [58]. In this scenario,\npure altermagnetic orders cannot be distinguished based\non the second-order response, as all yield a finite signal.\nThese findings unambiguously establish the presence\nof a nonlinear Hall response in AM at second order and\nalso offer an all-electrical approach to resolve pure alter-\nmagnetic orderings. As shown in Fig. 4, the electric field\ninduced SAHE increases when the strength of RSOC λ\ndecreases. Still, we recall that no SAHE is possible for\nλ = 0 due to preserved P. At first glance, this may seem\nnon-intuitive. However, we note that for finite λ, RSOC\nopens gaps at the band-touching points in altermagnetic\nmetals. When λ is small, these gaps are narrow, leading\nto an enhanced BCP (see Eq. (6)), which in turn am-\nplifies the field-induced SAHE. The evolution of the two\nspin-split FSs with λ in various doping limits are shown\nin the SM [58].\nIn summary, we demonstrate that Rashba-coupled hy-\nbrid AMs provide a viable platform for electric field in-\nduced SAHE that arises due to the nontrivial QM of the\nBloch bands. Although combined C4T symmetry forbids\nboth the linear anomalous Hall effect and the usual BCD-\ndriven second-order signal, an external dc field lowers the\nsymmetry and generates a finite field-induced BCD via\nthe nontrivial QM of the occupied Bloch states.\nThe\nintricate angular variation of the resulting second-order\nHall conductivity χAH(θ, ϕ) provides substantial exper-\nimental control over the second-order Hall current, al-\nlowing it to be toggled on or off by merely adjusting the\n\n5\nrelative orientation between the external dc field Edc and\nac probe field Eω. In certain doping regimes, its magni-\ntude can even probe the underlying altermagnetic form\nfactor, peaking in the dxy (B2g) limit and decreasing to-\nward dx2−y2 (B1g) configuration for Edc ⊥Eω, which\nthen provides a purely electrical means to distinguish the\ntwo.\nThese predictions are directly testable in candidate\nmaterial platforms such as epitaxial RuO2 [32–34] and\nMnTe [35–37] thin films, where strain or surface termina-\ntion can realize hybrid B1g–B2g textures and provide the\nrequired inversion-symmetry breaking at interfaces that\ngenerate interfacial RSOC. To observe the SAHE, an ex-\nperimental setup similar to the one used in Refs. [28, 29]\ncould be employed.\nMore broadly, our results demon-\nstrate that quantum geometry can enable nonlinear re-\nsponses in systems where symmetry otherwise suppresses\nintrinsic BC effects, enabling functionalities such as on-\nchip frequency doubling and rectification in zero-moment\nmagnets.\nA.B and A.M.B.-S. acknowledge financial support\nfrom the Swedish Research Council (Vetenskapsr˚adet)\ngrant no. 2022-03963 and the European Union through\nthe\nEuropean\nResearch\nCouncil\n(ERC)\nunder\nthe\nEuropean Union’s Horizon 2020 research and inno-\nvation programme (ERC-2022-CoG, Grant agreement\nno. 101087096). Views and opinions expressed are how-\never those of the author(s) only and do not necessar-\nily reflect those of the European Union or the Eu-\nropean Research Council Executive Agency.\nNeither\nthe European Union nor the granting authority can\nbe held responsible for them.\nB.S. acknowledges fi-\nnancial support from Swedish Research Council (grant\nno. 2022-04309), STINT Mobility Grant for Internation-\nalization (grant no. MG2022-9386) and DST-SPARC, In-\ndia (Ref. No. SPARC/2019-2020/P1879/SL). A.M. ac-\nknowledges the computational resources provided by the\nNational Academic Infrastructure for Supercomputing\nin Sweden (NAISS) at NSC and PDC (NAISS 2024/3-\n40) partially funded by the Swedish Research Council\nthrough grant agreement no. 2022-06725 and at UPP-\nMAX (NAISS 2025/2-203). B.S. also acknowledges the\nallocation of supercomputing hours granted by the Eu-\nroHPC JU Development Access call in LUMI-C super-\ncomputer (grant no. 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We also examine a doping\nregime in which both pure and hybrid altermagnetic orders feature Fermi pockets with similar curvature, allowing for\na direct comparison with the main text, where the results correspond to a doping range in which the altermagnetic\norders exhibit either Fermi surfaces with the same curvature or with opposite curvature.\nI.\nFIELD-INDUCED SECOND-ORDER HALL RESPONSE FOR PURE ALTERMAGNETIC ORDERS\nIn this section we show the electric field induced second-order Hall response for pure altermagnetic orders. For our\nmodel Hamiltonian Eq. (1) in the main text we consider hybrid altermagnetic order, i.e., a combination of dx2−y2\nand dxy symmetries by keeping α = 0.5. Here we instead consider the results for pure dx2−y2 and dxy altermagnetic\norder. In Fig. S1 and Fig. S2, we show the components of the Berry connection polarizability tensor (BCP) for one\nof the two bands and the field-induced Berry curvature for three different orientations of the symmetry-breaking field\nEdc for α = 1, i.e., for pure dx2−y2 symmetry and for α = 0, i.e., for pure dxy symmetry, respectively. These figures\nare equivalent to Fig. 1 in the main text.\n−π\n2\n0\nπ\n2\nky\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−20\n0\n20\n−50\n0\n50\nFIG. S1. Berry connection polarizability (BCP) tensor components for one of the band : (a) G1\nxx, (b) G1\nxy, and (c) G1\nyy in\nthe first Brillouin zone. Field-induced Berry curvature for the same band , ΩE(k), for dc-field orientations (d) θ = 0, (e) π/4,\nand (f) π/2, with orange arrows indicating electric field directions. Calculations performed with t = 1.0, tam = 0.5t, α = 1.0,\nµ = 0.3t, and λ = 0.08t.\nThe k-resolved distribution of ΩE shows a clear dipolar structure, which corresponds directly to a finite field-induced\nBCD. Not only the magnitude but also the direction of ΩE and, thus, the resulting BCD can be externally tuned by\nchanging the direction of Edc. The corresponding momentum-resolved components of the Fermi-weighted dipole of\nBC for the two bands for α = 1 and α = 0, with Edc oriented along the x = y direction, are shown in Fig. S4 and\nFig. S5, respectively, similar to Fig. 2 in the main text. For α = 1, the system possesses one electron pocket and one\nhole pocket, similar to α = 0.5, while for α = 0, it has two hole pockets. Figure S6 shows the resulting second-order\nconductivity χAH(θ, ϕ) as a function of ϕ for various values of θ for the two pure orders. The second-order response\nχAH(θ, ϕ) for α = 0 is more than an order of magnitude higher than that for α = 1. When the Fermi surfaces (FSs)\nare of opposite types, as in the case for α = 0.5 or 1, their contributions tend to cancel each other, resulting in a small\nresponse. In contrast, when both FSs have same types of curvature as in the case for α = 1, their contributions add\nconstructively, leading to a higher overall response.\n\n8\n−π\n2\n0\nπ\n2\nky\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−π\n2\n0\nπ\n2\nkx\nEdc\n−20\n0\n20\n−50\n0\n50\nFIG. S2. Berry connection polarizability (BCP) tensor components for one of the band : (a) G1\nxx, (b) G1\nxy, and (c) G1\nyy in\nthe first Brillouin zone. Field-induced Berry curvature for the same band , ΩE(k), for dc-field orientations (d) θ = 0, (e) π/4,\nand (f) π/2, with orange arrows indicating electric field directions. Calculations performed with t = 1.0, tam = 0.5t, α = 0.0,\nµ = 0.3t, and λ = 0.08t.\nΓ\n(0, 0)\nY\n(0, π)\nS2\n(−π, π)\nΓ\n(0, 0)\nS1\n(π, π)\nX\n(π, 0)\nΓ\n(0, 0)\n−4\n−2\n0\n2\n4\nE\n(a)\nBand 1\nBand 2\nΓ\n(0, 0)\nY\n(0, π)\nS2\n(−π, π)\nΓ\n(0, 0)\nS1\n(π, π)\nX\n(π, 0)\nΓ\n(0, 0)\n(b)\nBand 1\nBand 2\nΓ\n(0, 0)\nY\n(0, π)\nS2\n(−π, π)\nΓ\n(0, 0)\nS1\n(π, π)\nX\n(π, 0)\nΓ\n(0, 0)\n(c)\nBand 1\nBand 2\nFIG. S3. Band structure along high-symmetry paths for (a) α=0 (pure dxy), (b) α=0.5 (hybrid), and (c) α=1 (pure dx2−y2).\nBlue and red lines represent the two spin-split bands band 1 and band 2, respectively. The dashed line indicates the Fermi\nlevel. Parameters: t = 1.0, µ = 0.3t, λ = 0.08t, tam = 0.5t.\nII.\nTOPOGRAPHY OF FERMI SURFACES\nIn Fig. S7 and Fig. S8, we present the spin-split Fermi surfaces for various altermagnetic orders across different\ndoping regimes. In the absence of Rashba spin–orbit coupling (RSOC) λ, band-touching points are present, but a\ngap opens at these points as soon as λ is tuned to a finite value. The second-order Hall response increases with\ndecreasing RSOC λ, as shown in Fig. S9. This is because for a smaller value of λ, the energy gaps between bands are\nreduced, leading to an enhanced quantum metric or Berry curvature polarization (BCP), which in turn results in a\nstronger field-induced second-order response. In Fig. S9(a), we show the variation of χAH with α for µ = −3t keeping\nEdc ⊥Eω. In contrast to the results in Fig. 4 in the main text, we find that χAH is finite similar in magnitude for all\nα. In this doping limit, there are also two electron pockets for all α in contrast to the doping limit considered in the\nmain text. In this case, the contribution of the two similar Fermi surfaces add constructively to give an overall finite\nresponse for any α. In Fig. S9(b), the same plot is shown for Edc ∥Eω, with an overall small χAH, showing that also\nin this case there is large tunability with field-directions.\n\n9\n−5.0\n−2.5\n0.0\n2.5\n5.0\n−π\n2\n0\nπ\n2\nky\n(a)\n(b)\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\n(c)\n−π\n2\n0\nπ\n2\nkx\n(d)\nFIG. S4. Momentum distribution of the Fermi-function-weighted derivatives of the field-induced Berry curvature in log-scale\nfor the two bands, (a) f0 ∂kxΩE\n1 , (b) f0 ∂kyΩE\n1 , and for band 2, (c) f0 ∂kxΩE\n2 , (d) f0 ∂kyΩE\n2 , evaluated at θ = π/4. Parameters\nare the same as in Fig. S1.\n\n10\n−5.0\n−2.5\n0.0\n2.5\n5.0\n−π\n2\n0\nπ\n2\nky\n(a)\n(b)\n−π\n2\n0\nπ\n2\nkx\n−π\n2\n0\nπ\n2\nky\n(c)\n−π\n2\n0\nπ\n2\nkx\n(d)\nFIG. S5. Momentum distribution of the Fermi-function-weighted derivatives of the field-induced Berry curvature in log-scale\nfor the two bands, (a) f0 ∂kxΩE\n1 , (b) f0 ∂kyΩE\n1 , and for band 2, (c) f0 ∂kxΩE\n2 , (d) f0 ∂kyΩE\n2 , evaluated at θ = π/4. Parameters\nare the same as in Fig. S2.\n0\nπ\n2\nπ\n3π\n2\n2π\nφ\n−1\n0\n1\nχAH\n(a) α = 0.0\nθ = 0\nθ = π\n6\nθ = π\n4\nθ = π\n2\nθ = 3π\n4\n0\nπ\n2\nπ\n3π\n2\n2π\nφ\n−0.04\n−0.02\n0.00\n0.02\n0.04\n(b) α = 1.0\nFIG. S6. Second-order Hall conductivity χAH as a function of angle ϕ, set by the ac driving field Eω, for several different angles\nθ, set by the symmetry-breaking static field Edc, with the x-axis for (a) α = 0, (b) α = 1. χAH is scaled by the prefactor\nA =\ne3τ\n2(1+iωτ)ℏ2 and the parameters are the same as in Fig. S3.\n\n11\n−π/2\n0\nπ/2\nky\n(a)\nα = 0.0, λ = 0.00\n(b)\nα = 0.5, λ = 0.00\n(c)\nα = 1.0, λ = 0.00\n−π/2\n0\nπ/2\n−π/2\n0\nπ/2\nky\n(d)\nα = 0.0, λ = 0.10\n−π/2\n0\nπ/2\nkx\n(e)\nα = 0.5, λ = 0.10\n−π/2\n0\nπ/2\n(f)\nα = 1.0, λ = 0.10\n−1.0\n−0.5\n0.0\n0.5\n1.0\nFIG. S7. Spin-split Fermi surfaces (a,d) α = 0; (b,e) α = 0.5 and (c,f) α = 1.0. Upper row is for λ = 0 and lower row for\nλ = 0.1t. Chemical potential is set to µ = 0.3t, same as the main text.\n\n12\n−π/2\n0\nπ/2\nky\n(a)\nα = 0.0, λ = 0.00\n(b)\nα = 0.5, λ = 0.00\n(c)\nα = 1.0, λ = 0.00\n−π/2\n0\nπ/2\n−π/2\n0\nπ/2\nky\n(d)\nα = 0.0, λ = 0.10\n−π/2\n0\nπ/2\nkx\n(e)\nα = 0.5, λ = 0.10\n−π/2\n0\nπ/2\n(f)\nα = 1.0, λ = 0.10\n−1.0\n−0.5\n0.0\n0.5\n1.0\nFIG. S8. Spin-split Fermi surfaces (a,d) α = 0; (b,e) α = 0.5 and (c,f) α = 1.0. Upper row is for λ = 0 and lower row for\nλ = 0.1t. Chemical potential is set to µ = −3.0t.\n0.0\n0.5\n1.0\nα\n0.15\n0.20\n0.25\n0.30\nχAH\n(a) θ = 0, φ = π\n2\nλ = 0.08\nλ = 0.1\nλ = 0.15\n0.0\n0.5\n1.0\nα\n−0.02\n−0.01\n0.00\n0.01\n0.02\n(b) θ = 0, φ = π\nλ = 0.08\nλ = 0.1\nλ = 0.15\nFIG. S9. Second-order Hall conductivity χAH as a function of α for different RSOC λ values and field orientations: (a) θ = 0,\nϕ = π/2 and (b) θ = 0, ϕ = π with t = 1, tam = 0.5t, and µ = −3t (same as Fig. S3).\n"},"cleaned":{"kind":"string","value":"Electric field controlled second-order anomalous Hall effect in altermagnets Arnob Mukherjee, Biplab Sanyal, Annica M. Black-Schaffer, and Ankita Bhattacharya∗ -516, S-75120 Uppsala, Sweden (Dated: October 17, 2025) Altermagnets are a recently discovered class of compensated magnets with momentum-dependent spin splittings and unusual transport properties, even without a net magnetization. In the presence of combined four-fold rotation and time-reversal (C4T ) symmetry, linear and also second-order, driven by a Berry curvature dipole, anomalous Hall responses are forbidden in any pure d-wave altermagnet. Nevertheless, here we find that the nontrivial quantum metric of the occupied Bloch states allows for an electric field induced Berry curvature dipole, which generates a strong and tunable second-order Hall current, enabling it to be switched on or off by simply adjusting the relative orientation between the symmetry-reducing dc field and the ac probe field. Specifically, we investigate the electric field induced second-order anomalous Hall response in a two-dimensional Rashba-coupled hybrid altermagnet that interpolates between dx2-y2 (B1g) and dxy (B2g) altermagnet symmetry, motivated by recent proposals for mixed-symmetry states. Crucially, the nonlinear signal is highly sensitive to the underlying symmetry of the altermagnetic order at specific doping levels, offering a purely electrical method to distinguish distinct altermagnetic orders. Our results position hybrid altermagnets as a promising platform for controllable nonlinear transport and spintronic applications. The discovery of altermagnets (AMs) has revealed a new class of magnetic order that bridges ferromagnets (FMs) and antiferromagnets (AFMs), exhibiting momentumdependent spin splitting but zero net magnetization [17]. Unlike conventional (N ́eel) AFMs that preserve timereversal symmetry (T ) through spin-compensated sublattices related by translation or inversion (P), AM breaks global T while their sublattices are instead connected by crystalline point-group operations, such as rotations (Cn) or mirrors (M) [7, 8]. Consequently, AMs have been shown to host exotic electronic features, such as symmetry-protected nodal lines, Weyl planes [9], and momentum-locked spin textures [10], all while suppressing stray magnetic fields thus, making AMs promising candidates for advanced spintronic and topological devices [11, 12]. In AMs, the Berry curvature (BC) exhibits quadrupolar or higher-order multipolar symmetry in momentum space, resulting in a vanishing momentum-space average and, consequently, the absence of any BC-driven linear anomalous Hall response [13, 14], despite broken T . Consequently, the leading anomalous Hall response may only emerge at second order. Recently, such secondorder anomalous Hall effect (SAHE) has attracted considerable interest due to its fundamental connection to the quantum geometry of Bloch states and its potential for promising technological applications [15-22]. The primary driver for this SAHE is the dipole moment of the asymmetric distribution of the BC over the occupied state, namely the Berry curvature dipole (BCD) [15, 22]. In addition to requiring broken P, SAHE is subject to strict crystallographic symmetry constraints [15]. In dwave AMs, both T and four-fold rotational (C4) symmetries are broken individually, but the combined C4T is ∗ preserved. As a consequence, recent works have shown that the second-order anomalous Hall response is forbidden in AMs and subsequently, only nonlinear Hall responses at the third order have so far been considered [13, 23-26]. Through a parallel development, recent theoretical [27] and experimental [28, 29] works have demonstrated that the SAHE can also occur in systems where the inherent BCD is symmetry forbidden, if instead mediated by the Berry connection polarizability (BCP) or, equivalently, the band-normalized quantum metric (QM), i.e., the real part of the quantum geometric tensor. [18, 30, 31]. An external dc electric field can induce a finite field-induced BCD through the nontrivial QM of the occupied Bloch bands, making the SAHE present even in systems where the inherent BCD-induced SAHE is absent and, importantly, also provide greater external tunability. Notably, this electric-field driven SAHE is distinct from the QMdipole-induced intrinsic (independent of relaxation time) SAHE, reported in PT -symmetric AFM [18]. In this work we consider two-dimensional (2D) hybrid AMs. While prototypical AMs such as RuO2 [32-34] and MnTe [35-37] realize distinct dx2-y2 (B1g) and dxy (B2g) spin-splitting, respectively [7, 32, 38], theoretical classifications have emphasized that these two order parameters, belonging to different irreducible representations of the tetragonal point group D4h, can in principle coexist once the crystal symmetry is reduced to one of its lower-symmetry subgroups [39, 40]. Moreover, lowering the crystal symmetry, through strain [41-44] or surface termination [40, 45], may induce mixing between otherwise pure altermagnetic orders. Experimental techniques such as nanoscale domain engineering [36, 37] also allow for controlled tuning between these symmetry limits. Motivated by these developments, we consider a hybrid altermagnetic order that continuously interpolates between the B1g and B2g irreducible representations, cap16 Oct 2025 2 turing the possibility of mixed-symmetry states in realistic materials. We also consider AMs that inherently host Rashba spin-orbit coupling (RSOC) due to common structural asymmetry [46, 47], which ensures the broken P required for any finite SAHE [15]. Using hybrid AMs with RSOC as realistic platform for AMs, we theoretically establish the existence of an electric field induced BCD-assisted SAHE in AMs. In particular, this exploits the BCP or equivalently, the QM, to overcome the C4T constraints that otherwise prevents an inherent BCD-induced SAHE. Quite remarkably, we also find that the electric field induced SAHE varies substantially in magnitude between the dx2-y2 and dxy limits in certain doping regimes, thus enabling a purely electrical route to distinguish the two altermagnetic orders. Our findings establish AMs as a powerful platform to realize highly tunable nonlinear Hall transport already at second order, driven by quantum geometric effects. Model hybrid altermagnet (AM).- To investigate the SAHE in hybrid AMs, we introduce a minimal two-band Hamiltonian defined on a 2D square lattice. In this model, we phenomenologically mix the B1g and B2g components of the altermagnetic orderings to capture the essential physics of a hybrid AM. The tight-binding Hamiltonian is given by [48-50] H(k) = ε(k) σ0 + h(k) · σ, (1) where σν denote the Pauli matrices for ν = 1, 2, 3 and the 2 × 2 identity matrix for ν = 0 acting in the spinsublattice subspace. The components of the Hamiltonian are ε(k) = -2t (cos kx + cos ky) -μ, (2) h(k) = {-λ sin ky, λ sin kx, tam α (cos kx -cos ky) + tam (1 -α) sin kx sin ky} (3) where t denotes the nearest-neighbor hopping amplitude, and μ is the chemical potential, λ denotes the RSOC strength, and tam is the strength of the altermagnetic spin splitting . We parameterize the relative weight of the two altermagnetic orders by 0 ≤α ≤1. In two dimensions, AMs inherently exhibit RSOC due to structural asymmetry [46, 47]. The case α = 1 corresponds to a pure dx2-y2 AM order arising from nearest-neighbor spin-dependent hopping, while α = 0 yields pure dxy AM order governed by next-nearest-neighbor spin-dependent hopping. Intermediate values of α thus describe a hybrid AM that seamlessly interpolates between the two irreducible representations B1g and B2g of the point group D4h. Such hybridization significantly enriches the spin splitting . The square lattice Hamiltonian in Eq. (1) may be relevant for tetragonal AMs such as RuO2 [32-34] and MnTe [35-37], and is directly relevant to quasi-2D layered compounds like Ca3Ru2O7 that host both altermagnetism and RSOC [41]. Additionally, it naturally accommodates the dx2-y2 (B1g) and dxy (B2g) [2, 3] orders whose hybridization we study. Second-order anomalous Hall effect (SAHE).- Although the Hamiltonian in Eq. (1) breaks T , it exhibits no anomalous Hall response at linear order in an applied ac electric field Eω with frequency ω, owing to the presence of combined C4T symmetry. Consequently, the leading anomalous Hall transport may at most arise at second order in Eω. Within semi-classical Boltzmann theory, the second-order anomalous Hall current can be expressed as jAH i = χAH ijk Eω j Eω k , with the second-order transverse conductivity tensor χAH ijk [15] χAH ijk = εilk e3τ 2ħ2(1 + iωτ) X n Z ddk (2π)d f0 (∂kjΩn l ), (4) where εilk is the Levi-Civita symbol and f0 is the equilibrium Fermi-Dirac distribution function and Ωn l is the l-th component of BC of the nth band. Here, τ is the relaxation time that makes the SAHE an extrinsic effect. The integral is the first moment of the BC over the occupied states, known as the Berry curvature dipole (BCD) [15] Djl = X n Z ddk (2π)d f0 (∂kjΩn l ). (5) For the BCD, or equivalently χAH ijk , to be nonzero, inversion symmetry P must be broken, otherwise, the momentum derivative ∂kjΩn l (k) transforms as an odd function of momentum, causing its Brillouin zone integral to vanish. In Eq. (1), the Rashba term λ explicitly breaks P. Additionally, the BCD obeys a severe symmetry constraint, namely the presence of two or more mirror lines in a crystal forces the SAHE to still vanish, while a single mirror line forces it to be orthogonal to that mirror plane [15, 51, 52]. Thus, mirror symmetries Mx,y, when present, constrain BCD and thus, the SAHE. Pure dx2-y2 altermagnetic order, i.e., when α = 1 in Eq. (1), breaks both Mx and My mirror symmetries, while purely dxy, i.e., α = 0, preserves both Mx,y. The presence of both Mx,y thus directly prohibits any BCDinduced SAHE for α = 0. In contrast, the absence of both mirror planes for α = 1, in principle, should allow SAHE. However, the presence of combined C4T symmetry in AMs has been shown to completely forbid BCD-driven SAHE for any values of α, thus forcing the leading order Hall transport to be third order [13, 14, 53, 54]. Next we will show that the nontrivial QM of the Bloch states actually still enables an extrinsic second-order transverse response in the presence of a dc electric field in AMs. Electric field induced SAHE.- In the presence of a symmetry-reducing dc electromagnetic field Edc, the BC of the Bloch electrons is modified due to the positional shift of Bloch electrons arising from interband mixing [17]. This results in a field-induced Berry connection that is expressed as AE = ↔ G Edc, where ↔ G is a second-rank tensor, called the BCP. The component Gn ab of the BCP tensor is defined as [17, 55, 56] Gn ab(k) = 2 Re X m̸=n Anm a (k) Amn b (k) εn(k) -εm(k) . (6) 3 -π 2 0 π 2 ky -π 2 0 π 2 kx -π 2 0 π 2 ky Edc -π 2 0 π 2 kx Edc -π 2 0 π 2 kx Edc -20 0 20 -50 0 50 FIG. 1. Berry connection polarizability (BCP) tensor components for one of the band: (a) G1 xx, (b) G1 xy, and (c) G1 yy in the first Brillouin zone. Field-induced Berry curvature for the same band , ΩE(k), for dc-field orientations (d) θ = 0, (e) π/4, and (f) π/2, with range arrows indicating electric field directions. Calculations performed with t = 1.0, tam = 0.5t, α = 0.5, μ = 0.3t, and λ = 0.08t. Here, the interband Berry connection is Anm a (k) = ⟨un|i∂a|um⟩, where ∂a ≡∂ka and |un⟩is the periodic part of the n-th unperturbed Bloch state, with εn the corresponding band energy. Notably, Gn ab(k) is closely linked with the QM, gn ab = 2 P m̸=n Re[Anm a (k) Amn b (k)], the real part of the quantum geometric tensor [30]. A nontrivial BCP, or equivalently QM, generates in turn a fieldinduced BC ΩE n = ∇k×AE n , which, similar to the original BC of the Bloch band, acts like a pseudo-magnetic field in k-space which yields a SAHE, just as the original BC in Eq. (4). The field-induced BC is especially important in those systems where either the intrinsic BC vanishes or the original BCD-induced SAHE is forbidden due to the symmetry constraints, as discussed above for AMs. In typical Hall transport measurements, the applied electric field and the measured current are both restricted to the x-y plane. Then, an external electric field applied at an angle θ relative to the x-axis, such that Edc = Edc(cos θ, sin θ), induces a field-induced BC of the form [27, 55, 57], ΩE nz = Edc[(∂kxGn yx -∂kyGn xx) cos θ (7) + (∂kxGn yy -∂kyGn xy) sin θ], which is dependent on both the magnitude and direction of Edc. This field-induced BC can lead to a finite fieldinduced BCD following Eq. (5) with an angular dependence DE(θ) = DE xz(θ), DE yz(θ) . Due to the lowered symmetry in the presence of Edc, a field-induced BCD then gives rise to the SAHE. Finally, plugging the angular dependence of the BCD into Eq. (4), and recasting it through χAH = j2ω/(Eω)2, we arrive at the second-harmonic current j2ω [27, 28] j2ω = - e3τ 2(1 + iωτ)ħ2 (ˆz × Eω) [DE(θ) · Eω], (8) where Eω is the in-plane ac Hall probe field that makes an angle φ with the x-axis and satisfies the condition Eω ≪ Edc. As seen from Eq. (8), the maximum Hall response is obtained when Eω ∥DE(θ), while it vanishes for Eω ⊥ DE(θ). Overall, the angular dependence j2ω is intricate as it not only varies with θ through DE(θ) but also with φ, thus providing a high degree of external tunability of the SAHE. In 2D, only the two components χyxx and χxyy of the tensor χAH ijk are finite and independent. Thus χAH simplifies to [18, 27] χAH(θ, φ) = χyxx(θ) cos φ -χxyy(θ) sin φ. (9) SAHE in altermagnets.- To illustrate how the electric field induced SAHE emerges in AMs described by Eq. (1), we start by showcasing the k-resolved distribution of the components of the BCP tensor, or equivalently, bandnormalized QM for one of the two bands, setting α = 0.5 in Fig. 1 (a-c). For the other band, the contributions are exactly equal in magnitude but opposite in sign. For pure altermagnetic orders, i.e., for α = 0 or α = 1, the components of BCP tensors are shown in the Supplementary Material (SM) [58]. The corresponding band structures for these three α values are shown in the SM [58]. Consistent with expectations, the BCP values exhibit maxima at the band-touching points, due to the denominator in Eq. (6). The presence of a finite BCP leads to a finite BC, in accordance with Eq. (7), when an external dc electric field Edc is applied. For Edc oriented along the -5.0 -2.5 0.0 2.5 5.0 -π 2 0 π 2 ky (a) (b) -π 2 0 π 2 kx -π 2 0 π 2 ky (c) -π 2 0 π 2 kx (d) FIG. 2. Momentum distribution of the Fermi-functionweighted derivatives of the field-induced Berry curvature in log-scale for the two bands, (a) f0 ∂kxΩE 1 , (b) f0 ∂kyΩE 1 , and for band 2, (c) f0 ∂kxΩE 2 , (d) f0 ∂kyΩE 2 , evaluated at θ = π/4. Parameters are the same as in Fig. 1. 4 0 π 2 π 3π 2 2π φ -0.02 0.00 0.02 χAH θ = 0 θ = π 6 θ = π 4 θ = π 2 θ = 3π 4 FIG. 3. Second-order Hall conductivity χAH as a function of angle φ, set by the ac driving field Eω, for several different angles θ, set by the symmetry-breaking static field Edc, with the x-axis. χAH is scaled by the prefactor A = e3τ 2(1+iωτ)ħ2 and the parameters are the same as in Fig. 1. x (θ = 0), x = y (θ = π/4), and y(θ = π/2), the fieldinduced BC ΩE are shown in Fig. 1 (d)-(f), respectively. The k-resolved distribution of ΩE shows a clear dipolar structure. Not only the magnitude, but also the direction of the ΩE and, thus, also the resulting BCD, can be externally tuned by changing the direction of Edc [27]. Once a finite and field-tunable BCD is confirmed, it is straightforward to express χAH in terms of Dxz and Dyz following Eq. (4-5). Figure 2 shows the momentumresolved components of the Fermi-weighted Berry curvature dipole, i.e., the integrand of Eq. (5) for both bands again for α = 0.5, with Edc oriented along the x = y direction. These are then integrated over the Brillouin zone and summed over bands to yield the finite fieldinduced BCD. Finally, following Eq. (4), with it angular dependence in Eq. (9), we find an overall nonzero secondorder Hall response. The complex angular variation of the resulting second-order Hall conductivity χAH(θ, φ) in a hybrid AM is depicted in Fig. 3. It offers significant experimental control over the second-order Hall current, enabling it to be switched on or off simply by adjusting the relative orientation of Edc and Eω. These behaviors are similarly observed for the pure altermagnetic order, i.e., for α = 1 and α = 0, see the SM [58]. Remarkably, we find that the electric field induced SAHE exhibits a strong sensitivity in magnitude between different altermagnetic orders, at least for a range of doping levels, set by μ. The response is maximized at α = 0, corresponding to a pure dxy-type altermagnetic order when Edc ⊥Eω, but as α increases, the response χAH decreases and becomes negligible for α →1, i.e., pure dx2-y2 order, see Fig. 4(a). In contrast, when Edc ∥Eω, χAH is found to be vanishingly small for both α = 0 and α = 1 as shown in Fig. 4(b). Thus, simply tuning the direction of either Edc or Eω, provides a straightforward way to conclusively distinguish between pure orders dxy and dx2-y2. For the parameter choices used in the presented results, this doping range is found to be 0.0 0.5 1.0 α 0.0 0.5 1.0 1.5 χAH (a) θ = 0, φ = π 2 λ = 0.08 λ = 0.1 λ = 0.15 0.0 0.5 1.0 α -0.8 -0.6 -0.4 -0.2 0.0 (b) θ = 0, φ = π λ = 0.08 λ = 0.1 λ = 0.15 FIG. 4. Second-order Hall conductivity χAH as a function of α for different RSOC λ values and field orientations: (a) θ = 0, φ = π/2 and (b) θ = 0, φ = π with t = 1, tam = 0.5t, and μ = 0.3t (same as Fig. 1). -t < μ < t. We have also tested our results for smaller and larger tam, which shifts the corresponding doping range, although it is still clearly present. In this doping regime, the system exhibits two hole-like Fermi pockets for α = 0, whereas for α = 0.5 and α = 1, it features one electron and one hole Fermi pocket, see SM [58]. When the Fermi surfaces (FSs) have the opposite curvature, their contributions to Hall transport tend to cancel each other, resulting in a nearly vanishing response. In contrast, when both FSs have the same curvature, their contributions add constructively, leading to a finite overall response. For example, in the doping regime with two electron pockets (regardless of α), the field-induced SAHE remains nonzero for both pure and hybrid altermagnetic symmetries, see the SM [58]. In this scenario, pure altermagnetic orders cannot be distinguished based on the second-order response, as all yield a finite signal. These findings unambiguously establish the presence of a nonlinear Hall response in AM at second order and also offer an all-electrical approach to resolve pure altermagnetic orderings. As shown in Fig. 4, the electric field induced SAHE increases when the strength of RSOC λ decreases. Still, we recall that no SAHE is possible for λ = 0 due to preserved P. At first glance, this may seem non-intuitive. However, we note that for finite λ, RSOC opens gaps at the band-touching points in altermagnetic metals. When λ is small, these gaps are narrow, leading to an enhanced BCP (see Eq. (6)), which in turn amplifies the field-induced SAHE. The evolution of the two spin-split FSs with λ in various doping limits are shown in the SM [58]. In summary, we demonstrate that Rashba-coupled hybrid AMs provide a viable platform for electric field induced SAHE that arises due to the nontrivial QM of the Bloch bands. Although combined C4T symmetry forbids both the linear anomalous Hall effect and the usual BCDdriven second-order signal, an external dc field lowers the symmetry and generates a finite field-induced BCD via the nontrivial QM of the occupied Bloch states. The intricate angular variation of the resulting second-order Hall conductivity χAH(θ, φ) provides substantial experimental control over the second-order Hall current, allowing it to be toggled on or off by merely adjusting the 5 relative orientation between the external dc field Edc and ac probe field Eω. In certain doping regimes, its magnitude can even probe the underlying altermagnetic form factor, peaking in the dxy (B2g) limit and decreasing toward dx2-y2 (B1g) configuration for Edc ⊥Eω, which then provides a purely electrical means to distinguish the two. These predictions are directly testable in candidate material platforms such as epitaxial RuO2 [32-34] and MnTe [35-37] thin films, where strain or surface termination can realize hybrid B1g-B2g textures and provide the required inversion-symmetry breaking at interfaces that generate interfacial RSOC. To observe the SAHE, an experimental setup similar to the one used in Refs. [28, 29] could be employed. More broadly, our results demonstrate that quantum geometry can enable nonlinear responses in systems where symmetry otherwise suppresses intrinsic BC effects, enabling functionalities such as onchip frequency doubling and rectification in zero-moment magnets. A.B and A.M.B.-S. acknowledge financial support from the Swedish Research Council (Vetenskapsr ̊adet) grant no. 2022-03963 and the European Union through the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (ERC-2022-CoG, Grant agreement no. 101087096). Views and opinions expressed are however those of the author(s) only and do not necessarily reflect those of the European Union or the European Research Council Executive Agency. Neither the European Union nor the granting authority can be held responsible for them. B.S. acknowledges financial support from Swedish Research Council (grant no. 2022-04309), STINT Mobility Grant for Internationalization (grant no. MG2022-9386) and DST-SPARC, India (Ref. No. SPARC/2019-2020/P1879/SL). 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We also examine a doping regime in which both pure and hybrid altermagnetic orders feature Fermi pockets with similar curvature, allowing for a direct comparison with the main text, where the results correspond to a doping range in which the altermagnetic orders exhibit either Fermi surfaces with the same curvature or with opposite curvature. I. FIELD-INDUCED SECOND-ORDER HALL RESPONSE FOR PURE ALTERMAGNETIC ORDERS In this section we show the electric field induced second-order Hall response for pure altermagnetic orders. For our model Hamiltonian Eq. (1) in the main text we consider hybrid altermagnetic order, i.e., a combination of dx2-y2 and dxy symmetries by keeping α = 0.5. Here we instead consider the results for pure dx2-y2 and dxy altermagnetic order. In Fig. S1 and Fig. S2, we show the components of the Berry connection polarizability tensor (BCP) for one of the two bands and the field-induced Berry curvature for three different orientations of the symmetry-breaking field Edc for α = 1, i.e., for pure dx2-y2 symmetry and for α = 0, i.e., for pure dxy symmetry, respectively. These figures are equivalent to Fig. 1 in the main text. -π 2 0 π 2 ky -π 2 0 π 2 kx -π 2 0 π 2 ky Edc -π 2 0 π 2 kx Edc -π 2 0 π 2 kx Edc -20 0 20 -50 0 50 FIG. S1. Berry connection polarizability (BCP) tensor components for one of the band : (a) G1 xx, (b) G1 xy, and (c) G1 yy in the first Brillouin zone. Field-induced Berry curvature for the same band , ΩE(k), for dc-field orientations (d) θ = 0, (e) π/4, and (f) π/2, with orange arrows indicating electric field directions. Calculations performed with t = 1.0, tam = 0.5t, α = 1.0, μ = 0.3t, and λ = 0.08t. The k-resolved distribution of ΩE shows a clear dipolar structure, which corresponds directly to a finite field-induced BCD. Not only the magnitude but also the direction of ΩE and, thus, the resulting BCD can be externally tuned by changing the direction of Edc. The corresponding momentum-resolved components of the Fermi-weighted dipole of BC for the two bands for α = 1 and α = 0, with Edc oriented along the x = y direction, are shown in Fig. S4 and Fig. S5, respectively, similar to Fig. 2 in the main text. For α = 1, the system possesses one electron pocket and one hole pocket, similar to α = 0.5, while for α = 0, it has two hole pockets. Figure S6 shows the resulting second-order conductivity χAH(θ, φ) as a function of φ for various values of θ for the two pure orders. The second-order response χAH(θ, φ) for α = 0 is more than an order of magnitude higher than that for α = 1. When the Fermi surfaces (FSs) are of opposite types, as in the case for α = 0.5 or 1, their contributions tend to cancel each other, resulting in a small response. In contrast, when both FSs have same types of curvature as in the case for α = 1, their contributions add constructively, leading to a higher overall response. 8 -π 2 0 π 2 ky -π 2 0 π 2 kx -π 2 0 π 2 ky Edc -π 2 0 π 2 kx Edc -π 2 0 π 2 kx Edc -20 0 20 -50 0 50 FIG. S2. Berry connection polarizability (BCP) tensor components for one of the band : (a) G1 xx, (b) G1 xy, and (c) G1 yy in the first Brillouin zone. Field-induced Berry curvature for the same band , ΩE(k), for dc-field orientations (d) θ = 0, (e) π/4, and (f) π/2, with orange arrows indicating electric field directions. Calculations performed with t = 1.0, tam = 0.5t, α = 0.0, μ = 0.3t, and λ = 0.08t. Γ (0, 0) Y (0, π) S2 (-π, π) Γ (0, 0) S1 (π, π) X (π, 0) Γ (0, 0) -4 -2 0 2 4 E (a) Band 1 Band 2 Γ (0, 0) Y (0, π) S2 (-π, π) Γ (0, 0) S1 (π, π) X (π, 0) Γ (0, 0) (b) Band 1 Band 2 Γ (0, 0) Y (0, π) S2 (-π, π) Γ (0, 0) S1 (π, π) X (π, 0) Γ (0, 0) (c) Band 1 Band 2 FIG. S3. Band structure along high-symmetry paths for (a) α=0 (pure dxy), (b) α=0.5 (hybrid), and (c) α=1 (pure dx2-y2). Blue and red lines represent the two spin-split bands band 1 and band 2, respectively. The dashed line indicates the Fermi level. Parameters: t = 1.0, μ = 0.3t, λ = 0.08t, tam = 0.5t. II. TOPOGRAPHY OF FERMI SURFACES In Fig. S7 and Fig. S8, we present the spin-split Fermi surfaces for various altermagnetic orders across different doping regimes. In the absence of Rashba spin-orbit coupling (RSOC) λ, band-touching points are present, but a gap opens at these points as soon as λ is tuned to a finite value. The second-order Hall response increases with decreasing RSOC λ, as shown in Fig. S9. This is because for a smaller value of λ, the energy gaps between bands are reduced, leading to an enhanced quantum metric or Berry curvature polarization (BCP), which in turn results in a stronger field-induced second-order response. In Fig. S9(a), we show the variation of χAH with α for μ = -3t keeping Edc ⊥Eω. In contrast to the results in Fig. 4 in the main text, we find that χAH is finite similar in magnitude for all α. In this doping limit, there are also two electron pockets for all α in contrast to the doping limit considered in the main text. In this case, the contribution of the two similar Fermi surfaces add constructively to give an overall finite response for any α. In Fig. S9(b), the same plot is shown for Edc ∥Eω, with an overall small χAH, showing that also in this case there is large tunability with field-directions. 9 -5.0 -2.5 0.0 2.5 5.0 -π 2 0 π 2 ky (a) (b) -π 2 0 π 2 kx -π 2 0 π 2 ky (c) -π 2 0 π 2 kx (d) FIG. S4. Momentum distribution of the Fermi-function-weighted derivatives of the field-induced Berry curvature in log-scale for the two bands, (a) f0 ∂kxΩE 1 , (b) f0 ∂kyΩE 1 , and for band 2, (c) f0 ∂kxΩE 2 , (d) f0 ∂kyΩE 2 , evaluated at θ = π/4. Parameters are the same as in Fig. S1. 10 -5.0 -2.5 0.0 2.5 5.0 -π 2 0 π 2 ky (a) (b) -π 2 0 π 2 kx -π 2 0 π 2 ky (c) -π 2 0 π 2 kx (d) FIG. S5. Momentum distribution of the Fermi-function-weighted derivatives of the field-induced Berry curvature in log-scale for the two bands, (a) f0 ∂kxΩE 1 , (b) f0 ∂kyΩE 1 , and for band 2, (c) f0 ∂kxΩE 2 , (d) f0 ∂kyΩE 2 , evaluated at θ = π/4. Parameters are the same as in Fig. S2. 0 π 2 π 3π 2 2π φ -1 0 1 χAH (a) α = 0.0 θ = 0 θ = π 6 θ = π 4 θ = π 2 θ = 3π 4 0 π 2 π 3π 2 2π φ -0.04 -0.02 0.00 0.02 0.04 (b) α = 1.0 FIG. S6. Second-order Hall conductivity χAH as a function of angle φ, set by the ac driving field Eω, for several different angles θ, set by the symmetry-breaking static field Edc, with the x-axis for (a) α = 0, (b) α = 1. χAH is scaled by the prefactor A = e3τ 2(1+iωτ)ħ2 and the parameters are the same as in Fig. S3. 11 -π/2 0 π/2 ky (a) α = 0.0, λ = 0.00 (b) α = 0.5, λ = 0.00 (c) α = 1.0, λ = 0.00 -π/2 0 π/2 -π/2 0 π/2 ky (d) α = 0.0, λ = 0.10 -π/2 0 π/2 kx (e) α = 0.5, λ = 0.10 -π/2 0 π/2 (f) α = 1.0, λ = 0.10 -1.0 -0.5 0.0 0.5 1.0 FIG. S7. Spin-split Fermi surfaces (a,d) α = 0; (b,e) α = 0.5 and (c,f) α = 1.0. Upper row is for λ = 0 and lower row for λ = 0.1t. Chemical potential is set to μ = 0.3t, same as the main text. 12 -π/2 0 π/2 ky (a) α = 0.0, λ = 0.00 (b) α = 0.5, λ = 0.00 (c) α = 1.0, λ = 0.00 -π/2 0 π/2 -π/2 0 π/2 ky (d) α = 0.0, λ = 0.10 -π/2 0 π/2 kx (e) α = 0.5, λ = 0.10 -π/2 0 π/2 (f) α = 1.0, λ = 0.10 -1.0 -0.5 0.0 0.5 1.0 FIG. S8. Spin-split Fermi surfaces (a,d) α = 0; (b,e) α = 0.5 and (c,f) α = 1.0. Upper row is for λ = 0 and lower row for λ = 0.1t. Chemical potential is set to μ = -3.0t. 0.0 0.5 1.0 α 0.15 0.20 0.25 0.30 χAH (a) θ = 0, φ = π 2 λ = 0.08 λ = 0.1 λ = 0.15 0.0 0.5 1.0 α -0.02 -0.01 0.00 0.01 0.02 (b) θ = 0, φ = π λ = 0.08 λ = 0.1 λ = 0.15 FIG. S9. Second-order Hall conductivity χAH as a function of α for different RSOC λ values and field orientations: (a) θ = 0, φ = π/2 and (b) θ = 0, φ = π with t = 1, tam = 0.5t, and μ = -3t (same as Fig. S3)."}}},{"rowIdx":92,"cells":{"id":{"kind":"string","value":"2510.14889"},"raw":{"kind":"string","value":"Detecting Early and Implicit Suicidal Ideation via Longitudinal and\nInformation Environment Signals on Social Media\nSoorya Ram Shimgekar1, Ruining Zhao1, Agam Goyal1, Violeta J. Rodriguez1,\nPaul A. Bloom2, Hari Sundaram1, Koustuv Saha1\n1University of Illinois Urbana-Champaign, {sooryas2, ruining, agamg2, vjrodrig, hs1, ksaha2}@illinois.edu\n2Columbia University, New York State Psychiatric Institute, paul.bloom@nyspi.columbia.edu\nAbstract\nOn social media, many individuals experienc-\ning suicidal ideation (SI) do not disclose their\ndistress explicitly. Instead, signs may surface\nindirectly through everyday posts or peer inter-\nactions. Detecting such implicit signals early\nis critical but remains challenging. We frame\nearly and implicit SI as a forward-looking pre-\ndiction task and develop a computational frame-\nwork that models a user’s information envi-\nronment, consisting of both their longitudinal\nposting histories as well as the discourse of\ntheir socially proximal peers. We adopted a\ncomposite network centrality measure to iden-\ntify top neighbors of a user, and temporally\naligned the user’s and neighbors’ interactions—\nintegrating the multi-layered signals in a fine-\ntuned DeBERTa-v3 model. In a Reddit study of\n1,000 (500 Case and 500 Control) users, our ap-\nproach improves early and implicit SI detection\nby 15% over individual-only baselines. These\nfindings highlight that peer interactions offer\nvaluable predictive signals and carry broader\nimplications for designing early detection sys-\ntems that capture indirect as well as masked\nexpressions of risk in online environments.\n1\nIntroduction\nMore than 703,000 individuals die by suicide each\nyear worldwide, making it a major global public\nhealth concern (WHO, 2025). Suicidal ideation\n(SI) refers to a spectrum of cognitive and behav-\nioral manifestations related to suicide, ranging from\npassive thoughts of death to active planning and\nengagement in self-injurious actions with intent to\ndie (Joiner, 2005). While early detection of these\nsigns is critical, psychiatry and mental health ser-\nvices have long struggled to detect risk before indi-\nviduals disclose it explicitly (Calear and Batterham,\n2019; Hom et al., 2017). A recent meta-analysis\nrevealed that the estimated prevalence of SI dis-\nclosure is only 46.3%, inferring that the majority\nof people who experience suicidal thoughts do not\ndisclose them, making early detection critical for\ntimely intervention (Hallford et al., 2023).\nOnline platforms have become vital spaces\nwhere signs of SI may surface. People use on-\nline mental health communities to share distress,\nseek help, and connect with peers (De Choud-\nhury and De, 2014). This has created opportuni-\nties for both researchers and clinicians to under-\nstand suicide risk through language, social inter-\nactions, and online behavioral cues. Prior work\nhas shown the value of linguistic cues for identify-\ning mental health risks (Coppersmith et al., 2014;\nDe Choudhury et al., 2013; Guntuku et al., 2017).\nIn particular, within the context of SI, prior work\nhas computationally modeled the language of SI\non social media (Burnap et al., 2015; Saha et al.,\n2019; Alghazzawi et al., 2025; Zhang et al., 2025;\nNaseem et al., 2025). More recent computational\napproaches leveraged longitudinal and multimodal\ndata, as well as social network analyses, to antici-\npate suicide-related behaviors (Shen et al., 2020).\nYet, critical limitations persist. Prior research has\nprimarily examined posts where SI is explicitly con-\nveyed, such as direct or indirect references to self-\nharm or suicidal thoughts within suicide-related fo-\nrums or discussions (Ji, 2022a; Bloom et al., 2025).\nThese approaches presuppose that individuals ar-\nticulate their distress, yet, many at-risk individuals\nneither disclose SI nor exhibit overt warning signs,\nbut instead mask distress within seemingly ordi-\nnary discourse (McGillivray et al., 2022; Mérelle\net al., 2018; Podlogar et al., 2022). Our work targets\nthe detection of these undisclosed SI, characterized\nby subtle, contextually obscured, or socially dis-\ntributed indicators, to potentially enable earlier and\nmore effective intervention.\nTo address the above gap, our work is guided by\nthe research question (RQ): How can early signals\nof SI be detected in social media activity, par-\nticularly in the absence of explicit disclosures?\nTo address the RQ, we conceptualize early and im-\n1\narXiv:2510.14889v1 [cs.SI] 16 Oct 2025\n\nplicit SI as a forward-looking prediction problem\nthat requires modeling both an individual’s longi-\ntudinal behavior and the broader social context in\nwhich that behavior unfolds. We develop a frame-\nwork that jointly captures temporal dynamics in\na user’s posting history and the conversations of\nsocially proximal peers, enabling a richer repre-\nsentation of early warning signals. This idea of\nusing socially proximal peers to understand an in-\ndividual’s mental state is derived from many prior\nacclaimed psychology works (La Greca and Har-\nrison, 2005; Copeland, 2021; Victor et al., 2019).\nWithin this framework, we pursue three aims:\nAim 1: Examine whether users’ longitudinal post-\ning patterns, in conjunction with commentary on\nthose posts, reveal early signals of implicit SI.\nAim 2: Examine how the surrounding informa-\ntion environment, including peer interactions and\nexposure to content, contributes as an additional\npredictive feature for implicit SI.\nAim 3: Identify the linguistic markers that underlie\ninteraction types, in terms of how specific language\ncues are associated with these interactions.\nWe conducted our study on Reddit, focusing on\na sample of 1,000 users divided into 500 Case and\n500 Control users (who never participated in men-\ntal health conversations). Our predictive framework\njointly modeled each user’s full posting timeline\nalong with the discourse of their most socially prox-\nimal neighbors, identified through network central-\nity, and fine-tuned a DeBERTa-v3 model (He et al.,\n2021) to embed both individual and peer interac-\ntion signals into a unified representation.\nOur findings show that incorporating peer inter-\nactions within the information environment, along-\nside users’ longitudinal posting histories, signifi-\ncantly enhances detection of early and implicit SI,\nimproving performance by 15% compared to mod-\nels based only on longitudinal user data. Overall,\nthis paper contributes: 1) a framework for detecting\nimplicit SI; 2) a method for systematically inte-\ngrating environment information through neighbor\ninteractions; and 3) empirical evidence that this\nintegration improves early detection.\n2\nRelated Work\n2.1\nSuicidal Ideation and Social Context\nPsychological theory views SI as the product of\nintertwined social and cognitive factors. The Inter-\npersonal Theory of Suicide (Joiner, 2005) posits\nthat suicidal desire arises from perceived burden-\nsomeness and thwarted belongingness, while ca-\npability develops through repeated exposure to\npain or fear. Early computational studies of de-\npression and SI focused on overt signals such as\nself-disclosure or negative sentiment (Coppersmith\net al., 2014; De Choudhury et al., 2013), but later\nwork showed that many at-risk individuals express\ndistress through subtle cues, motivating methods\nthat model temporal and semantic dynamics of be-\nhavior (Guntuku et al., 2017; Benton et al., 2017;\nSaha and De Choudhury, 2017).\nA range of methods have been proposed to detect\nmental health risk signals beyond surface cues. For\ninstance, (Fatima et al., 2021) introduced DASenti-\nmental, a semi-supervised model combining bag-\nof-words and semantic networks, while (Trotzek\net al., 2018) showed that convolutional networks\nwith linguistic metadata enable earlier detection.\nMore recent work leverages large language mod-\nels, with GPT-3.5/4 using chain-of-thought prompt-\ning on diaries (Shin et al., 2024) and reasoning-\nguided LLMs improving interpretability (Teng\net al., 2025). Temporal dynamics remain critical,\nfrom emotional “phases” in user timelines (Sawh-\nney et al., 2021a) to transformer-based models en-\nriched with temporal signals (Sawhney et al., 2020).\nSocial context is equally important: hyperbolic em-\nbeddings of user histories and peer interactions\nenhance prediction (Sawhney et al., 2021b), peer\nnetworks and conversational responses influence\ntrajectories (Wyman et al., 2019; De Choudhury\nand Kiciman, 2017), and longitudinal patterns re-\nveal precursors of SI (De Choudhury et al., 2016).\nComplementary directions include clinical-domain\ndatasets such as ScAN (Rawat et al., 2022), auto-\nmated counseling support with PsyGUARD (Qiu\net al., 2024), and calls to model underlying intent\nrather than surface disclosure (Ji, 2022b).\nTogether, this work affirms that SI arises from\npsychological distress, temporal dynamics, and so-\ncial context, demanding models that go beyond sur-\nface cues. Yet most approaches still rely on explicit\ndisclosures or static timelines, overlooking how\nevolving language interacts with peer responses.\nOur framework addresses this by jointly modeling\nusers’ longitudinal histories and post-level com-\nmentary, enabling early detection of implicit SI.\n2.2\nSocial Media and Mental Health\nOnline platforms have become key venues for\nmental-health self-disclosure (De Choudhury and\n2\n\nDe, 2014), with communities like Reddit fostering\ntargeted support and a sense of belonging (Saha and\nSharma, 2020; De Choudhury, 2015; Shimgekar\net al., 2025; Kim et al., 2023). Moderated peer-\nsupport spaces reduce isolation and help people\ndiscuss stigmatized experiences (Johnson et al.,\n2022). Social support, especially emotional and\ninformational, has been shown to improve well-\nbeing both offline and online (Cutrona and Trout-\nman, 1986; De Choudhury and Kiciman, 2017;\nSaha and Sharma, 2020). Language plays a central\nrole: psycholinguistic research links specific lin-\nguistic markers to mental-health outcomes (Chung\nand Pennebaker, 2007; Pennebaker et al., 2001),\nand computational studies have used these cues to\ndetect distress and model support dynamics (Chan-\ncellor et al., 2016; Guntuku et al., 2017; Chancellor\nand De Choudhury, 2020). Prior work has also\nestablished the construct validity of these measure-\nments (Saha et al., 2022).\nRecent NLP work has focused on interpretable\nand fine-grained modeling of mental health on so-\ncial media. Symptom-based approaches such as\nPSYSYM (Zhang et al., 2022; Chen et al., 2023)\nlink online language to clinically meaningful cate-\ngories of disorders. Depression severity has been\nquantified through semantic similarity to symptom\ndescriptors (Pérez et al., 2022), while large lan-\nguage models now enable explainable detection\nwith interpretable rationales (Wang et al., 2024).\nAnalyses of pre and post diagnosis language shifts,\nhighlight the temporal dynamics of distress ex-\npression (Alhamed et al., 2024). Supportive lan-\nguage marked by adaptability, immediacy, and emo-\ntionality predicts better outcomes (Althoff et al.,\n2016; Saha and Sharma, 2020), and automatic\nempathy-detection models scale such insights to\npeer-support settings (Sharma et al., 2020). For SI,\nmachine-learning approaches have identified risk\nsignals in social media language alongside emo-\ntional patterns that precede suicide attempts (Cop-\npersmith et al., 2016; De Choudhury et al., 2016).\nWhile online data shows promise for early risk\ndetection, most methods isolate either individual\nlanguage or specific interactions. Our approach in-\nstead models full posting timelines alongside peer\ninfluences, capturing risk even without explicit SI\ndisclosures. Our methodological framework identi-\nfies early indicators without requiring references to\nself-harm or participation in such spaces.\nFigure 1: Distribution of SI probability for the same set\nof users before their last non suicidal post and their first\npost in r/SuicideWatch.\n3\nData\nWe used data from r/SuicideWatch, a semi-\nanonymous Reddit community focused on SI,\nalongside posts and comments from other sub-\nreddits. From the PushShift archive (Baumgart-\nner et al., 2020) of April 2019 (18.3M posts,\n138.5M comments), the data includes 10,037 posts\nand 38,130 comments from r/SuicideWatch. Prior\nwork has leveraged Reddit for SI (De Choudhury\net al., 2016; De Choudhury and Kiciman, 2017;\nShimgekar et al., 2025) and broader mental health\nstudies (Sharma and De Choudhury, 2018; Saha\net al., 2020; De Choudhury and De, 2014).\nConstructing Case and Control datasets. We iden-\ntified two user cohorts. The first cohort comprises\n500 Case individuals, defined as individuals who\nhave made at least one post on r/SuicideWatch. The\nsecond cohort consists of 500 Control individuals,\nwho never participated in any subreddit related to\nmental health. We identified the Control users by\nreferencing the taxonomy of mental health-related\nsubreddits from prior work (Sharma and De Choud-\nhury, 2018).\nFor the Case group, all posts and comments\nbefore each user’s first r/SuicideWatch disclosure\nwere labeled positive (1). On average, users made\n10.07 posts before disclosure, transitioning within\n1.5 days of their last non-r/SuicideWatch post. To\nconstruct a balanced Control group, we sampled\nthe first 10 posts from each user (matching Case\naverages), labeling them negative (0). The dataset\nwas then split 75:25 by users into train and test sets,\nensuring no overlap.\nA zero-shot NLI model (Laurer et al., 2023) re-\nvealed significantly higher SI probabilities in users’\nfirst r/SuicideWatch posts, indicating that entry\nmarks a key turning point (Figure 1). The model,\n3\n\nPost 1\n(r/random)\nComment 2\n(r/random2)\nPost 3\n(r/random3)\nPost 4\n(r/SuicideWatch)\nNeighbors'\nPosts\nR1, R2\nTop \nNeighbors\nU\nUser \"U\"\nUser-User Graph\nof \"U\"\nOthers'\nReplies\nR1, R2\nNeighbors'\nPosts\nR1, R2\nOthers'\nReplies\nR1, R2\nNeighbors'\nPosts\nR1, R2\nOthers'\nReplies\nR1, R2\nNeighbor\nInteraction\nImmediate \nInteraction\nInput Data\nFormalization\nFinetuned\nModel\nEarly and\nImplicit Suicide\nDetection\n[yes/no]\nFigure 2: Illustration of user interactions (Immediate/Neighbor): Immediate interactions include users’ self-posts,\nself-comments, and received replies, while neighbor interactions represent top neighbor posts and self-posts. Top\nneighbors identified via NeighborScore\nbased on DeBERTa-v3-base, was trained on 1.3M\nhypothesis–premise pairs from eight NLI datasets\n(e.g., MultiNLI, FEVER-NLI, LingNLI, DocNLI)\nto capture long-range reasoning. Using zero-shot\nlabels (SI, non-SI), it assigned SI probabilities to\nposts, showing that users’ first r/SuicideWatch posts\nexhibit stronger suicidal self-disclosure than their\nprior posts. While these probabilities may partly\nreflect the act of joining r/SuicideWatch, the linguis-\ntic patterns align more with distress and self-harm\nintent, supporting their use as an operational proxy\nfor heightened suicidal self-disclosure.\n4\nMethods\nTo address our RQ on detecting early signals of\nSI in social media activity, particularly when ex-\nplicit disclosures are absent, we conceptualized\nimplicit SI as a forward-looking prediction prob-\nlem of the likelihood that a user would eventu-\nally make a SI disclosure, proxied by their first\npost on r/SuicideWatch. Our framework models\na user’s longitudinal activity and social context\nvia two interaction types: (1) immediate inter-\nactions—self-posts, self-comments, and received\ncomments, and (2) neighbor interactions—self-\nposts and top neighbors’ posts. This captures both\nactive engagement and passive exposure, reflect-\ning how users interact with and are influenced by\ncontent they relate to (De Choudhury and Kiciman,\n2017).\nOur methodological framework consists of five\ncomponents—1) Timeline construction, 2) Neigh-\nboring user detection, 3) Input data formalization,\nand 4) implicit SI signal classification with mod-\neling, 5) linguistic analysis of interaction types,\nwhich we describe below:\n4.1\nTimeline Construction\nFor each user U, we analyzed their full post and\ncomment history for SI signals, ordered chronolog-\nically from earliest to latest.\n4.2\nNeighboring user detection\nFor a user U, we found its top neighbors by the\nfollowing four steps:\nStep 1: Initial User Collection\nWe first collected\nall users who interacted with the user U, defined\nas either U commenting on their posts/comments\nor them commenting on U’s posts/comments. This\nneighbor-identification procedure was then applied\nrecursively to a maximum depth d=3, ensuring that\nboth direct and indirect neighbors of U were cap-\ntured.\nStep 2: User-User Graph\nBased on all the initial\ncollected users, we constructed a user-user graph\nwhere each Node represents an individual user in\nthe network, and an undirected Edge connects two\nusers if either of the users has commented on the\nother’s post. The weight of the edge was quantified\nas the total number of comment-based interactions,\nwith higher weights indicating stronger ties.\nStep\n3:\nTop\nNeighbor\nDetection\nFrom\nthe user-user graph, we identified the top-\nn (n=10) neighbors for U. We ranked the\nneighbors\nusing\na\nNeighborScore,\na\ncom-\nbined\ncentrality\nscore\nS(n),\ndefined\nas:\nS(n) = Cin-degree(n) + Cout-degree(n)\n+ Ccloseness(n) + Ceigenvector(n)\n+ Cbetweenness(n) + CPageRank(n)\nCin-degree and Cout-degree capture normalized con-\nnectivity, Cbetweenness measures shortest-path cen-\ntrality, Ccloseness denotes proximity, Ceigenvector\n4\n\nTable 1: Model performance metrics on different data\ncombinations for Models M1–M4 (epochs=20).\nData Used\nAcc.\nF1\nPrec.\nRec.\nM1\nSelf-posts\n0.86\n0.85\n0.88\n0.84\nM2\nSelf-posts + Self-comments\n0.81\n0.83\n0.78\n0.91\nM3\nSelf-posts + Self-comments + Others’ comments\n0.80\n0.79\n0.79\n0.80\nM4\nSelf-posts + Top neighbor posts\n0.95\n0.96\n0.93\n0.99\nTable 2: Model performance on showing the importance\nof choosing the best neighbors (epochs=20).\nData Used\nAcc.\nF1 Prec. Rec.\nM4 Self-posts + top neighbor posts\n0.95 0.96\n0.93 0.99\nM5 Self-posts + worst neighbor posts 0.75 0.76\n0.78 0.75\nM6 Self-posts + non-neighbor posts\n0.68 0.71\n0.67 0.78\nreflects influence via important neighbors, and\nCPageRank estimates probabilistic importance. The\naggregated NeighborScore identifies peers with\nstrong direct ties and broader network influence\naround U. As some centralities are correlated, S(n)\napproximates overall neighbor prominence rather\nthan precise influence. Performance variation by\nneighbor depth and count is shown in Appendix Ta-\nble A9.\n4.3\nInput Data Formalization\nWe structured the input to language models to cap-\nture U’s timeline, others’ replies, and peer interac-\ntions. The overall data formalization, distinguish-\ning immediate and neighbor interactions, is shown\nin Figure 2.\nImmediate Interaction\nWe aggregated content\nfrom a user’s timeline, along with others’ replies,\nand then used all of these textual content for fine-\ntuning the language model. To train our models, we\nused different combinations of features—1) U’s\nself-posts, 2) U’s self-posts and self-comments,\nand 3) U’s self-posts, self-comments, and others’\nreplies to U’s posts or comments.\nNeighbor Interaction\nNeighbor Interaction cap-\ntures signals from proximate peers. For this pur-\npose, we temporally aligned the timelines of U and\ntheir top-n neighbors. Then, at each timestamp i,\nwe selected ten posts by the top neighbors closest\nin time to U’s post. We aggregated the neighbors’\nposts with U’s posts and embedded them into a\ndense vector representation. This approach led to\nthe fourth type of model, where we included fea-\ntures from both immediate interactions above, as\nwell as the top neighbors’ posts.\nTable 3: Model performance excluding neighbors with\nr/SuicideWatch posts. Results remain comparable to\nusing all neighbors, indicating that broader neighbor\ninteractions capture key signals for implicit SI detection\n(20 epochs).\nData Used\nAcc.\nF1 Prec. Rec.\nM4 Self-posts + top neighbor posts\n0.95 0.96\n0.93 0.99\nM7 Self-posts + filtered neighbor posts 0.89 0.90\n0.87 0.93\n4.4\nImplicit SI signal classification with\nmodeling\nFor our study, we leveraged Microsoft’s Deberta-\nv3-large model (He et al., 2021)—a 418M param-\neter transformer with 24 layers that processes se-\nquences up to 512 tokens. We fine-tuned this model\nusing CLS for global representation and SEP for\nsegment boundaries on an Nvidia A100 GPU. We\nframed our problem of detecting implicit SI as a\nbinary classification task—labeling each input xi\nas yi ∈{0, 1} for absence or presence of risk. We\ntokenized text into subword embeddings with at-\ntention masks, padded or truncated to 512 tokens,\nand encoded them to obtain a pooled [CLS] vector\n(1024-dim). A linear layer then mapped this vector\nto logits, followed by softmax for class probabil-\nities. We optimized the model with cross-entropy\nloss using AdamW (learning rate 2 × 10−5, weight\ndecay 0.01) for 20 epochs with batch size 8, and\nevaluated after each epoch on a held-out validation\nset using accuracy, precision, recall, and F1-score.\nResults from fine-tuning alternative base models\nare provided in Appendix Table A2.\n4.5\nLinguistic Analysis of Interaction Types\nWe analyzed lexical, topical, and psycholinguistic\npatterns across users’ self-posts, self-comments,\nreplies, and top-neighbor posts. For lexical analy-\nsis, we used the Sparse Additive Generative Model\n(SAGE) (Eisenstein et al., 2011) to identify dis-\ncriminative unigrams and bigrams distinguishing\nimmediate and neighbor interactions. SAGE uses\nmultinomial models with adaptive regularization to\nbalance frequent and rare terms. For topical analy-\nsis, we employed BERTopic (Grootendorst, 2022)\non Case and Control data, varying topic numbers\n(k=2–15) and achieving the highest coherence at\nk=12. After removing one outlier, eleven topics\nremained (Table 4) whose topic names were as-\nsigned by our clinical psychologist coauthor. Their\nnormalized proportions captured topical variation\nacross interaction types Table 5. For psycholinguis-\ntic analysis, we examined the occurrences of the af-\n5\n\n20\n40\n60\n80\nuser posts(t)\n0.90\n0.91\n0.92\n0.93\n0.94\neval_accuracy\nt=88, 0.943\n(a) Accuracy\n20\n40\n60\n80\nuser posts(t)\n0.91\n0.92\n0.93\n0.94\n0.95\neval_f1\nt=88, 0.948\n(b) F1 Score\n20\n40\n60\n80\nuser posts(t)\n0.925\n0.930\n0.935\n0.940\n0.945\neval_precision\nt=73, 0.948\n(c) Precision\n20\n40\n60\n80\nuser posts(t)\n0.89\n0.90\n0.91\n0.92\n0.93\n0.94\n0.95\neval_recall\nt=44, 0.956\n(d) Recall\nFigure 3: Performance metrics on varying the number of input user posts (t). The plots show that the performance\npeaks at t=88.\n2\n4\n6\n8 10 12 14\nneighbor post(n)\n0.88\n0.90\n0.92\n0.94\n0.96\neval_accuracy\nn=7, 0.966\n(a) Accuracy\n2\n4\n6\n8 10 12 14\nneighbor post(n)\n0.88\n0.90\n0.92\n0.94\n0.96\neval_f1\nn=7, 0.970\n(b) F1 Score\n2 4 6 8 101214\nneighbor post(n)\n0.84\n0.86\n0.88\n0.90\n0.92\n0.94\n0.96\n0.98\neval_precision\nn=10, 0.978\n(c) Precision\n2\n4\n6\n8 10 12 14\nneighbor post(n)\n0.92\n0.93\n0.94\n0.95\n0.96\n0.97\n0.98\n0.99\neval_recall\nn=7, 0.990\n(d) Recall\nFigure 4: Performance metrics on varying the number of neighbor posts (n). The plots show that the performance\npeaks at around n=7.\nfect category of keywords as per the well-validated\nLinguistic Inquiry and Word Count (LIWC) lexi-\ncon (Tausczik and Pennebaker, 2010).\n5\nResults\n5.1\nModel Performance (Immediate\nInteraction vs. Neighbor Interaction):\nTable 1 summarizes the performance comparison\nof models with combinations of immediate inter-\nactions and neighbor interactions feature set. The\nbaseline model using only a target user’s posts per-\nforms strongly (Accuracy = 0.86, F1 = 0.85),\nshowing that self-authored text alone captures\nsalient linguistic markers of future SI, consistent\nwith prior temporal analyses on Reddit and Face-\nbook (Coppersmith et al., 2018; De Choudhury\net al., 2016). Adding the user’s self-comments\nboosts recall (0.84 →0.91) but lowers precision\n(0.88 →0.78), suggesting broader coverage but\nadded noise in terms of precision. For example, a\ncomment from a Case user reads, “Are they allowed\nto hit me? [..] I need to stay strong,” including both\nthemes of physical abuse as well as optimism. Addi-\ntionally, including replies to user’s posts/comments\nfurther reduces accuracy (0.81 →0.80) and F1\n(0.83 →0.79). Such replies often convey empa-\nthy or advice that mask the user’s true mindset\n(Gkotsis et al., 2017), as in one response: “I think\ntalking to a professional would help, because he\ncan understand you and give you advice.” We note\nthat the top-neighbor interaction approach achieves\nthe highest performance (Accuracy=0.95, F1=0.96,\nPrecision=0.93, Recall=0.99), with high recall re-\nducing false negatives, which is critical for SI risk\ndetection (Franklin et al., 2017). This suggests that\nfeatures reflecting social exposure and information\nenvironment context provide valuable predictive\nsignals for implicit SI.\n5.2\nDetermining Optimal Post Counts for\nRobust Prediction\nWe evaluated how many posts from users (Case and\nControl) and their top neighbors optimize implicit\nSI detection. First, varying the number of neigh-\nbor posts (n) showed an improvement in accuracy\nfrom 0.87 to 0.97, with near-perfect recall (0.99) at\nn = 7, after which the gains plateaued (Figure 4).\nWith n = 7 fixed, varying self-posts (t) revealed\nunstable performance at low t (e.g., recall 0.87 at\nt = 1), stabilizing for t ≥40; the best results oc-\ncurred at t = 88 (accuracy 0.94, F1 0.95, precision\n0.95, recall 0.96) (Figure 3). These findings indi-\ncate that combining a user’s full history with a few\n6\n\nTable 4: Clinically informed topics identified from posts and comments via topic modeling, with corresponding\nexplanations and keywords.\nTopic Theme\nExplanation\nKeywords\nSelf-Harm Tools\nMentions of knives, blades, or cutting instruments indicating\nself-injury ideation\nknife, blade, cut, sharpen\nDiet / Body Image\nDiscussions of calories, weight, dieting, or eating habits reflect-\ning body concerns\ncalori, weight, diet, eat\nPhysical Pain / Health\nMentions of pain, medication, or suffering over time\npain, aspirin, suffer, longer\nWar / Military\nDiscussions related to military, soldiers, or historical conflicts\nsoldier, german, armi, soviet\nAcademic / Mental Stress\nMentions of depression, PhD, or mental strain\ndepress, phd, ggoddamn, eurozon\nCringe / Embarrassment\nMentions of awkwardness, social discomfort, or cringing situa-\ntions\ncri, cringi, crusad, crimin\nBurn / Injury\nMentions of burns, cuts, or physical injury\nburn, cut, hurt, degre\nSelf-Harm / Coping Strategies\nMentions of harming oneself or suicide coping mechanisms\nharm, self, suicidebyword, cope\nViolent Ideation\nMentions of killing or violent intent in extreme contexts\nkill, killin, zaibatsu, meeee\nConfusion / Uncertainty\nExpressions of perplexity or inability to understand situations\nconfus, percent, stranger, 12ish\nCute / Affection\nPositive, playful, or affectionate language\ncute, soo, aww, omgggg\nTable 5: Normalized topic proportions across different\ncontent sources. ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05.\nTopic\nUser’s\nSelf-\nPosts\nUser’s\nSelf-\nComments\nOther’s\nCom-\nments\nTop\nNeigh-\nbor’s\nPosts\nKruskal-\nWallis\nH-stat.\nSelf-Harm Tools\n0.002\n0.002\n0.002\n0.006\n44.08***\nDiet & Body Image\n0.005\n0.003\n0.003\n0.003\n46.36***\nPhysical\nPain\n&\nHealth\n0.003\n0.002\n0.002\n0.003\n51.55***\nWar & Military\n0.013\n0.003\n0.001\n0.002\n7.45*\nAcademic Stress\n0.001\n0.001\n0.002\n0.002\n43.81***\nCringe & Embarrass-\nment\n0.000\n0.001\n0.001\n0.002\n39.49***\nBurn & Injury\n0.000\n0.000\n0.000\n0.001\n1.84\nCoping & Self-Harm\n0.001\n0.001\n0.001\n0.002\n22.81***\nViolent Ideation\n0.001\n0.001\n0.000\n0.002\n26.13***\nConfusion & Uncer-\ntainty\n0.001\n0.001\n0.001\n0.000\n12.62**\nCute / Affection\n0.000\n0.001\n0.001\n0.002\n22.93***\nneighbor posts yields a robust SI predictor.\n5.3\nRole of Top Neighbor Posts in Capturing\nSI Signals\nTo understand the superior performance of top-\nneighbor-based modeling, we analyzed lexical, top-\nical, and psycholinguistic patterns in users’ self-\nposts, self-comments, replies, and posts of their\ntop neighbors. Others’ replies to a user’s posts\nor comments provide a limited discriminative sig-\nnal: common bigrams such as “feel like”, “year\nold”, and “every day” appear across both Case\nand Control data, reflecting supportive or neutral\ndiscourse (Zirikly et al., 2019). Topic modeling\nshows that clinically relevant themes (e.g., Self-\nHarm Tools, Coping/Self-Harm, Violent Ideation)\nare rare in others’ replies, while users’ own posts\nexhibit only moderate and inconsistent signals (Ta-\nble 4, Table 5). In contrast, top neighbor posts con-\nsistently reflect mental-health-related themes that\nare highly predictive of SI (Kirtley et al., 2021). For\nCase users, neighbors frequently post about Self-\nHarm Tools, Coping / Self-Harm, Violent Ideation,\nPhysical Pain / Health, and Academic / Mental\nStress, with bigrams such as “mental health”, “sui-\ncide prevention”, and “emotional support” appear-\ning frequently. Neighbors of Control users, by com-\nparison, focus on neutral or unrelated topics (Ta-\nble 4, Table 5). Table 6 shows that differences in\naffective categories (e.g., negative affect, anger, and\nsadness) are statistically significant, indicating that\nincluding neighbor posts enhances the distinction\nbetween Case and Control users’ content. There-\nfore, top-neighbor posts provide richer social con-\ntext than replies or isolated user history. Combining\nthem with user history yields a more robust model\nfor early detection of early and implicit SI.\n5.4\nRobustness Tests:\nTo ensure that our findings are not artifacts of\nspecific modeling or sampling choices, we con-\nducted additional robustness tests by varying the\ntype of neighbors included—for instance, incor-\nporating non-top neighbors into our models. For\nthis purpose, we built additional models M5 with\nthe lowest-ranked neighbors and M6 with non-\nneighbors—a random selection of other users—of a\ntarget user. Table 2 compares the performance—we\nnote both M4 and M5 perform significantly poorly.\nTherefore, the performance boost by including the\ntop neighbor posts is not by chance, and rather an\nimportant element in detecting implicit SI (Cero\net al., 2024). Moreover, Table 3 demonstrates that\nthe model retains strong predictive performance\neven after filtering out all top neighbors who have\nposted in r/SuicideWatch. This finding indicates\nthat successful detection does not rely solely on\nneighbors’ direct SI-related language, but rather on\nbroader contextual cues captured from high-quality\ninteractions. Appendix Table A4 shows linguistic\nanalyses aligned with Table 2, Table A3 highlights\n“Affect” differences linked to Table 3, and addi-\n7\n\nTable 6: Comparison of LIWC Affect categories across M1–M4. Columns show normalized category counts for\nCase and Control groups and their differences. ***p<0.001, *p<0.01, p<0.05.\nCategory\nM1 (Immediate)\nM2 (Immediate)\nM3 (Immediate)\nM4 (Neighbor)\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nPos. Affect\n0.049\n0.053\n-0.004\n38.80***\n0.080\n0.074\n0.006\n416.94***\n0.060\n0.057\n0.003\n45.86***\n0.048\n0.053\n-0.005\n23.57***\nNeg. Affect\n0.029\n0.025\n0.004\n74.947***\n0.035\n0.030\n0.005\n651.17***\n0.029\n0.026\n0.003\n100.07***\n0.029\n0.021\n0.007\n169.87***\nAnxiety\n0.004\n0.003\n0.001\n125.379***\n0.003\n0.002\n0.001\n358.217***\n0.002\n0.002\n0.000\n176.606***\n0.003\n0.002\n0.000\n62.663***\nAnger\n0.009\n0.006\n0.003\n50.654***\n0.012\n0.010\n0.002\n322.855***\n0.010\n0.008\n0.002\n77.809***\n0.011\n0.006\n0.005\n130.857***\nSadness\n0.007\n0.007\n0.000\n71.131***\n0.008\n0.007\n0.001\n376.545***\n0.006\n0.005\n0.001\n136.379***\n0.009\n0.005\n0.004\n181.551***\ntional model comparisons with LIWC results are\nin Appendix Table A1 (Tables A5–A8).\n6\nDiscussion and Implications\nA key takeaway of our study is that, early and im-\nplicit SI is best detected by modeling both an indi-\nvidual’s longitudinal online activity and their sur-\nrounding information environment, capturing the\ninherently relational nature of SI expressions (Am-\nmerman and Jacobucci, 2023). Neighbor-based\nmodeling is particularly effective, as top neigh-\nbors, weighted via a NeighborScore, capture direct\nbehavior and network-level influences, consistent\nwith epidemiological evidence and classic effects\nlike Werther and Papageno (Wyman et al., 2019;\nPhillips, 1974; Niederkrotenthaler et al., 2010;\nYuan et al., 2023). Neighbor posts provide com-\nplementary signals beyond the user’s self-content,\nshowing that exposure to neighbors’ expressions of\ndistress or coping is associated with detection.\nNeighbor-informed models frame suicidality\nwithin a social-ecological perspective (Bronfen-\nbrenner, 1979), reflecting social contagion pro-\ncesses where suicidal behaviors and ideation can\nspread through networks, especially when peers\ndisclose distress (Wyman et al., 2019; Gould et al.,\n2003). Online environments may amplify these\neffects, as individuals in high-risk networks can\nshow subtle precursors before explicit disclosure.\nNetwork theories suggest that peers model behav-\nior and influence interpretations of distress (Cero\net al., 2024; Bearman and Moody, 2004; Christakis\nand Fowler, 2025), explaining why neighbor in-\nteractions reflect relational signals of implicit SI.\nA user’s longitudinal posting history remains es-\nsential, tracing gradual shifts in tone, sentiment,\nand discourse (De Choudhury et al., 2016). How-\never, interactional content must be selective: in-\ncluding all replies adds noise and weakens risk sig-\nnals (Schmidt et al., 2024). As shown in Table 6, the\ndifference between the LIWC “Affect” categories is\nconsistently higher in M4 where neighbors’ posts\nare added. This shows that including posts from the\nneighbors add a clear distinction signal between\nthe Case and Control data, making it better for\nthe model to predict (Lu and Tu, 2024; Guo et al.,\n2012). These findings suggest avenues for theoreti-\ncal and methodological extensions. Beyond the In-\nterpersonal Theory of Suicide (Joiner, 2005), mod-\nels such as the Integrated Motivational Volitional\nModel (O’Connor and Kirtley, 2018) and the Three-\nStep Theory (Klonsky and May, 2015) may help\nidentify subtle motivational or volitional cues in\nlanguage and interactions. The strong influence\nof social context further supports contagion and\nnetwork theories, indicating that implicit suicidal\nideation emerges from both individual cognition\nand broader interpersonal environments. Overall,\nthis work enhances detection accuracy while fram-\ning suicide risk as a relational process in digital\nsocial environments. By showing that peer-network\ninteractions substantially improve predictive power,\nit bridges computational modeling with network-\ninformed prevention strategies.\n7\nConclusion\nOur study demonstrates that early and implicit SI\nis best detected when a user’s longitudinal activ-\nity is interpreted within their social context. Subtle\nlinguistic and interactional cues often emerge well\nbefore explicit disclosures, enabling identification\nof risk trajectories at an early stage. By incorporat-\ning posts from socially proximal peers, our frame-\nwork operationalizes theories of suicide contagion\nand social influence, yielding substantial gains in\npredictive performance. Notably, while a user’s\nown posts and comments provide moderate insight,\neven a small set of strategically selected neighbor\nposts carries a significantly strong predictive sig-\nnal. These findings highlight that suicide risk is\ninherently relational: effective detection requires\nselectively integrating self-content and socially dis-\ntributed cues. Beyond improving accuracy, this re-\nlational framing supports scalable, context-aware,\nand ethically responsible early-warning systems\nthat align with both clinical theory and the dynam-\nics of online communities.\n8\n\n8\nLimitations\nOur study has limitations that also show interesting\nfuture directions. Methodologically, our models\nexclusively rely on textual content, overlooking\nmultimodal signals such as images, videos, emojis,\nGIFs, or external links that often convey emotional\nstates, coping strategies, or distress. Incorporating\nthese modalities could improve sensitivity to subtle\nrisk signals, enhance interpretability, and provide\na more holistic understanding of online behaviors\nassociated with suicidal ideation. Similarly, our\ncurrent approach treats peer interactions homoge-\nneously, though peers can exert positive, neutral,\nor negative influences. Modeling these distinctions\ncould capture the functional impact of social con-\ntext on risk trajectories and guide ethically respon-\nsible interventions.\nFinally, robustness and generalizability can be\nimproved through replication across platforms (e.g.,\nTwitter, TikTok, Discord), multilingual and cross-\ncultural settings, and naturalistic populations. Tem-\nporal weighting, trajectory-based risk modeling,\nand interpretability techniques such as attention\nvisualization, SHAP, or counterfactuals could clar-\nify key linguistic and network features while en-\nhancing actionable insights. Future work should\nintegrate multimodal inputs, refine social context\nmodeling, and maintain rigorous ethical oversight\nto ensure predictive models support vulnerable in-\ndividuals responsibly and effectively.\n9\nEthical Considerations\nReflexivity\nThis paper used publicly accessible\nsocial media discussions on Reddit and did not re-\nquire direct interactions with individuals, thereby\nnot requiring ethics board approval. However, we\nare committed to the ethics of the research and we\nfollowed practices to secure the privacy of individ-\nuals in our dataset. Our research team comprises\nresearchers holding diverse gender, racial, and cul-\ntural backgrounds, including people of color and\nimmigrants, and hold interdisciplinary research ex-\npertise. This team consists of computer scientists\nwith expertise in social computing, NLP, and HCI,\nand psychologists with expertise in clinical psy-\nchology, adolescent depression and suicide, and\ndigital health interventions. One psychologist coau-\nthor specializes in suicide etiology, suicide preven-\ntion, and crisis intervention, and another psychol-\nogist coauthor is a clinical psychologist with over\n16 years of experience spanning adult and adoles-\ncent inpatient care and crisis suicide helplines. To\nensure validity and prevent misrepresentation, our\nfindings were reviewed and corroborated by our\npsychologist coauthors. However, our work is not\nintended to replace the clinical evaluation of an\nindividual undergoing suicidal thoughts and should\nnot be taken out of context to conduct mental health\nassessments.\nRisk of Misinterpretation and Harm with Au-\ntomated Detection\nOur study leverages compu-\ntational methods to identify early and implicit sui-\ncidal ideation (SI) in online communities. While\nthese approaches provide opportunities for early\nintervention, they also present serious ethical chal-\nlenges. Automated systems may misinterpret sub-\ntle linguistic signals or social cues, leading to false\npositives or false negatives. Misclassification can\nresult in unwarranted labeling of individuals as at\nrisk, potentially causing stigma, anxiety, or social\nconsequences. Conversely, failing to detect gen-\nuine risk may deny individuals timely support. We\nemphasize that suicidal ideation manifests hetero-\ngeneously across individuals, and even carefully\ndesigned models cannot fully capture the nuanced\ncontext of personal distress.\nPrivacy, Consent, and Data Sensitivity\nGiven\nthe sensitive nature of suicidal thoughts, the pri-\nvacy and confidentiality of users are paramount.\nOur approach relies on publicly available text, but\nthe inclusion of social network information, even\naggregated, raises concerns about inadvertent expo-\nsure of personal interactions. Ethical deployment\nrequires anonymization, secure storage, and strict\naccess controls. Users should be informed about\npotential data usage and retention, and explicit con-\nsent should be prioritized wherever feasible.\nSocial Context and Potential Misuse\nBy mod-\neling social context and peer interactions, our work\noperationalizes theories of social influence in sui-\ncidal ideation. However, this introduces additional\nethical considerations. Insights derived from peer\ninteractions could be misused if exploited for non-\nclinical purposes, such as targeted advertising or\nprofiling of vulnerable individuals. Careful gover-\nnance and ethical oversight are necessary to ensure\nthat social context is leveraged solely for support-\nive, preventive interventions rather than for com-\nmercial or punitive applications.\nTransparency, Interpretability, and Responsi-\nble Use\nEthical deployment also requires trans-\n9\n\nparency and interpretability. Systems identifying SI\nmust clearly communicate their scope, limitations,\nand the fact that outputs are probabilistic, not diag-\nnostic. Stakeholders, including clinicians, platform\nmoderators, and potentially affected users, should\nunderstand how predictions are generated, espe-\ncially when interventions are triggered. Responsi-\nble use also involves integrating human-in-the-loop\nframeworks, where trained professionals augment\ncomputational predictions to avoid over-reliance\non automated systems.\nFuture Directions for Ethical AI\nFuture re-\nsearch should explore multimodal detection meth-\nods while carefully balancing privacy and ethical\nconstraints. Integrating images, URLs, or videos\nmay improve sensitivity, but must be approached\nwith strict ethical safeguards. Additionally, system-\natically categorizing peer influence, positive, neu-\ntral, or negative, could enhance the interpretability\nand safety of predictions, ensuring interventions are\ninformed by context rather than raw social activ-\nity. Overall, ethical design in computational mental\nhealth requires prioritizing user welfare, minimiz-\ning harm, and embedding accountability at every\nstage of model development and deployment.\n10\nAI Involvement Disclosure\nCertain sections of the manuscript were refined us-\ning AI-assisted writing tools (e.g., ChatGPT, Gram-\nmarly). All analyses, scientific content, and experi-\nments were written solely by the authors.\nReferences\nDaniyal Alghazzawi, Hayat Ullah, Naila Tabassum,\nSahar K Badri, and Muhammad Zubair Asghar.\n2025. 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In\nProceedings of the sixth workshop on computational\nlinguistics and clinical psychology, pages 24–33.\n13\n\nA\nAppendix\nTable A1: Model Performance on Remaining Data Combinations (epochs=20).\nModel\nData Used\nAccuracy\nF1\nPrecision\nRecall\nM8\nTop neighbor’s posts only\n0.83\n0.82\n0.92\n0.74\nM9\nTarget user posts + comments + top neighbor posts\n0.91\n0.92\n0.91\n0.92\nM10\nTarget user posts + comments + others’ comments + top neighbor\nposts\n0.71\n0.64\n0.75\n0.56\nM11\nFiltered neighbor posts only (no post/comments in r/SuicideWatch)\n0.78\n0.77\n0.73\n0.82\nTable A2: Performance comparison of transformer models on the classification task.\nModel\nAccuracy\nF1\nPrecision\nRecall\nmicrosoft/deberta-v3-large\n0.955\n0.946\n0.945\n0.945\nroberta-large\n0.944\n0.931\n0.944\n0.919\nbert-base-uncased\n0.933\n0.921\n0.897\n0.946\ngoogle/electra-large-discriminator\n0.944\n0.935\n0.900\n0.973\nTable A3: Comparison of LIWC affect categories between M4 and M7 showing how the LIWC Affect category\nvaries when considering all top neighbors vs. filtered top neighbors. Columns show mean proportions for users\nshowing suicidal ideation (SI) and not, their difference, and the Kruskal–Wallis H-statistic. Significance codes:\n***p<0.001, **p<0.01, *p<0.05.\nCategory\nM4\nM7\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nposemo\n0.048\n0.053\n-0.005\n23.568***\n0.050\n0.053\n-0.003\n17.482***\nnegemo\n0.029\n0.021\n0.007\n169.871***\n0.032\n0.025\n0.007\n72.531***\nanx\n0.003\n0.002\n0.000\n62.663***\n0.003\n0.002\n0.001\n63.084***\nanger\n0.011\n0.006\n0.005\n130.857***\n0.010\n0.006\n0.004\n51.436***\nsad\n0.009\n0.005\n0.004\n181.551***\n0.008\n0.007\n0.001\n71.131***\n14\n\nTable A4: Comparison of LIWC affect categories across M4, M5, and M6 showing how the Affect LIWC category\nvaries across various neighbors (top, worst, random). Columns show mean proportions for users showing suicidal\nideation (SI) and not, their difference, and the Kruskal–Wallis H-statistic. Significance codes: ***p<0.001, **p<0.01,\n*p<0.05.\nCategory\nM4\nM5\nM6\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nCase\nControl\nDiff.\nH\nposemo\n0.048\n0.053\n-0.005\n23.568***\n0.047\n0.049\n-0.003\n38.611***\n0.048\n0.047\n0.001\n42.425***\nnegemo\n0.029\n0.021\n0.007\n169.871***\n0.029\n0.025\n0.004\n74.947***\n0.030\n0.024\n0.006\n59.078***\nanx\n0.003\n0.002\n0.000\n62.663***\n0.004\n0.002\n0.002\n53.518***\n0.003\n0.002\n0.001\n125.397***\nanger\n0.011\n0.006\n0.005\n130.857***\n0.009\n0.006\n0.003\n40.154***\n0.009\n0.006\n0.002\n138.704***\nsad\n0.009\n0.005\n0.004\n181.551***\n0.007\n0.007\n0.000\n71.131***\n0.007\n0.008\n-0.001\n36.640***\nTable A5: Comparison of LIWC affect categories for M8. Columns show mean proportions for suicidal and non-\nsuicidal users, their difference, and the Kruskal–Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01,\n∗p < 0.05.\nCategory\nM8\nCase\nControl\nDiff\nH\nposemo\n0.053\n0.053\n0.000\n0.409\nnegemo\n0.026\n0.025\n0.001\n4.805*\nanx\n0.001\n0.002\n-0.001\n3.323\nanger\n0.009\n0.006\n0.003\n9.523**\nsad\n0.005\n0.007\n-0.001\n3.638\nTable A6: Comparison of LIWC affect categories for M9. Columns show mean proportions for suicidal and non-\nsuicidal users, their difference, and the Kruskal–Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01,\n∗p < 0.05.\nCategory\nM9\nCase\nControl\nDiff\nH\nposemo\n0.050\n0.053\n-0.003\n17.482***\nnegemo\n0.032\n0.025\n0.007\n72.531***\nanx\n0.003\n0.002\n0.001\n63.084***\nanger\n0.010\n0.006\n0.004\n51.436***\nsad\n0.008\n0.007\n0.001\n71.131***\nTable A7: Comparison of LIWC affect categories for M10. Columns show mean proportions for suicidal and non-\nsuicidal users, their difference, and the Kruskal–Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01,\n∗p < 0.05.\nCategory\nM10\nCase\nControl\nDiff\nH\nposemo\n0.070\n0.067\n0.003\n349.268***\nnegemo\n0.031\n0.028\n0.003\n538.074***\nanx\n0.003\n0.002\n0.000\n403.691***\nanger\n0.010\n0.009\n0.001\n281.206***\nsad\n0.007\n0.006\n0.001\n361.992***\nTable A8: Comparison of LIWC affect categories for M11. Columns show mean proportions for suicidal and non-\nsuicidal users, their difference, and the Kruskal–Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01,\n∗p < 0.05.\nCategory\nM11\nCase\nControl\nDiff\nH\nposemo\n0.053\n0.053\n0.000\n0.409\nnegemo\n0.026\n0.025\n0.001\n4.805*\nanx\n0.001\n0.002\n-0.001\n3.323\nanger\n0.009\n0.006\n0.003\n9.523**\nsad\n0.005\n0.007\n-0.001\n3.638\n15\n\nTable A9: Performance metrics for different depth and neighbor configurations. Depth refers to the number of\nrecursive depth considered, and Neighbors refers to the number of top neighbor posts included.\nDepth\nNeighbors\nAccuracy\nF1\nPrecision\nRecall\n1\n1\n0.98\n0.97\n0.96\n0.98\n1\n2\n0.97\n0.96\n0.95\n0.97\n1\n3\n0.96\n0.95\n0.95\n0.96\n1\n4\n0.96\n0.95\n0.94\n0.96\n1\n5\n0.95\n0.94\n0.94\n0.95\n2\n1\n0.97\n0.96\n0.95\n0.97\n2\n2\n0.96\n0.95\n0.94\n0.96\n2\n3\n0.95\n0.94\n0.93\n0.95\n2\n4\n0.95\n0.94\n0.93\n0.95\n2\n5\n0.94\n0.93\n0.92\n0.94\n3\n1\n0.96\n0.95\n0.94\n0.96\n3\n2\n0.95\n0.94\n0.93\n0.95\n3\n3\n0.94\n0.93\n0.92\n0.94\n3\n4\n0.94\n0.93\n0.92\n0.94\n3\n5\n0.95\n0.94\n0.93\n0.95\n4\n1\n0.95\n0.94\n0.93\n0.95\n4\n2\n0.94\n0.93\n0.92\n0.94\n4\n3\n0.93\n0.92\n0.91\n0.93\n4\n4\n0.93\n0.92\n0.91\n0.93\n4\n5\n0.92\n0.91\n0.90\n0.92\n5\n1\n0.94\n0.93\n0.92\n0.94\n5\n2\n0.93\n0.92\n0.91\n0.93\n5\n3\n0.92\n0.91\n0.90\n0.92\n5\n4\n0.92\n0.91\n0.90\n0.92\n5\n5\n0.91\n0.90\n0.89\n0.91\n16\n"},"cleaned":{"kind":"string","value":"Detecting Early and Implicit Suicidal Ideation via Longitudinal and Information Environment Signals on Social Media Soorya Ram Shimgekar1, Ruining Zhao1, Agam Goyal1, Violeta J. Rodriguez1, Paul A. Bloom2, Hari Sundaram1, Koustuv Saha1 1 -Champaign, {sooryas2, ruining, agamg2, vjrodrig, hs1, 2Columbia University, New York State Psychiatric Institute, Abstract On social media, many individuals experiencing suicidal ideation (SI) do not disclose their distress explicitly. Instead, signs may surface indirectly through everyday posts or peer interactions. Detecting such implicit signals early is critical but remains challenging. We frame early and implicit SI as a forward-looking prediction task and develop a computational framework that models a user's information environment, consisting of both their longitudinal posting histories as well as the discourse of their socially proximal peers. We adopted a composite network centrality measure to identify top neighbors of a user, and temporally aligned the user's and neighbors' interactionsintegrating the multi-layered signals in a finetuned DeBERTa-v3 model. In a Reddit study of 1,000 (500 Case and 500 Control) users, our approach improves early and implicit SI detection by 15% over individual-only baselines. These findings highlight that peer interactions offer valuable predictive signals and carry broader implications for designing early detection systems that capture indirect as well as masked expressions of risk in online environments. 1 Introduction More than 703,000 individuals die by suicide each year worldwide, making it a major global public health concern (WHO, 2025). Suicidal ideation (SI) refers to a spectrum of cognitive and behavioral manifestations related to suicide, ranging from passive thoughts of death to active planning and engagement in self-injurious actions with intent to die (Joiner, 2005). While early detection of these signs is critical, psychiatry and mental health services have long struggled to detect risk before individuals disclose it explicitly (Calear and Batterham, 2019; Hom et al., 2017). A recent meta-analysis revealed that the estimated prevalence of SI disclosure is only 46.3%, inferring that the majority of people who experience suicidal thoughts do not disclose them, making early detection critical for timely intervention (Hallford et al., 2023). Online platforms have become vital spaces where signs of SI may surface. People use online mental health communities to share distress, seek help, and connect with peers (De Choudhury and De, 2014). This has created opportunities for both researchers and clinicians to understand suicide risk through language, social interactions, and online behavioral cues. Prior work has shown the value of linguistic cues for identifying mental health risks (Coppersmith et al., 2014; De Choudhury et al., 2013; Guntuku et al., 2017). In particular, within the context of SI, prior work has computationally modeled the language of SI on social media (Burnap et al., 2015; Saha et al., 2019; Alghazzawi et al., 2025; Zhang et al., 2025; Naseem et al., 2025). More recent computational approaches leveraged longitudinal and multimodal data, as well as social network analyses, to anticipate suicide-related behaviors (Shen et al., 2020). Yet, critical limitations persist. Prior research has primarily examined posts where SI is explicitly conveyed, such as direct or indirect references to selfharm or suicidal thoughts within suicide-related forums or discussions (Ji, 2022a; Bloom et al., 2025). These approaches presuppose that individuals articulate their distress, yet, many at-risk individuals neither disclose SI nor exhibit overt warning signs, but instead mask distress within seemingly ordinary discourse (McGillivray et al., 2022; Mérelle et al., 2018; Podlogar et al., 2022). Our work targets the detection of these undisclosed SI, characterized by subtle, contextually obscured, or socially distributed indicators, to potentially enable earlier and more effective intervention. To address the above gap, our work is guided by the research question (RQ): How can early signals of SI be detected in social media activity, particularly in the absence of explicit disclosures? To address the RQ, we conceptualize early and im1 16 Oct 2025 plicit SI as a forward-looking prediction problem that requires modeling both an individual's longitudinal behavior and the broader social context in which that behavior unfolds. We develop a framework that jointly captures temporal dynamics in a user's posting history and the conversations of socially proximal peers, enabling a richer representation of early warning signals. This idea of using socially proximal peers to understand an individual's mental state is derived from many prior acclaimed psychology works (La Greca and Harrison, 2005; Copeland, 2021; Victor et al., 2019). Within this framework, we pursue three aims: Aim 1: Examine whether users' longitudinal posting patterns, in conjunction with commentary on those posts, reveal early signals of implicit SI. Aim 2: Examine how the surrounding information environment, including peer interactions and exposure to content, contributes as an additional predictive feature for implicit SI. Aim 3: Identify the linguistic markers that underlie interaction types, in terms of how specific language cues are associated with these interactions. We conducted our study on Reddit, focusing on a sample of 1,000 users divided into 500 Case and 500 Control users (who never participated in mental health conversations). Our predictive framework jointly modeled each user's full posting timeline along with the discourse of their most socially proximal neighbors, identified through network centrality, and fine-tuned a DeBERTa-v3 model (He et al., 2021) to embed both individual and peer interaction signals into a unified representation. Our findings show that incorporating peer interactions within the information environment, alongside users' longitudinal posting histories, significantly enhances detection of early and implicit SI, improving performance by 15% compared to models based only on longitudinal user data. Overall, this paper contributes: 1) a framework for detecting implicit SI; 2) a method for systematically integrating environment information through neighbor interactions; and 3) empirical evidence that this integration improves early detection. 2 Related Work 2.1 Suicidal Ideation and Social Context Psychological theory views SI as the product of intertwined social and cognitive factors. The Interpersonal Theory of Suicide (Joiner, 2005) posits that suicidal desire arises from perceived burdensomeness and thwarted belongingness, while capability develops through repeated exposure to pain or fear. Early computational studies of depression and SI focused on overt signals such as self-disclosure or negative sentiment (Coppersmith et al., 2014; De Choudhury et al., 2013), but later work showed that many at-risk individuals express distress through subtle cues, motivating methods that model temporal and semantic dynamics of behavior (Guntuku et al., 2017; Benton et al., 2017; Saha and De Choudhury, 2017). A range of methods have been proposed to detect mental health risk signals beyond surface cues. For instance, (Fatima et al., 2021) introduced DASentimental, a semi-supervised model combining bagof-words and semantic networks, while (Trotzek et al., 2018) showed that convolutional networks with linguistic metadata enable earlier detection. More recent work leverages large language models, with GPT-3.5/4 using chain-of-thought prompting on diaries (Shin et al., 2024) and reasoningguided LLMs improving interpretability (Teng et al., 2025). Temporal dynamics remain critical, from emotional \"phases\" in user timelines (Sawhney et al., 2021a) to transformer-based models enriched with temporal signals (Sawhney et al., 2020). Social context is equally important: hyperbolic embeddings of user histories and peer interactions enhance prediction (Sawhney et al., 2021b), peer networks and conversational responses influence trajectories (Wyman et al., 2019; De Choudhury and Kiciman, 2017), and longitudinal patterns reveal precursors of SI (De Choudhury et al., 2016). Complementary directions include clinical-domain datasets such as ScAN (Rawat et al., 2022), automated counseling support with PsyGUARD (Qiu et al., 2024), and calls to model underlying intent rather than surface disclosure (Ji, 2022b). Together, this work affirms that SI arises from psychological distress, temporal dynamics, and social context, demanding models that go beyond surface cues. Yet most approaches still rely on explicit disclosures or static timelines, overlooking how evolving language interacts with peer responses. Our framework addresses this by jointly modeling users' longitudinal histories and post-level commentary, enabling early detection of implicit SI. 2.2 Social Media and Mental Health Online platforms have become key venues for mental-health self-disclosure (De Choudhury and 2 De, 2014), with communities like Reddit fostering targeted support and a sense of belonging (Saha and Sharma, 2020; De Choudhury, 2015; Shimgekar et al., 2025; Kim et al., 2023). Moderated peersupport spaces reduce isolation and help people discuss stigmatized experiences (Johnson et al., 2022). Social support, especially emotional and informational, has been shown to improve wellbeing both offline and online (Cutrona and Troutman, 1986; De Choudhury and Kiciman, 2017; Saha and Sharma, 2020). Language plays a central role: psycholinguistic research links specific linguistic markers to mental-health outcomes (Chung and Pennebaker, 2007; Pennebaker et al., 2001), and computational studies have used these cues to detect distress and model support dynamics (Chancellor et al., 2016; Guntuku et al., 2017; Chancellor and De Choudhury, 2020). Prior work has also established the construct validity of these measurements (Saha et al., 2022). Recent NLP work has focused on interpretable and fine-grained modeling of mental health on social media. Symptom-based approaches such as PSYSYM (Zhang et al., 2022; Chen et al., 2023) link online language to clinically meaningful categories of disorders. Depression severity has been quantified through semantic similarity to symptom descriptors (Pérez et al., 2022), while large language models now enable explainable detection with interpretable rationales (Wang et al., 2024). Analyses of pre and post diagnosis language shifts, highlight the temporal dynamics of distress expression (Alhamed et al., 2024). Supportive language marked by adaptability, immediacy, and emotionality predicts better outcomes (Althoff et al., 2016; Saha and Sharma, 2020), and automatic empathy-detection models scale such insights to peer-support settings (Sharma et al., 2020). For SI, machine-learning approaches have identified risk signals in social media language alongside emotional patterns that precede suicide attempts (Coppersmith et al., 2016; De Choudhury et al., 2016). While online data shows promise for early risk detection, most methods isolate either individual language or specific interactions. Our approach instead models full posting timelines alongside peer influences, capturing risk even without explicit SI disclosures. Our methodological framework identifies early indicators without requiring references to self-harm or participation in such spaces. Figure 1: Distribution of SI probability for the same set of users before their last non suicidal post and their first post in r/SuicideWatch. 3 Data We used data from r/SuicideWatch, a semianonymous Reddit community focused on SI, alongside posts and comments from other subreddits. From the PushShift archive (Baumgartner et al., 2020) of April 2019 (18.3M posts, 138.5M comments), the data includes 10,037 posts and 38,130 comments from r/SuicideWatch. Prior work has leveraged Reddit for SI (De Choudhury et al., 2016; De Choudhury and Kiciman, 2017; Shimgekar et al., 2025) and broader mental health studies (Sharma and De Choudhury, 2018; Saha et al., 2020; De Choudhury and De, 2014). Constructing Case and Control datasets. We identified two user cohorts. The first cohort comprises 500 Case individuals, defined as individuals who have made at least one post on r/SuicideWatch. The second cohort consists of 500 Control individuals, who never participated in any subreddit related to mental health. We identified the Control users by referencing the taxonomy of mental health-related subreddits from prior work (Sharma and De Choudhury, 2018). For the Case group, all posts and comments before each user's first r/SuicideWatch disclosure were labeled positive (1). On average, users made 10.07 posts before disclosure, transitioning within 1.5 days of their last non-r/SuicideWatch post. To construct a balanced Control group, we sampled the first 10 posts from each user (matching Case averages), labeling them negative (0). The dataset was then split 75:25 by users into train and test sets, ensuring no overlap. A zero-shot NLI model (Laurer et al., 2023) revealed significantly higher SI probabilities in users' first r/SuicideWatch posts, indicating that entry marks a key turning point (Figure 1). The model, 3 Post 1 (r/random) Comment 2 (r/random2) Post 3 (r/random3) Post 4 (r/SuicideWatch) Neighbors' Posts R1, R2 Top Neighbors U User \"U\" User-User Graph of \"U\" Others' Replies R1, R2 Neighbors' Posts R1, R2 Others' Replies R1, R2 Neighbors' Posts R1, R2 Others' Replies R1, R2 Neighbor Interaction Immediate Interaction Input Data Formalization Finetuned Model Early and Implicit Suicide Detection [yes/no] Figure 2: Illustration of user interactions (Immediate/Neighbor): Immediate interactions include users' self-posts, self-comments, and received replies, while neighbor interactions represent top neighbor posts and self-posts. Top neighbors identified via NeighborScore based on DeBERTa-v3-base, was trained on 1.3M hypothesis-premise pairs from eight NLI datasets (e.g., MultiNLI, FEVER-NLI, LingNLI, DocNLI) to capture long-range reasoning. Using zero-shot labels (SI, non-SI), it assigned SI probabilities to posts, showing that users' first r/SuicideWatch posts exhibit stronger suicidal self-disclosure than their prior posts. While these probabilities may partly reflect the act of joining r/SuicideWatch, the linguistic patterns align more with distress and self-harm intent, supporting their use as an operational proxy for heightened suicidal self-disclosure. 4 Methods To address our RQ on detecting early signals of SI in social media activity, particularly when explicit disclosures are absent, we conceptualized implicit SI as a forward-looking prediction problem of the likelihood that a user would eventually make a SI disclosure, proxied by their first post on r/SuicideWatch. Our framework models a user's longitudinal activity and social context via two interaction types: (1) immediate interactions-self-posts, self-comments, and received comments, and (2) neighbor interactions-selfposts and top neighbors' posts. This captures both active engagement and passive exposure, reflecting how users interact with and are influenced by content they relate to (De Choudhury and Kiciman, 2017). Our methodological framework consists of five components-1) Timeline construction, 2) Neighboring user detection, 3) Input data formalization, and 4) implicit SI signal classification with modeling, 5) linguistic analysis of interaction types, which we describe below: 4.1 Timeline Construction For each user U, we analyzed their full post and comment history for SI signals, ordered chronologically from earliest to latest. 4.2 Neighboring user detection For a user U, we found its top neighbors by the following four steps: Step 1: Initial User Collection We first collected all users who interacted with the user U, defined as either U commenting on their posts/comments or them commenting on U's posts/comments. This neighbor-identification procedure was then applied recursively to a maximum depth d=3, ensuring that both direct and indirect neighbors of U were captured. Step 2: User-User Graph Based on all the initial collected users, we constructed a user-user graph where each Node represents an individual user in the network, and an undirected Edge connects two users if either of the users has commented on the other's post. The weight of the edge was quantified as the total number of comment-based interactions, with higher weights indicating stronger ties. Step 3: Top Neighbor Detection From the user-user graph, we identified the topn (n=10) neighbors for U. We ranked the neighbors using a NeighborScore, a combined centrality score S(n), defined as: S(n) = Cin-degree(n) + Cout-degree(n) + Ccloseness(n) + Ceigenvector(n) + Cbetweenness(n) + CPageRank(n) Cin-degree and Cout-degree capture normalized connectivity, Cbetweenness measures shortest-path centrality, Ccloseness denotes proximity, Ceigenvector 4 Table 1: Model performance metrics on different data combinations for Models M1-M4 (epochs=20). Data Used Acc. F1 Prec. Rec. M1 Self-posts 0.86 0.85 0.88 0.84 M2 Self-posts + Self-comments 0.81 0.83 0.78 0.91 M3 Self-posts + Self-comments + Others' comments 0.80 0.79 0.79 0.80 M4 Self-posts + Top neighbor posts 0.95 0.96 0.93 0.99 Table 2: Model performance on showing the importance of choosing the best neighbors (epochs=20). Data Used Acc. F1 Prec. Rec. M4 Self-posts + top neighbor posts 0.95 0.96 0.93 0.99 M5 Self-posts + worst neighbor posts 0.75 0.76 0.78 0.75 M6 Self-posts + non-neighbor posts 0.68 0.71 0.67 0.78 reflects influence via important neighbors, and CPageRank estimates probabilistic importance. The aggregated NeighborScore identifies peers with strong direct ties and broader network influence around U. As some centralities are correlated, S(n) approximates overall neighbor prominence rather than precise influence. Performance variation by neighbor depth and count is shown in Appendix Table A9. 4.3 Input Data Formalization We structured the input to language models to capture U's timeline, others' replies, and peer interactions. The overall data formalization, distinguishing immediate and neighbor interactions, is shown in Figure 2. Immediate Interaction We aggregated content from a user's timeline, along with others' replies, and then used all of these textual content for finetuning the language model. To train our models, we used different combinations of features-1) U's self-posts, 2) U's self-posts and self-comments, and 3) U's self-posts, self-comments, and others' replies to U's posts or comments. Neighbor Interaction Neighbor Interaction captures signals from proximate peers. For this purpose, we temporally aligned the timelines of U and their top-n neighbors. Then, at each timestamp i, we selected ten posts by the top neighbors closest in time to U's post. We aggregated the neighbors' posts with U's posts and embedded them into a dense vector representation. This approach led to the fourth type of model, where we included features from both immediate interactions above, as well as the top neighbors' posts. Table 3: Model performance excluding neighbors with r/SuicideWatch posts. Results remain comparable to using all neighbors, indicating that broader neighbor interactions capture key signals for implicit SI detection (20 epochs). Data Used Acc. F1 Prec. Rec. M4 Self-posts + top neighbor posts 0.95 0.96 0.93 0.99 M7 Self-posts + filtered neighbor posts 0.89 0.90 0.87 0.93 4.4 Implicit SI signal classification with modeling For our study, we leveraged Microsoft's Debertav3-large model (He et al., 2021)-a 418M parameter transformer with 24 layers that processes sequences up to 512 tokens. We fine-tuned this model using CLS for global representation and SEP for segment boundaries on an Nvidia A100 GPU. We framed our problem of detecting implicit SI as a binary classification task-labeling each input xi as yi ∈{0, 1} for absence or presence of risk. We tokenized text into subword embeddings with attention masks, padded or truncated to 512 tokens, and encoded them to obtain a pooled [CLS] vector (1024-dim). A linear layer then mapped this vector to logits, followed by softmax for class probabilities. We optimized the model with cross-entropy loss using AdamW (learning rate 2 × 10-5, weight decay 0.01) for 20 epochs with batch size 8, and evaluated after each epoch on a held-out validation set using accuracy, precision, recall, and F1-score. Results from fine-tuning alternative base models are provided in Appendix Table A2. 4.5 Linguistic Analysis of Interaction Types We analyzed lexical, topical, and psycholinguistic patterns across users' self-posts, self-comments, replies, and top-neighbor posts. For lexical analysis, we used the Sparse Additive Generative Model (SAGE) (Eisenstein et al., 2011) to identify discriminative unigrams and bigrams distinguishing immediate and neighbor interactions. SAGE uses multinomial models with adaptive regularization to balance frequent and rare terms. For topical analysis, we employed BERTopic (Grootendorst, 2022) on Case and Control data, varying topic numbers (k=2-15) and achieving the highest coherence at k=12. After removing one outlier, eleven topics remained (Table 4) whose topic names were assigned by our clinical psychologist coauthor. Their normalized proportions captured topical variation across interaction types Table 5. For psycholinguistic analysis, we examined the occurrences of the af5 20 40 60 80 user posts(t) 0.90 0.91 0.92 0.93 0.94 eval_accuracy t=88, 0.943 (a) Accuracy 20 40 60 80 user posts(t) 0.91 0.92 0.93 0.94 0.95 eval_f1 t=88, 0.948 (b) F1 Score 20 40 60 80 user posts(t) 0.925 0.930 0.935 0.940 0.945 eval_precision t=73, 0.948 (c) Precision 20 40 60 80 user posts(t) 0.89 0.90 0.91 0.92 0.93 0.94 0.95 eval_recall t=44, 0.956 (d) Recall Figure 3: Performance metrics on varying the number of input user posts (t). The plots show that the performance peaks at t=88. 2 4 6 8 10 12 14 neighbor post(n) 0.88 0.90 0.92 0.94 0.96 eval_accuracy n=7, 0.966 (a) Accuracy 2 4 6 8 10 12 14 neighbor post(n) 0.88 0.90 0.92 0.94 0.96 eval_f1 n=7, 0.970 (b) F1 Score 2 4 6 8 101214 neighbor post(n) 0.84 0.86 0.88 0.90 0.92 0.94 0.96 0.98 eval_precision n=10, 0.978 (c) Precision 2 4 6 8 10 12 14 neighbor post(n) 0.92 0.93 0.94 0.95 0.96 0.97 0.98 0.99 eval_recall n=7, 0.990 (d) Recall Figure 4: Performance metrics on varying the number of neighbor posts (n). The plots show that the performance peaks at around n=7. fect category of keywords as per the well-validated Linguistic Inquiry and Word Count (LIWC) lexicon (Tausczik and Pennebaker, 2010). 5 Results 5.1 Model Performance (Immediate Interaction vs. Neighbor Interaction): Table 1 summarizes the performance comparison of models with combinations of immediate interactions and neighbor interactions feature set. The baseline model using only a target user's posts performs strongly (Accuracy = 0.86, F1 = 0.85), showing that self-authored text alone captures salient linguistic markers of future SI, consistent with prior temporal analyses on Reddit and Facebook (Coppersmith et al., 2018; De Choudhury et al., 2016). Adding the user's self-comments boosts recall (0.84 →0.91) but lowers precision (0.88 →0.78), suggesting broader coverage but added noise in terms of precision. For example, a comment from a Case user reads, \"Are they allowed to hit me? [..] I need to stay strong,\" including both themes of physical abuse as well as optimism. Additionally, including replies to user's posts/comments further reduces accuracy (0.81 →0.80) and F1 (0.83 →0.79). Such replies often convey empathy or advice that mask the user's true mindset (Gkotsis et al., 2017), as in one response: \"I think talking to a professional would help, because he can understand you and give you advice.\" We note that the top-neighbor interaction approach achieves the highest performance (Accuracy=0.95, F1=0.96, Precision=0.93, Recall=0.99), with high recall reducing false negatives, which is critical for SI risk detection (Franklin et al., 2017). This suggests that features reflecting social exposure and information environment context provide valuable predictive signals for implicit SI. 5.2 Determining Optimal Post Counts for Robust Prediction We evaluated how many posts from users (Case and Control) and their top neighbors optimize implicit SI detection. First, varying the number of neighbor posts (n) showed an improvement in accuracy from 0.87 to 0.97, with near-perfect recall (0.99) at n = 7, after which the gains plateaued (Figure 4). With n = 7 fixed, varying self-posts (t) revealed unstable performance at low t (e.g., recall 0.87 at t = 1), stabilizing for t ≥40; the best results occurred at t = 88 (accuracy 0.94, F1 0.95, precision 0.95, recall 0.96) (Figure 3). These findings indicate that combining a user's full history with a few 6 Table 4: Clinically informed topics identified from posts and comments via topic modeling, with corresponding explanations and keywords. Topic Theme Explanation Keywords Self-Harm Tools Mentions of knives, blades, or cutting instruments indicating self-injury ideation knife, blade, cut, sharpen Diet / Body Image Discussions of calories, weight, dieting, or eating habits reflecting body concerns calori, weight, diet, eat Physical Pain / Health Mentions of pain, medication, or suffering over time pain, aspirin, suffer, longer War / Military Discussions related to military, soldiers, or historical conflicts soldier, german, armi, soviet Academic / Mental Stress Mentions of depression, PhD, or mental strain depress, phd, ggoddamn, eurozon Cringe / Embarrassment Mentions of awkwardness, social discomfort, or cringing situations cri, cringi, crusad, crimin Burn / Injury Mentions of burns, cuts, or physical injury burn, cut, hurt, degre Self-Harm / Coping Strategies Mentions of harming oneself or suicide coping mechanisms harm, self, suicidebyword, cope Violent Ideation Mentions of killing or violent intent in extreme contexts kill, killin, zaibatsu, meeee Confusion / Uncertainty Expressions of perplexity or inability to understand situations confus, percent, stranger, 12ish Cute / Affection Positive, playful, or affectionate language cute, soo, aww, omgggg Table 5: Normalized topic proportions across different content sources. ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05. Topic User's SelfPosts User's SelfComments Other's Comments Top Neighbor's Posts KruskalWallis H-stat. Self-Harm Tools 0.002 0.002 0.002 0.006 44.08*** Diet & Body Image 0.005 0.003 0.003 0.003 46.36*** Physical Pain & Health 0.003 0.002 0.002 0.003 51.55*** War & Military 0.013 0.003 0.001 0.002 7.45* Academic Stress 0.001 0.001 0.002 0.002 43.81*** Cringe & Embarrassment 0.000 0.001 0.001 0.002 39.49*** Burn & Injury 0.000 0.000 0.000 0.001 1.84 Coping & Self-Harm 0.001 0.001 0.001 0.002 22.81*** Violent Ideation 0.001 0.001 0.000 0.002 26.13*** Confusion & Uncertainty 0.001 0.001 0.001 0.000 12.62** Cute / Affection 0.000 0.001 0.001 0.002 22.93*** neighbor posts yields a robust SI predictor. 5.3 Role of Top Neighbor Posts in Capturing SI Signals To understand the superior performance of topneighbor-based modeling, we analyzed lexical, topical, and psycholinguistic patterns in users' selfposts, self-comments, replies, and posts of their top neighbors. Others' replies to a user's posts or comments provide a limited discriminative signal: common bigrams such as \"feel like\", \"year old\", and \"every day\" appear across both Case and Control data, reflecting supportive or neutral discourse (Zirikly et al., 2019). Topic modeling shows that clinically relevant themes (e.g., SelfHarm Tools, Coping/Self-Harm, Violent Ideation) are rare in others' replies, while users' own posts exhibit only moderate and inconsistent signals (Table 4, Table 5). In contrast, top neighbor posts consistently reflect mental-health-related themes that are highly predictive of SI (Kirtley et al., 2021). For Case users, neighbors frequently post about SelfHarm Tools, Coping / Self-Harm, Violent Ideation, Physical Pain / Health, and Academic / Mental Stress, with bigrams such as \"mental health\", \"suicide prevention\", and \"emotional support\" appearing frequently. Neighbors of Control users, by comparison, focus on neutral or unrelated topics (Table 4, Table 5). Table 6 shows that differences in affective categories (e.g., negative affect, anger, and sadness) are statistically significant, indicating that including neighbor posts enhances the distinction between Case and Control users' content. Therefore, top-neighbor posts provide richer social context than replies or isolated user history. Combining them with user history yields a more robust model for early detection of early and implicit SI. 5.4 Robustness Tests: To ensure that our findings are not artifacts of specific modeling or sampling choices, we conducted additional robustness tests by varying the type of neighbors included-for instance, incorporating non-top neighbors into our models. For this purpose, we built additional models M5 with the lowest-ranked neighbors and M6 with nonneighbors-a random selection of other users-of a target user. Table 2 compares the performance-we note both M4 and M5 perform significantly poorly. Therefore, the performance boost by including the top neighbor posts is not by chance, and rather an important element in detecting implicit SI (Cero et al., 2024). Moreover, Table 3 demonstrates that the model retains strong predictive performance even after filtering out all top neighbors who have posted in r/SuicideWatch. This finding indicates that successful detection does not rely solely on neighbors' direct SI-related language, but rather on broader contextual cues captured from high-quality interactions. Appendix Table A4 shows linguistic analyses aligned with Table 2, Table A3 highlights \"Affect\" differences linked to Table 3, and addi7 Table 6: Comparison of LIWC Affect categories across M1-M4. Columns show normalized category counts for Case and Control groups and their differences. ***p<0.001, *p<0.01, p<0.05. Category M1 (Immediate) M2 (Immediate) M3 (Immediate) M4 (Neighbor) Case Control Diff. H Case Control Diff. H Case Control Diff. H Case Control Diff. H Pos. Affect 0.049 0.053 -0.004 38.80*** 0.080 0.074 0.006 416.94*** 0.060 0.057 0.003 45.86*** 0.048 0.053 -0.005 23.57*** Neg. Affect 0.029 0.025 0.004 74.947*** 0.035 0.030 0.005 651.17*** 0.029 0.026 0.003 100.07*** 0.029 0.021 0.007 169.87*** Anxiety 0.004 0.003 0.001 125.379*** 0.003 0.002 0.001 358.217*** 0.002 0.002 0.000 176.606*** 0.003 0.002 0.000 62.663*** Anger 0.009 0.006 0.003 50.654*** 0.012 0.010 0.002 322.855*** 0.010 0.008 0.002 77.809*** 0.011 0.006 0.005 130.857*** Sadness 0.007 0.007 0.000 71.131*** 0.008 0.007 0.001 376.545*** 0.006 0.005 0.001 136.379*** 0.009 0.005 0.004 181.551*** tional model comparisons with LIWC results are in Appendix Table A1 (Tables A5-A8). 6 Discussion and Implications A key takeaway of our study is that, early and implicit SI is best detected by modeling both an individual's longitudinal online activity and their surrounding information environment, capturing the inherently relational nature of SI expressions (Ammerman and Jacobucci, 2023). Neighbor-based modeling is particularly effective, as top neighbors, weighted via a NeighborScore, capture direct behavior and network-level influences, consistent with epidemiological evidence and classic effects like Werther and Papageno (Wyman et al., 2019; Phillips, 1974; Niederkrotenthaler et al., 2010; Yuan et al., 2023). Neighbor posts provide complementary signals beyond the user's self-content, showing that exposure to neighbors' expressions of distress or coping is associated with detection. Neighbor-informed models frame suicidality within a social-ecological perspective (Bronfenbrenner, 1979), reflecting social contagion processes where suicidal behaviors and ideation can spread through networks, especially when peers disclose distress (Wyman et al., 2019; Gould et al., 2003). Online environments may amplify these effects, as individuals in high-risk networks can show subtle precursors before explicit disclosure. Network theories suggest that peers model behavior and influence interpretations of distress (Cero et al., 2024; Bearman and Moody, 2004; Christakis and Fowler, 2025), explaining why neighbor interactions reflect relational signals of implicit SI. A user's longitudinal posting history remains essential, tracing gradual shifts in tone, sentiment, and discourse (De Choudhury et al., 2016). However, interactional content must be selective: including all replies adds noise and weakens risk signals (Schmidt et al., 2024). As shown in Table 6, the difference between the LIWC \"Affect\" categories is consistently higher in M4 where neighbors' posts are added. This shows that including posts from the neighbors add a clear distinction signal between the Case and Control data, making it better for the model to predict (Lu and Tu, 2024; Guo et al., 2012). These findings suggest avenues for theoretical and methodological extensions. Beyond the Interpersonal Theory of Suicide (Joiner, 2005), models such as the Integrated Motivational Volitional Model (O'Connor and Kirtley, 2018) and the ThreeStep Theory (Klonsky and May, 2015) may help identify subtle motivational or volitional cues in language and interactions. The strong influence of social context further supports contagion and network theories, indicating that implicit suicidal ideation emerges from both individual cognition and broader interpersonal environments. Overall, this work enhances detection accuracy while framing suicide risk as a relational process in digital social environments. By showing that peer-network interactions substantially improve predictive power, it bridges computational modeling with networkinformed prevention strategies. 7 Conclusion Our study demonstrates that early and implicit SI is best detected when a user's longitudinal activity is interpreted within their social context. Subtle linguistic and interactional cues often emerge well before explicit disclosures, enabling identification of risk trajectories at an early stage. By incorporating posts from socially proximal peers, our framework operationalizes theories of suicide contagion and social influence, yielding substantial gains in predictive performance. Notably, while a user's own posts and comments provide moderate insight, even a small set of strategically selected neighbor posts carries a significantly strong predictive signal. These findings highlight that suicide risk is inherently relational: effective detection requires selectively integrating self-content and socially distributed cues. Beyond improving accuracy, this relational framing supports scalable, context-aware, and ethically responsible early-warning systems that align with both clinical theory and the dynamics of online communities. 8 8 Limitations Our study has limitations that also show interesting future directions. Methodologically, our models exclusively rely on textual content, overlooking multimodal signals such as images, videos, emojis, GIFs, or external links that often convey emotional states, coping strategies, or distress. Incorporating these modalities could improve sensitivity to subtle risk signals, enhance interpretability, and provide a more holistic understanding of online behaviors associated with suicidal ideation. Similarly, our current approach treats peer interactions homogeneously, though peers can exert positive, neutral, or negative influences. Modeling these distinctions could capture the functional impact of social context on risk trajectories and guide ethically responsible interventions. Finally, robustness and generalizability can be improved through replication across platforms (e.g., Twitter, TikTok, Discord), multilingual and crosscultural settings, and naturalistic populations. Temporal weighting, trajectory-based risk modeling, and interpretability techniques such as attention visualization, SHAP, or counterfactuals could clarify key linguistic and network features while enhancing actionable insights. Future work should integrate multimodal inputs, refine social context modeling, and maintain rigorous ethical oversight to ensure predictive models support vulnerable individuals responsibly and effectively. 9 Ethical Considerations Reflexivity This paper used publicly accessible social media discussions on Reddit and did not require direct interactions with individuals, thereby not requiring ethics board approval. However, we are committed to the ethics of the research and we followed practices to secure the privacy of individuals in our dataset. Our research team comprises researchers holding diverse gender, racial, and cultural backgrounds, including people of color and immigrants, and hold interdisciplinary research expertise. This team consists of computer scientists with expertise in social computing, NLP, and HCI, and psychologists with expertise in clinical psychology, adolescent depression and suicide, and digital health interventions. One psychologist coauthor specializes in suicide etiology, suicide prevention, and crisis intervention, and another psychologist coauthor is a clinical psychologist with over 16 years of experience spanning adult and adolescent inpatient care and crisis suicide helplines. To ensure validity and prevent misrepresentation, our findings were reviewed and corroborated by our psychologist coauthors. However, our work is not intended to replace the clinical evaluation of an individual undergoing suicidal thoughts and should not be taken out of context to conduct mental health assessments. Risk of Misinterpretation and Harm with Automated Detection Our study leverages computational methods to identify early and implicit suicidal ideation (SI) in online communities. While these approaches provide opportunities for early intervention, they also present serious ethical challenges. Automated systems may misinterpret subtle linguistic signals or social cues, leading to false positives or false negatives. Misclassification can result in unwarranted labeling of individuals as at risk, potentially causing stigma, anxiety, or social consequences. Conversely, failing to detect genuine risk may deny individuals timely support. We emphasize that suicidal ideation manifests heterogeneously across individuals, and even carefully designed models cannot fully capture the nuanced context of personal distress. Privacy, Consent, and Data Sensitivity Given the sensitive nature of suicidal thoughts, the privacy and confidentiality of users are paramount. Our approach relies on publicly available text, but the inclusion of social network information, even aggregated, raises concerns about inadvertent exposure of personal interactions. Ethical deployment requires anonymization, secure storage, and strict access controls. Users should be informed about potential data usage and retention, and explicit consent should be prioritized wherever feasible. Social Context and Potential Misuse By modeling social context and peer interactions, our work operationalizes theories of social influence in suicidal ideation. However, this introduces additional ethical considerations. Insights derived from peer interactions could be misused if exploited for nonclinical purposes, such as targeted advertising or profiling of vulnerable individuals. Careful governance and ethical oversight are necessary to ensure that social context is leveraged solely for supportive, preventive interventions rather than for commercial or punitive applications. Transparency, Interpretability, and Responsible Use Ethical deployment also requires trans9 parency and interpretability. Systems identifying SI must clearly communicate their scope, limitations, and the fact that outputs are probabilistic, not diagnostic. Stakeholders, including clinicians, platform moderators, and potentially affected users, should understand how predictions are generated, especially when interventions are triggered. Responsible use also involves integrating human-in-the-loop frameworks, where trained professionals augment computational predictions to avoid over-reliance on automated systems. Future Directions for Ethical AI Future research should explore multimodal detection methods while carefully balancing privacy and ethical constraints. Integrating images, URLs, or videos may improve sensitivity, but must be approached with strict ethical safeguards. Additionally, systematically categorizing peer influence, positive, neutral, or negative, could enhance the interpretability and safety of predictions, ensuring interventions are informed by context rather than raw social activity. Overall, ethical design in computational mental health requires prioritizing user welfare, minimizing harm, and embedding accountability at every stage of model development and deployment. 10 AI Involvement Disclosure Certain sections of the manuscript were refined using AI-assisted writing tools (e.g., ChatGPT, Grammarly). All analyses, scientific content, and experiments were written solely by the authors. References Daniyal Alghazzawi, Hayat Ullah, Naila Tabassum, Sahar K Badri, and Muhammad Zubair Asghar. 2025. Explainable ai-based suicidal and non-suicidal ideations detection from social media text with enhanced ensemble technique. Scientific Reports, 15(1):1111. Falwah Alhamed, Julia Ive, and Lucia Specia. 2024. Classifying social media users before and after depression diagnosis via their language usage: A dataset and study. 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Model Data Used Accuracy F1 Precision Recall M8 Top neighbor's posts only 0.83 0.82 0.92 0.74 M9 Target user posts + comments + top neighbor posts 0.91 0.92 0.91 0.92 M10 Target user posts + comments + others' comments + top neighbor posts 0.71 0.64 0.75 0.56 M11 Filtered neighbor posts only (no post/comments in r/SuicideWatch) 0.78 0.77 0.73 0.82 Table A2: Performance comparison of transformer models on the classification task. Model Accuracy F1 Precision Recall microsoft/deberta-v3-large 0.955 0.946 0.945 0.945 roberta-large 0.944 0.931 0.944 0.919 bert-base-uncased 0.933 0.921 0.897 0.946 google/electra-large-discriminator 0.944 0.935 0.900 0.973 Table A3: Comparison of LIWC affect categories between M4 and M7 showing how the LIWC Affect category varies when considering all top neighbors vs. filtered top neighbors. Columns show mean proportions for users showing suicidal ideation (SI) and not, their difference, and the Kruskal-Wallis H-statistic. Significance codes: ***p<0.001, **p<0.01, *p<0.05. Category M4 M7 Case Control Diff. H Case Control Diff. H posemo 0.048 0.053 -0.005 23.568*** 0.050 0.053 -0.003 17.482*** negemo 0.029 0.021 0.007 169.871*** 0.032 0.025 0.007 72.531*** anx 0.003 0.002 0.000 62.663*** 0.003 0.002 0.001 63.084*** anger 0.011 0.006 0.005 130.857*** 0.010 0.006 0.004 51.436*** sad 0.009 0.005 0.004 181.551*** 0.008 0.007 0.001 71.131*** 14 Table A4: Comparison of LIWC affect categories across M4, M5, and M6 showing how the Affect LIWC category varies across various neighbors (top, worst, random). Columns show mean proportions for users showing suicidal ideation (SI) and not, their difference, and the Kruskal-Wallis H-statistic. Significance codes: ***p<0.001, **p<0.01, *p<0.05. Category M4 M5 M6 Case Control Diff. H Case Control Diff. H Case Control Diff. H posemo 0.048 0.053 -0.005 23.568*** 0.047 0.049 -0.003 38.611*** 0.048 0.047 0.001 42.425*** negemo 0.029 0.021 0.007 169.871*** 0.029 0.025 0.004 74.947*** 0.030 0.024 0.006 59.078*** anx 0.003 0.002 0.000 62.663*** 0.004 0.002 0.002 53.518*** 0.003 0.002 0.001 125.397*** anger 0.011 0.006 0.005 130.857*** 0.009 0.006 0.003 40.154*** 0.009 0.006 0.002 138.704*** sad 0.009 0.005 0.004 181.551*** 0.007 0.007 0.000 71.131*** 0.007 0.008 -0.001 36.640*** Table A5: Comparison of LIWC affect categories for M8. Columns show mean proportions for suicidal and nonsuicidal users, their difference, and the Kruskal-Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05. Category M8 Case Control Diff H posemo 0.053 0.053 0.000 0.409 negemo 0.026 0.025 0.001 4.805* anx 0.001 0.002 -0.001 3.323 anger 0.009 0.006 0.003 9.523** sad 0.005 0.007 -0.001 3.638 Table A6: Comparison of LIWC affect categories for M9. Columns show mean proportions for suicidal and nonsuicidal users, their difference, and the Kruskal-Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05. Category M9 Case Control Diff H posemo 0.050 0.053 -0.003 17.482*** negemo 0.032 0.025 0.007 72.531*** anx 0.003 0.002 0.001 63.084*** anger 0.010 0.006 0.004 51.436*** sad 0.008 0.007 0.001 71.131*** Table A7: Comparison of LIWC affect categories for M10. Columns show mean proportions for suicidal and nonsuicidal users, their difference, and the Kruskal-Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05. Category M10 Case Control Diff H posemo 0.070 0.067 0.003 349.268*** negemo 0.031 0.028 0.003 538.074*** anx 0.003 0.002 0.000 403.691*** anger 0.010 0.009 0.001 281.206*** sad 0.007 0.006 0.001 361.992*** Table A8: Comparison of LIWC affect categories for M11. Columns show mean proportions for suicidal and nonsuicidal users, their difference, and the Kruskal-Wallis H statistic. Significance codes: ∗∗∗p < 0.001, ∗∗p < 0.01, ∗p < 0.05. Category M11 Case Control Diff H posemo 0.053 0.053 0.000 0.409 negemo 0.026 0.025 0.001 4.805* anx 0.001 0.002 -0.001 3.323 anger 0.009 0.006 0.003 9.523** sad 0.005 0.007 -0.001 3.638 15 Table A9: Performance metrics for different depth and neighbor configurations. Depth refers to the number of recursive depth considered, and Neighbors refers to the number of top neighbor posts included. Depth Neighbors Accuracy F1 Precision Recall 1 1 0.98 0.97 0.96 0.98 1 2 0.97 0.96 0.95 0.97 1 3 0.96 0.95 0.95 0.96 1 4 0.96 0.95 0.94 0.96 1 5 0.95 0.94 0.94 0.95 2 1 0.97 0.96 0.95 0.97 2 2 0.96 0.95 0.94 0.96 2 3 0.95 0.94 0.93 0.95 2 4 0.95 0.94 0.93 0.95 2 5 0.94 0.93 0.92 0.94 3 1 0.96 0.95 0.94 0.96 3 2 0.95 0.94 0.93 0.95 3 3 0.94 0.93 0.92 0.94 3 4 0.94 0.93 0.92 0.94 3 5 0.95 0.94 0.93 0.95 4 1 0.95 0.94 0.93 0.95 4 2 0.94 0.93 0.92 0.94 4 3 0.93 0.92 0.91 0.93 4 4 0.93 0.92 0.91 0.93 4 5 0.92 0.91 0.90 0.92 5 1 0.94 0.93 0.92 0.94 5 2 0.93 0.92 0.91 0.93 5 3 0.92 0.91 0.90 0.92 5 4 0.92 0.91 0.90 0.92 5 5 0.91 0.90 0.89 0.91 16"}}},{"rowIdx":93,"cells":{"id":{"kind":"string","value":"2510.14886"},"raw":{"kind":"string","value":"Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems\nJer´onimo Duarte,1 Ignacio Garc´ıa-Mata,1 and Diego A. Wisniacki2\n1Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR),\nFacultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata\n& CONICET, Funes 3350 (7600) Mar del Plata, Argentina\n2Departamento de F´ısica “J. J. Giambiagi” and IFIBA, FCEyN,\nUniversidad de Buenos Aires, 1428 Buenos Aires, Argentina\n(Dated: October 17, 2025)\nThe out-of-time-order correlator (OTOC) quantifies information scrambling in quantum systems\nand serves as a key diagnostic of quantum chaos. In one-body systems with a classical counterpart,\nthe relaxation of the OTOC is governed by Ruelle-Pollicott resonances. For many-body systems\nlacking a semiclassical limit, recent studies have identified an analogous role played by the Liouvillian\nspectrum of weakly open extensions of the dynamics, where the slowest decay rate?the Liouvillian\ngap?encodes relaxation. Here we study the kicked Ising spin chain and show that the long-time\nexponential decay of the OTOC in the isolated system occurs at a rate equal to twice this intrinsic\ngap. This correspondence persists even in crossover regimes between integrability and chaos, demon-\nstrating that the Liouvillian spectrum provides a unified framework for understanding relaxation\nand irreversibility in closed many-body quantum systems.\nI.\nINTRODUCTION\nUnderstanding relaxation dynamics and the emergence\nof effective irreversibility in quantum many-body systems\nis a central problem at the interface of quantum chaos,\nstatistical mechanics, and open-system theory. In clas-\nsical chaotic dynamics, the intermediate time decay of\ncorrelations is governed by the Ruelle-Pollicott (RP) res-\nonances [1–3] – isolated singularities of the analytically\ncontinued spectrum of the Frobenius?Perron operator –\nwhich generate effective irreversibility despite the under-\nlying deterministic and reversible motion. In quantum\none-body systems with a classical limit, an analogous\ncorrespondence is well established through spectral and\noperator-truncation approaches [4–6] or through weakly\ndissipative formulations [7–9].\nIn contrast, identifying the quantum analogue of RP\nresonances in generic many-body systems remains an\nopen question.\nSeveral works [10–12] have introduced\ncoarse-grained formulations of the operator space, re-\nvealing isolated eigenvalues inside the unit disk that can\nbe interpreted as quantum RP resonances. While con-\nceptually appealing, this approach becomes impracti-\ncal for large systems due to the exponential growth of\nthe operator space and the difficulty of defining an ap-\npropriate coarse-graining.\nA more recent development\nexploits translational invariance and quasi-momentum\ndecomposition [13], enabling the identification of reso-\nnances within each symmetry sector.\nAn alternative line of research focuses on the Liou-\nvillian spectrum of weakly open quantum systems [14].\nWhen a system is coupled to an environment, its relax-\nation dynamics are governed by the eigenvalues of the\ncorresponding Lindblad superoperator, with the slowest\nnonzero decay rate defining the Liouvillian gap. In ex-\nactly solvable models [15–17], this gap has been shown\nto coincide with the leading RP resonance of the corre-\nsponding isolated dynamics.\nThe magnitude of the leading RP resonance determines\nthe approach to equilibrium of dynamical observables,\nincluding the out-of-time-order correlator (OTOC). The\nOTOC measures the spread of local perturbations and\nserves as a sensitive probe of information scrambling in\nquantum systems. Although the butterfly effect in quan-\ntum chaos is not universal [18], the relaxation of the\nOTOC at late times provides a robust signature of under-\nlying dynamics. In one-body systems, the crossover from\nintegrability to chaos can be characterized through the\nOTOC decay rate [19], a result that extends to systems\nwith mixed classical dynamics [20].\nHere we extend this connection to many-body systems.\nWe study the kicked Ising spin chain, a paradigmatic\nmodel that exhibits both integrable and chaotic regimes\ndepending on its parameters.\nSpecifically, we analyze\nthree complementary quantities: (i) the Liouvillian gap ¯g\nextracted from a weakly dissipative version of the model,\n(ii) standard chaos indicators from spectral statistics and\neigenstate delocalization, and (iii) the intermediate time\nexponential decay rate α of the OTOC in the isolated\nchain. We show that α ≈2¯g even in transitional regimes\nbetween integrability and chaos, establishing the Liou-\nvillian gap as a qualitative (and quantitative) measure of\nintegrability in quantum many-body systems.\nII.\nMANY-BODY SYSTEM: KICKED ISING\nSPIN CHAIN\nTo investigate the interplay between relaxation dy-\nnamics, quantum chaos, and dissipation, we consider the\nkicked Ising spin chain [21, 22], a paradigmatic model\nof many-body Floquet dynamics. This system is partic-\nularly well suited to our study because its parameters\nallow for a continuous interpolation between integrable\nand chaotic behavior.\narXiv:2510.14886v1 [quant-ph] 16 Oct 2025\n\n2\nThe model is defined by the time-periodic Hamiltonian\nˆH(t) = ˆHfree + ˆHkick τ\n∞\nX\nn=−∞\nδ(t −nτ),\n(1)\nwhere the “free” and “kick” components take the form\n\n\n\n\n\n\n\n\n\nˆHfree = −J\nL−2\nP\ni=0\nˆσz\ni ˆσz\ni+1 −hz\nL−1\nP\ni=0\nˆσz\ni ,\nˆHkick = −hx\nL−1\nP\ni=0\nˆσx\ni .\n(2)\nHere, ˆσα\ni\n(α = x, z) denote Pauli operators acting on\nsite i, J is the nearest-neighbor coupling, and hx, hz\nare transverse and longitudinal magnetic fields, respec-\ntively. The dynamics is stroboscopic with period τ, and\nthe evolution over one period is governed by the Floquet\noperator\nˆUF = ˆUfree ˆUkick,\n(3)\nwhere\n\n\n\nˆUfree = exp(−i ˆHfreeτ),\nˆUkick = exp(−i ˆHkickτ).\n(4)\nThis Floquet representation conveniently describes both\nunitary and, when extended, weakly dissipative evolu-\ntions.\nThroughout this work we set J = hz = τ = 1 and\nvary the transverse field hx, which controls the degree of\nchaoticity. These values ensure nontrivial dynamics while\nreducing the parameter space to a single effective control\nvariable. All simulations are performed for system sizes\nup to L = 12 spins.\nTo understand the structure of the spectrum, it is es-\nsential to discuss the symmetries present in the model.\nSince we work with a Floquet operator, the relevant spec-\ntrum consists of quasienergies. Throughout this study\nwe impose open boundary conditions (OBC). The sys-\ntem possesses an external reflection symmetry ˆR, which\nacts as\nˆR |m1, m2, . . . , mL−2, mL−1⟩= |mL−1, mL−2, . . . , m2, m1⟩,\nwhere |m1, m2, . . . , mL−1⟩denotes a computational-basis\nstate. Because [ ˆUF , ˆR] = 0, the Hilbert space decomposes\ninto two invariant subspaces corresponding to the eigen-\nvalues +1 and −1 of ˆR, which we refer to as the even\nand odd parity sectors. This symmetry decomposition\nallows us to analyze each sector independently, avoiding\nspectral mixing and enabling a cleaner identification of\nchaotic behavior and Liouvillian relaxation modes.\nIII.\nQUANTUM CHAOS DIAGNOSTICS\nTo characterize the degree of quantum chaos in the\nkicked Ising spin chain, we employ two complementary\ndiagnostics:\nspectral statistics and eigenstate delocal-\nization.\nThese indicators enable us to distinguish be-\ntween integrable, chaotic, and intermediate regimes, pro-\nviding essential context for interpreting the behavior of\nthe OTOC and its connection to the Liouvillian gap.\nA well-established signature of quantum chaos is the\nstatistical distribution of level spacings [23].\nChaotic\nsystems exhibit Wigner-Dyson distributions predicted\nby random matrix theory (RMT), with the specific\nensemble—in our case, the Circular Orthogonal Ensem-\nble (COE)—determined by the symmetries of the sys-\ntem [24]. In contrast, integrable systems display Poisson\nstatistics reflecting the presence of extensive conserved\nquantities [25].\nRather than computing the full level-spacing distribu-\ntion, we employ the ratio of adjacent spacings [26], which\nprovides a practical and robust diagnostic. For the Flo-\nquet operator satisfying\nˆUF |ψn⟩= eiφn |ψn⟩,\nn = 1, 2, . . . , D,\n(5)\nwhere φn are the quasienergies, |ψn⟩are the eigenstates,\nand D is the Hilbert space dimension, we compute\nrn = min(δn, δn−1)\nmax(δn, δn−1),\nδn = φn+1 −φn.\n(6)\nThe mean value ⟨r⟩exhibits two distinct limits: ⟨r⟩COE ≈\n0.5307 for COE statistics and ⟨r⟩P ≈0.3863 for Poisson\nstatistics [27, 28]. To interpolate between these limits,\nwe define the normalized parameter\nη =\n⟨r⟩−⟨r⟩P\n⟨r⟩COE −⟨r⟩P\n,\n(7)\nwhich provides a continuous measure ranging from η →0\nin the integrable regime to η →1 in the fully chaotic\nregime.\nA complementary diagnostic is provided by eigenstate\ndelocalization, quantified through the participation ra-\ntio (PR) [29]. Consider an eigenstate |ψi⟩expanded in\na reference basis {|ϕj⟩}D−1\nj=0 as |ψi⟩= P\nj aij |ϕj⟩. The\nparticipation ratio is defined as\nξE(i) =\n\n\nD−1\nX\nj=0\n|aij|4\n\n\n−1\n.\n(8)\nThis quantity measures how extended an eigenstate is in\nthe chosen basis. Small values indicate localization, while\nhigh values signal delocalization.\nFor chaotic systems\nconsistent with RMT, the coefficients |aij|2 behave as\nindependent random variables, leading to typical values\nξdeloc\nE\n≈D/3, where D is the Hilbert space dimension [30,\n31]. In our numerical analysis, we compute the average\nof ξE(i) over all eigenstates.\nThese spectral and eigenstate-based chaos indicators\nprovide a quantitative map of the dynamical regime of\nthe system.\nWe analyze their behavior as a function\n\n3\nof the transverse field strength hx, which controls the\nkick strength in the Floquet dynamics. To avoid spec-\ntral mixing between symmetry sectors, all analyses are\nperformed within the even-parity subspace, ensuring a\nclearer characterization of the underlying dynamical be-\nhavior.\nThis characterization serves as a reference for\nstudying the Liouvillian gap and exploring potential con-\nnections between unitary and dissipative indicators of\nquantum chaos.\nIV.\nOUT-OF-TIME-ORDER CORRELATOR\nThe out-of-time-order correlator (OTOC) quantifies\nthe scrambling of quantum information and is defined\nas the thermal expectation value of the squared commu-\ntator between two operators evaluated at different times:\nC(t) =\n\u001ch\nˆW(t), ˆV\ni h\nˆW(t), ˆV\ni†\u001d\n,\n(9)\nwhere ˆW(t) is the Heisenberg picture evolution of ˆW. Ex-\npectation values are taken over the infinite-temperature\nensemble, ⟨·⟩= Tr(·)/D, with D the Hilbert-space di-\nmension. This ensemble is appropriate for Floquet sys-\ntems or Hamiltonians with bounded spectra, where mi-\ncrocanonical and infinite-temperature averages coincide\nin the thermodynamic limit.\nOriginally introduced in studies of superconductiv-\nity [32], the OTOC has become a central tool for char-\nacterizing quantum chaos and information scrambling in\nmany-body systems [33]. The OTOC gained attention\nwhen upper bound for the exponential growth was found\n[34], setting the bar of fast scramblers like black holes[35]\nand strongly correlated fermionic systems[36–38]. In sys-\ntems with a classical limit, the growth rate coincides with\nthe classical Lyapunov exponent [19, 39–41].\nAt later\ntimes, quantum interference leads to saturation around a\nconstant plateau with residual fluctuations that depend\non the underlying dynamics [42].\nWe focus on local Hermitian and unitary operators ˆW\nand ˆV acting on distinct sites of the spin chain. A conve-\nnient choice is given by Pauli operators ˆσµ\ni (µ = x, y, z)\nacting on site i. At infinite temperature, the OTOC takes\nthe explicit form\nCµν(l, t) = 1\n2\nD\n[ˆσµ\n0 (t), ˆσν\nl ]2E\n= 1 −1\nD Re {Tr [ˆσµ\n0 (t)ˆσν\nl ˆσµ\n0 (t)ˆσν\nl ]} .\n(10)\nWe define\nOµν\n1 (l, t) = 1\nD Re {Tr [ˆσµ\n0 (t)ˆσν\nl ˆσµ\n0 (t)ˆσν\nl ]} ,\n(11)\nso that Cµν(l, t) = 1−Oµν\n1 (l, t). In what follows we focus\non the case µ = ν = z, denoted Czz(l, t), which probes\nthe scrambling and relaxation of local ˆσz operators.\nAt intermediate times, the approach to saturation of\nC(t), at (what we will call intermediate times) is reflected\nby an exponential decay of O1(t), which can be present\neven in systems which are not completely chaotic [20].\nFor one-body, completely chaotic systems, the decay is\ngiven by the classical RP resonances[19].\nFigure 1 illustrates the time evolution of |Ozz\n1 (l, t)| for\nl = 1 and a transverse field hx deep in the chaotic regime.\nThe data, obtained for L = 12, display a clear exponen-\ntial decay at intermediate times . An exponential fit of\nthe form |Ozz\n1 (1, t)| ∝e−αt yields the decay rate α, which\nwe compare below with twice the Liouvillian relaxation\nrate extracted from the weakly dissipative extension of\nthe model.\n0\n10\n20\n30\n40\n50\n60\nt\n10−4\n10−3\n10−2\n10−1\n100\n|Ozz\n1 |\nFIG. 1. The absolute value of Ozz\n1 (1, t) for a transverse field\nhx in the chaotic regime. The data corresponds to a spin chain\nof size L = 12 and hx = 0.8168. The blue line with circles\nrepresents Ozz\n1 (1, t), while the dashed line is the exponential\nfit to the intermediate time behavior of |Ozz\n1 (1, t)|, with a\ndecay exponent α = 0.2888.\nV.\nLIOUVILLIAN GAP\nTo investigate the relation between the decay rate dis-\ncussed in the previous section and the spectral properties\nof the system, we follow Ref. [14]. In this approach, the\nsystem is extended to a weakly open setting described by\na Liouvillian superoperator\nL = −i[H, · ] + γD,\n(12)\nwhere D is a dissipator and γ the dissipation strength.\nBuilding on the kicked Ising model defined above, we\nintroduce bulk dephasing and study the corresponding\nrelaxation dynamics through the time-periodic Lindblad\nmaster equation\ndρ\ndt = −i [ ˆH(t), ρ] + γ\nL−1\nX\ni=0\n\u0010\nσz\ni ρ σz\ni −\n1\n2{σz\ni σz\ni , ρ}\n\u0011\n≡L(t) ρ,\n(13)\n\n4\nwhere γ > 0 controls the dephasing strength and ˆH(t) =\nˆH(t + τ) is the Floquet Hamiltonian.\nThe Liouvillian superoperator L(t) has a complex spec-\ntrum λα with negative real parts. For time-independent\ngenerators (L) there is one zero mode (λ0 = 0) corre-\nsponding to the steady state, plus decaying modes with\nRe λα < 0. The Liouvillian gap\ng = −max\nα̸=0 Re λα\n(14)\ncharacterizes the slowest non-trivial relaxation rate. In\nperiodically driven systems it is convenient to define the\nFloquet map\nUF = T e\nR τ\n0 L(t) dt\n(15)\nwhere T denotes time ordering. The eigenvalues of UF\nare eλατ, allowing one to extract g equivalently from the\nFloquet-Liouvillian spectrum.\n0.00\n0.05\n0.10\n0.15\n0.20\nγ\n0.0\n0.2\n0.4\n0.6\ng\n¯g\nFIG. 2. Liouvillian gap g(γ) as a function of the dissipation\nstrength γ for system sizes L = 6 (blue circles), L = 8 (orange\nsquares), and L = 10 (green triangles) and hx = 0.8168. The\ndashed line indicate quadratic fit. For L = 10, the extrapo-\nlated value of ¯g is 0.1429\nIn chaotic many-body systems, it has been shown that\ntaking the limits L →∞first and γ →0, the gap con-\nverges to a nonzero limit\n¯g = lim\nγ→0 lim\nL→∞g,\n(16)\nwhich coincides with the real part of the leading Ruelle-\nPollicott resonance of the corresponding isolated chain\n[14].\nTo extract ¯g in the weak dissipation regime, we com-\npute g(γ) for several small values of γ and perform a\nquadratic fit in γ. Following Eq. (16), the fit is applied\nto the largest available system size. The eigenvalues of\nthe Floquet-Liouvillian spectrum, from which g(γ) is ob-\ntained, are computed using the Arnoldi-Lindblad method\ndescribed in Appendix A. This procedure provides the ex-\ntrapolated value ¯g ≡limγ→0 limL→∞g(γ), characterizing\nthe intrinsic relaxation of the isolated chain.\nFigure 2 shows the Liouvillian gap g(γ) as a function\nof the dissipation strength for system sizes L = 6, 8, 10\nat hx = 0.8168. Dashed lines indicate quadratic fits, and\nfor L = 10 the extrapolated value ¯g = 0.1429. Remark-\nably, the corresponding OTOC decay rate obtained in\nthe isolated chain satisfies α ≃2¯g, demonstrating that\nthe Liouvillian relaxation rate of the weakly open system\npredicts the intermediate time OTOC decay in the closed\ndynamics.\nParity-resolved analysis – We analyze the Liouvillian\nspectrum further within each parity subspace.\nAcross\nthe integrable-to-chaotic transition, the gaps of the two\nsubspaces intersect when plotted as a function of the\ndissipation strength. As a result, the smallest nonzero\neigenvalue – and hence the global Liouvillian gap – orig-\ninates from the even subspace in certain parameter re-\ngions, while in others it comes from the odd subspace.\nThis leads to an effective switching between sectors, so\nthat the extracted global Liouvillian gap combines contri-\nbutions from both subspaces depending on which hosts\nthe slowest decaying mode. Importantly, this does not\nimply that the Arnoldi-Lindblad method is superior to\nthe approach in Ref. [14]; rather, it allows us to probe\nparameter regimes?particularly the crossover from inte-\ngrable to chaotic dynamics?where the behavior of the Li-\nouvillian gaps can be resolved in detail.\nTo carry out this analysis, we explicitly constructed\nboth the Lindbladian superoperator and the spatial in-\nversion superoperator. By diagonalizing the latter, we\nobtained its eigenvalues and eigenvectors, which allowed\nus to separate the Hilbert space into parity-even and\nparity-odd subspaces. Since the Lindbladian commutes\nwith the spatial inversion operator, it can be represented\nin this eigenbasis as a block-diagonal matrix, with one\nblock corresponding to each parity sector.\nRestricting\nthe dynamics to each block separately, we extracted the\neigenvalues of the Lindbladian within each subspace, and\nthereby determined the corresponding parity-resolved Li-\nouvillian gaps.\nFigure 3 compares the Liouvillian gaps computed\nwithin each parity subspace?parity-even (blue empty cir-\ncles) and parity-odd (orange empty squares)?with the\nglobal gap obtained via the Arnoldi-Lindblad method\n(green crosses).\nSolid lines serve as guides to the eye\nfor the subspace gaps. As the dissipation parameter is\nvaried, the dominant eigenvalue can switch between sub-\nspaces: in certain regions the even sector exhibits the\nsmallest eigenvalue, while in others the odd sector be-\ncomes dominant. The Arnoldi method naturally captures\nthis transition, since it always selects the eigenvalue with\nthe smallest real part across all subspaces.\nMoreover,\nas the system size increases, the spectra of the two par-\nity sectors become increasingly separated, making such\ncrossings less probable.\nThis behavior highlights that\nthe Arnoldi approach consistently identifies the slowest\n\n5\n0.00\n0.05\n0.10\n0.0\n0.2\n0.4\ng\n(a)\n0.00\n0.02\n0.04\n0.06\nγ\n0.00\n0.01\n0.02\ng\n(b)\nFIG. 3.\nComparison of Liouvillian gaps for two values of the\ntransverse field hx. (a) hx = 0.7854. The parity-even sub-\nspace is shown in blue empty circles, the parity-odd subspace\nin orange empty squares, and the global gap obtained via the\nArnoldi-Lindblad method in green crosses. (b) hx = 0.2749.\nThe parity- even subspace is shown in blue empty circles,\nthe parity- odd subspace in orange empty squares, and the\nglobal gap obtained via the Arnoldi-Lindblad method in green\ncrosses.\ndecaying mode of the global dynamics, regardless of the\nunderlying sectoral structure.\nThe case analyzed in Fig. 3(a) corresponds to the\nfully chaotic regime, where the Liouvillian gap curves\nfor the two parity subspaces are clearly separated and\nno crossings occur, reflecting a well-defined hierarchy of\ndecay rates between the sectors. In contrast, Fig. 3(b)\nshows an intermediate regime between integrability and\nchaos, where the curves partially overlap and crossings\nemerge, indicating that the slowest decaying mode can\noriginate from either subspace depending on the dissi-\npation strength. Both regimes are further illustrated in\nFig. 4, highlighting how the Arnoldi-Lindblad method ef-\nfectively captures the dominant relaxation dynamics and\nsmoothly interpolates between different contributions as\nthe system transitions from integrable to chaotic behav-\nior.\nVI.\nRELATION BETWEEN OTOC DECAY, THE\nLIOUVILLIAN GAP AND CHAOS PARAMETERS\nIn this section, we bring together the chaos indicators,\nOTOC decay exponents, and the Liouvillian gap?into\na single comparative framework.\nPreviously, we have\nshown how to compute the normalized level?spacing ra-\ntio η and participation ratio ¯ξE across different transverse\nfields hx, how to extract the intermediate time exponen-\ntial decay exponent α of the local correlator Ozz\n1 (l = 1, t),\nand how to obtain the Liouvillian gap ¯g via quadratic ex-\ntrapolation in the weak?dissipation limit for L = 10.\n0.0\n0.5\n1.0\nη, ¯ξE\n(a)\n0.5\n1.0\nhx\n0.0\n0.1\n0.2\n0.3\nα, 2¯g\n(b)\nFIG. 4. (a) Chaos indicators η and ¯ξE as functions of the\ntransverse field hx, computed from the Floquet spectrum in\nthe parity-even subspace for system size L = 12. Blue upward\ntriangles denote η and cyan diamonds denote ¯ξE. (b) Compar-\nison between the OTOC decay exponent α (red crosses) and\ntwice the Liouvillian gap 2¯g (orange empty circles) as func-\ntions of hx. The Liouvillian gap ¯g is extrapolated for L = 10,\nwhile α is obtained from the isolated chain with L = 12. The\ntwo panels together illustrate how both dynamical and spec-\ntral quantities capture the transition between integrable and\nchaotic regimes.\nFigure 4(a) displays the chaos indicators η (blue up-\nward triangles) and ¯ξE (cyan diamonds) as functions of\nthe transverse field hx.\nBoth quantities are computed\nfrom the unitary Floquet spectrum at system size L = 12,\nand?owing to the symmetries of the model?are analyzed\nwithin the parity-even subspace defined with respect to\nthe external reflection operator. These indicators clearly\ncapture the crossover from integrability to chaos.\nFigure 4(b) shows the comparison between the OTOC\n\n6\ndecay exponent α (red crosses) and twice the Liouvillian\ngap 2¯g (orange empty circles) as functions of hx.\nFor\neach value of hx, α is obtained by fitting |Ozz\n1 (1, t)| to an\nexponential decay at intermediate times in the isolated\nchain of size L = 12, while ¯g is extrapolated from g(γ)\nfor L = 10. The two quantities closely track each other\nacross both fully chaotic and transitional regimes.\nOverall, this figure highlights that the OTOC decay\nexponent remains approximately equal to twice the Li-\nouvillian gap for all hx values where scrambling-induced\nrelaxation persists. For hx > 1.00, the correspondence\nbecomes less pronounced, likely due to finite-size ef-\nfects?larger system sizes would be required to achieve\nquantitative convergence of both α and ¯g in this regime.\nNevertheless, even at these modest sizes, a clear qualita-\ntive and quantitative agreement is observed, underscor-\ning the robustness of the relationship between the OTOC\ndecay rate and the Liouvillian gap.\nIt is worth noting that the decay exponent α and the\nLiouvillian gap ¯g qualitatively track the transition be-\ntween dynamical regimes.\nA similar qualitative (and\nquantitative) behavior of the OTOC decay was previ-\nously reported for quantum maps in Ref. [20].\nVII.\nCONCLUSIONS\nWe have investigated the connection between the decay\nof the out-of-time-order correlator (OTOC), the Liouvil-\nlian gap, and spectral indicators of chaos in many-body\nquantum systems, focusing on the kicked Ising spin chain.\nOur central finding is that the intermediate time expo-\nnential decay rate of the OTOC for generic, symmetry-\nunresolved operators equals twice the Liouvillian gap of\nthe corresponding weakly open extension of the dynam-\nics. This correspondence, previously established for sys-\ntems with a semiclassical limit, is shown here to per-\nsist across the entire crossover from integrable to chaotic\nregimes, demonstrating that the Liouvillian spectrum\nprovides a unifying description of relaxation and irre-\nversibility in quantum many-body dynamics.\nA\ndetailed\nparity-resolved\nanalysis\nconfirms\nthat\nthe\nintrinsic\nLiouvillian\ngap\nextracted\nfrom\nthe\nArnoldi?Lindblad method reproduces the global relax-\nation rate of the system, even when the dominant decay\nmode shifts between symmetry sectors. This robustness\nhighlights that both the OTOC and the Liouvillian gap\nrespond coherently to changes in the underlying dynam-\nical regime, providing consistent signatures of the tran-\nsition from integrability to chaos.\nThe practicality of the Arnoldi?Lindblad approach lies\nin its ability to access the smallest Liouvillian eigenval-\nues without constructing the full superoperator, enabling\nefficient computation of the Liouvillian gap and its ex-\ntrapolation to the weak-dissipation limit. In contrast to\nthe momentum-resolved analysis of Ref. [13] for systems\nwith periodic boundary conditions, our framework cap-\ntures the global relaxation properties of the chain under\nopen boundaries, providing a complementary perspective\nthat links spectral relaxation modes directly to informa-\ntion scrambling.\nOverall, our results establish a quantitative bridge be-\ntween unitary and dissipative descriptions of quantum\nrelaxation, extending the Liouvillian?OTOC correspon-\ndence beyond the fully chaotic regime. This connection\nsuggests that Liouvillian spectroscopy may serve as a\ngeneral diagnostic tool for intermediate time dynamics\nand operator spreading in isolated quantum systems.\nACKNOWLEDGMENTS\nWe acknowledge support from Argentinian Agen-\ncia I+D+i (Grants No.\nPICT-2020-SERIEA-00740\nand PICT-2020-SERIEA-01082).\nI.G-M received sup-\nport from the French-Argentinian International Research\nProject Complex Quantum Systems (COQSYS), funded\nby CNRS. D.A.W. received support from CONICET\n(Grant No. PIP 11220200100568CO), UBACyT (Grant\nNo. 20020220300049BA) and PREI-UNAM.\nAppendix A: Arnoldi-Lindblad time evolution\nTo compute the Liouvillian spectrum efficiently, we\nemploy the Arnoldi–Lindblad approach of Ref. [43]. This\nmethod allows one to access the smallest-magnitude\neigenvalues of the Floquet-Lindbladian without con-\nstructing the full superoperator, whose matrix represen-\ntation has dimensions D2×D2, with D being the Hilbert-\nspace dimension. As a result, explicitly building the Li-\nouvillian quickly becomes computationally prohibitive as\nthe system size increases.\nWe consider a periodically driven open quantum sys-\ntem whose stroboscopic dynamics is governed by the\nLindblad master equation of period T,\n∂tˆρ(t) = L(t) ˆρ(t),\nL(t) = L(t + T).\n(A1)\nThe evolution over one period is formally given by the\ntime-ordered exponential\nˆρ(T) = T\n\"\nexp\n Z T\n0\nL(t′) dt′\n!#\nˆρ(0) ≡F(T, 0) ˆρ(0),\n(A2)\nwhere F(T, 0) denotes the Floquet map associated with\none full period of evolution.\nBy periodicity,\nF(t, 0) = [F(T, 0)]n F(t −nT, 0),\n(A3)\nfor integer n.\nDefining the Floquet Liouvillian LF\nthrough\nF(T, 0) = exp(LF T),\n(A4)\n\n7\none obtains\nF ˆρF\nj = φF\nj ˆρF\nj = eλF\nj T ˆρF\nj\n⇐⇒\nLF ˆρF\nj = λF\nj ˆρF\nj .\n(A5)\nThe\neigenvalues\nλF\nj\nform\nthe\nFloquet-Liouvillian\nspectrum,\nfrom\nwhich\nthe\nLiouvillian\ngap\ng\n=\n−maxα̸=0 Re λF\nα is extracted.\nIn practice, we generate the Krylov subspace\nKn = {ˆρ(0), ˆρ(T), ˆρ(2T), . . . , ˆρ(nT)}\n= {ˆρ(0), F ˆρ(0), F2ˆρ(0), . . . , Fnˆρ(0)}.\n(A6)\ndirectly from time evolution under the Lindblad equa-\ntion, without explicitly constructing L or F. 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Chen and T. Zhou, Operator scrambling and quantum\nchaos (2018).\n[42] E. M. Fortes, I. Garc´ıa-Mata, R. A. Jalabert, and D. A.\nWisniacki, Gauging classical and quantum integrability\nthrough out-of-time-ordered correlators, Phys. Rev. E\n100, 042201 (2019).\n[43] F. Minganti and D. Huybrechts, Arnoldi-Lindblad time\nevolution: Faster-than-the-clock algorithm for the spec-\ntrum of time-independent and Floquet open quantum\nsystems, Quantum 6, 649 (2022).\n"},"cleaned":{"kind":"string","value":"Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems Jer ́onimo Duarte,1 Ignacio Garc ́ıa-Mata,1 and Diego A. Wisniacki2 1Instituto de Investigaciones F ́ısicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata & CONICET, Funes 3350 (7600) Mar del Plata, Argentina 2Departamento de F ́ısica \"J. J. Giambiagi\" and IFIBA, FCEyN, Universidad de Buenos Aires, 1428 Buenos Aires, Argentina (Dated: October 17, 2025) The out-of-time-order correlator (OTOC) quantifies information scrambling in quantum systems and serves as a key diagnostic of quantum chaos. In one-body systems with a classical counterpart, the relaxation of the OTOC is governed by Ruelle-Pollicott resonances. For many-body systems lacking a semiclassical limit, recent studies have identified an analogous role played by the Liouvillian spectrum of weakly open extensions of the dynamics, where the slowest decay rate?the Liouvillian gap?encodes relaxation. Here we study the kicked Ising spin chain and show that the long-time exponential decay of the OTOC in the isolated system occurs at a rate equal to twice this intrinsic gap. This correspondence persists even in crossover regimes between integrability and chaos, demonstrating that the Liouvillian spectrum provides a unified framework for understanding relaxation and irreversibility in closed many-body quantum systems. I. INTRODUCTION Understanding relaxation dynamics and the emergence of effective irreversibility in quantum many-body systems is a central problem at the interface of quantum chaos, statistical mechanics, and open-system theory. In classical chaotic dynamics, the intermediate time decay of correlations is governed by the Ruelle-Pollicott (RP) resonances [1-3] - isolated singularities of the analytically continued spectrum of the Frobenius?Perron operator - which generate effective irreversibility despite the underlying deterministic and reversible motion. In quantum one-body systems with a classical limit, an analogous correspondence is well established through spectral and operator-truncation approaches [4-6] or through weakly dissipative formulations [7-9]. In contrast, identifying the quantum analogue of RP resonances in generic many-body systems remains an open question. Several works [10-12] have introduced coarse-grained formulations of the operator space, revealing isolated eigenvalues inside the unit disk that can be interpreted as quantum RP resonances. While conceptually appealing, this approach becomes impractical for large systems due to the exponential growth of the operator space and the difficulty of defining an appropriate coarse-graining. A more recent development exploits translational invariance and quasi-momentum decomposition [13], enabling the identification of resonances within each symmetry sector. An alternative line of research focuses on the Liouvillian spectrum of weakly open quantum systems [14]. When a system is coupled to an environment, its relaxation dynamics are governed by the eigenvalues of the corresponding Lindblad superoperator, with the slowest nonzero decay rate defining the Liouvillian gap. In exactly solvable models [15-17], this gap has been shown to coincide with the leading RP resonance of the corresponding isolated dynamics. The magnitude of the leading RP resonance determines the approach to equilibrium of dynamical observables, including the out-of-time-order correlator (OTOC). The OTOC measures the spread of local perturbations and serves as a sensitive probe of information scrambling in quantum systems. Although the butterfly effect in quantum chaos is not universal [18], the relaxation of the OTOC at late times provides a robust signature of underlying dynamics. In one-body systems, the crossover from integrability to chaos can be characterized through the OTOC decay rate [19], a result that extends to systems with mixed classical dynamics [20]. Here we extend this connection to many-body systems. We study the kicked Ising spin chain, a paradigmatic model that exhibits both integrable and chaotic regimes depending on its parameters. Specifically, we analyze three complementary quantities: (i) the Liouvillian gap ̄g extracted from a weakly dissipative version of the model, (ii) standard chaos indicators from spectral statistics and eigenstate delocalization, and (iii) the intermediate time exponential decay rate α of the OTOC in the isolated chain. We show that α ≈2 ̄g even in transitional regimes between integrability and chaos, establishing the Liouvillian gap as a qualitative (and quantitative) measure of integrability in quantum many-body systems. II. MANY-BODY SYSTEM: KICKED ISING SPIN CHAIN To investigate the interplay between relaxation dynamics, quantum chaos, and dissipation, we consider the kicked Ising spin chain [21, 22], a paradigmatic model of many-body Floquet dynamics. This system is particularly well suited to our study because its parameters allow for a continuous interpolation between integrable and chaotic behavior. 16 Oct 2025 2 The model is defined by the time-periodic Hamiltonian ˆH(t) = ˆHfree + ˆHkick τ ∞ X n=-∞ δ(t -nτ), (1) where the \"free\" and \"kick\" components take the form ˆHfree = -J L-2 P i=0 ˆσz i ˆσz i+1 -hz L-1 P i=0 ˆσz i , ˆHkick = -hx L-1 P i=0 ˆσx i . (2) Here, ˆσα i (α = x, z) denote Pauli operators acting on site i, J is the nearest-neighbor coupling, and hx, hz are transverse and longitudinal magnetic fields, respectively. The dynamics is stroboscopic with period τ, and the evolution over one period is governed by the Floquet operator ˆUF = ˆUfree ˆUkick, (3) where ˆUfree = exp(-i ˆHfreeτ), ˆUkick = exp(-i ˆHkickτ). (4) This Floquet representation conveniently describes both unitary and, when extended, weakly dissipative evolutions. Throughout this work we set J = hz = τ = 1 and vary the transverse field hx, which controls the degree of chaoticity. These values ensure nontrivial dynamics while reducing the parameter space to a single effective control variable. All simulations are performed for system sizes up to L = 12 spins. To understand the structure of the spectrum, it is essential to discuss the symmetries present in the model. Since we work with a Floquet operator, the relevant spectrum consists of quasienergies. Throughout this study we impose open boundary conditions (OBC). The system possesses an external reflection symmetry ˆR, which acts as ˆR |m1, m2, . . . , mL-2, mL-1⟩= |mL-1, mL-2, . . . , m2, m1⟩, where |m1, m2, . . . , mL-1⟩denotes a computational-basis state. Because [ ˆUF , ˆR] = 0, the Hilbert space decomposes into two invariant subspaces corresponding to the eigenvalues +1 and -1 of ˆR, which we refer to as the even and odd parity sectors. This symmetry decomposition allows us to analyze each sector independently, avoiding spectral mixing and enabling a cleaner identification of chaotic behavior and Liouvillian relaxation modes. III. QUANTUM CHAOS DIAGNOSTICS To characterize the degree of quantum chaos in the kicked Ising spin chain, we employ two complementary diagnostics: spectral statistics and eigenstate delocalization. These indicators enable us to distinguish between integrable, chaotic, and intermediate regimes, providing essential context for interpreting the behavior of the OTOC and its connection to the Liouvillian gap. A well-established signature of quantum chaos is the statistical distribution of level spacings [23]. Chaotic systems exhibit Wigner-Dyson distributions predicted by random matrix theory (RMT), with the specific ensemble-in our case, the Circular Orthogonal Ensemble (COE)-determined by the symmetries of the system [24]. In contrast, integrable systems display Poisson statistics reflecting the presence of extensive conserved quantities [25]. Rather than computing the full level-spacing distribution, we employ the ratio of adjacent spacings [26], which provides a practical and robust diagnostic. For the Floquet operator satisfying ˆUF |ψn⟩= eiφn |ψn⟩, n = 1, 2, . . . , D, (5) where φn are the quasienergies, |ψn⟩are the eigenstates, and D is the Hilbert space dimension, we compute rn = min(δn, δn-1) max(δn, δn-1), δn = φn+1 -φn. (6) The mean value ⟨r⟩exhibits two distinct limits: ⟨r⟩COE ≈ 0.5307 for COE statistics and ⟨r⟩P ≈0.3863 for Poisson statistics [27, 28]. To interpolate between these limits, we define the normalized parameter η = ⟨r⟩-⟨r⟩P ⟨r⟩COE -⟨r⟩P , (7) which provides a continuous measure ranging from η →0 in the integrable regime to η →1 in the fully chaotic regime. A complementary diagnostic is provided by eigenstate delocalization, quantified through the participation ratio (PR) [29]. Consider an eigenstate |ψi⟩expanded in a reference basis {|φj⟩}D-1 j=0 as |ψi⟩= P j aij |φj⟩. The participation ratio is defined as ξE(i) = D-1 X j=0 |aij|4 -1 . (8) This quantity measures how extended an eigenstate is in the chosen basis. Small values indicate localization, while high values signal delocalization. For chaotic systems consistent with RMT, the coefficients |aij|2 behave as independent random variables, leading to typical values ξdeloc E ≈D/3, where D is the Hilbert space dimension [30, 31]. In our numerical analysis, we compute the average of ξE(i) over all eigenstates. These spectral and eigenstate-based chaos indicators provide a quantitative map of the dynamical regime of the system. We analyze their behavior as a function 3 of the transverse field strength hx, which controls the kick strength in the Floquet dynamics. To avoid spectral mixing between symmetry sectors, all analyses are performed within the even-parity subspace, ensuring a clearer characterization of the underlying dynamical behavior. This characterization serves as a reference for studying the Liouvillian gap and exploring potential connections between unitary and dissipative indicators of quantum chaos. IV. OUT-OF-TIME-ORDER CORRELATOR The out-of-time-order correlator (OTOC) quantifies the scrambling of quantum information and is defined as the thermal expectation value of the squared commutator between two operators evaluated at different times: C(t) = h ˆW(t), ˆV i h ˆW(t), ˆV i† , (9) where ˆW(t) is the Heisenberg picture evolution of ˆW. Expectation values are taken over the infinite-temperature ensemble, ⟨·⟩= Tr(·)/D, with D the Hilbert-space dimension. This ensemble is appropriate for Floquet systems or Hamiltonians with bounded spectra, where microcanonical and infinite-temperature averages coincide in the thermodynamic limit. Originally introduced in studies of superconductivity [32], the OTOC has become a central tool for characterizing quantum chaos and information scrambling in many-body systems [33]. The OTOC gained attention when upper bound for the exponential growth was found [34], setting the bar of fast scramblers like black holes[35] and strongly correlated fermionic systems[36-38]. In systems with a classical limit, the growth rate coincides with the classical Lyapunov exponent [19, 39-41]. At later times, quantum interference leads to saturation around a constant plateau with residual fluctuations that depend on the underlying dynamics [42]. We focus on local Hermitian and unitary operators ˆW and ˆV acting on distinct sites of the spin chain. A convenient choice is given by Pauli operators ˆσμ i (μ = x, y, z) acting on site i. At infinite temperature, the OTOC takes the explicit form Cμν(l, t) = 1 2 D [ˆσμ 0 (t), ˆσν l ]2E = 1 -1 D Re {Tr [ˆσμ 0 (t)ˆσν l ˆσμ 0 (t)ˆσν l ]} . (10) We define Oμν 1 (l, t) = 1 D Re {Tr [ˆσμ 0 (t)ˆσν l ˆσμ 0 (t)ˆσν l ]} , (11) so that Cμν(l, t) = 1-Oμν 1 (l, t). In what follows we focus on the case μ = ν = z, denoted Czz(l, t), which probes the scrambling and relaxation of local ˆσz operators. At intermediate times, the approach to saturation of C(t), at (what we will call intermediate times) is reflected by an exponential decay of O1(t), which can be present even in systems which are not completely chaotic [20]. For one-body, completely chaotic systems, the decay is given by the classical RP resonances[19]. Figure 1 illustrates the time evolution of |Ozz 1 (l, t)| for l = 1 and a transverse field hx deep in the chaotic regime. The data, obtained for L = 12, display a clear exponential decay at intermediate times . An exponential fit of the form |Ozz 1 (1, t)| ∝e-αt yields the decay rate α, which we compare below with twice the Liouvillian relaxation rate extracted from the weakly dissipative extension of the model. 0 10 20 30 40 50 60 t 10-4 10-3 10-2 10-1 100 |Ozz 1 | FIG. 1. The absolute value of Ozz 1 (1, t) for a transverse field hx in the chaotic regime. The data corresponds to a spin chain of size L = 12 and hx = 0.8168. The blue line with circles represents Ozz 1 (1, t), while the dashed line is the exponential fit to the intermediate time behavior of |Ozz 1 (1, t)|, with a decay exponent α = 0.2888. V. LIOUVILLIAN GAP To investigate the relation between the decay rate discussed in the previous section and the spectral properties of the system, we follow Ref. [14]. In this approach, the system is extended to a weakly open setting described by a Liouvillian superoperator L = -i[H, · ] + γD, (12) where D is a dissipator and γ the dissipation strength. Building on the kicked Ising model defined above, we introduce bulk dephasing and study the corresponding relaxation dynamics through the time-periodic Lindblad master equation dρ dt = -i [ ˆH(t), ρ] + γ L-1 X i=0 σz i ρ σz i - 1 2{σz i σz i , ρ} ≡L(t) ρ, (13) 4 where γ > 0 controls the dephasing strength and ˆH(t) = ˆH(t + τ) is the Floquet Hamiltonian. The Liouvillian superoperator L(t) has a complex spectrum λα with negative real parts. For time-independent generators (L) there is one zero mode (λ0 = 0) corresponding to the steady state, plus decaying modes with Re λα 1.00, the correspondence becomes less pronounced, likely due to finite-size effects?larger system sizes would be required to achieve quantitative convergence of both α and ̄g in this regime. Nevertheless, even at these modest sizes, a clear qualitative and quantitative agreement is observed, underscoring the robustness of the relationship between the OTOC decay rate and the Liouvillian gap. It is worth noting that the decay exponent α and the Liouvillian gap ̄g qualitatively track the transition between dynamical regimes. A similar qualitative (and quantitative) behavior of the OTOC decay was previously reported for quantum maps in Ref. [20]. VII. CONCLUSIONS We have investigated the connection between the decay of the out-of-time-order correlator (OTOC), the Liouvillian gap, and spectral indicators of chaos in many-body quantum systems, focusing on the kicked Ising spin chain. Our central finding is that the intermediate time exponential decay rate of the OTOC for generic, symmetryunresolved operators equals twice the Liouvillian gap of the corresponding weakly open extension of the dynamics. This correspondence, previously established for systems with a semiclassical limit, is shown here to persist across the entire crossover from integrable to chaotic regimes, demonstrating that the Liouvillian spectrum provides a unifying description of relaxation and irreversibility in quantum many-body dynamics. A detailed parity-resolved analysis confirms that the intrinsic Liouvillian gap extracted from the Arnoldi?Lindblad method reproduces the global relaxation rate of the system, even when the dominant decay mode shifts between symmetry sectors. This robustness highlights that both the OTOC and the Liouvillian gap respond coherently to changes in the underlying dynamical regime, providing consistent signatures of the transition from integrability to chaos. The practicality of the Arnoldi?Lindblad approach lies in its ability to access the smallest Liouvillian eigenvalues without constructing the full superoperator, enabling efficient computation of the Liouvillian gap and its extrapolation to the weak-dissipation limit. In contrast to the momentum-resolved analysis of Ref. [13] for systems with periodic boundary conditions, our framework captures the global relaxation properties of the chain under open boundaries, providing a complementary perspective that links spectral relaxation modes directly to information scrambling. Overall, our results establish a quantitative bridge between unitary and dissipative descriptions of quantum relaxation, extending the Liouvillian?OTOC correspondence beyond the fully chaotic regime. This connection suggests that Liouvillian spectroscopy may serve as a general diagnostic tool for intermediate time dynamics and operator spreading in isolated quantum systems. ACKNOWLEDGMENTS We acknowledge support from Argentinian Agencia I+D+i (Grants No. PICT-2020-SERIEA-00740 and PICT-2020-SERIEA-01082). I.G-M received support from the French-Argentinian International Research Project Complex Quantum Systems (COQSYS), funded by CNRS. D.A.W. received support from CONICET (Grant No. PIP 11220200100568CO), UBACyT (Grant No. 20020220300049BA) and PREI-UNAM. Appendix A: Arnoldi-Lindblad time evolution To compute the Liouvillian spectrum efficiently, we employ the Arnoldi-Lindblad approach of Ref. [43]. This method allows one to access the smallest-magnitude eigenvalues of the Floquet-Lindbladian without constructing the full superoperator, whose matrix representation has dimensions D2×D2, with D being the Hilbertspace dimension. As a result, explicitly building the Liouvillian quickly becomes computationally prohibitive as the system size increases. We consider a periodically driven open quantum system whose stroboscopic dynamics is governed by the Lindblad master equation of period T, ∂tˆρ(t) = L(t) ˆρ(t), L(t) = L(t + T). (A1) The evolution over one period is formally given by the time-ordered exponential ˆρ(T) = T \" exp Z T 0 L(t′) dt′ !# ˆρ(0) ≡F(T, 0) ˆρ(0), (A2) where F(T, 0) denotes the Floquet map associated with one full period of evolution. By periodicity, F(t, 0) = [F(T, 0)]n F(t -nT, 0), (A3) for integer n. Defining the Floquet Liouvillian LF through F(T, 0) = exp(LF T), (A4) 7 one obtains F ˆρF j = φF j ˆρF j = eλF j T ˆρF j ⇐⇒ LF ˆρF j = λF j ˆρF j . (A5) The eigenvalues λF j form the Floquet-Liouvillian spectrum, from which the Liouvillian gap g = -maxα̸=0 Re λF α is extracted. In practice, we generate the Krylov subspace Kn = {ˆρ(0), ˆρ(T), ˆρ(2T), . . . , ˆρ(nT)} = {ˆρ(0), F ˆρ(0), F2ˆρ(0), . . . , Fnˆρ(0)}. (A6) directly from time evolution under the Lindblad equation, without explicitly constructing L or F. 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However, stringent\nrequirements on computation time and the risk of numerical\ninstability can limit the use of optimization-based planners in\nsafety-critical scenarios. This work presents an optimization-\nfree planning framework called STITCHER that stitches short\ntrajectory segments together with graph search to compute long-\nrange, expressive, and near-optimal trajectories in real-time.\nSTITCHER outperforms modern optimization-based planners\nthrough our innovative planning architecture and several al-\ngorithmic developments that make real-time planning possible.\nExtensive simulation testing is performed to analyze the al-\ngorithmic components that make up STITCHER, along with\na thorough comparison with two state-of-the-art optimization\nplanners. Simulation tests show that safe trajectories can be\ncreated within a few milliseconds for paths that span the entirety\nof two 50 m x 50 m environments. Hardware tests with a custom\nquadrotor verify that STITCHER can produce trackable paths in\nreal-time while respecting nonconvex constraints, such as limits\non tilt angle and motor forces, which are otherwise hard to\ninclude in optimization-based planners.\nIndex Terms—Trajectory planning, aerial systems, motion\nprimitives, graph search, collision avoidance.\nI. INTRODUCTION\nP\nLANNING collision-free, dynamically feasible trajecto-\nries in real-time through complex environments is crucial\nfor many autonomous systems. As a result, trajectory planning\nhas garnered significant interest from the research community,\nbut meeting the reliability requirements for safety-critical\nreal-world applications remains challenging. Specifically, few\nmethods have guarantees regarding trajectory optimality and\ntime/memory complexity without sacrificing trajectory length,\ncomputation time, or expressiveness. Our approach addresses\nthis gap by combining optimal control with graph search to\ngenerate near-optimal trajectories over long distances in real-\ntime, resulting in a framework that provides strong guarantees\non path quality and algorithm complexity.\nOptimization-based trajectory planning has emerged as the\nprimary framework for autonomous systems that must navigate\ncomplex environments. This is because constraints and perfor-\nmance objectives can be naturally stated in the optimization\nproblem. Most approaches can be broadly classified by their\nuse of continuous or integer variables. Continuous variable\nmethods employ gradient descent to jointly optimize over\nAuthors are with the VECTR Laboratory, University of California, Los\nAngeles, Los Angeles, CA, USA. {hjlevy, btlopez}@ucla.edu\nFig. 1: A trajectory (colored based on speed) generated by our proposed\nalgorithm called STITCHER through a perlin noise environment. STITCHER\nsearches over candidates motion primitives (white) to find a safe trajectory\nin real-time with time and memory complexity guarantees. Indoor flight\nexperiments were performed to verify dynamic feasibility of trajectory plans.\nthe coefficients of basis functions (e.g., polynomials) and\nwaypoint arrival times while imposing obstacle and state\nconstraints [1]–[6]. Integer variable methods require that the\nfree space of the environment be represented as the union of\nconvex sets (continuous variable methods have also used this\nrepresentation, e.g., [5], [6]) and solve a mixed-integer pro-\ngram for a collision-free trajectory [7]–[10]. Despite continued\ninnovations, these methods lack a priori time complexity\nbounds and often scale very poorly with trajectory length; this\nis especially true for integer programming approaches.\nA computationally efficient alternative to optimization-\nbased trajectory planning is the use of so-called motion\nprimitives: a library of short length or duration trajectories\nthat can be efficiently computed and evaluated [11]–[14]. To\neffectively use motion primitives, a planner must operate in\na receding horizon fashion, i.e., continuously replan, because\nmotion primitives are inherently near-sighted with their short\nlength or duration. This can introduce several performance\nissues, e.g., myopic behavior or unexpressive trajectories, that\nare exacerbated in large, complex environments. Subsequent\nwork has attempted to pose the problem as a graph search\nwith nodes and edges being desired states (position, velocity,\netc.) and motion primitives, respectively [15]–[18]. While this\nallows for the creation of long-range trajectories, search times\ncan be extremely high (seconds) because of the graph size. An\nadmissible search heuristic can be used to reduce the number\nof node expansions required to find a solution, also known as\nsearch effort, while preserving optimality of the graph [19].\nHowever, designing such a search heuristic is non-trivial.\nWe propose a new trajectory planning algorithm called\nSTITCHER that can perform real-time motion primitive\narXiv:2510.14893v2 [cs.RO] 20 Oct 2025\n\nsearches across long distances in complex environments.\nSTITCHER utilizes an innovative three-stage planning ar-\nchitecture to generate smooth and expressive trajectories by\nstitching motion primitives together through graph search.\nThe first two stages are designed to expedite the motion\nprimitive search in the final stage by constructing a compact\nbut expressive search graph and search heuristic. Specifically,\ngiven a set of waypoints computed in the first stage, we\ncreate a velocity graph by representing nodes as sampled\nvelocities at each waypoint, and edges as quick-to-generate\nminimum-time trajectories. We employ dynamic programming\non this graph, using the Bellman equation to compute the\ncost-to-go for each node. Critically, the cost-to-go is then\nused as an admissible heuristic to efficiently guide the motion\nprimitive search in the third stage. We also leverage a greedy\ngraph pre-processing step to form a compact motion primitive\ngraph. We prove all graphs are finite and that the proposed\nheuristic is admissible. These technical results are critical\nas they guarantee i) a priori time and memory complexity\nbounds and ii) trajectory optimality with respect to the graph\ndiscretization. To further reduce computation time, we improve\nthe collision checking procedure from [13], by leveraging\nthe known free space from previous nearest-neighbor queries,\nbypassing the rigidity and computational complexity of free\nspace decomposition. Additionally, we show that employing a\nsimple sampling procedure in the final search stage is effective\nat pruning candidate trajectories that violate complex state\nor actuator constraints. STITCHER was extensively tested in\ntwo simulation environments to evaluate algorithmic innova-\ntions and assess its performance against two state-of-the-art\noptimization-based planners [5], [9]. STITCHER is shown to\ngenerate high-quality, dynamically feasible trajectories over\nlong distances (over 50 m) with computation times in the\nmilliseconds, and consistently generates trajectories faster than\nthe state-of-the-art with comparable execution times.\nThis paper substantially builds upon previous work [20] by\nproviding a new in-depth analysis of the algorithmic inno-\nvations used to achieve real-time planning times, along with\nhardware experiments to demonstrate real-world feasibility.\nWe include a parameter sensitivity analysis to evaluate the\nperformance of STITCHER with different hyperparameters,\nsuch as using different state discretization sets. A new study on\nthe proposed search heuristic is conducted to examine the ef-\nfects of varied edge costs and acceleration constraints on both\nadmissibility and search effort (node expansions). The impact\nof the greedy graph pre-processing step on solution quality\nand computation time is evaluated through comparisons with\nan exhaustive, i.e., non-greedy, graph creation. We also showed\nSTITCHER can adhere to highly nonconvex constraints, such\nas individual motor constraints for a quadrotor, with no\nnoticeable increase computation time. Comparison with state-\nof-the-art algorithms in simulation was also expanded, incor-\nporating new metrics to better evaluate performance regarding\noptimized waypoints and the sampled set of states. Finally,\nwe flew the trajectories designed by STITCHER in hardware\non a custom quadrotor to show that the trajectories were\ndynamically feasible, adhered to physical constraints, and\ncould be tracked via standard geometric cascaded control.\nII. RELATED WORKS\nA. Optimization-based Planning\nDesigning high quality trajectories using online optimiza-\ntion has become a popular planning strategy as a performance\nindex and constraints can be systematically incorporated into\nan optimization problem. Optimization-based trajectory plan-\nners can be categorized using several criteria, but the clearest\ndelineation is whether the method uses continuous or integer\nvariables. For methods that use only continuous variables,\nthe work by [2] reformulated the quadratic program in [1]\nto jointly optimize over polynomial endpoint derivatives and\narrival times for a trajectory passing through waypoints. Col-\nlisions were handled by adding intermediate waypoints and\nredoing the trajectory optimization if the original trajectory\nwas in collision. Oleynikova et al. [3] represented obstacles\nusing an Euclidean Signed Distance Field (ESDF) which was\nincorporated into a nonconvex solver as a soft constraint. Zhou\net al. [4] used a similar penalty-based method but introduced a\ntopological path search to escape local minima. An alternative\napproach is to decompose the occupied space or free space\ninto convex polyhedra [7], [21], [22] which can be easily\nincorporated as constraints in an optimization. The methods\nin [5], [6] treat these constraints as soft while efficiently\noptimizing over polynomial trajectory segments that must pass\nnear waypoints. One can also use the free-space polyhedra to\nformulate a mixed-integer program [8]–[10], [23] to bypass the\nnonconvexity introduced by having unknown waypoint arrival\ntimes, but at the expense of poor scalability with trajectory\nlength and number of polyhedra. Marcucci et al. [24] addresses\nscalability concerns of [10] by solving a sequence of convex\nproblems instead of one large-scale optimization but requires\nan offline process for generating collision-free convex sets.\nB. Motion Primitives\nMotion primitive planners have been proposed as an alterna-\ntive to optimization-based planners to address computational\ncomplexity and numerical instability concerns. The underlying\nidea of motion primitive planners is to select trajectories\nonline from a precomputed, finite library of trajectories. Initial\nwork on motion primitives for quadrotors leveraged differen-\ntial flatness and known solutions to specific optimal control\nproblems to efficiently compute point-to-point trajectories in\nreal-time [11], [25]. Later work employed motion primitives\nfor receding horizon collision avoidance where primitives\nwere efficiently generated online by sampling desired final\nstates, and selected at each planning step based on safety and\ntrajectory cost [12]–[14], [26]–[29]. Howard et al. [26] first\nintroduced this idea of searching over feasible trajectories of\na car with a model predictive control framework. Subsequent\nworks extended this methodology to quadcopters using depth\nimages [12], [14], point clouds [13], [27], or ESDFs [28] for\nmotion primitive evaluations and collision avoidance. While\ncomputationally efficient, the behavior of these planners can be\nmyopic, leading to suboptimal behavior in complex environ-\nments which limit their use for planning long-term trajectories.\n\nC. Motion Primitive Search\nOne way to address nearsightedness is to perform a graph\nsearch over motion primitives, i.e., stitch motion primitives\ntogether. This can be achieved by extending traditional graph\nsearch algorithms [30]–[32], which typically use coarse dis-\ncrete action sets, to using a lattice of motion primitives [15]–\n[17], [33]–[35]. Graph search algorithms are an attractive\nmethod for planning due to inherent guarantees of complete-\nness, optimality1, and bounded time and memory complexity\n[36]. The works by Liu et al. [15], [16] were some of the\nfirst works to successfully showcase a search-based algorithm\nusing a lattice of motion primitives for use on quadcopters.\nHowever, these methods can be computationally expensive as\ngenerating high-quality trajectories relies on generating a large\nnumber of motion primitives for sufficient density. Jarin et al.\n[37] addresses computation concerns by improving upon the\nsampling of different motion primitives, inspired by a mini-\nmum dispersion sampling method [38]. Another way to narrow\nthe search space is by utilizing a geometric path as a prior\nand constraining motion primitives to pass through waypoints\nfrom the path. Recently, [18], [39]–[41] proposed an efficient\nmotion primitive search in velocity space using minimum-time\ninput-constrained trajectories from a double integrator model\nrestricted to pass through a set of waypoints. The search can\nbe done in real-time but the resulting bang-bang acceleration\nprofile is dynamically infeasible for quadrotors, leading to\npoor tracking performance. An additional smoothing step, e.g.,\nmodel predictive contouring control, is required to achieve\nsufficient trajectory smoothness [40]–[42].\nD. Search Heuristics\nFast graph search speed while retaining optimality guar-\nantees can be achieved by employing an admissible search\nheuristic [36] to guide the search to the goal state. Constructing\nan informative and admissible heuristic, however, is non-\ntrivial. Much of the previous work in motion primitive search\noverlooks the importance of the heuristic by generating a\nweak approximation to the goal [17], using an inadmissible\nheuristic which forfeits optimality guarantees [16], or pro-\nceeding without a heuristic [18]. As a result, motion primitive\nsearch algorithms to date scale poorly in large environments\nand for large planning horizons, making them unsuitable for\nsystems with limited onboard computational resources. Paden\net al. [43] proposed a method to systematically construct\nadmissible heuristics for use in kinodynamic planning using\nsum-of-squares (SOS) programming. However, the resulting\nsize of the SOS program requires heuristic calculations to be\nperformed offline. Other strategies involve learning a search\nheuristic or cost-to-go [44]–[47]. Kim et al. [44] uses a neural\nnetwork to approximate graph node distances and provides\na sub-optimality bound on the solution. Reinforcement and\nimitation learning have also been proposed for learning search\nheuristics [45]–[47], but these works focus on minimizing\nnode expansions rather than ensuring admissibility, sacrificing\nthe optimality guarantees of graph search.\n1In the context of graph search, optimality refers to resolution optimality,\ni.e., optimality with respect to the discretized state space.\nIII. PROBLEM FORMULATION\nThis work is concerned with solving the following trajectory\nplanning problem:\nmin\nu ∈U\nJ = r(T) +\nZ T\n0\nq(x, u) dt\ns.t.\n˙x = Ax + Bu\nx ∈Xs, x /∈Xobst, u ∈U\nx(0) = x0, x(T) = xf,\n(1)\nwhere x ∈Rn is the state that must satisfy state Xs and\nobstacle (collision) Xobst constraints, u ∈Rm is the control\ninput that must satisfy actuator constraints U, A ∈Rn×n and\nB ∈Rn×m govern the system’s dynamics and are assumed to\ntake a multi-axis chain of integrators form, and r : R+ →R+\nand q : Rn × Rm →R+ are the terminal and stage cost,\nrespectively. The goal is to find an optimal final time T ∗and\nfeasible optimal state trajectory x∗(t) with a corresponding\ncontrol input sequence u∗(t) for t ∈[0 T ∗] that steers the\nsystem from an initial state x0 to a desired final state xf\nthat minimizes the cost functional J. While the dynamics\nare linear in (1), many nonlinear systems can be placed into\nthe linear control affine form if they are differentially flat,\ne.g., VTOL vehicles like quadrotors, capturing a large class\nof systems of interest. In many cases, the state vector can\nbe x = (r, v, a, . . . , r(p−1))⊤and the control input can be\nu = r(p) where r = (x, y, z)⊤is the position of the vehicle\nin some reference frame.\nA. Background: Motion Primitives\nWe define motion primitives to be closed-form solutions\nto specific optimal control problems. In this work, we will\nrestrict our attention to the following two optimal control\nproblems: the input-constrained minimum-time problem for\na double integrator and the linear quadratic minimum time\nproblem for a p-th order integrator. We will briefly review\neach optimal control problem and the structure of its solution.\nThe formulations will be presented for a single axis, but can\nbe repeated for all three position coordinates.\nMinimum-Time Double Integrator: Given an initial state\n(s0, v0) ∈R2 and desired final state (sf, vf) ∈R2, the\nminimum-time double integrator optimal control problem is\nmin\nu\nJ = T\n(2)\ns.t.\n¨s = u, |u| ≤umax\ns(0) = s0, v(0) = v0\ns(T) = sf, v(T) = vf,\nwhere the final time T is free. From Pontryagin’s minimum\nprinciple, the optimal control solution is known to have\na bang-bang control profile. The switching times can be\nefficiently computed by solving a quadratic equation. It is\nrequired that each coordinate axis trajectory all have the\nsame execution time, so the limiting minimum-time horizon is\nT ∗= max{Tx, Ty, Tz}. The limiting minimum-time horizon\nT ∗, is then applied as a known variable for the non-limiting\n\nFig. 2: System architecture describing the three planning stages. Stage 1: A sparse geometric path is found via an A* search through the voxelized environment.\nStage 2: A velocity state is then introduced at each waypoint and dynamic programming is used to recursively solve for the cost-to-go at each node. Stage\n3: A full motion primitive search informed by the previous stages is performed, and checks for collisions are completed to yield the final trajectory.\naxes. This allows one to then solve a quadratic equation for a\nnew bound ¯u ≤umax on the control input.\nLinear Quadratic Minimum-Time p-th Order Integrator:\nSmooth trajectories can be generated by solving the linear\nquadratic minimum-time (LQMT) optimal control problem,\nmin\nT, u\nJ = ρ T +\nZ T\n0\nu2 dt\n(3)\ns.t.\ns(p) = u\ns(0) = s0, v(0) = v0, . . . , s(p−1)(0) = s(p−1)\n0\ns(T) = sf, v(T) = vf, s(k−1)(T) free for 3 ≤k ≤p\nwhere ρ > 1 penalizes the final time. The final time T and\nall terminal states except position and velocity are free. The\noptimal trajectory is a polynomial in time so the cost functional\ncan be expressed analytically in terms of T and the known\nboundary conditions. The final time can be found efficiently\nusing a root-finding algorithm such as QR algorithm [48].\nState constraints are omitted from (3) as it is more efficient\nto prune many candidate trajectories once the final time is\nknown, as discussed in Section IV-D.\nIV. METHODOLOGY\nSTITCHER, detailed in Algorithm 1, generates a full-\nstate trajectory by stitching collision-free, dynamically feasible\ntrajectory segments together through graph search. At its core,\nSTITCHER searches over closed-form solutions, i.e., motion\nprimitives, to optimal control problems of the form discussed\nabove. These solutions serve as a basis set for the solution\nspace to (1). To achieve real-time performance, STITCHER\nutilizes a three stage planning process (see Fig. 2). In Stage 1\n(left), A* algorithm is used to produce a sparse geometric\npath, i.e., waypoints, in the free space of the environment\n(line 3); this is standard in many planning frameworks. In\nStage 2 (middle), nodes representing sampled velocities at the\nwaypoints are formed into a velocity graph where dynamic\nprogramming is used to compute the minimum time path from\nAlgorithm 1: STITCHER Trajectory Planner\n1 input: P ←point cloud, ns ←start, ng ←goal;\n2 output: s∗(t);\n// extract waypoints and path features\n3 w, q, H ←getGeometricPath(P, ns, ng);\n4 G ←buildVelocityGraph(w, q, H);\n// get heuristic from recursive cost-to-go\n5 h(n) ←dynamicProgramming(G, ns, ng);\n6 Gmp ←buildFullStateGraph(G);\n7 function planPath(P, Gmp, ns, ng):\n8 ncurr = ns;\n9 while ncurr ̸= ng do\n// get node with lowest cost g(n)+h(n)\n10\nncurr = OPEN.pop();\n11\nCLOSED.insert(ncurr);\n12\nif ncurr = ng then\n13\nbreak;\n14\nend\n15\nEncurr ←getSuccessors(ncurr, Gmp);\n16\nfor e in Encurr do\n// collision and state constraint check\n17\npruneMotionPrimitive(e, P);\n18\nOPEN.insert(ϕ(n, e));\n19\nend\n20 end\n21 s∗(t) ←getFinalMotionPrimitives(ncurr, CLOSED);\neach node to the desired final state using a control-constrained\ndouble integrator model (lines 4-5). This step is critical for\nconstructing an admissible heuristic to guide the full motion\nprimitive search, and is one of the key innovations that enables\nreal-time performance. Note that the optimal “path” in velocity\nspace is never used; computing the cost-to-go is the primary\nobjective as it serves as an admissible heuristic for motion\nprimitive search as shown in Section V-B. In Stage 3 (right),\nan A* search is performed using the heuristic from Stage\n2 over motion primitives of a p-th order integrator where\np ≥2 . A greedy pre-processing step is used to construct a\ncompact motion primitive graph (line 6), ensuring the search\nremains real-time. At this stage, position and all higher-order\nderivatives are considered, yielding a full state trajectory that\ncan be tracked by the system (lines 7-21). The remainder of\n\nthis section expands upon each component of STITCHER.\nA. Stage 1: Forward Geometric Path Search\nSTITCHER requires a sequence of waypoints that essen-\ntially guides the motion primitive search by limiting the size\nof the search space. This can be done by generating a collision-\nfree geometric path (see Fig. 2 left) through the environment\nwith A* search or any other discrete graph search algorithm\nwhere the environment is represented as a 3D voxel grid\nwhere each grid cell contains occupancy information. Let the\ncollision-free, geometric path generated by a discrete graph\nsearch algorithm be composed of points O = {o1, o2, ..., oH}\nwhere oi ∈R3. The set of points O is further pruned to\ncreate a sparse set of waypoints W = {w1, w2, ..., wN} where\nN ≤H and wi ∈R3. Sparsification is done by finding\nthe minimal set of points in O that can be connected with\ncollision-free line segments. The geometric path search is used\nin line 3 of Algorithm 1.\nB. State 2: Backward Velocity Search\nThe ordered waypoint set W found in Stage 1 only provides\na collision-free geometric path through the environment. In\nother words, the velocity, acceleration, and higher-order states\nnecessary for tracking control are not specified. We propose\ncreating a velocity graph (see Fig. 2 middle) where each node\nin the graph is defined by a position and velocity. The positions\nare restricted to waypoint locations and M velocities are\nsampled at each waypoint. More explicitly, for each waypoint\nwi ∈W, we sample a set of velocities V = {v1, ..., vM},\nwhere V is composed of candidate velocity magnitudes Vm\nand directions Vd. The choice of Vm and Vd can impact the\nSTITCHER’s performance in terms of path optimality and\ncomputational complexity; this point will be discussed in more\ndetail in Section VI.\nWith the ordered waypoint W and sampled velocity V sets,\nwe create a velocity graph G = (N, E), where node n ∈N\nis a given position and sampled velocity, i.e., n = (wi, vj)\nwith wi ∈W and vj ∈V, and edge e ∈E is the double\nintegrator control-constrained minimum-time motion primitive\nr(t) from (2) that connects neighboring nodes. Recall that the\nsolution to (2) is fully determined by having an initial and final\nposition and velocity pair which is precisely how each node\nin N is defined. At this stage, collision and state constraints\nare not enforced to prevent candidate trajectories from being\neliminated prematurely.\nWe recursively compute and store the ranked list of cost-\nto-go’s Vd : N × E →R+ for each node n ∈N and all\nconnecting edges e ∈En of n where\nVd(n, e) = ℓ(n, e) + V ∗\nd\n\u0000ϕ(n, e)\n\u0001\n∀e ∈En,\n(4)\nwith the optimal cost-to-go V ∗\nd (n) = mine∈En Vd(n, e), the\ncost of taking edge e from node n being ℓ(n, e), and the node\nreached by taking edge e being ϕ(n, e). The cost of taking\nan edge is given by ℓ(n, e) = T ∗\nd (n, e), where T ∗\nd (n, e) is the\nminimum-time of trajectory r(t) connecting the states of node\nn to the states of ϕ(n, e). Minimizing (4) is the well-known\nBellman equation, which is guaranteed to return the optimal\nFig. 3: The achievable mass-normalized thrust (nonconvex) of a aerial VTOL\nvehicle with limits on minimum thrust fmin, maximum thrust fmax, and\nmaximum tilt θmax.\ncost-to-go. In Section V-B, we prove V ∗\nd (n) for each node in\na graph G is an admissible heuristic for an A* search over a\nbroad class of motion primitives. Building and searching the\nvelocity graph are shown in lines 4-5 of Algorithm 1.\nC. Stage 3: Forward Motion Primitive Search\nThe cost-to-go’s computed in Stage 2 for the sampled\nvelocities at each waypoint serve as an admissible heuristic\n(formally defined later in Definition 1) that guides an efficient\nA* search over motion primitives. The motion primitives can\nbe generated using any chain of integrators model of order at\nleast two so long as i) the initial and final position and veloc-\nities match those used to construct the velocity graph G and\nii) the allowable acceleration is maximally bounded by umax\ngiven in (2). It is important to note that the bound on allowable\nacceleration can be easily satisfied with user defined umax or\nsimply by applying the box constraint ∥a∥∞≤umax. The mo-\ntion primitive search graph is denoted as Gmp = (Nmp, Emp)\nwhere Nmp is the set of nodes, each corresponding to a state\nvector, and Emp is the set of edges, each corresponding to\na motion primitive that connecting neighboring nodes. A*\nsearch is used to meet real-time constraints where the search\nminimizes the cost f(n) = g(n) + h(n) where n ∈Nmp is\nthe current node, g : Nmp →R+ is the cost from the start\nnode ns to node n, and h : Nmp →R+ is the estimated cost\nfrom the current node n to the goal node ng. In the context of\noptimal control, g is the cost accrued, i.e., the cost functional\nJ, for a path from ns to n whereas h is the estimated cost-\nto-go, i.e., the estimated value function V ∗, from n to ng. In\nthis stage, collision and state constraints are checked for each\ncandidate motion primitive to ensure safety. The methodology\nfor both is discussed in Section IV-D. Each step of the motion\nprimitive A* search is shown in lines 6-21 of Algorithm 1.\nD. Pruning Infeasible & In-Collision Motion Primitives\nSTITCHER guarantees safety by pruning motion primitives\nfrom the final search that violate constraints or are in colli-\nsion. For state and actuator constraints, many optimization-\nbased planning approaches approximate the true physical\nconstraints of the system with simple convex constraints, e.g.,\n∥v∥∞≤vmax, ∥a∥∞≤amax, etc., to reduce computational\ncomplexity. When polynomials are used to represent the\noptimal trajectory, imposing a convex hull constraint on the\npolynomial is one method for enforcing such state constraints\n\nFig. 4: Removing redundant collision checks. (a): Motion primitive r0(t)\nchecks for collisions using [13]. (b): Sampled points of r1(t) are checked to\nlie within obstacle-free regions derived from r0(t) calculations.\n[9], [17]. However, many of these approximations are made\nonly to simplify the resulting optimization problem and do not\naccurately reflect the actual physical constraint, which can lead\nto conservatism. STITCHER has the freedom to use a variety\nof methods to enforce state and actuator constraints. For the\nexamples shown in this work, we uniformly sample candidate\ntrajectories in time to check for constraint violations as it was\nfound to be effective and efficient. Sampling avoids mistakenly\neliminating safe trajectories, and the observed computation\ntime under the current discretization was better than using\nconvex hulls. Critically, sampling allows for the inclusion of\nmore complex constraints, such as those that couple multiple\naxes. Examples are\nThrust Magnitude: 0 < fmin ≤∥f∥2 ≤fmax\nThrust Tilt Angle: ∥f∥2 cos(θmax) ≤fz\nLinear Velocity: ∥v∥2 ≤vmax\nAngular Velocity: ∥ω∥2 ≤ωmax,\n(5)\nwhere f is the mass-normalized thrust, θ is the thrust tilt angle,\nand ω is the angular velocity. Note that differential flatness\ncan be leveraged to express the angular velocity constraint\nin terms of derivatives of position. Figure 3 depicts the\nachievable mass-normalized thrust of a VTOL vehicle given\nthrust and tilt constraints in (5). The constraints are nonconvex\nmaking it difficult to include in real-time optimization-based\nplanners without some form of relaxation, e.g., as in [49] for\na double integrator, which is tight, or a more conservative\nrelaxation. Even more direct system constraints which limit\nthe maximum force exerted by individual motors are highly\nnonconvex functions of the flat variables and their derivatives.\nWhile the majority of this paper showcases results that en-\nforce constraints (5), we include a case study of STITCHER\nconstraining individual thrusters in Section VI-D.\nAn efficient collision checking strategy was devised by\nconstructing a safe set of spheres resulting from a sampling-\nbased collision checking approach developed in [13] (see\nFig. 4). The core idea from [13] is that a trajectory can be\nintelligently sampled for collisions by estimating the next\npossible “time-of-collision” along the trajectory by combin-\ning obstacle proximity and the vehicle’s maximum speed.\nLeveraging this idea, further computation time savings can\nbe achieved by storing and reusing nearest neighbor queries.\nAlgorithm 2 details our strategy which takes in a k-d tree\ndata structure filled with points from a point cloud P of\nAlgorithm 2: Collision Check\n1 input: T ←k-d tree, r(t) ←motion primitive,\n2 output: bool collision\n3 if S = ∅then\n// initial collision check using k-d tree\n4\ncollision, S ←collisionCheckMap(r(t), T );\n5\nreturn collision\n6 end\n7 τ ←0;\n8 dmin ←∞;\n9 while τ ≤T do\n10\nd ←calcDistToSphereCenters(c, r(τ))\n11\nfor i = 1 to |S| do\n12\nif di < Ri & di ≤dmin then\n13\ndmin ←di;\n14\nk ←i;\n15\nend\n16\nend\n17\nif dmin < ∞then\n// update sample time\n18\nτ ←τ + (Rk −dk)/vmax;\n19\nelse\n// point outside spheres, use k-d tree\n20\ncollision, S ←collisionCheckMap(r(t), T );\n21\nend\n22 end\n23 return collision\nthe environment. For the first candidate motion primitive\nconnecting two successive waypoints, we use the strategy from\n[13] to intelligently sample for collisions while also storing\nthe resulting set of safe, obstacle-free spheres S, defined by\ncenter and radius vectors, c and R (line 4). For subsequent\nmotion primitives between the same waypoint pair, a nearest\nneighbor query is only done if the primitive is expected\nto leave the initial set of obstacle-free spheres. For a point\nfound to be within a certain sphere (lines 12-15), the next\npossible “time-of-collision” is when the trajectory intersects\nthe edge of the sphere, which can estimated by assuming the\ntrajectory is emanating radially from the center of the sphere\nat maximum velocity (line 18). The process is repeated until\nthe final time horizon T is reached. Unlike spherical safety\ncorridors, our safe set is only used as a means to avoid repeated\ncalculation, and allows for on-the-fly addition of collision-\nfree spheres. In other words, our approach does not restrict\nsolutions to remain within convex sets centered along the\ngeometric path. STITCHER thus has the flexibility to create\nand check candidate trajectories without being restricted to\npre-defined safety spheres.\nE. Motion Primitive Search Graph with Triple Integrator\nIn many applications, a triple integrator model for gen-\nerating motion primitives is sufficient because the resulting\ntrajectory is smooth enough for most aerial vehicles to track\nas discontinuities in jerk typically do not severely degrade\ntracking. This was verified through hardware experiments\ndiscussed in Section VII. Motion primitives in our formulation\n(3) are derived imposing a free terminal acceleration.\nConstructing a motion primitive search graph where nodes\nare a waypoint-velocity-acceleration tuple drastically increases\nboth computation and memory consumption as the graph\n\nFig. 5: Difference in motion primitive graphs with free terminal accelerations employing an (a): exhaustive search, (b): greedy search, and (c): greedy graph\npre-processing step. Green edges constitute the optimal path, blue edges are the least cost parent saved in memory, grey edges are those evaluated but not\nsaved and red dashed edges are connections that no longer exist. Greedy pre-processing (c) leads to some disconnections for graphs with over 3 waypoints,\nbut maintains more connections than a greedy search (b), and does not suffer from exponential edge growth like the graph of an exhaustive search (a).\nsize depends on both the number of sampled velocities and\naccelerations. If the acceleration at each node, i.e., the fi-\nnal acceleration, af, is free, the number of edges grows\nexponentially with respect to the number of waypoints (See\nFig. 5a). Our formulation employs a greedy pre-processing\nstep in which the motion primitive search graph Gmp is\nidentical in size to the velocity graph G (graph size detailed in\nSection V-A). This formulation offers an advantage in terms of\ncomputational efficiency, as a full state trajectory is generated\nwhile the graph size is restricted by only the number of\nsampled velocities. Excluding acceleration information from\ngraph creation assumes that the optimal stitched trajectory is\nonly weakly dependent on acceleration at each waypoint. It is\nimportant to note the difference between a greedy algorithm\nFig. 5b and our greedy graph pre-processing step Fig. 5c.\nWhile the pre-processing step does limit edges that could be\ngenerated in the exhaustive graph, it maintains more than a\ngreedy algorithm. Further comparisons of the relative solution\ncost and computation speed using STITCHER’s greedy pre-\nprocessing step versus the exhaustive search were conducted\nin Section VI-B.\nV. THEORETICAL ANALYSIS\nIn this section we prove STITCHER has bounded time\nand memory complexity by showing the velocity and motion\nprimitive graphs are finite. We also show STITCHER is\ncomplete and optimal by proving the heuristic used in the\nmotion primitive search is admissible.\nA. Velocity Graph Complexity\nThe following proposition proves the size of the velocity\ngraph G is finite and solely depends on the number of\nwaypoints and sampled velocities; a property that also holds\nfor the motion primitive graph Gmp by extension. This result is\ncritical as a finite graph yields known time complexity for the\nmotion primitive search. In other words, an upper bound can\nbe placed on the computation time of the planner given known\nquantities. This is in contrast to optimization-based methods\nwhere the time complexity depends on the number of iterations\nrequired to converge—which cannot be known a priori—so\nthe time to compute a trajectory via optimization does not\nhave an a priori bound.\nProposition 1. For N waypoints and M sampled velocities,\nthe number of nodes |N| and edges |E| in graph G is\n|N| = (N −2)M + 2,\n(6)\n|E| = (N −3)M 2 + 2M\nfor N > 2.\n(7)\nProof. Using Fig. 2 (middle), the start and goal nodes con-\ntribute 2 nodes to the graph G. For intermediate waypoints,\ngiven M sampled velocities, there are M nodes per waypoint.\nAs a result, |N| = (N −2)M +2 which is (6). For each edge,\nwe consider the transition to successive waypoints. Ignoring\nthe trivial N = 2 case where |E| = 1, there are M connections\nbetween the start node and next waypoint, which also has M\nnodes. The same applies for connecting waypoint wN−1 to\nthe goal node, resulting in a total of 2M edges. For all other\nintermediate waypoint pairs, M nodes connect to M nodes at\nthe next waypoint so there are M 2 edges. The total number\nof edges is then (7).\nCorollary 1. The size of the motion primitive graph Gmp using\nLinear Quadratic Minimum Time (LQMT) motion primitives\nwith free terminal acceleration for a triple integrator is\nidentical to the velocity graph G.\nProof. The proof is immediate since the terminal acceleration\nis free so N and M are identical for both graphs.\nRemark 1. Corollary 1 can be generalized to any motion\nprimitive search graph where the primitives are solutions to\nan optimal control problem with the dynamics being a chain of\nintegrators and all terminal state derivatives of second order\nor higher are free. Note that this assumes the greedy graph\npre-processing step still yields adequate performance.\nB. Admissible Heuristic for Motion Primitive Search\nIt is well known that heuristics can be used to expedite\nsearching through a graph by incentivizing the search to\nprioritize exploring promising nodes. For example, in A*\nsearch, the next node explored is selected based on minimizing\nthe cost f(n) = g(n) + h(n), where g is the stage cost to get\n\nfrom the start node ns to node n, and h is a heuristic estimate\nof the remaining cost to reach the goal node ng. A* search is\nguaranteed to find an optimal solution so long as the heuristic\nfunction h is admissible (see Definition 1) [36]. Below, we\nprove the cost-to-go V ∗for each node in the velocity graph\nG calculated in Stage 2, i.e., the minimum-time to goal for a\ndouble integrator, is an admissible heuristic for an A* search\nover motion primitives of any higher-order chain of integrators.\nDefinition 1 ([36]). A function h : N →R is an admissible\nheuristic if for all n ∈N then h(n) ≤h∗(n), where h∗is the\noptimal cost from n to the goal node ng.\nProposition 2. Consider the optimal control problem\nmin\nT, u\nJ = ρ T +\nZ T\n0\nq(r, v, . . . , u) dt\n(8)\ns.t.\nr(p) = u, c(a) ≤0\nr(0) = r0, v(0) = v0, . . . , r(p−1)(0) = r(p−1)\n0\nr(T) = rf, v(T) = vf, r(k−1)(T) free for 3 ≤k ≤p\nwhere q is a positive definite function, the system is at least\nsecond order (p ≥2), and the position and velocities boundary\nconditions are identical to those of (2), with all other boundary\nconstraints free to specify. If umax in (2) is the maximum\npossible acceleration achievable in a given axis imposed by\nc(a) ≤0, then the optimal cost-to-go V ∗from the initial\nconditions for (8) satisfies V ∗≥ρ T ∗\nd where T ∗\nd is the optimal\nfinal time for (2).\nProof. First, consider the case when p = 2. For a given axis,\nif umax is chosen so that it exceeds the allowable acceleration\nimposed by c(a) ≤0, e.g., ux,max ≥maxax c(a) (see Fig. 3),\nthen the optimal final time T ∗for (8) will always be greater\nthan that of (2) even when q = 0. Specifically, when q = 0,\none can show the optimal final time for (2) increases as umax\ndecreases. Moreover, T ∗\nd for (2) is guaranteed to exist and be\nunique [50]. Hence, by appropriate selection of umax, we can\nensure T ∗≥T ∗\nd always, where equality holds when q = 0 and\nc(a) is a box constraint. If q ̸= 0, then it immediately follows\nthat T ∗> T ∗\nd because q is positive definite by construction.\nNow consider the case when p > 2. We can deduce V ∗> ρ T ∗\nd\nby contradiction. Specifically, assume T ∗= T ∗\nd for p > 2.\nThis would require a to be discontinuous in order to match\nthe bang-bang acceleration profile of (2). However, (8) is a\ncontinuous-time linear system that will not exhibit discrete\nbehaviors, e.g., jumps, so it is mathematically impossible to\ngenerate an optimal control sequence where the acceleration\nprofiles for Eqs. (2) and (8) will be identical. It can then be\nconcluded V ∗> ρ T ∗\nd for p > 2. Therefore, V ∗≥ρ T ∗\nd for\np ≥2, as desired.\nRemark 2. Proposition 2 also holds when inequality state or\nactuator constraints in (8) are present, and when the terminal\ndesired states are specified rather than free.\nThe main result of this section can now be stated.\nTheorem 1. The optimal cost-to-go for the minimum-time\ninput-constrained double integrator optimal control problem\n(2) is an admissible heuristic for motion primitive search\nFig. 6: Simulation test environments. (a): Willow Garage environment. (b):\nPerlin Noise environment.\nwhere the primitives are solutions to the optimal control\nproblem of the form (8).\nProof. Let G = (N, E) be a graph with nodes being sampled\nvelocities at waypoints and edges being the time-optimal\ntrajectories using an input-constrained double integrator. Fur-\nther, let Gmp = (Nmp, Emp) be a graph with nodes being\nsampled velocities, accelerations, etc. at waypoints and edges\nbeing trajectories that are solutions to (8). Using the Bellman\nequation, the optimal cost-to-go V ∗\nmp(n) for any n ∈Nmp\ncan be computed recursively. Using Proposition 2, V ∗\nmp(n) ≥\nV ∗\nd (n′) by induction where V ∗\nd is the optimal cost-to-go for\nthe minimum-time input-constrained double integrator with\nn′ ∈N. Recognizing N ⊆Nmp, V ∗\nd (n′) can be rewritten\nas V ∗\nd (n). Setting h∗(n) = V ∗\nmp(n) and h(n) = V ∗\nd (n), it can\nbe concluded h(n) ≤h∗(n). Therefore, by Definition 1, the\noptimal cost-to-go computed for G is an admissible heuristic\nfor the motion primitive search over Gmp.\nThe importance of Theorem 1 follows from the well-known\nresult that searching a graph with an admissible heuristic is\nguaranteed to return the optimal path through the graph [36],\nand can significantly improve search efficiency because not\nevery node in the graph has to be explored. The effectiveness\nof the proposed heuristic both in terms of path quality and\nsearch times is analyzed in Section VI.\nVI. SIMULATION RESULTS\nThis section contains an analysis of STITCHER. First, we\nconduct a parameter sensitivity study to determine a suitable\nsampled velocity set (direction and speed) using a modified\nversion of Dijkstra’s algorithm that has access to a dense set\nof velocities and no constraints on computation time. Second,\nSTITCHER is compared to a non-greedy algorithm variant to\ncharacterize the impact of our greedy graph pre-processing\nstep on solution cost and computation. Third, we investigate\nthe effectiveness of the heuristic proposed in Section V-B\nin reducing the number of edges generated by STITCHER.\nFourth, a study of STITCHER constraining individual mo-\ntor forces is presented to demonstrate the flexibility of our\nmethod in adhering to complex actuator constraints. Fifth,\nwe characterize the average computation time of the different\ncomponents that make up STITCHER. Lastly, STITCHER is\ncompared to two optimization-based modern planners [5], [9]\ncapable of running in real-time.\nSimulation experiments were run in a Perlin Noise environ-\nment and the Willow Garage environment, both with a volume\n\nFig. 7: Velocity directions sampled at waypoint wi, where ˆai is normal to\nthe separating hyperplane Hi and ˆqi is the heading toward wi+1.\nTABLE I: Frequency of Velocity Directions in Final Trajectory.\nZenith Angle\n70◦\n80◦\n90◦\n100◦\n110◦\nFrequency\n0%\n13%\n79%\n8%\n0%\nAzimuth Angle\n-50◦\n-20◦\n-10◦\n0◦\n10◦\n20◦\nFrequency\n4%\n4%\n17%\n42%\n29%\n4%\nof approximately 50 × 50 × 5 m (see Fig. 6). Geometric paths\nwith N = 4, 6, 8 waypoints were found for different start\nand end locations in each environment. For all experiments,\nwe imposed the following constraints assuming agile drone\nflight and reasonable thrust limits: fmin = 0.85 m/s2, fmax =\n18.75 m/s2, θmax = 60◦, ωmax = 6 rad/s, vmax = 10 m/s,\nand a time penalty ρ = 1000. All reported times are from\ntests conducted on an 11th generation Intel i7 CPU.\nA. Parameter Sensitivity Analysis: Sampled Velocity Set\nSTITCHER requires a discrete velocity set V composed of a\nset of magnitudes and directions. As shown in Section IV-B,\nthe size of the velocity graph scales quadratically with the\nnumber of sampled velocities, so it is desirable to choose V\nto be small without degrading path quality. Hence, this section\nfocuses on constructing a velocity sample set V that optimizes\nthe trade-off between computation time and path quality as\nmeasured by execution time. We conducted two studies to\ndetermine i) the predominate sampled velocity directions used\nby an offline version of Dijkstra’s algorithm and ii) the trade-\noff between path cost and the number of sampled speeds. In\nthe following comparison, Dijkstra’s algorithm was modified\nto search over a dense set of velocities with 3611 terminal\nvelocities sampled per waypoint; the method is referred to as\nDense Dijkstra. Dijkstra’s algorithm is a complete and optimal\nsearch algorithm with respect to the set of actions [36], making\nit a suitable benchmark.\n1) Sampled Direction: Dense Dijkstra was used to statisti-\ncally identify velocity directions set Vd commonly employed\nin a variety of test cases. Using Fig. 7, we define angles\nwith respect to the hyperplane H with normal vector ˆa at\nthe plane of symmetry between path segments connecting two\nwaypoints. The search was given velocity direction zenith and\nazimuth angles in the range [0◦, 180◦] and [-90◦, 90◦] sampled\nin 10◦increments.\nTable I shows the frequency of velocity directions chosen\nby Dense Dijkstra across all motion primitives for six different\npath length trials in the Perlin Noise and Willow Garage\nenvironment. The velocities chosen by Dense Dijkstra align\nwith the normal vector ˆa of the hyperplane 80% of the time.\nFig. 8: Analysis of speed discretization on execution and planning time. As the\ndiscretization of our method is increased, we converge to the Dense Dijkstra\nsolution. We use Vm = 5 for our experiments as it achieves significant\ncomputational advantages while retaining suitable performance.\nFig. 9: Speed profiles of STITCHER using varied speed discretizations. As\nthe number of speed samples increases, STITCHER converges to the optimal\nsolution approximated by Dense Dijkstra.\nFrom these results, sampling at the center and boundaries of\na 20◦cone centered around a given ˆa will yield a suitable set\nof velocity directions.\n2) Sampled Speed: Using the velocity direction set Vd\nidentified in the previous test, the performance of Dense\nDijkstra was compared to STITCHER with different sampled\nspeed sets Vm. Each set Vm consists of k discrete speeds\nsampled from the interval [0, vmax], such that |Vm| = k. In\norder to ensure a fair comparison, these sets must be subsets\nof those used by Dense Dijkstra. In other words, we ensure\nk ≤K, where K denotes the number of speeds sampled in\nDense Dijkstra, and our sampled sets are as evenly distributed\nas possible within the interval.\nFigure 8 shows the relative execution time (left) and plan-\nning times (right) for different sizes of Vm compared to\nDense Dijkstra. As expected, the execution time increases as\nthe sampled speed set becomes sparser because the planner\nhas fewer speed options available. However, the observed\nincrease in execution time is at most 8% even with the\nsparsest speed set tested (|Vm| = 5). Critically, the minimal\nincrease in execution time is accompanied by a significant\nreduction in planning time: a speed-up of four orders of\nmagnitude with the sparsest sampled speed set tested. Al-\nthough a sample speed set size of |Vm| = 11 yields nearly\nidentical execution times to the dense set while offering three\norders of magnitude improvement in planning time, adequate\nperformance is achieved when |Vm| = 5. Hence, we use the\nset Vm = {0, 0.25 vmax, 0.5 vmax, 0.75 vmax, vmax} for the\nremainder of our analysis. A representative speed profile for\ndifferent sized speed sets is shown in Fig. 9.\n\nFig. 10: Evaluation of the effect of a greedy pre-processing step on solution\nquality and computation speed. Results are from a monte carlo simulation of\n50 random realizations for trajectories with between 2 and 10 waypoints.\nFig. 11: The final speed profile given by a motion primitive search with (a)\nan exhaustive exponentially growing graph and (b) STITCHER.\nB. Greediness Benchmarking\nSTITCHER employs a greedy graph pre-processing step in\norder to generate a compact final search graph which enables\nreal-time trajectory computation. It is important to note that\nthis greedy pre-processing step results in a graph that does\nnot exhaustively account for all possible acceleration states\nat a given waypoint (see Fig. 5). Hence, we evaluated the\nextent to which this greedy pre-processing step leads to sub-\noptimal solutions by performing a Monte Carlo simulation\ncomparing the path cost of STITCHER to that of an exhaus-\ntive exponentially-growing graph. The experiment includes 50\nrealizations of randomly generated initial and final positions,\nwith the number of waypoints ranging between 2 and 10.\nFigure 10 shows the relative solution cost (left) and\nplanning times (right) resulting from searching over the\nexponentially-growing graph and running STITCHER. On\naverage, STITCHER achieved a 0.5% difference in solution\ncost, with the maximum difference being 7%. Additionally,\nSTITCHER achieves approximately 66% speed-up in compu-\ntation time as a result of the reduced graph size. Excluding\nsolution cases with 2 or 3 waypoints where the exponential\ngrowth of the graph including terminal acceleration states\ncannot be observed, STITCHER achieves an average of 82%\nspeed-up. Figure 11 shows the speed profile of a case in\nwhich STITCHER achieved a solution cost that was 2%\ndifferent from that of the exponential graph. Qualitatively, the\ntrajectories are very similar with only a minor deviation at\nthe end of the trajectory. Therefore, the greedy graph pre-\nprocessing step results in a negligible increase in solution cost,\nbut a substantial improvement in computation time.\nFig. 12: Final trajectories (red) through a six waypoint path in (a): Perlin\nNoise and (b): Willow Garage environment. The trajectory options (white)\nwhich inform the heuristic are more likely to be in collision in the Willow\nGarage environment due to tight corridors.\nC. Heuristic Benchmarking\nThe quality of the heuristic used to guide the motion prim-\nitive search phase of STITCHER can be quantified by com-\nparing the number of edges, i.e., motion primitives, generated\nby STITCHER with that of Dijkstra’s algorithm. The velocity\nmagnitude and direction sets were kept constant across both\nplanners with |Vm| = 11 and |Vd| = 3. The number of edges\ncreated is a better evaluation metric than the nodes explored\nbecause the main source of computation time comes from\ngenerating motion primitives and checking them for constraint\nviolations. The effectiveness of the search heuristic depends\non two main factors: how accurately it approximates the true\ncost-to-go, and whether the heuristic is admissible under the\nspecified constraints. The edge cost used in the final search\naffects the tightness of the heuristic approximation, while the\nacceleration constraint imposed to prune motion primitives can\ninfluence admissibility. Therefore, we assess the effectiveness\nof STITCHER’s heuristic with various graph edge costs and\nacceleration constraints to provide a general understanding of\nhow search performance is impacted.\n1) Varying Edge Cost: We evaluate the quality of the\nheuristic with two different edge costs: (1) equivalent to the\nLQMT performance index and (2) execution time. Table II\nshows the percent reduction in the number of edges created\nfor STITCHER compared to Dijkstra’s algorithm. Using the\nLQMT edge cost in the final search, STITCHER creates 7%\nfewer edges on average compared to Dijkstra’s algorithm in\nthe Perlin Noise environment and 5% in the Willow Garage\nenvironment. Greater edge reduction can be achieved by\ndefining the edge cost to be the trajectory execution time.\nRecall that the search heuristic is the minimum time required\nto reach the goal using a double integrator model. Therefore,\nby using an edge cost soley defined by trajectory duration, the\nheuristic now more closely approximates the cost-to-go. This\ntighter approximation is reflected in the results of Table II,\nwhere STITCHER generates an average of 20% fewer edges\nin Perlin Noise and 13% fewer in Willow Garage. The reduced\neffectiveness of the heuristic in Willow Garage was attributed\nto having more tight corridors, so more motion primitives\nwere in collision (see Fig. 12). Therefore, while environment\ndependent, using a search heuristic that closely approximates\nthe remaining cost to the goal greatly reduces search effort.\n2) Varying Acceleration Constraint: Varying acceleration\nconstraints may also lead to differing performance of the\n\nTABLE II: Heuristic Evaluation with Varied Edge Costs and Acceleration Constraints via Percent Reduction in Generated Edges Compared to Dijkstra’s.\nEdge\nCost\nAcceleration\nConstraint\nMap\nPerlin Noise\nWillow Garage\nN = 4\nN = 6\nN = 8\nN = 4\nN = 6\nN = 8\n% Red.\n% Red.\n% Red.\n% Red.\n% Red.\n% Red.\nLQMT\nAdmissible Truncated Cone\n8\n8\n7\n0\n3\n13\nTime\nAdmissible Truncated Cone\n22\n21\n15\n0.3\n13\n26\nTime\nAdmissible Box\n27\n34\n28\n2\n20\n24\nTime\nInadmissible Truncated Cone\n45\n44\n22\n9\n38\n39\nFig. 13: Different acceleration constraints used for heuristic study: (a) admis-\nsible truncated cone, (b) admissible box, and (c) inadmissible truncated cone.\nGreen dashed outlines are Stage 2 constraints used for heuristic creation while\nsolid black outlines denote constraints imposed for the final trajectory.\nheuristic as this constraint directly affects admissibility. We\ntested three different acceleration constraints: (1) admissible\ntruncated cone, (2) admissible box, and (3) inadmissible trun-\ncated cone (See Fig. 13). The truncated cone constraint comes\nfrom the resulting achievable mass-normalized thrust volume\nwith constraints on the magnitude and tilt. This constraint c(a)\nis admissible when it is maximally bounded by the maximum\ncontrol input umax imposed during heuristic generation in\nStage 2 (see Fig. 13a), while the inadmissible variant applies\na smaller umax, e.g., ux,max ≤maxax c(a) (see Fig. 13c).\nAn admissible box constraint is an acceleration constraint that\nperfectly matches that of Stage 2 (see Fig. 13b). Table II\nshows that improved performance can be achieved by applying\nan admissible box constraint on acceleration compared to the\nadmissible truncated cone. In this case, STITCHER achieves\napproximately a 30% reduction in edges in the Perlin Noise\nenvironment and a 15% reduction in the Willow Garage\ncompared to Dijkstra’s algorithm. Because the box constraint\non acceleration is equivalent to that used in the heuristic\ngeneration, the STITCHER algorithm guides the search away\nfrom branches that are likely to exceed acceleration con-\nstraints. Further improvement in search speed may be observed\nwhile using an inadmissible truncated cone constraint, and is\nreflected in Table II, where this algorithm variant achieves the\nhighest percent reduction among all STITCHER variations.\nBy allowing the final motion primitive search to have a\nlarger allowance in achievable acceleration, the true edge\ncost may be smaller than that estimated by the heuristic.\nThe use of inadmissible heuristics in graph search foregos\ndesirable optimality guarantees, but can more rapidly motivate\nthe search toward the goal as a larger weighting is placed on\nthe heuristic cost.\n3) Summary: This study showed that the proposed heuristic\nis effective, but its performance depends on the environment,\ngraph edge cost, and the chosen acceleration constraint. For\nthe remaining experiments, we used an LQMT edge cost\nto ensure fair comparisons to state-of-the-art algorithms, and\napply an admissible truncated cone acceleration constraint to\nretain graph search optimality guarantees and accurately reflect\nthe true physical limits of quadrotor systems.\nD. Constrained Motor Forces\nWe present a case study demonstrating STITCHER’s ability\nto generate trajectories that satisfy individual thruster limits for\na quadrotor, which also extends to lander vehicles. We assume\nthe vehicle dynamics takes the form\nm¨r = R T −mg\nJ ˙ω = τ −ω × Jω,\nwhere r is the vehicle’s position vector, R is the rotation\nmatrix that represents the vehicle orientation with respect to an\ninertial frame, T = T ˆez is the thrust vector directed along the\nbody z-axis, g is the gravity vector, ω is the angular velocity\nvector, J is the inertia tensor, and τ is the body torque vector.\nFor the case study, we assume a quadrotor with a mass m =\n0.5 kg, moment of inertia J = diag(0.01, 0.01, 0.01) kg m2,\narm length l = 12.5 cm, motor drag coefficient c = 0.2 m,\nminimum single-motor thrust Fmin = 0.15 N, and maximum\nsingle-motor thrust Fmax = 3 N. Each motor thrust Fi is\nconstrained by\n0 < Fmin ≤Fi ≤Fmax,\n∀i ∈{1, 2, 3, 4}.\nAt any given time, individual motor thrusts Fi are found by\nsolving a linear system that depends on motor placement and\nphysical vehicle properties, i.e., solving\n\n\nT\nτx\nτy\nτz\n\n= M\n\n\nF1\nF2\nF3\nF4\n\n,\nfor some invertible matrix M ∈R4×4 with T denoting the\ntotal thrust T = m∥f∥2. The torque τ = [τx, τy, τz]⊤along\nthe trajectory is τ = J ˙ω + ω × Jω and can be found via\ndifferential flatness, with components of ω and ˙ω given by\n\n\nωy\n−ωx\n0\n\n= R⊤˙ˆ\nf,\n\n\n˙ωy\n−˙ωx\n0\n\n= R⊤¨ˆ\nf −ω × R⊤˙ˆ\nf,\nwhere ˆ\nf is the normalized thrust vector.\nWe conducted two experiments in the Willow Garage envi-\nronment, where the tight corridors may make individual motor\nconstraint satisfaction more difficult. As shown in Fig. 14, the\ntrajectories achieve high speeds while strictly obeying the per-\nmotor limits on thrust. These results showcase the versatility\n\n(a) Individual Motor Constraint Case Study 1.\n(b) Individual Motor Constraint Case Study 2.\nFig. 14: Case studies in the Willow Garage environment of trajectory plans constraining individual motors by the maximum thrust limits given by manufacturer\nspecifications. Left: Trajectory colored by speed. Right: The corresponding thrust profile of individual motors depicting constraint satisfaction.\nof our framework in adhering to highly nonconvex constraints\nthat are directly limited by hardware.\nE. Planning Time Composition Analysis\nFigure 15 shows the computation time of the different\ncomponents of STITCHER averaged over six tested trials.\nThe average time to perform the velocity graph search is\n1.8 ms, compared to 2.2 ms for the motion primitive search.\nAlthough both searches have the same graph size, it is\nimportant to note that the motion primitives from (3) are\nmore time-consuming to generate than the minimum time\ntrajectories used in the velocity graph. Therefore, the similar\ncomputation times arise from the search heuristic reducing the\nnumber of edges generated. The low computation time of the\nvelocity search further indicates its effectiveness in computing\nan informative admissible heuristic for the motion primitive\nsearch. Finally, constraint-checking with uniform samples at\nevery 0.1 s averaged only 0.3 ms, showing the efficacy of the\nrelatively simple approach.\nF. Comparison with State-of-the-Art\nWe compared STITCHER with two state-of-the-art algo-\nrithms: GCOPTER [5] and FASTER [9]. GCOPTER performs\nan online optimization by incorporating state constraints into\nFig. 15: The average contribution of different path planning components.\nthe cost functional and running a quasi-newton method, while\nFASTER solves a mixed integer quadratic program online.\nBoth algorithms rely on a sparse geometric path to form\nsafe flight corridors, but do not enforce final trajectories to\npass through waypoints. We evaluate each planner by time\n(planning time versus execution time) and failure (constraint\nviolation or incomplete/no path found). For the Perlin Noise\nenvironment, the path lengths were 12.5 m, 30 m, and 55 m\nwhile the path lengths for Willow Garage were 20 m, 25 m,\nand 30 m with N = 4, 6, 8 waypoints for both environments.\n1) Time Analysis: Table III compares the planning times\nand trajectory execution times of each planner given the six\ndifferent geometric paths. STITCHER’s planning times are\n\nTABLE III: State-of-the-Art Comparison Time Analysis.\nMap\nN\nPlanning time (ms)\nExecution time (s)\n[9]\n[5]\nOurs\n[9]\n[5]\nOurs\nPerlin\nNoise\n4\n109\n21.1\n3.82\n2.98\n3.51\n3.67\n6\n207\n59.4\n10.4\n4.42\n4.42\n5.75\n8\n1020\n116\n16.9\n7.62\n7.34\n9.62\nWillow\nGarage\n4\n240\n38.2\n8.27\n4.40\n4.37\n4.53\n6\n5240\n53.2\n20.9\n8.25\n5.97\n6.30\n8\n10400\n84.4\n27.1\n10.5\nFAILED\n7.95\nTABLE IV: Effect of Different Waypoints and Sampling in Perlin Noise.\nN\nPlanning time (ms)\nExecution time (s)\nSTITCHER\nSTITCHER\n[5]\nS\nW\nSW\n[5]\nS\nW\nSW\n4\n27.2\n14.8\n3.01\n10.5\n3.51\n3.44\n3.61\n3.56\n6\n72.0\n44.7\n13.8\n42.4\n4.42\n5.43\n5.38\n4.90\n8\n130\n64.1\n24.6\n86.9\n7.34\n9.13\n8.99\n9.26\nAbbreviations: S – speed sample set expanded to |Vm| = 9; W – optimized\nwaypoints from [5] used; SW – combination of S and W.\nTABLE V: State-of-the-Art Comparison Failure Analysis.\nMap\nNo Path Found\n(%)\nState Violation\n(%)\nCollisions\n(%)\n[9]\n[5]\nOurs\n[9]\n[5]\nOurs\n[9]\n[5]\nOurs\nPerlin\nNoise\n0\n0\n0\n6\n0\n0\n4\n4\n0\nWillow\nGarage\n18\n2\n2\n8\n0\n0\n0\n24\n0\nfaster than those measured for FASTER and GCOPTER for\nevery test, and are up to 7x faster than GCOPTER and 400x\nfaster than FASTER. It is important to note the computation\ntimes for the motion primitive search planner [16] were omit-\nted from Table III because planning times exceeded several\nseconds. In some cases GCOPTER and FASTER achieved\nlower execution times. This can be attributed to their ability to\ntreat waypoints as soft constraints, i.e., the trajectory is only\nrequired to pass nearby a waypoint rather than through it, as\nwell as the chosen resolution of state samples in STITCHER.\nThe difference is most noticeable in the Perlin Noise en-\nvironment where more free space is available for waypoint\nadjustment. Table IV compares the planning and execution\ntimes of GCOPTER [5] to different algorithm variants of\nSTITCHER evaluating the effect of sampling and waypoint\nselection in the Perlin Noise environment. Case S increases\nthe sampled speed set to |Vm| = 9, Case W utilizes the\noptimized waypoints from GCOPTER, and Case SW is a\ncombination of both strategies. All the case variants reduce\nSTITCHER’s baseline execution time, motivating future work\nin incorporating waypoint flexibility.\n2) Failure Analysis: A Monte Carlo simulation composed\nof 50 realizations was conducted to evaluate the different\nmodes of failure experienced by each planner. Table V com-\npares the rate at which each planner does not find a path,\ngenerates a trajectory violating state constraints or gener-\nates a trajectory in collision. The “No Path Found” metric\nincludes a numerical solver not returning a solution, or if\nthe solution does not reach the goal. Using either FASTER\nor GCOPTER, this can occur due to numerical instability\nor when a feasible solution is not within the calculated set\nof safe corridors. Conversely, in STITCHER’s graph search\nframework, a graph disconnection occurs when a feasible\nFig. 16: Example of mass-normalized thrust profile strictly satisfying con-\nstraints in Willow Garage environment.\nFig. 17: Control and estimation architecture for custom quadrotor platform\nused in hardware experiments. The vehicle is equipped with an OrangePi 3.0\ncomputer, Teensy 4.0 microcontroller, and an InvenSense ICM-20948 IMU.\nAn external motion capture system provides pose measurements at 100Hz.\nsolution is not within the discrete set of sampled states.\nAcross all test cases, STITCHER’s motion primitive graph\ndisconnects only once, achieving the lowest rate of failure\namong the state-of-the-art planners. In the Willow Garage\nenvironment, where narrow corridors make collisions more\nlikey, the number of failed solutions by FASTER and collisions\nby GCOPTER significantly increases. In contrast, STITCHER\nnever violates constraints (state, control, or obstacles) because\nall constraints are strictly enforced. As an example, Fig. 16 is\na representative mass-normalized thrust profile generated by\nSTITCHER which remains within the valid limits.\nVII. HARDWARE RESULTS\nThe custom quadrotor used for hardware experiments is\nshown in Figure 17. Onboard hardware includes an Orange Pi\n3.0 computer, Teensy 4.0 microcontroller, and an InvenSense\nICM-20948 IMU. The system was flown indoors in a 4m x 4m\nx 4m drone cage equipped with an OptiTrack motion capture\nsystem that accurately measures the pose of the vehicle.\nA. Control and Estimation Architecture\nThe control and state estimation architecture used in the\nhardware experiments is shown in Fig. 17. We employed the\nstandard cascaded geometric controller [51], [52] to track the\ntrajectory generated by STITCHER with the desired acceler-\nation and jerk as feedforward. The outer loop PID position\ntracking controller ran on the Orange Pi flight computer, and\nthe inner loop quaternion-based attitude tracking controller\nfrom [53] ran on the Teensy microcontroller; the two pro-\ncessors communicate via serial. The outer loop control rate\nwas 100Hz while the inner loop control rate was 500Hz. For\nestimation, position and orientation measurements from an\n\nFig. 18: Motion trail of quadrotor executing trajectory from Experiment 1 in\nthe virtual Random Forest 1 environment through 4x4x4 meter drone cage.\nTABLE VI: Trajectory Tracking Error.\nRMSE\nExperiment\nRandom\nForest 1\nRandom\nForest 2\nObstacle\nCourse\nWindowed\nWall\nPos. (cm)\n9.86\n13.3\n10.9\n8.59\nTilt (deg)\n4.98\n6.01\n6.28\n6.41\nexternal motion capture system were sent wirelessly at 100Hz\nto the onboard Orange Pi computer, and fused with an IMU\nto estimate position, velocity, orientation, angular velocity,\nand IMU biases with a nonlinear geometric observer [54].\nEstimates were generated by the observer at 100Hz and used\nby the outer loop position controller.\nB. Flight Experiments\nIndoor flight experiments were conducted to evaluate the dy-\nnamic feasibility of the trajectories planned with STITCHER.\nThe trajectories were first generated with virtual obstacles\nand then flown using our custom quadrotor (see Fig. 18).\nIn Fig. 19, planned trajectories generated by STITCHER\nare displayed alongside the actual trajectories flown by the\nquadrotor in four experiments. The environments are named:\nRandom Forest 1 (Fig. 19a), Random Forest 2 (Fig. 19b),\nObstacle Course (Fig. 19c), and Windowed Wall (Fig. 19d).\nThe desired path lengths for each test in their respective\nenvironment were 9.6 m, 13.1 m, 8.7 m, and 8.5 m.\nTable VI reports the root mean square errors (RMSE) of\nposition and tilt for each experiment. Across all experiments,\nthe maximum RMSE is 6.41 degrees in tilt and 13.3 cm in\nposition. Further, the average ratio between position RMSE to\nthe total path length was only 1.07% over all the tests. These\nflight experiments show that STITCHER generates trajectories\nthat satisfy constraints and can be tracked with low error\nusing a standard cascaded geometric controller. Additionally,\nwhile STITCHER may use any p−th order integrator with\np ≥2 to model system dynamics, the results show that\nusing a triple integrator model is sufficient for quadrotor\ntracking control despite discontinuities in jerk at waypoints.\nFigure 20 compares the actual and desired profiles of the\nmass-normalized thrust (top), position (middle), and attitude\n(bottom) through Random Forest 1. The quadrotor remains\nwithin the physical limits dictated by the thrust magnitude and\ntilt constraints, and the motor commands executed by the drone\nclosely match the desired mass-normalized thrust. The position\nand attitude error profiles further show that for the duration\nof the flight, the maximum deviations in position and tilt are\nless than 20 cm and 20 degrees, respectively. These results\ndemonstrate that STITCHER generates trajectories that safely\nand accurately express complex system-level constraints.\nVIII. CONCLUSIONS\nIn this work, we presented STITCHER, a motion primitive\nsearch planning algorithm that utilizes a novel three-stage\nplanning architecture to design constrained trajectories in real-\ntime over long distances. We proved the search graph is finite,\nand the proposed search heuristic is admissible, so STITCHER\nis guaranteed to i) have a priori bounded time and memory\ncomplexity and ii) generate optimal trajectories with respect\nto the sampled set of states. Real-time search speeds were\nachieved through our novel heuristic crafting technique, greedy\ngraph pre-processing method, and non-conservative constraint\nand collision checking procedure. Our extensive simulation\nstudy showed the trade-off in terms of path quality and compu-\ntation time for different sampled velocity sets, the effectiveness\nof the proposed heuristic with varied edge costs and state\nconstraints, a case study imposing individual thruster limits,\nand the average computation times of the components that\nmake up STITCHER. We also found that our greedy motion\nprimitive graph pre-processing step has a negligible effect on\nsolution cost compared to the observed computation speed\nup owing to the reduced graph size. Importantly, STITCHER\nwas shown to consistently generate trajectories faster than\ntwo state-of-the-art optimization-based planners while never\nviolating constraints. Hardware experiments further proved\nthat our planner is effective in generating trajectories suitable\nfor position and attitude tracking control while remaining\nwithin set physical limits. 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Modern trajectory planning techniques primarily use numerical optimization, as they enable the systematic computation of high-quality, expressive trajectories that satisfy various constraints. However, stringent requirements on computation time and the risk of numerical instability can limit the use of optimization-based planners in safety-critical scenarios. This work presents an optimizationfree planning framework called STITCHER that stitches short trajectory segments together with graph search to compute longrange, expressive, and near-optimal trajectories in real-time. STITCHER outperforms modern optimization-based planners through our innovative planning architecture and several algorithmic developments that make real-time planning possible. Extensive simulation testing is performed to analyze the algorithmic components that make up STITCHER, along with a thorough comparison with two state-of-the-art optimization planners. Simulation tests show that safe trajectories can be created within a few milliseconds for paths that span the entirety of two 50 m x 50 m environments. Hardware tests with a custom quadrotor verify that STITCHER can produce trackable paths in real-time while respecting nonconvex constraints, such as limits on tilt angle and motor forces, which are otherwise hard to include in optimization-based planners. Index Terms-Trajectory planning, aerial systems, motion primitives, graph search, collision avoidance. I. INTRODUCTION P LANNING collision-free, dynamically feasible trajectories in real-time through complex environments is crucial for many autonomous systems. As a result, trajectory planning has garnered significant interest from the research community, but meeting the reliability requirements for safety-critical real-world applications remains challenging. Specifically, few methods have guarantees regarding trajectory optimality and time/memory complexity without sacrificing trajectory length, computation time, or expressiveness. Our approach addresses this gap by combining optimal control with graph search to generate near-optimal trajectories over long distances in realtime, resulting in a framework that provides strong guarantees on path quality and algorithm complexity. Optimization-based trajectory planning has emerged as the primary framework for autonomous systems that must navigate complex environments. This is because constraints and performance objectives can be naturally stated in the optimization problem. Most approaches can be broadly classified by their use of continuous or integer variables. Continuous variable methods employ gradient descent to jointly optimize over Authors are with the VECTR Laboratory, . {hjlevy, Fig. 1: A trajectory (colored based on speed) generated by our proposed algorithm called STITCHER through a perlin noise environment. STITCHER searches over candidates motion primitives (white) to find a safe trajectory in real-time with time and memory complexity guarantees. Indoor flight experiments were performed to verify dynamic feasibility of trajectory plans. the coefficients of basis functions (e.g., polynomials) and waypoint arrival times while imposing obstacle and state constraints [1]-[6]. Integer variable methods require that the free space of the environment be represented as the union of convex sets (continuous variable methods have also used this representation, e.g., [5], [6]) and solve a mixed-integer program for a collision-free trajectory [7]-[10]. Despite continued innovations, these methods lack a priori time complexity bounds and often scale very poorly with trajectory length; this is especially true for integer programming approaches. A computationally efficient alternative to optimizationbased trajectory planning is the use of so-called motion primitives: a library of short length or duration trajectories that can be efficiently computed and evaluated [11]-[14]. To effectively use motion primitives, a planner must operate in a receding horizon fashion, i.e., continuously replan, because motion primitives are inherently near-sighted with their short length or duration. This can introduce several performance issues, e.g., myopic behavior or unexpressive trajectories, that are exacerbated in large, complex environments. Subsequent work has attempted to pose the problem as a graph search with nodes and edges being desired states (position, velocity, etc.) and motion primitives, respectively [15]-[18]. While this allows for the creation of long-range trajectories, search times can be extremely high (seconds) because of the graph size. An admissible search heuristic can be used to reduce the number of node expansions required to find a solution, also known as search effort, while preserving optimality of the graph [19]. However, designing such a search heuristic is non-trivial. We propose a new trajectory planning algorithm called STITCHER that can perform real-time motion primitive 20 Oct 2025 searches across long distances in complex environments. STITCHER utilizes an innovative three-stage planning architecture to generate smooth and expressive trajectories by stitching motion primitives together through graph search. The first two stages are designed to expedite the motion primitive search in the final stage by constructing a compact but expressive search graph and search heuristic. Specifically, given a set of waypoints computed in the first stage, we create a velocity graph by representing nodes as sampled velocities at each waypoint, and edges as quick-to-generate minimum-time trajectories. We employ dynamic programming on this graph, using the Bellman equation to compute the cost-to-go for each node. Critically, the cost-to-go is then used as an admissible heuristic to efficiently guide the motion primitive search in the third stage. We also leverage a greedy graph pre-processing step to form a compact motion primitive graph. We prove all graphs are finite and that the proposed heuristic is admissible. These technical results are critical as they guarantee i) a priori time and memory complexity bounds and ii) trajectory optimality with respect to the graph discretization. To further reduce computation time, we improve the collision checking procedure from [13], by leveraging the known free space from previous nearest-neighbor queries, bypassing the rigidity and computational complexity of free space decomposition. Additionally, we show that employing a simple sampling procedure in the final search stage is effective at pruning candidate trajectories that violate complex state or actuator constraints. STITCHER was extensively tested in two simulation environments to evaluate algorithmic innovations and assess its performance against two state-of-the-art optimization-based planners [5], [9]. STITCHER is shown to generate high-quality, dynamically feasible trajectories over long distances (over 50 m) with computation times in the milliseconds, and consistently generates trajectories faster than the state-of-the-art with comparable execution times. This paper substantially builds upon previous work [20] by providing a new in-depth analysis of the algorithmic innovations used to achieve real-time planning times, along with hardware experiments to demonstrate real-world feasibility. We include a parameter sensitivity analysis to evaluate the performance of STITCHER with different hyperparameters, such as using different state discretization sets. A new study on the proposed search heuristic is conducted to examine the effects of varied edge costs and acceleration constraints on both admissibility and search effort (node expansions). The impact of the greedy graph pre-processing step on solution quality and computation time is evaluated through comparisons with an exhaustive, i.e., non-greedy, graph creation. We also showed STITCHER can adhere to highly nonconvex constraints, such as individual motor constraints for a quadrotor, with no noticeable increase computation time. Comparison with stateof-the-art algorithms in simulation was also expanded, incorporating new metrics to better evaluate performance regarding optimized waypoints and the sampled set of states. Finally, we flew the trajectories designed by STITCHER in hardware on a custom quadrotor to show that the trajectories were dynamically feasible, adhered to physical constraints, and could be tracked via standard geometric cascaded control. II. RELATED WORKS A. Optimization-based Planning Designing high quality trajectories using online optimization has become a popular planning strategy as a performance index and constraints can be systematically incorporated into an optimization problem. Optimization-based trajectory planners can be categorized using several criteria, but the clearest delineation is whether the method uses continuous or integer variables. For methods that use only continuous variables, the work by [2] reformulated the quadratic program in [1] to jointly optimize over polynomial endpoint derivatives and arrival times for a trajectory passing through waypoints. Collisions were handled by adding intermediate waypoints and redoing the trajectory optimization if the original trajectory was in collision. Oleynikova et al. [3] represented obstacles using an Euclidean Signed Distance Field (ESDF) which was incorporated into a nonconvex solver as a soft constraint. Zhou et al. [4] used a similar penalty-based method but introduced a topological path search to escape local minima. An alternative approach is to decompose the occupied space or free space into convex polyhedra [7], [21], [22] which can be easily incorporated as constraints in an optimization. The methods in [5], [6] treat these constraints as soft while efficiently optimizing over polynomial trajectory segments that must pass near waypoints. One can also use the free-space polyhedra to formulate a mixed-integer program [8]-[10], [23] to bypass the nonconvexity introduced by having unknown waypoint arrival times, but at the expense of poor scalability with trajectory length and number of polyhedra. Marcucci et al. [24] addresses scalability concerns of [10] by solving a sequence of convex problems instead of one large-scale optimization but requires an offline process for generating collision-free convex sets. B. Motion Primitives Motion primitive planners have been proposed as an alternative to optimization-based planners to address computational complexity and numerical instability concerns. The underlying idea of motion primitive planners is to select trajectories online from a precomputed, finite library of trajectories. Initial work on motion primitives for quadrotors leveraged differential flatness and known solutions to specific optimal control problems to efficiently compute point-to-point trajectories in real-time [11], [25]. Later work employed motion primitives for receding horizon collision avoidance where primitives were efficiently generated online by sampling desired final states, and selected at each planning step based on safety and trajectory cost [12]-[14], [26]-[29]. Howard et al. [26] first introduced this idea of searching over feasible trajectories of a car with a model predictive control framework. Subsequent works extended this methodology to quadcopters using depth images [12], [14], point clouds [13], [27], or ESDFs [28] for motion primitive evaluations and collision avoidance. While computationally efficient, the behavior of these planners can be myopic, leading to suboptimal behavior in complex environments which limit their use for planning long-term trajectories. C. Motion Primitive Search One way to address nearsightedness is to perform a graph search over motion primitives, i.e., stitch motion primitives together. This can be achieved by extending traditional graph search algorithms [30]-[32], which typically use coarse discrete action sets, to using a lattice of motion primitives [15]- [17], [33]-[35]. Graph search algorithms are an attractive method for planning due to inherent guarantees of completeness, optimality1, and bounded time and memory complexity [36]. The works by Liu et al. [15], [16] were some of the first works to successfully showcase a search-based algorithm using a lattice of motion primitives for use on quadcopters. However, these methods can be computationally expensive as generating high-quality trajectories relies on generating a large number of motion primitives for sufficient density. Jarin et al. [37] addresses computation concerns by improving upon the sampling of different motion primitives, inspired by a minimum dispersion sampling method [38]. Another way to narrow the search space is by utilizing a geometric path as a prior and constraining motion primitives to pass through waypoints from the path. Recently, [18], [39]-[41] proposed an efficient motion primitive search in velocity space using minimum-time input-constrained trajectories from a double integrator model restricted to pass through a set of waypoints. The search can be done in real-time but the resulting bang-bang acceleration profile is dynamically infeasible for quadrotors, leading to poor tracking performance. An additional smoothing step, e.g., model predictive contouring control, is required to achieve sufficient trajectory smoothness [40]-[42]. D. Search Heuristics Fast graph search speed while retaining optimality guarantees can be achieved by employing an admissible search heuristic [36] to guide the search to the goal state. Constructing an informative and admissible heuristic, however, is nontrivial. Much of the previous work in motion primitive search overlooks the importance of the heuristic by generating a weak approximation to the goal [17], using an inadmissible heuristic which forfeits optimality guarantees [16], or proceeding without a heuristic [18]. As a result, motion primitive search algorithms to date scale poorly in large environments and for large planning horizons, making them unsuitable for systems with limited onboard computational resources. Paden et al. [43] proposed a method to systematically construct admissible heuristics for use in kinodynamic planning using sum-of-squares (SOS) programming. However, the resulting size of the SOS program requires heuristic calculations to be performed offline. Other strategies involve learning a search heuristic or cost-to-go [44]-[47]. Kim et al. [44] uses a neural network to approximate graph node distances and provides a sub-optimality bound on the solution. Reinforcement and imitation learning have also been proposed for learning search heuristics [45]-[47], but these works focus on minimizing node expansions rather than ensuring admissibility, sacrificing the optimality guarantees of graph search. 1In the context of graph search, optimality refers to resolution optimality, i.e., optimality with respect to the discretized state space. III. PROBLEM FORMULATION This work is concerned with solving the following trajectory planning problem: min u ∈U J = r(T) + Z T 0 q(x, u) dt s.t. ̇x = Ax + Bu x ∈Xs, x /∈Xobst, u ∈U x(0) = x0, x(T) = xf, (1) where x ∈Rn is the state that must satisfy state Xs and obstacle (collision) Xobst constraints, u ∈Rm is the control input that must satisfy actuator constraints U, A ∈Rn×n and B ∈Rn×m govern the system's dynamics and are assumed to take a multi-axis chain of integrators form, and r : R+ →R+ and q : Rn × Rm →R+ are the terminal and stage cost, respectively. The goal is to find an optimal final time T ∗and feasible optimal state trajectory x∗(t) with a corresponding control input sequence u∗(t) for t ∈[0 T ∗] that steers the system from an initial state x0 to a desired final state xf that minimizes the cost functional J. While the dynamics are linear in (1), many nonlinear systems can be placed into the linear control affine form if they are differentially flat, e.g., VTOL vehicles like quadrotors, capturing a large class of systems of interest. In many cases, the state vector can be x = (r, v, a, . . . , r(p-1))⊤and the control input can be u = r(p) where r = (x, y, z)⊤is the position of the vehicle in some reference frame. A. Background: Motion Primitives We define motion primitives to be closed-form solutions to specific optimal control problems. In this work, we will restrict our attention to the following two optimal control problems: the input-constrained minimum-time problem for a double integrator and the linear quadratic minimum time problem for a p-th order integrator. We will briefly review each optimal control problem and the structure of its solution. The formulations will be presented for a single axis, but can be repeated for all three position coordinates. Minimum-Time Double Integrator: Given an initial state (s0, v0) ∈R2 and desired final state (sf, vf) ∈R2, the minimum-time double integrator optimal control problem is min u J = T (2) s.t. ̈s = u, |u| ≤umax s(0) = s0, v(0) = v0 s(T) = sf, v(T) = vf, where the final time T is free. From Pontryagin's minimum principle, the optimal control solution is known to have a bang-bang control profile. The switching times can be efficiently computed by solving a quadratic equation. It is required that each coordinate axis trajectory all have the same execution time, so the limiting minimum-time horizon is T ∗= max{Tx, Ty, Tz}. The limiting minimum-time horizon T ∗, is then applied as a known variable for the non-limiting Fig. 2: System architecture describing the three planning stages. Stage 1: A sparse geometric path is found via an A* search through the voxelized environment. Stage 2: A velocity state is then introduced at each waypoint and dynamic programming is used to recursively solve for the cost-to-go at each node. Stage 3: A full motion primitive search informed by the previous stages is performed, and checks for collisions are completed to yield the final trajectory. axes. This allows one to then solve a quadratic equation for a new bound ̄u ≤umax on the control input. Linear Quadratic Minimum-Time p-th Order Integrator: Smooth trajectories can be generated by solving the linear quadratic minimum-time (LQMT) optimal control problem, min T, u J = ρ T + Z T 0 u2 dt (3) s.t. s(p) = u s(0) = s0, v(0) = v0, . . . , s(p-1)(0) = s(p-1) 0 s(T) = sf, v(T) = vf, s(k-1)(T) free for 3 ≤k ≤p where ρ > 1 penalizes the final time. The final time T and all terminal states except position and velocity are free. The optimal trajectory is a polynomial in time so the cost functional can be expressed analytically in terms of T and the known boundary conditions. The final time can be found efficiently using a root-finding algorithm such as QR algorithm [48]. State constraints are omitted from (3) as it is more efficient to prune many candidate trajectories once the final time is known, as discussed in Section IV-D. IV. METHODOLOGY STITCHER, detailed in Algorithm 1, generates a fullstate trajectory by stitching collision-free, dynamically feasible trajectory segments together through graph search. At its core, STITCHER searches over closed-form solutions, i.e., motion primitives, to optimal control problems of the form discussed above. These solutions serve as a basis set for the solution space to (1). To achieve real-time performance, STITCHER utilizes a three stage planning process (see Fig. 2). In Stage 1 (left), A* algorithm is used to produce a sparse geometric path, i.e., waypoints, in the free space of the environment (line 3); this is standard in many planning frameworks. In Stage 2 (middle), nodes representing sampled velocities at the waypoints are formed into a velocity graph where dynamic programming is used to compute the minimum time path from Algorithm 1: STITCHER Trajectory Planner 1 input: P ←point cloud, ns ←start, ng ←goal; 2 output: s∗(t); // extract waypoints and path features 3 w, q, H ←getGeometricPath(P, ns, ng); 4 G ←buildVelocityGraph(w, q, H); // get heuristic from recursive cost-to-go 5 h(n) ←dynamicProgramming(G, ns, ng); 6 Gmp ←buildFullStateGraph(G); 7 function planPath(P, Gmp, ns, ng): 8 ncurr = ns; 9 while ncurr ̸= ng do // get node with lowest cost g(n)+h(n) 10 ncurr = OPEN.pop(); 11 CLOSED.insert(ncurr); 12 if ncurr = ng then 13 break; 14 end 15 Encurr ←getSuccessors(ncurr, Gmp); 16 for e in Encurr do // collision and state constraint check 17 pruneMotionPrimitive(e, P); 18 OPEN.insert(φ(n, e)); 19 end 20 end 21 s∗(t) ←getFinalMotionPrimitives(ncurr, CLOSED); each node to the desired final state using a control-constrained double integrator model (lines 4-5). This step is critical for constructing an admissible heuristic to guide the full motion primitive search, and is one of the key innovations that enables real-time performance. Note that the optimal \"path\" in velocity space is never used; computing the cost-to-go is the primary objective as it serves as an admissible heuristic for motion primitive search as shown in Section V-B. In Stage 3 (right), an A* search is performed using the heuristic from Stage 2 over motion primitives of a p-th order integrator where p ≥2 . A greedy pre-processing step is used to construct a compact motion primitive graph (line 6), ensuring the search remains real-time. At this stage, position and all higher-order derivatives are considered, yielding a full state trajectory that can be tracked by the system (lines 7-21). The remainder of this section expands upon each component of STITCHER. A. Stage 1: Forward Geometric Path Search STITCHER requires a sequence of waypoints that essentially guides the motion primitive search by limiting the size of the search space. This can be done by generating a collisionfree geometric path (see Fig. 2 left) through the environment with A* search or any other discrete graph search algorithm where the environment is represented as a 3D voxel grid where each grid cell contains occupancy information. Let the collision-free, geometric path generated by a discrete graph search algorithm be composed of points O = {o1, o2, ..., oH} where oi ∈R3. The set of points O is further pruned to create a sparse set of waypoints W = {w1, w2, ..., wN} where N ≤H and wi ∈R3. Sparsification is done by finding the minimal set of points in O that can be connected with collision-free line segments. The geometric path search is used in line 3 of Algorithm 1. B. State 2: Backward Velocity Search The ordered waypoint set W found in Stage 1 only provides a collision-free geometric path through the environment. In other words, the velocity, acceleration, and higher-order states necessary for tracking control are not specified. We propose creating a velocity graph (see Fig. 2 middle) where each node in the graph is defined by a position and velocity. The positions are restricted to waypoint locations and M velocities are sampled at each waypoint. More explicitly, for each waypoint wi ∈W, we sample a set of velocities V = {v1, ..., vM}, where V is composed of candidate velocity magnitudes Vm and directions Vd. The choice of Vm and Vd can impact the STITCHER's performance in terms of path optimality and computational complexity; this point will be discussed in more detail in Section VI. With the ordered waypoint W and sampled velocity V sets, we create a velocity graph G = (N, E), where node n ∈N is a given position and sampled velocity, i.e., n = (wi, vj) with wi ∈W and vj ∈V, and edge e ∈E is the double integrator control-constrained minimum-time motion primitive r(t) from (2) that connects neighboring nodes. Recall that the solution to (2) is fully determined by having an initial and final position and velocity pair which is precisely how each node in N is defined. At this stage, collision and state constraints are not enforced to prevent candidate trajectories from being eliminated prematurely. We recursively compute and store the ranked list of costto-go's Vd : N × E →R+ for each node n ∈N and all connecting edges e ∈En of n where Vd(n, e) = l(n, e) + V ∗ d φ(n, e) ∀e ∈En, (4) with the optimal cost-to-go V ∗ d (n) = mine∈En Vd(n, e), the cost of taking edge e from node n being l(n, e), and the node reached by taking edge e being φ(n, e). The cost of taking an edge is given by l(n, e) = T ∗ d (n, e), where T ∗ d (n, e) is the minimum-time of trajectory r(t) connecting the states of node n to the states of φ(n, e). Minimizing (4) is the well-known Bellman equation, which is guaranteed to return the optimal Fig. 3: The achievable mass-normalized thrust (nonconvex) of a aerial VTOL vehicle with limits on minimum thrust fmin, maximum thrust fmax, and maximum tilt θmax. cost-to-go. In Section V-B, we prove V ∗ d (n) for each node in a graph G is an admissible heuristic for an A* search over a broad class of motion primitives. Building and searching the velocity graph are shown in lines 4-5 of Algorithm 1. C. Stage 3: Forward Motion Primitive Search The cost-to-go's computed in Stage 2 for the sampled velocities at each waypoint serve as an admissible heuristic (formally defined later in Definition 1) that guides an efficient A* search over motion primitives. The motion primitives can be generated using any chain of integrators model of order at least two so long as i) the initial and final position and velocities match those used to construct the velocity graph G and ii) the allowable acceleration is maximally bounded by umax given in (2). It is important to note that the bound on allowable acceleration can be easily satisfied with user defined umax or simply by applying the box constraint ∥a∥∞≤umax. The motion primitive search graph is denoted as Gmp = (Nmp, Emp) where Nmp is the set of nodes, each corresponding to a state vector, and Emp is the set of edges, each corresponding to a motion primitive that connecting neighboring nodes. A* search is used to meet real-time constraints where the search minimizes the cost f(n) = g(n) + h(n) where n ∈Nmp is the current node, g : Nmp →R+ is the cost from the start node ns to node n, and h : Nmp →R+ is the estimated cost from the current node n to the goal node ng. In the context of optimal control, g is the cost accrued, i.e., the cost functional J, for a path from ns to n whereas h is the estimated costto-go, i.e., the estimated value function V ∗, from n to ng. In this stage, collision and state constraints are checked for each candidate motion primitive to ensure safety. The methodology for both is discussed in Section IV-D. Each step of the motion primitive A* search is shown in lines 6-21 of Algorithm 1. D. Pruning Infeasible & In-Collision Motion Primitives STITCHER guarantees safety by pruning motion primitives from the final search that violate constraints or are in collision. For state and actuator constraints, many optimizationbased planning approaches approximate the true physical constraints of the system with simple convex constraints, e.g., ∥v∥∞≤vmax, ∥a∥∞≤amax, etc., to reduce computational complexity. When polynomials are used to represent the optimal trajectory, imposing a convex hull constraint on the polynomial is one method for enforcing such state constraints Fig. 4: Removing redundant collision checks. (a): Motion primitive r0(t) checks for collisions using [13]. (b): Sampled points of r1(t) are checked to lie within obstacle-free regions derived from r0(t) calculations. [9], [17]. However, many of these approximations are made only to simplify the resulting optimization problem and do not accurately reflect the actual physical constraint, which can lead to conservatism. STITCHER has the freedom to use a variety of methods to enforce state and actuator constraints. For the examples shown in this work, we uniformly sample candidate trajectories in time to check for constraint violations as it was found to be effective and efficient. Sampling avoids mistakenly eliminating safe trajectories, and the observed computation time under the current discretization was better than using convex hulls. Critically, sampling allows for the inclusion of more complex constraints, such as those that couple multiple axes. Examples are Thrust Magnitude: 0 2. (7) Proof. Using Fig. 2 (middle), the start and goal nodes contribute 2 nodes to the graph G. For intermediate waypoints, given M sampled velocities, there are M nodes per waypoint. As a result, |N| = (N -2)M +2 which is (6). For each edge, we consider the transition to successive waypoints. Ignoring the trivial N = 2 case where |E| = 1, there are M connections between the start node and next waypoint, which also has M nodes. The same applies for connecting waypoint wN-1 to the goal node, resulting in a total of 2M edges. For all other intermediate waypoint pairs, M nodes connect to M nodes at the next waypoint so there are M 2 edges. The total number of edges is then (7). Corollary 1. The size of the motion primitive graph Gmp using Linear Quadratic Minimum Time (LQMT) motion primitives with free terminal acceleration for a triple integrator is identical to the velocity graph G. Proof. The proof is immediate since the terminal acceleration is free so N and M are identical for both graphs. Remark 1. Corollary 1 can be generalized to any motion primitive search graph where the primitives are solutions to an optimal control problem with the dynamics being a chain of integrators and all terminal state derivatives of second order or higher are free. Note that this assumes the greedy graph pre-processing step still yields adequate performance. B. Admissible Heuristic for Motion Primitive Search It is well known that heuristics can be used to expedite searching through a graph by incentivizing the search to prioritize exploring promising nodes. For example, in A* search, the next node explored is selected based on minimizing the cost f(n) = g(n) + h(n), where g is the stage cost to get from the start node ns to node n, and h is a heuristic estimate of the remaining cost to reach the goal node ng. A* search is guaranteed to find an optimal solution so long as the heuristic function h is admissible (see Definition 1) [36]. Below, we prove the cost-to-go V ∗for each node in the velocity graph G calculated in Stage 2, i.e., the minimum-time to goal for a double integrator, is an admissible heuristic for an A* search over motion primitives of any higher-order chain of integrators. Definition 1 ([36]). A function h : N →R is an admissible heuristic if for all n ∈N then h(n) ≤h∗(n), where h∗is the optimal cost from n to the goal node ng. Proposition 2. Consider the optimal control problem min T, u J = ρ T + Z T 0 q(r, v, . . . , u) dt (8) s.t. r(p) = u, c(a) ≤0 r(0) = r0, v(0) = v0, . . . , r(p-1)(0) = r(p-1) 0 r(T) = rf, v(T) = vf, r(k-1)(T) free for 3 ≤k ≤p where q is a positive definite function, the system is at least second order (p ≥2), and the position and velocities boundary conditions are identical to those of (2), with all other boundary constraints free to specify. If umax in (2) is the maximum possible acceleration achievable in a given axis imposed by c(a) ≤0, then the optimal cost-to-go V ∗from the initial conditions for (8) satisfies V ∗≥ρ T ∗ d where T ∗ d is the optimal final time for (2). Proof. First, consider the case when p = 2. For a given axis, if umax is chosen so that it exceeds the allowable acceleration imposed by c(a) ≤0, e.g., ux,max ≥maxax c(a) (see Fig. 3), then the optimal final time T ∗for (8) will always be greater than that of (2) even when q = 0. Specifically, when q = 0, one can show the optimal final time for (2) increases as umax decreases. Moreover, T ∗ d for (2) is guaranteed to exist and be unique [50]. Hence, by appropriate selection of umax, we can ensure T ∗≥T ∗ d always, where equality holds when q = 0 and c(a) is a box constraint. If q ̸= 0, then it immediately follows that T ∗> T ∗ d because q is positive definite by construction. Now consider the case when p > 2. We can deduce V ∗> ρ T ∗ d by contradiction. Specifically, assume T ∗= T ∗ d for p > 2. This would require a to be discontinuous in order to match the bang-bang acceleration profile of (2). However, (8) is a continuous-time linear system that will not exhibit discrete behaviors, e.g., jumps, so it is mathematically impossible to generate an optimal control sequence where the acceleration profiles for Eqs. (2) and (8) will be identical. It can then be concluded V ∗> ρ T ∗ d for p > 2. Therefore, V ∗≥ρ T ∗ d for p ≥2, as desired. Remark 2. Proposition 2 also holds when inequality state or actuator constraints in (8) are present, and when the terminal desired states are specified rather than free. The main result of this section can now be stated. Theorem 1. The optimal cost-to-go for the minimum-time input-constrained double integrator optimal control problem (2) is an admissible heuristic for motion primitive search Fig. 6: Simulation test environments. (a): Willow Garage environment. (b): Perlin Noise environment. where the primitives are solutions to the optimal control problem of the form (8). Proof. Let G = (N, E) be a graph with nodes being sampled velocities at waypoints and edges being the time-optimal trajectories using an input-constrained double integrator. Further, let Gmp = (Nmp, Emp) be a graph with nodes being sampled velocities, accelerations, etc. at waypoints and edges being trajectories that are solutions to (8). Using the Bellman equation, the optimal cost-to-go V ∗ mp(n) for any n ∈Nmp can be computed recursively. Using Proposition 2, V ∗ mp(n) ≥ V ∗ d (n′) by induction where V ∗ d is the optimal cost-to-go for the minimum-time input-constrained double integrator with n′ ∈N. Recognizing N ⊆Nmp, V ∗ d (n′) can be rewritten as V ∗ d (n). Setting h∗(n) = V ∗ mp(n) and h(n) = V ∗ d (n), it can be concluded h(n) ≤h∗(n). Therefore, by Definition 1, the optimal cost-to-go computed for G is an admissible heuristic for the motion primitive search over Gmp. The importance of Theorem 1 follows from the well-known result that searching a graph with an admissible heuristic is guaranteed to return the optimal path through the graph [36], and can significantly improve search efficiency because not every node in the graph has to be explored. The effectiveness of the proposed heuristic both in terms of path quality and search times is analyzed in Section VI. VI. SIMULATION RESULTS This section contains an analysis of STITCHER. First, we conduct a parameter sensitivity study to determine a suitable sampled velocity set (direction and speed) using a modified version of Dijkstra's algorithm that has access to a dense set of velocities and no constraints on computation time. Second, STITCHER is compared to a non-greedy algorithm variant to characterize the impact of our greedy graph pre-processing step on solution cost and computation. Third, we investigate the effectiveness of the heuristic proposed in Section V-B in reducing the number of edges generated by STITCHER. Fourth, a study of STITCHER constraining individual motor forces is presented to demonstrate the flexibility of our method in adhering to complex actuator constraints. Fifth, we characterize the average computation time of the different components that make up STITCHER. Lastly, STITCHER is compared to two optimization-based modern planners [5], [9] capable of running in real-time. Simulation experiments were run in a Perlin Noise environment and the Willow Garage environment, both with a volume Fig. 7: Velocity directions sampled at waypoint wi, where ˆai is normal to the separating hyperplane Hi and ˆqi is the heading toward wi+1. TABLE I: Frequency of Velocity Directions in Final Trajectory. Zenith Angle 70◦ 80◦ 90◦ 100◦ 110◦ Frequency 0% 13% 79% 8% 0% Azimuth Angle -50◦ -20◦ -10◦ 0◦ 10◦ 20◦ Frequency 4% 4% 17% 42% 29% 4% of approximately 50 × 50 × 5 m (see Fig. 6). Geometric paths with N = 4, 6, 8 waypoints were found for different start and end locations in each environment. For all experiments, we imposed the following constraints assuming agile drone flight and reasonable thrust limits: fmin = 0.85 m/s2, fmax = 18.75 m/s2, θmax = 60◦, ωmax = 6 rad/s, vmax = 10 m/s, and a time penalty ρ = 1000. All reported times are from tests conducted on an 11th generation Intel i7 CPU. A. Parameter Sensitivity Analysis: Sampled Velocity Set STITCHER requires a discrete velocity set V composed of a set of magnitudes and directions. As shown in Section IV-B, the size of the velocity graph scales quadratically with the number of sampled velocities, so it is desirable to choose V to be small without degrading path quality. Hence, this section focuses on constructing a velocity sample set V that optimizes the trade-off between computation time and path quality as measured by execution time. We conducted two studies to determine i) the predominate sampled velocity directions used by an offline version of Dijkstra's algorithm and ii) the tradeoff between path cost and the number of sampled speeds. In the following comparison, Dijkstra's algorithm was modified to search over a dense set of velocities with 3611 terminal velocities sampled per waypoint; the method is referred to as Dense Dijkstra. Dijkstra's algorithm is a complete and optimal search algorithm with respect to the set of actions [36], making it a suitable benchmark. 1) Sampled Direction: Dense Dijkstra was used to statistically identify velocity directions set Vd commonly employed in a variety of test cases. Using Fig. 7, we define angles with respect to the hyperplane H with normal vector ˆa at the plane of symmetry between path segments connecting two waypoints. The search was given velocity direction zenith and azimuth angles in the range [0◦, 180◦] and [-90◦, 90◦] sampled in 10◦increments. Table I shows the frequency of velocity directions chosen by Dense Dijkstra across all motion primitives for six different path length trials in the Perlin Noise and Willow Garage environment. The velocities chosen by Dense Dijkstra align with the normal vector ˆa of the hyperplane 80% of the time. Fig. 8: Analysis of speed discretization on execution and planning time. As the discretization of our method is increased, we converge to the Dense Dijkstra solution. We use Vm = 5 for our experiments as it achieves significant computational advantages while retaining suitable performance. Fig. 9: Speed profiles of STITCHER using varied speed discretizations. As the number of speed samples increases, STITCHER converges to the optimal solution approximated by Dense Dijkstra. From these results, sampling at the center and boundaries of a 20◦cone centered around a given ˆa will yield a suitable set of velocity directions. 2) Sampled Speed: Using the velocity direction set Vd identified in the previous test, the performance of Dense Dijkstra was compared to STITCHER with different sampled speed sets Vm. Each set Vm consists of k discrete speeds sampled from the interval [0, vmax], such that |Vm| = k. In order to ensure a fair comparison, these sets must be subsets of those used by Dense Dijkstra. In other words, we ensure k ≤K, where K denotes the number of speeds sampled in Dense Dijkstra, and our sampled sets are as evenly distributed as possible within the interval. Figure 8 shows the relative execution time (left) and planning times (right) for different sizes of Vm compared to Dense Dijkstra. As expected, the execution time increases as the sampled speed set becomes sparser because the planner has fewer speed options available. However, the observed increase in execution time is at most 8% even with the sparsest speed set tested (|Vm| = 5). Critically, the minimal increase in execution time is accompanied by a significant reduction in planning time: a speed-up of four orders of magnitude with the sparsest sampled speed set tested. Although a sample speed set size of |Vm| = 11 yields nearly identical execution times to the dense set while offering three orders of magnitude improvement in planning time, adequate performance is achieved when |Vm| = 5. Hence, we use the set Vm = {0, 0.25 vmax, 0.5 vmax, 0.75 vmax, vmax} for the remainder of our analysis. A representative speed profile for different sized speed sets is shown in Fig. 9. Fig. 10: Evaluation of the effect of a greedy pre-processing step on solution quality and computation speed. Results are from a monte carlo simulation of 50 random realizations for trajectories with between 2 and 10 waypoints. Fig. 11: The final speed profile given by a motion primitive search with (a) an exhaustive exponentially growing graph and (b) STITCHER. B. Greediness Benchmarking STITCHER employs a greedy graph pre-processing step in order to generate a compact final search graph which enables real-time trajectory computation. It is important to note that this greedy pre-processing step results in a graph that does not exhaustively account for all possible acceleration states at a given waypoint (see Fig. 5). Hence, we evaluated the extent to which this greedy pre-processing step leads to suboptimal solutions by performing a Monte Carlo simulation comparing the path cost of STITCHER to that of an exhaustive exponentially-growing graph. The experiment includes 50 realizations of randomly generated initial and final positions, with the number of waypoints ranging between 2 and 10. Figure 10 shows the relative solution cost (left) and planning times (right) resulting from searching over the exponentially-growing graph and running STITCHER. On average, STITCHER achieved a 0.5% difference in solution cost, with the maximum difference being 7%. Additionally, STITCHER achieves approximately 66% speed-up in computation time as a result of the reduced graph size. Excluding solution cases with 2 or 3 waypoints where the exponential growth of the graph including terminal acceleration states cannot be observed, STITCHER achieves an average of 82% speed-up. Figure 11 shows the speed profile of a case in which STITCHER achieved a solution cost that was 2% different from that of the exponential graph. Qualitatively, the trajectories are very similar with only a minor deviation at the end of the trajectory. Therefore, the greedy graph preprocessing step results in a negligible increase in solution cost, but a substantial improvement in computation time. Fig. 12: Final trajectories (red) through a six waypoint path in (a): Perlin Noise and (b): Willow Garage environment. The trajectory options (white) which inform the heuristic are more likely to be in collision in the Willow Garage environment due to tight corridors. C. Heuristic Benchmarking The quality of the heuristic used to guide the motion primitive search phase of STITCHER can be quantified by comparing the number of edges, i.e., motion primitives, generated by STITCHER with that of Dijkstra's algorithm. The velocity magnitude and direction sets were kept constant across both planners with |Vm| = 11 and |Vd| = 3. The number of edges created is a better evaluation metric than the nodes explored because the main source of computation time comes from generating motion primitives and checking them for constraint violations. The effectiveness of the search heuristic depends on two main factors: how accurately it approximates the true cost-to-go, and whether the heuristic is admissible under the specified constraints. The edge cost used in the final search affects the tightness of the heuristic approximation, while the acceleration constraint imposed to prune motion primitives can influence admissibility. Therefore, we assess the effectiveness of STITCHER's heuristic with various graph edge costs and acceleration constraints to provide a general understanding of how search performance is impacted. 1) Varying Edge Cost: We evaluate the quality of the heuristic with two different edge costs: (1) equivalent to the LQMT performance index and (2) execution time. Table II shows the percent reduction in the number of edges created for STITCHER compared to Dijkstra's algorithm. Using the LQMT edge cost in the final search, STITCHER creates 7% fewer edges on average compared to Dijkstra's algorithm in the Perlin Noise environment and 5% in the Willow Garage environment. Greater edge reduction can be achieved by defining the edge cost to be the trajectory execution time. Recall that the search heuristic is the minimum time required to reach the goal using a double integrator model. Therefore, by using an edge cost soley defined by trajectory duration, the heuristic now more closely approximates the cost-to-go. This tighter approximation is reflected in the results of Table II, where STITCHER generates an average of 20% fewer edges in Perlin Noise and 13% fewer in Willow Garage. The reduced effectiveness of the heuristic in Willow Garage was attributed to having more tight corridors, so more motion primitives were in collision (see Fig. 12). Therefore, while environment dependent, using a search heuristic that closely approximates the remaining cost to the goal greatly reduces search effort. 2) Varying Acceleration Constraint: Varying acceleration constraints may also lead to differing performance of the TABLE II: Heuristic Evaluation with Varied Edge Costs and Acceleration Constraints via Percent Reduction in Generated Edges Compared to Dijkstra's. Edge Cost Acceleration Constraint Map Perlin Noise Willow Garage N = 4 N = 6 N = 8 N = 4 N = 6 N = 8 % Red. % Red. % Red. % Red. % Red. % Red. LQMT Admissible Truncated Cone 8 8 7 0 3 13 Time Admissible Truncated Cone 22 21 15 0.3 13 26 Time Admissible Box 27 34 28 2 20 24 Time Inadmissible Truncated Cone 45 44 22 9 38 39 Fig. 13: Different acceleration constraints used for heuristic study: (a) admissible truncated cone, (b) admissible box, and (c) inadmissible truncated cone. Green dashed outlines are Stage 2 constraints used for heuristic creation while solid black outlines denote constraints imposed for the final trajectory. heuristic as this constraint directly affects admissibility. We tested three different acceleration constraints: (1) admissible truncated cone, (2) admissible box, and (3) inadmissible truncated cone (See Fig. 13). The truncated cone constraint comes from the resulting achievable mass-normalized thrust volume with constraints on the magnitude and tilt. This constraint c(a) is admissible when it is maximally bounded by the maximum control input umax imposed during heuristic generation in Stage 2 (see Fig. 13a), while the inadmissible variant applies a smaller umax, e.g., ux,max ≤maxax c(a) (see Fig. 13c). An admissible box constraint is an acceleration constraint that perfectly matches that of Stage 2 (see Fig. 13b). Table II shows that improved performance can be achieved by applying an admissible box constraint on acceleration compared to the admissible truncated cone. In this case, STITCHER achieves approximately a 30% reduction in edges in the Perlin Noise environment and a 15% reduction in the Willow Garage compared to Dijkstra's algorithm. Because the box constraint on acceleration is equivalent to that used in the heuristic generation, the STITCHER algorithm guides the search away from branches that are likely to exceed acceleration constraints. Further improvement in search speed may be observed while using an inadmissible truncated cone constraint, and is reflected in Table II, where this algorithm variant achieves the highest percent reduction among all STITCHER variations. By allowing the final motion primitive search to have a larger allowance in achievable acceleration, the true edge cost may be smaller than that estimated by the heuristic. The use of inadmissible heuristics in graph search foregos desirable optimality guarantees, but can more rapidly motivate the search toward the goal as a larger weighting is placed on the heuristic cost. 3) Summary: This study showed that the proposed heuristic is effective, but its performance depends on the environment, graph edge cost, and the chosen acceleration constraint. For the remaining experiments, we used an LQMT edge cost to ensure fair comparisons to state-of-the-art algorithms, and apply an admissible truncated cone acceleration constraint to retain graph search optimality guarantees and accurately reflect the true physical limits of quadrotor systems. D. Constrained Motor Forces We present a case study demonstrating STITCHER's ability to generate trajectories that satisfy individual thruster limits for a quadrotor, which also extends to lander vehicles. We assume the vehicle dynamics takes the form m ̈r = R T -mg J ̇ω = τ -ω × Jω, where r is the vehicle's position vector, R is the rotation matrix that represents the vehicle orientation with respect to an inertial frame, T = T ˆez is the thrust vector directed along the body z-axis, g is the gravity vector, ω is the angular velocity vector, J is the inertia tensor, and τ is the body torque vector. For the case study, we assume a quadrotor with a mass m = 0.5 kg, moment of inertia J = diag(0.01, 0.01, 0.01) kg m2, arm length l = 12.5 cm, motor drag coefficient c = 0.2 m, minimum single-motor thrust Fmin = 0.15 N, and maximum single-motor thrust Fmax = 3 N. Each motor thrust Fi is constrained by 0 < Fmin ≤Fi ≤Fmax, ∀i ∈{1, 2, 3, 4}. At any given time, individual motor thrusts Fi are found by solving a linear system that depends on motor placement and physical vehicle properties, i.e., solving T τx τy τz = M F1 F2 F3 F4 , for some invertible matrix M ∈R4×4 with T denoting the total thrust T = m∥f∥2. The torque τ = [τx, τy, τz]⊤along the trajectory is τ = J ̇ω + ω × Jω and can be found via differential flatness, with components of ω and ̇ω given by ωy -ωx 0 = R⊤ ̇ˆ f, ̇ωy - ̇ωx 0 = R⊤ ̈ˆ f -ω × R⊤ ̇ˆ f, where ˆ f is the normalized thrust vector. We conducted two experiments in the Willow Garage environment, where the tight corridors may make individual motor constraint satisfaction more difficult. As shown in Fig. 14, the trajectories achieve high speeds while strictly obeying the permotor limits on thrust. These results showcase the versatility (a) Individual Motor Constraint Case Study 1. (b) Individual Motor Constraint Case Study 2. Fig. 14: Case studies in the Willow Garage environment of trajectory plans constraining individual motors by the maximum thrust limits given by manufacturer specifications. Left: Trajectory colored by speed. Right: The corresponding thrust profile of individual motors depicting constraint satisfaction. of our framework in adhering to highly nonconvex constraints that are directly limited by hardware. E. Planning Time Composition Analysis Figure 15 shows the computation time of the different components of STITCHER averaged over six tested trials. The average time to perform the velocity graph search is 1.8 ms, compared to 2.2 ms for the motion primitive search. Although both searches have the same graph size, it is important to note that the motion primitives from (3) are more time-consuming to generate than the minimum time trajectories used in the velocity graph. Therefore, the similar computation times arise from the search heuristic reducing the number of edges generated. The low computation time of the velocity search further indicates its effectiveness in computing an informative admissible heuristic for the motion primitive search. Finally, constraint-checking with uniform samples at every 0.1 s averaged only 0.3 ms, showing the efficacy of the relatively simple approach. F. Comparison with State-of-the-Art We compared STITCHER with two state-of-the-art algorithms: GCOPTER [5] and FASTER [9]. GCOPTER performs an online optimization by incorporating state constraints into Fig. 15: The average contribution of different path planning components. the cost functional and running a quasi-newton method, while FASTER solves a mixed integer quadratic program online. Both algorithms rely on a sparse geometric path to form safe flight corridors, but do not enforce final trajectories to pass through waypoints. We evaluate each planner by time (planning time versus execution time) and failure (constraint violation or incomplete/no path found). For the Perlin Noise environment, the path lengths were 12.5 m, 30 m, and 55 m while the path lengths for Willow Garage were 20 m, 25 m, and 30 m with N = 4, 6, 8 waypoints for both environments. 1) Time Analysis: Table III compares the planning times and trajectory execution times of each planner given the six different geometric paths. STITCHER's planning times are TABLE III: State-of-the-Art Comparison Time Analysis. Map N Planning time (ms) Execution time (s) [9] [5] Ours [9] [5] Ours Perlin Noise 4 109 21.1 3.82 2.98 3.51 3.67 6 207 59.4 10.4 4.42 4.42 5.75 8 1020 116 16.9 7.62 7.34 9.62 Willow Garage 4 240 38.2 8.27 4.40 4.37 4.53 6 5240 53.2 20.9 8.25 5.97 6.30 8 10400 84.4 27.1 10.5 FAILED 7.95 TABLE IV: Effect of Different Waypoints and Sampling in Perlin Noise. N Planning time (ms) Execution time (s) STITCHER STITCHER [5] S W SW [5] S W SW 4 27.2 14.8 3.01 10.5 3.51 3.44 3.61 3.56 6 72.0 44.7 13.8 42.4 4.42 5.43 5.38 4.90 8 130 64.1 24.6 86.9 7.34 9.13 8.99 9.26 Abbreviations: S - speed sample set expanded to |Vm| = 9; W - optimized waypoints from [5] used; SW - combination of S and W. TABLE V: State-of-the-Art Comparison Failure Analysis. Map No Path Found (%) State Violation (%) Collisions (%) [9] [5] Ours [9] [5] Ours [9] [5] Ours Perlin Noise 0 0 0 6 0 0 4 4 0 Willow Garage 18 2 2 8 0 0 0 24 0 faster than those measured for FASTER and GCOPTER for every test, and are up to 7x faster than GCOPTER and 400x faster than FASTER. It is important to note the computation times for the motion primitive search planner [16] were omitted from Table III because planning times exceeded several seconds. In some cases GCOPTER and FASTER achieved lower execution times. This can be attributed to their ability to treat waypoints as soft constraints, i.e., the trajectory is only required to pass nearby a waypoint rather than through it, as well as the chosen resolution of state samples in STITCHER. The difference is most noticeable in the Perlin Noise environment where more free space is available for waypoint adjustment. Table IV compares the planning and execution times of GCOPTER [5] to different algorithm variants of STITCHER evaluating the effect of sampling and waypoint selection in the Perlin Noise environment. Case S increases the sampled speed set to |Vm| = 9, Case W utilizes the optimized waypoints from GCOPTER, and Case SW is a combination of both strategies. All the case variants reduce STITCHER's baseline execution time, motivating future work in incorporating waypoint flexibility. 2) Failure Analysis: A Monte Carlo simulation composed of 50 realizations was conducted to evaluate the different modes of failure experienced by each planner. Table V compares the rate at which each planner does not find a path, generates a trajectory violating state constraints or generates a trajectory in collision. The \"No Path Found\" metric includes a numerical solver not returning a solution, or if the solution does not reach the goal. Using either FASTER or GCOPTER, this can occur due to numerical instability or when a feasible solution is not within the calculated set of safe corridors. Conversely, in STITCHER's graph search framework, a graph disconnection occurs when a feasible Fig. 16: Example of mass-normalized thrust profile strictly satisfying constraints in Willow Garage environment. Fig. 17: Control and estimation architecture for custom quadrotor platform used in hardware experiments. The vehicle is equipped with an OrangePi 3.0 computer, Teensy 4.0 microcontroller, and an InvenSense ICM-20948 IMU. An external motion capture system provides pose measurements at 100Hz. solution is not within the discrete set of sampled states. Across all test cases, STITCHER's motion primitive graph disconnects only once, achieving the lowest rate of failure among the state-of-the-art planners. In the Willow Garage environment, where narrow corridors make collisions more likey, the number of failed solutions by FASTER and collisions by GCOPTER significantly increases. In contrast, STITCHER never violates constraints (state, control, or obstacles) because all constraints are strictly enforced. As an example, Fig. 16 is a representative mass-normalized thrust profile generated by STITCHER which remains within the valid limits. VII. HARDWARE RESULTS The custom quadrotor used for hardware experiments is shown in Figure 17. Onboard hardware includes an Orange Pi 3.0 computer, Teensy 4.0 microcontroller, and an InvenSense ICM-20948 IMU. The system was flown indoors in a 4m x 4m x 4m drone cage equipped with an OptiTrack motion capture system that accurately measures the pose of the vehicle. A. Control and Estimation Architecture The control and state estimation architecture used in the hardware experiments is shown in Fig. 17. We employed the standard cascaded geometric controller [51], [52] to track the trajectory generated by STITCHER with the desired acceleration and jerk as feedforward. The outer loop PID position tracking controller ran on the Orange Pi flight computer, and the inner loop quaternion-based attitude tracking controller from [53] ran on the Teensy microcontroller; the two processors communicate via serial. The outer loop control rate was 100Hz while the inner loop control rate was 500Hz. For estimation, position and orientation measurements from an Fig. 18: Motion trail of quadrotor executing trajectory from Experiment 1 in the virtual Random Forest 1 environment through 4x4x4 meter drone cage. TABLE VI: Trajectory Tracking Error. RMSE Experiment Random Forest 1 Random Forest 2 Obstacle Course Windowed Wall Pos. (cm) 9.86 13.3 10.9 8.59 Tilt (deg) 4.98 6.01 6.28 6.41 external motion capture system were sent wirelessly at 100Hz to the onboard Orange Pi computer, and fused with an IMU to estimate position, velocity, orientation, angular velocity, and IMU biases with a nonlinear geometric observer [54]. Estimates were generated by the observer at 100Hz and used by the outer loop position controller. B. Flight Experiments Indoor flight experiments were conducted to evaluate the dynamic feasibility of the trajectories planned with STITCHER. The trajectories were first generated with virtual obstacles and then flown using our custom quadrotor (see Fig. 18). In Fig. 19, planned trajectories generated by STITCHER are displayed alongside the actual trajectories flown by the quadrotor in four experiments. The environments are named: Random Forest 1 (Fig. 19a), Random Forest 2 (Fig. 19b), Obstacle Course (Fig. 19c), and Windowed Wall (Fig. 19d). The desired path lengths for each test in their respective environment were 9.6 m, 13.1 m, 8.7 m, and 8.5 m. Table VI reports the root mean square errors (RMSE) of position and tilt for each experiment. Across all experiments, the maximum RMSE is 6.41 degrees in tilt and 13.3 cm in position. Further, the average ratio between position RMSE to the total path length was only 1.07% over all the tests. These flight experiments show that STITCHER generates trajectories that satisfy constraints and can be tracked with low error using a standard cascaded geometric controller. Additionally, while STITCHER may use any p-th order integrator with p ≥2 to model system dynamics, the results show that using a triple integrator model is sufficient for quadrotor tracking control despite discontinuities in jerk at waypoints. Figure 20 compares the actual and desired profiles of the mass-normalized thrust (top), position (middle), and attitude (bottom) through Random Forest 1. The quadrotor remains within the physical limits dictated by the thrust magnitude and tilt constraints, and the motor commands executed by the drone closely match the desired mass-normalized thrust. The position and attitude error profiles further show that for the duration of the flight, the maximum deviations in position and tilt are less than 20 cm and 20 degrees, respectively. These results demonstrate that STITCHER generates trajectories that safely and accurately express complex system-level constraints. VIII. CONCLUSIONS In this work, we presented STITCHER, a motion primitive search planning algorithm that utilizes a novel three-stage planning architecture to design constrained trajectories in realtime over long distances. We proved the search graph is finite, and the proposed search heuristic is admissible, so STITCHER is guaranteed to i) have a priori bounded time and memory complexity and ii) generate optimal trajectories with respect to the sampled set of states. Real-time search speeds were achieved through our novel heuristic crafting technique, greedy graph pre-processing method, and non-conservative constraint and collision checking procedure. Our extensive simulation study showed the trade-off in terms of path quality and computation time for different sampled velocity sets, the effectiveness of the proposed heuristic with varied edge costs and state constraints, a case study imposing individual thruster limits, and the average computation times of the components that make up STITCHER. We also found that our greedy motion primitive graph pre-processing step has a negligible effect on solution cost compared to the observed computation speed up owing to the reduced graph size. Importantly, STITCHER was shown to consistently generate trajectories faster than two state-of-the-art optimization-based planners while never violating constraints. Hardware experiments further proved that our planner is effective in generating trajectories suitable for position and attitude tracking control while remaining within set physical limits. Future work includes developing a receding horizon implementation for navigating through unknown environments, the use of imitation learning to improve search efficiency, and learning motion primitives for more general optimal control problems. Acknowledgments The authors would like to thank lab members Grace Kwak, Ryu Adams, James Row, and Qiyuan Wu for implementation and hardware support. REFERENCES [1] D. Mellinger and V. Kumar, \"Minimum snap trajectory generation and control for quadrotors,\" in IEEE International Conference on Robotics and Automation, 2011, pp. 2520-2525. [2] C. Richter, A. Bry, and N. Roy, \"Polynomial trajectory planning for aggressive quadrotor flight in dense indoor environments,\" in International Symposium of Robotics Research, 2016, pp. 649-666. [3] H. Oleynikova, M. 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Lopez, \"A contracting hierarchical observer for pose-inertial fusion,\" arXiv preprint , 2023."}}},{"rowIdx":95,"cells":{"id":{"kind":"string","value":"2510.14890"},"raw":{"kind":"string","value":"EM Approaches to Nonparametric\nEstimation for Mixture of Linear Regressions\nAndrew Welbaum\nDepartment of Statistics, George Mason University\nand\nWanli Qiao\nDepartment of Statistics, George Mason University\nOctober 17, 2025\nAbstract\nIn a mixture of linear regression model, the regression coefficients are treated as\nrandom vectors that may follow either a continuous or discrete distribution.\nWe\npropose two Expectation-Maximization (EM) algorithms to estimate this prior dis-\ntribution. The first algorithm solves a kernelized version of the nonparametric max-\nimum likelihood estimation (NPMLE). This method not only recovers continuous\nprior distributions but also accurately estimates the number of clusters when the\nprior is discrete. The second algorithm, designed to approximate the NPMLE, tar-\ngets prior distributions with a density. It also performs well for discrete priors when\ncombined with a post-processing step. We study the convergence properties of both\nalgorithms and demonstrate their effectiveness through simulations and applications\nto real datasets.\nKeywords: Mixture of linear regressions, EM algorithm, nonparametric maximum likeli-\nhood estimation, gradient ascent, mixture models, clustering\n1\narXiv:2510.14890v1 [stat.ME] 16 Oct 2025\n\n1\nIntroduction\nThe mixture of linear regression model extends the multiple linear regression by allow-\ning the coefficient vector to have a continuous or discrete prior distribution, and has a\nwide number of applications when the response and covariates can have individualized\nor clustered linear relation, including market segmentation (Wedel and Kamakura, 2000),\nmedical studies (Schlattmann, 2009), educational research (Ding, 2006), and various indus-\ntry and economic studies, such as housing construction (Quandt, 1972), wage prediction\n(Quandt and Ramsey, 1978), and climate-economic growth relationships (Yao and Song,\n2015). The model we consider in this article is as follows: There are n independent obser-\nvations (x1,y1),⋯,(xn,yn) ∈Rd × R generated from the model\nyi = x⊺\ni βi + σzi,\n(1)\nwhere β1,⋯,βn,z1,⋯,zn are independent of β1,⋯,βn\niid∼G∗and z1,⋯,zn\niid∼N(0,1), and\nσ > 0. Here, G∗is a probability distribution on Rd, which may be discrete or continuous.\nWe will assume that σ is known, but G∗, the true distribution, is not known and needs\nto be estimated. In practice, an unknown σ can be estimated using cross-validation.\nThe estimation of G∗can be further used to make statistical inference for individual βi\nthrough a plug-in approach. The posterior distribution of βi given (xi,yi) with respect to\na prior distribution G (which is G∗if the prior is correctly identified) is\nP(βi ∈A ∣xi,yi,G) = ∫A ϕσ (yi −x⊺\ni β)dG(β)\n∫Rd ϕσ (yi −x⊺\ni β)dG(β),\n(2)\nfor any measurable set A ⊂Rd, where ϕ is the standard normal density, and ϕσ(⋅) =\n(1/σ)ϕ(⋅/σ). The posterior mean of βi given (xi,yi) is\nE(βi ∣xi,yi,G) = ∫Rd ϕσ (yi −x⊺\ni β)βdG(β)\n∫Rd ϕσ (yi −x⊺\ni β)dG(β) .\n(3)\nWe can then obtain a point estimate of βi if we replace G by an estimator of G∗in\nE(βi ∣xi,yi,G).\n2\n\nWhen G∗is discrete, an additional interesting task is to cluster (xi,yi)’s based on this\nestimation. Suppose G∗(β) = ∑K\nj=1 πjδβj(β), where K is unknown, πj represents the weight\nof the component j such that ∑K\nk=1 πk = 1 and πk > 0, and δβj is the Dirac delta with point\nmass at βj. The posterior distribution in (2) when G = G∗takes the form of\nK\n∑\nj=1\n[\nπjϕσ (yi −x⊺\ni βj)\n∑K\nk=1 πkϕσ (yi −x⊺\ni βk)]δβj(β).\n(4)\nIn an oracle setting, if we knew the true prior distribution G∗= G, we could use this to\nidentify cluster membership of the points (xi,yi)’s by assigning (xi,yi) to the component\nj with the highest posterior probability, i.e.,\narg max\nj∈{1,...,K}\nπjϕσ (yi −x⊺\ni βj)\n∑K\nk=1 πkϕσ (yi −x⊺\ni βk).\n(5)\nIf the estimator of G∗is in the form of ∑\nˆ\nK\nj=1 ˆπjδˆβj, the above clustering rule is updated by\nreplacing {(πk,βk) ∶k = 1,...,K} with {(ˆπk, ˆβk) ∶k = 1,..., ˆK}.\nA nonparametric approach to estimating G∗is nonparametric maximum likelihood es-\ntimation (NPMLE), which does not assume a parametric form of G∗. The NPMLE for\nmixture models was first described by Kiefer and Wolfowitz (1956). Some recent NPMLE-\nrelated works include Jiang and Zhang (2009); Koenker and Mizera (2014); Dicker and\nZhao (2016); Jagabathula et al. (2020); Polyanskiy and Wu (2020); Saha and Guntuboyina\n(2020); Deb et al. (2021); Gu and Koenker (2022); and Jiang and Guntuboyina (2025). Of\nnote, Jiang and Zhang (2009) introduce a general maximum likelihood empirical Bayesian\nmethod for estimatng a vector of means in a mixture of Gaussians problem, assuming in-\ndependent and identically distributed errors. Their method aims to find the MLE of the\ndistribution of the unknown parameters with a discrete set of support points using an EM\nalgorithm. Koenker and Mizera (2014) introduce an interior point method that estimates\nthe nonparametric MLE of the Gaussian mixture problem. Also, Jiang and Guntuboyina\n(2025) describe an NPMLE approach to the mixture of linear regression problem using a\nmethod called Conditional Gradient Method (CGM).\n3\n\nWe now describe the NPMLE to estimate G∗. The log-likelihood function for model\n(1) with a prior distribution G given (x1,y1),⋯,(xn,yn) is\nG ↦L(G) ∶=\nn\n∑\ni=1\nlog f G\nxi(yi),\n(6)\nwhere\nf G\nxi(yi) = ∫ϕσ (yi −x⊺\ni β)dG(β), i = 1,⋯,n.\n(7)\nWe seek a maximizer of L over the set G of probability distributions G supported on X ⊂Rd.\nThat is, the NPMLE problem can be stated as\nˆG ∈arg max\nG∈G\nL(G).\n(8)\nIf X is compact or satisfies a regularity condition met by Rd, (Jiang and Guntuboyina,\n2025) establishes the existence of an NPMLE ˆG that is a probability measure supported on\nat most n points in X and is a consistent estimator of G∗in terms of the L´evy-Prokhorov\nmetric.\nIf G is the class of mixture of K probability distributions where K is known, that is,\nG = GK ∶= {G =\nK\n∑\nj=1\nπjδβj ∶\nK\n∑\nj=1\nπj = 1, πj > 0, βj ∈Rd},\nthen the model becomes a finite mixture of linear regressions, and this problem can be\nsolved by the regular EM algorithm for a known number of components (Leisch, 2004).\nThe modern EM algorithm was formalized in Dempster et al. (1977). DeSarbo and Cron\n(1988) introduced an EM algorithm for the cluster of linear regression problem with K\ncomponents, which extended previous work on estimating two-component models such as\nin Quandt (1972); Hosmer (1974); and Quandt and Ramsey (1978).\nIn the classic EM algorithm, the number of components, if it is unknown, must be\nestimated prior to initiating the algorithm. Some approaches to estimating this number\ninclude methods based on the incomplete and complete log-likelihoods, Fischer informa-\ntion matrix, Bayesian criteria, and information theory (e.g., AIC) (Hawkins et al., 2001;\n4\n\nMelnykov and Melnykov, 2012). On the other hand, methods based on NPMLE such as\nthe approach in Jiang and Guntuboyina (2025) do not appear to be able to accurately\ndetect the true number of components in the mixture model. Although the consistency\nfor NPMLE has been established in Jiang and Guntuboyina (2025), in practice, a true\ncluster may correspond to a few small clusters detected by NPMLE, where the sum of the\nestimated weights approximate the true weight of that single cluster (see Figure 1).\nFigure 1: Estimated regression lines for a mixture of linear regression model with three\ntrue components: y = 3 −x, y = 1 + 1.5x, and y = −1 + .5x with true weights (.3, .3, and .4),\nrespectively, and noise with σ = .5. Gray points are (xi,yi)’s of size n = 5000. Left panel:\nNPMLE solution estimated using the CGM method in Jiang and Guntuboyina (2025).\nThere are 10 estimated regression lines with weights ranging from 0.012 to 0.343, although\nthree clusters appear to be formed with aggregated weights 0.310, 0.296, and 0.393. Right\npanel: NPKMLE solution estimated by an EM algorithm developed in this paper. There\nare exactly 3 estimated regression lines with weights 0.314, 0.299 and 0.386, respectively.\nIn our approach, instead of maximizing (8) without imposing a form restriction on G,\nwe restrict the set G to Gkde, which is the set of kernel density estimators with a given\nkernel function and a bandwidth based on n points in Rd and solve\nˆGkde ∈arg max\nG∈Gkde\nL(G).\n(9)\n5\n\nWe call this the nonparametric kernel maximum likelihood estimator (NPKMLE). The\noptimization is reduced to finding the n points that define the KDE solution since the\nkernel function and the bandwidth are given.\nWe develop an EM algorithm that aims to find the solution to the above optimization\nproblem. The algorithm is able to estimate a mixture of linear regression model with a true\nprior distribution that is discrete or continuous, as opposed to the classical EM-algorithm’s\nrestriction to estimating the parameters of a mixture of regression model with a finite\nnumber of components. In the case of a discrete prior distribution, our algorithm does not\nrequire a prior estimate of the number of components, and is able to determine the true\nnumber of components automatically (see Figure 1). Hence the output of our algorithm\ncan be used for clustering in the finite mixture of linear regression model.\nIf the prior distribution G∗is known to have a density, we also develop another EM\nalgorithm that converges to NPMLE. This EM algorithm falls short when the density does\nnot exist. However, specific post-processing steps tailored for the underlying structure of\nG∗can remediate the issue, for example, when G∗is supported on a finite set or has a low\ndimensional manifold as it support.\nThis article is structured as follows: In Section 2, we present the mixture of linear\nregression problem and discuss our approaches to solving it. We introduce an EM algorithm\nthat can be used when the true distribution G∗has a density, as well as a different EM\nalgorithm when we drop this assumption. We also discuss the algorithms’ properties. In\nSection 3, we demonstrate the performance of our algorithms using simulations and show\ntheir utility in real-world applications. All the proofs are provided in the Appendix.\n2\nMethods\nIn the discussion below, we distinguish two cases between G∗has a density and G∗does\nnot necessarily have a density (e.g., when G∗is discrete with an unknown number of\n6\n\ncomponents), and describe an EM algorithm for each case.\nOur starting point is the NPMLE. Note that L(G) in (6) is the “incomplete-data”\nlog-likelihood function (meaning that the information of βi for each pair (xi,yi) is not\navailable). If G has a density g, then, abusing the notation L, which was used for L(G),\nthe log-likelihood can be written as\ng ↦L(g) ∶=\nn\n∑\ni=1\nlog [∫ϕσ (yi −x⊺\ni β)g(β)dβ].\n(10)\nThe corresponding “complete-data” log-likelihood function is\nL(G∣x,y,β) =\nn\n∑\ni=1\nlog [ϕσ (yi −x⊺\ni βi)g(βi)],\n(11)\nwhere x = (x1,⋯,xn), y = (y1,⋯,yn), and β = {β1,⋯,βn}. Here g should be understood\nas the density of G if it exists; If G is discrete such that G = ∑K\nj=1 πjδβj, then g(β) =\n∑K\nj=1 πj1βj(β), where 1βj is the indicator function at βj.\n2.1\nEM-NPMLE algorithm: when G∗has a density function\nWe first assume G∗is a continuous distribution for which a density function g∗exists. In\nthis case, the posterior density of βi given (xi,yi) with respect to g∗exists and is\nfi(β ∣xi,yi,g∗) =\nϕσ (yi −x⊺\ni β)g∗(β)\n∫Rd ϕσ (yi −x⊺\ni β)g∗(β)dβ .\n(12)\nWe provide an iterative algorithm to approximate g∗in what follows.\nWith an initial\nestimate of g∗, we can obtain the posterior density of βi given (xi,yi) with respect to this\ninitial estimate. Then it is a natural idea to use the average of the posterior densities\nacross all βi to update the estimate of g∗, and the process is continued iteratively until\nconvergence. More formally, with an initialization density g(0) for βi’s, for t = 0,1,2,⋯, the\nalgorithm iterates between\nf (t+1)\ni\n(β) ≡fi(β ∣xi,yi,g(t)) =\nϕσ (yi −x⊺\ni β)g(t)(β)\n∫Rd ϕσ (yi −x⊺\ni β)g(t)(β)dβ , i = 1,⋯,n,\n(13)\n7\n\nand\ng(t+1) = 1\nn\nn\n∑\ni=1\nf (t+1)\ni\n.\n(14)\nIt turns out that this simple but elegant algorithm can be interpreted as an EM al-\ngorithm, and it is inspired by a similar algorithm in the setting of mixture of distribu-\ntions (Vardi et al., 1985; Laird and Louis, 1991; Vardi and Lee, 1993; Chung and Lindsay,\n2015; Chae et al., 2018). We will call this an EM-NPMLE algorithm. To see this is an EM\nalgorithm, taking the expectation of L(g∣x,y,β) (we again abuse notation L) with respect\nto the posterior density of β given x,y,g(t), we get\nEβ∣x,y,g(t)L(g∣x,y,β) =\nn\n∑\ni=1\n∫Rd log [ϕσ (yi −x⊺\ni β)g(β)]ϕσ (yi −x⊺\ni β)g(t)(β)dβ\n∫Rd ϕσ (yi −x⊺\ni β)g(t)(β)dβ\n.\n(15)\nThis is the E-step in an EM algorithm. Let Gden be the space of distributions on Rd for\nwhich their density functions exist. We would like to find a maximizer of (15) in Gden.\nNote that ∫Rd g(u)du = 1 for G ∈Gden such that g is the density function of G, which is a\nrestriction in the optimization. Define the Lagrangian function for the optimization as\nF(g;g(t)) ∶= Eβ∣x,y,g(t)L(g∣x,y,β) −n[∫Rd g(u)du −1],\n(16)\nwhere n is the Lagrange multiplier. Notice that\narg max\nG∈Gden\nF(g;g(t)) = arg max\nG∈Gden\nEβ∣x,y,g(t)L(g∣x,y,β).\n(17)\nTo find the solution to (17), we can first take the derivative of F(g;g(t)) with respect to g,\nwhich is a functional derivative. In general, the functional derivative of a functional D(f)\nis defined as δD/δf such that\nlim\nϵ→0 [D[f + ϵη] −D[f]\nϵ\n] = ∫\nδD\nδf (x)η(x)dx,\n(18)\nwhere η is an arbitrary function and ϵ is a scalar value (Parr and Weitao, 1994).\n8\n\nTaking the functional derivative of F(g;g(t)) with respect to g, we obtain\nδF(g;g(t))\nδg\n=\nn\n∑\ni=1\nϕσ (yi −x⊺\ni β)g(t)(β)\ng(β)∫Rd ϕσ (yi −x⊺\ni β)g(t)(β)dβ −n.\n(19)\nBy letting δF(g;g(t))\nδg\n= 0 and setting the solution as g(t+1), we recover the algorithm in (14):\ng(t+1)(β) = 1\nn\nn\n∑\ni=1\nϕσ (yi −x⊺\ni β)g(t)(β)\n∫Rd ϕσ (yi −x⊺\ni β)g(t)(β)dβ .\n(20)\nThe following proposition states that (14) provides the solution to the M-step.\nProposition 1. In the above setting,\ng(t+1) = arg max\nG∈Gden\nEβ∣x,y,g(t)L(g∣x,y,β), t = 0,1,2,⋯,\n(21)\nthat is, g(t+1) is the unique maximizer of the expectation of the complete log-likelihood given\ng(t).\nThe output of this algorithm converges to NPMLE under mild assumptions, and the\nconvergence limit g(∞) can then be used as a final estimate for g∗. The following theorem\nis similar to Proposition 2 in Chae et al. (2018) and its proof is omitted. Recall that X is\nthe support of G∗.\nTheorem 2.1. Suppose for every i = 1,⋯,n, that β →ϕσ (yi −x⊺\ni β) is a continuous and\nstrictly positive map on X. In addition, assume that for each ϵ > 0 there exists a compact\nX0 ⊂X where sup\nβ∈X ∁\n0\nϕσ (yi −x⊺\ni β) < ϵ for all i = 1,⋯,n. If a unique NPMLE ˆG exists, then\nas t →∞, G(t) converges weakly to ˆG, where G(t) is the probability distribution function of\ng(t).\n2.2\nEM-NPKMLE algorithm: when G∗possibly does not have a\ndensity\nThe success of the EM-NPMLE method in Section 2.1 requires G∗to have a density, but\nmay not perform well when this is not true. For example, it is not expected to give a\n9\n\nconsistent estimate of the number of components when G∗has a finite number of support\npoints, because g(t+1) from (14) will always be a density, although it converges to the\nNPMLE, which has at most n support points (see Theorem 1 in Jiang and Guntuboyina\n(2025)). Figure 2 illustrates an example of the estimated regression lines with coefficients\nsampled from the output of the EM-NPMLE algorithm in Section 2.1.\nThe algorithm\ndeveloped in Jiang and Guntuboyina (2025) aiming to approximate the NPMLE also tends\nto overestimate the number of components in the mixture of linear regression model.\nFigure 2: The regression lines of different colors representing 100 β coefficients sampled\nfrom an estimated density generated from the output of the EM-NPMLE algorithm for the\nsame simulation as used in Figure 1.\nTo address this issue when G∗does not have a density function, for example, when it is\nonly supported on K points in Rd, where we emphasize that we do not assume that K is\nknown, we propose to restrict the maximization of the likelihood function in (8) to a subset\nGkde, which is the set of the kernel density estimates (KDEs) with a bandwidth h > 0 and a\n10\n\nkernel function V based on n points. The optimization problem becomes NPKMLE in (9).\nThe kernel function V is a probability density function on Rd and we specifically require\nthat it is differentiable and spherically symmetric such that V (x) = v(∥x∥2) for some v,\nwhere v ∶R≥0 →R≥0 is called the profile of V (Comaniciu and Meer, 2002). An example of\nV is the density function of the standard normal distribution on Rd. We can write\nGkde ≡Gkde(v,h,n) = { 1\nnhd\nn\n∑\nℓ=1\nv(∥⋅−˜βℓ∥2\nh2\n) ∶˜β1,..., ˜βn ∈Rd}.\n(22)\nHere we fix v and h so that each element in Gkde is determined by ˜β = {˜β1,..., ˜βn} ⊂Rd,\nand the corresponding distribution is denoted Gβ. The optimization in (9) is reduced to\nfinding the ˜β associated with the NPKMLE, with the solution denoted by ˆβ. The empirical\ndistribution based on ˆβ (which is discrete) is used to estimate G∗and cluster all (xi,yi)’s\nusing a plug-in approach as described in Section 1.\nThe above approach works even when the true distribution is discrete, because we can\nalways get a kernel density estimate using a sample, even from a discrete distribution.\nTo solve NPKMLE, we will again apply the EM idea.\nRecall that given a current\nestimate G(t) of G∗with density g(t), the expectation of “complete-data” log-likelihood in\n(11) with respect to the posterior distribution is given in (15). Since we require g(t) to be\na KDE in Gkde, G(t) is characterized by a set of n points in Rd denoted by β(t) = {β(t)\nℓ\n∈\nRd ∶ℓ= 1,⋯,n}, such that we can write G(t) = Gβ(t). This is the E-step of our algorithm.\nFor the M-step, we seek a g(t+1) such that\ng(t+1) ∈arg max\ng∈Gkde\nQ(G;G(t)),\n(23)\nwhere\nQ(G;G(t)) = Eβ∣x,y,G(t)L(G∣x,y,β).\n(24)\nIn other words, we would like to maximize the function in (15) with respect to g but require\ng to take the form of\n11\n\ng(β) = g(t+1)(β) =\n1\nnhd\nn\n∑\nℓ=1\nv(∥β −β(t+1)\nℓ\n∥2\nh2\n),\n(25)\nwhere {β(t+1)\nℓ\n∈Rd ∶ℓ= 1,⋯,n} =∶β(t+1) is a new set of points to be determined, which are\nviewed as an update from β(t). In what follows we also use β(t) as a vector by stacking all\nits elements together.\nSince any g ∈Gkde is only determined by n (unspecified) points in Rd, the optimization\nproblem in (23) is reduced to finding these n points. There is no closed form solution to\nthis optimization problem in general, and therefore a gradient-ascent type algorithm is pro-\nposed, which features adaptive step sizes that do not need to be chosen. The gradient-ascent\nalgorithm has no guarantee to converge to a global maximum, and hence the optimization\nin (23) is understood to find a local maximum. This leads to an EM algorithm involving\ntwo loops, as given in Algorithm 1 below, which we call EM-NPKMLE.\nAlgorithm 1 EM-NPKMLE algorithm\nInput: x,y,\nβ(0) = {β(0)\n1 ,⋯,β(0)\nn }\nFor t = 0,1,2,...\nν(0) = β(t)\nFor r = 0,1,2,...\nν(r+1) ←Ð ξ(ν(r);β(t),x,y), where ξ is given in (32)\nEnd\nβ(t+1) ←Ð ν(∞)\nEnd\nOutput: empirical distribution of {β(∞)\nj\n∶j = 1,⋯,n}\nThe key step in the algorithm is the update from ν(r) to ν(r+1), which can be indeed\nunderstood as a gradient ascent method because it can be expressed in the following form:\n12\n\nν(r+1) = ξ(ν(r);β(t),x,y) ∶= ν(r) +\n1\nC(ν(r),β(t),x,y)∇Q(Gν(r);G(t)),\n(26)\nwhere ν(r) is the current update approaching to β(t+1), ∇Q is the gradient of Q(Gν;G(t))\nwith respect to ν, and 1/C(ν(r),β(t),x,y) is an adaptive step size—see (31) below for the\ndefinition of C.\nWe now derive the result in (26). Denote\nvh(∥x∥2) ∶= v(∥x∥2/h2) and v′\nh(∥x∥2) ∶= v′(∥x∥2/h2).\nAlso, let w = −v′ and wh = −v′\nh. Plugging (25) into (15), we get\nQ(Gν;G(t))\n=Eβ∣x,y,G(t)L(Gν∣x,y,β)\n=\nn\n∑\ni=1\n∫Rd log [ϕσ (yi −x⊺\ni β)\n1\nnhd ∑n\nℓ=1 vh(∥β −νℓ∥2)]Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n,\n(27)\nwhere\nSi(β,β(t),x,y) = ϕσ (yi −x⊺\ni β)\nn\n∑\nℓ=1\nvh(∥β −β(t)\nℓ∥2).\n(28)\nTaking the derivative of Q with respect to νℓ, ℓ= 1,⋯,n, where νℓis the ℓth entry of ν, we\nhave\nζ(νℓ;β(t),x,y) ∶= ∂Q(Gν;G(t))\n∂νℓ\n=\nn\n∑\ni=1\n∫Rd\nwh(∥β−νℓ∥2)(β−νℓ)\n∑n\nm=1 vh(∥β−νm∥2) Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n, ℓ= 1,⋯,n.\n(29)\nSetting ζ(νℓ;β(t),x,y) = 0 we can write\nn\n∑\ni=1\n∫Rd\nwh(∥β−νℓ∥2)β\n∑n\nm=1 vh(∥β−νm∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶\n=A(νℓ;β(t),x,y)\n(30)\n=νℓ\nn\n∑\ni=1\n∫Rd\nwh(∥β−νℓ∥2)\n∑n\nm=1 vh(∥β−νm∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n, ℓ= 1,⋯,n\n´¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¸¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¹¶\n=C(νℓ,β(t),x,y)\n.\n(31)\n13\n\nOr equivalently,\nξ(νℓ;β(t),x,y) ∶= A(νℓ;β(t),x,y)\nC(νℓ,β(t),x,y) = νℓ, ℓ= 1,⋯,n.\n(32)\nHence, the gradient of Q(Gν;G(t)) with respect to ν is\n∇Q(Gν;G(t))\n(33)\n=( ζ(ν1;β(t),x,y),⋯,ζ(νn;β(t),x,y) )\n⊺\n=( C(ν1,β(t),x,y)[ξ(ν1;β(t),x,y) −ν1],⋯,C(νn,β(t),x,y)[ξ(νn;β(t),x,y) −νn] )\n⊺\n,\nwhich justifies (26). We now show that Algorithm 1 generates an increasing sequence of\n(27) as a function of ν(r). Here we require that the profile v is monotonically decreasing\nand function log(∑n\ni=1 v(xi)) is convex, which is satisfied when V is the Gaussian kernel\nsince the LogSumExp function is convex.1\nTheorem 2.2. In the current setting, the expectation of the complete log-likelihood function\nQ in (27) as a function of ν(r) is monotonically non-decreasing, i.e., for r = 0,1,⋯,\nQ(Gν(r+1);G(t)) ≥Q(Gν(r);G(t)),\nand Q(Gν(r);G(t)) converges as r →∞, where Gν(r+1), Gν(r), and G(t) are the distributions\nof the kernel density estimations of G using ν(r+1), ν(r), and β(t), respectively.\nThe convergence of ν(r) is not implied by the above result but will be assumed in what\nfollows. Now we show the monotonicity of the incomplete likelihood function in (6) as a\nfunction of the sequence G(t).\nTheorem 2.3. The incomplete likelihood function L(G) in (6) is bounded from above,\nand is a monotonically non-decreasing function of the sequence G(t), that is, L(G(t+1)) ≥\nL(G(t)), and hence L(G(t)) converges as t →∞.\n1See\nhttps://www.math.uwaterloo.ca/~hwolkowi/henry/teaching/w10/367.w10/367miscfiles/\npages48to60.pdf.\n14\n\nIn the EM-NPKMLE algorithm, we take the sequence νr in our Algorithm to conver-\ngence, which can be viewed as a complete M-step to locally maximize (23), and can be com-\nputationally expensive sometimes. As a lightweight variant, we may just run the gradient\nascent step in the inner loop of the algorithm one time instead of taking it to convergence,\nwhich is a generalized (or gradient) EM (GEM) algorithm, and is called GEM-NPKMLE.\nSimilar to the result in Theorem 2.3, the incomplete likelihood function L evaluated at the\ndistribution sequence G produced by GEM-NPKMLE algorithm is also non-decreasing and\nconvergent.\n2.3\nPost-processing EM-NPMLE: when G∗possibly does not have\na density\nAlthough the EM-NPMLE algorithm presented in Section 2.1 to approximate NPMLE is\nnot directly suitable for the case where the true prior distribution is discrete, if a clustering\nalgorithm is applied as a post-processing step to a random sample generated from the\ndensity g(t) for t sufficiently large produced in (14), we can still obtain a good result to\nestimate the components in the finite mixture of the linear regression model. The key\nobservation is that as t grows to infinity, g(t) gets closer to the NPMLE and the true prior\ndistribution, formed by peaks near the true support points (see Figure 2). The clustering\nalgorithm we choose to use in this postprocessing step is the mean shift algorithm, which\ndoes not require the specification of the number of clusters, and naturally uses local modes\n(peaks in density) as the central idea of clustering. The mean shift algorithm is a density-\nbased clustering algorithm that iteratively calculates a local weighted mean starting at a\nsample point. A kernel function acts as a weight function for this local weighted mean,\nand a bandwidth determines the level of smoothing. A sequence of local weighted means is\niteratively calculated until convergence, which can be shown to approximate the gradient\nflow of the kernel density estimator using the given sample. All sample points that converge\n15\n\nto the same local mode are considered part of the same cluster (Comaniciu et al., 2003).\nThe post-processing idea can be applied not only when we know the true prior distribu-\ntion is discrete, but can also be extended to some other settings when we have some specific\ninformation about the distribution. For example, an interesting scenario occurs when the\ntrue prior distribution is supported on a low-dimensional submanifold in Rd. In this case,\nthe distribution does not have a density with respect to the Lebesgue measure, and hence\ndoes not satisfy the assumptions when we apply the EM-NPMLE algorithm in Section 2.1.\nHowever, we can use a random sample from the output of this EM-NPMLE algorithm as\ninitialization and apply the subspace constrained mean shift (SCMS) algorithm (Ozertem\nand Erdogmus, 2011; Qiao and Polonik, 2021), which extends the mean shift algorithm to\nextract low-dimensional features called ridges from data—in fact, we can view a set of local\nmodes as a 0-dimensional submanifold for which the mean shift algorithm is applicable.\nLike the mean shift algorithm, the SCMS algorithm also requires a kernel (weight) func-\ntion and a smoothing bandwidth. However, the SCMS projects the gradient of the kernel\ndensity estimator onto a subspace spanned by a subset of the eigenvectors of the Hessian\nmatrix using the kernel density estimator. We demonstrate this approach in a simulation\nstudy in Section 3.1.2.\nWe emphasize that the specific post-processing procedure depends on the further knowl-\nedge or assumption about the true prior distribution. Jiang and Guntuboyina (2025) de-\nvelop an algorithm to approximate NPMLE, and in order to improve the estimation of the\nclusters when the prior distribution is discrete, they add a BIC trimming step, which has\na similar purpose as the mean shift post-processing step using the output of EM-NPMLE\nalgorithm in Section 2.1.\nIn practice, it can be difficult to decide whether the prior distribution is continuous or\ndiscrete (see the CO2 vs GDP real data example in Section 3.2), which can create challenges\nof determining the best post-processing steps to use. As a comparison, the EM-NPKMLE\n16\n\nalgorithm we present in Section 2.2 can be used to handle both discrete and continuous\nprior distributions without requiring a post-processing step.\n3\nNumerical Results\nFor the EM-NPMLE algorithm introduced in Section 2.1, it requires an initial density g(0),\nwhich can be chosen as a (non-informative) uniform distribution over a bounded subset in\nRd. For the EM-NPKMLE algorithm developed in Section 2.2, we need a set of initial points\nβ(0), which can be chosen by sampling from the same uniform distribution. Alternatively,\none can also use a random sample from the estimated density after running the EM-NPMLE\nalgorithm. This creates initial data points that reflect the underlying distribution G∗more\naccurately than assuming no prior knowledge of the distribution G∗and using samples\nfrom an arbitrary distribution (e.g., a uniform distribution) as the starting values. Our\nsimulation results below indicate that the choice of the initial points β(0) does not have\na strong influence on the results, as the performance of the algorithm is good for either\nchoice described above.\nWe used the Gaussian kernel as the kernel function V for EM-NPKMLE in all the sim-\nulations below. We propose using a bandwidth based on the maximal smoothing principle\nas described in Terrell (1990). Based on this principle, the bandwidth is chosen based on\nthe maximum smoothing consistent with the density’s estimated dispersion. We find this\napproach advantageous because by using such a bandwidth, we guard against choosing a\nsmaller bandwidth that may introduce spurious features, such as additional local modes in\nthe estimated distribution that do not exist in G∗. The bandwidth is defined as follows:\nhOS = U × [(d + 8)\nd+6\n2 πd/2R(V )\n16nΓ(d+8\n2 )d(d + 2) ]\n1\nd+4\n,\n(34)\nwhere R(V ) = ∫V 2, which is equal to 1/(4π) when V is a Gaussian kernel (Terrell, 1990;\nDavies, 2013). For U, we calculate the Gaussian-scaled interquartile range (IQR) (i.e.,\n17\n\nIQR/1.34, a robust estimator of the standard deviation of a normal density) of the values\nsampled from the estimated density in each dimension in our data and then take the mean,\nas proposed in Davies and Hazelton (2010). We emphasize that this bandwidth selection\nstrategy is used only when the initialization is based on sampling from the EM-NPMLE\noutput. Otherwise an adjustment may be needed for a different initilization.\nThe numerical study includes simulations where the true distribution is discrete and\ncontinuous. For the simulation using a discrete distribution, when we examined the per-\nformance of the EM-NPKMLE, we tested two different initialization procedures. We also\nran the GEM-NPKMLE with a single iteration in the inner loop, as discussed at the end of\nSection 2.2. We compared the EM-NPKMLE performance with the mean-shift algorithm\nas a post-processing step after sampling from the output density based on the EM-NPMLE\nalgorithm, and with the CGM developed in Jiang and Guntuboyina (2025). In addition,\nwe included a study verifying that cross-validation can effectively estimate σ that was as-\nsumed to be known. For the continuous distribution case, we generated one simulation and\ncompared the results visually using the EM–NPMKLE, the EM-NPMLE with and without\na post-processing step, and the CGM. We also implemented our EM algorithms in two real\napplications.\nWe used R version 4.1.2 (R Core Team, 2023) for all the numerical results except for the\nCGM method in Jiang and Guntuboyina (2025), where we used the Python code referenced\nin their paper.\n3.1\nSimulations\n3.1.1\nSimulation 1\nWe used the following model to test the algorithm, which has also been used in Jiang and\nGuntuboyina (2025). The mixture of linear regression model has three components with\n18\n\nprior weights .3, .3, and .4, respectively:\ny = 3 −x + ε,\ny = 1 + 1.5x + ε,\n(35)\ny = −1 + .5x + ε,\nwhere the noise ε has a N(0,σ2) distribution with σ = .5. The x values were generated\nfrom a uniform distribution on [−1,3]. We ran our algorithm on 200 simulated datasets\n(see Figure 3 for an example). We assume that σ is known until we present the result using\ncross-validation.\nFigure 3: An example of a simulated dataset from the model in (35) for σ = .5 and n=500\n(left panel) and the corresponding true components in the mixture of linear regression\nmodel represented by different colors (right panel).\nIn our simulation experiment, we used the uniform distribution over [−4,4]2 as the\ninitialization for the EM-NPMLE algorithm, to iteratively calculate g(t+1) using the ex-\npression in (14) until the L2 distance between g(t) and g(t+1) fell below a predefined small\nthreshold. The final estimated density was used to sample n points as the starting points\nfor the EM-NPKMLE algorithm. We stepped through sample sizes starting from 500 until\n10,000. The EM-NPKMLE algorithm produced a set of points that defines a KDE solution\n19\n\nto (23). Suppose its empirical distribution can be represented by ˆG = ∑\nˆ\nK\nj=1 ˆπjδˆβj where the\nˆβj’s are all distinct, that is, we aggregate the solution points according to their unique\nvalues. For a mixture of linear regression model that has K components with K unknown,\nthe true prior distribution G∗= ∑K\nj=1 πjδβj is estimated by ˆG, where each βj is estimated\nby the closest ˆβj, and each πj by the associated ˆπj. To cluster a pair of points (xi,yi), it\nis assigned to a class j for which the posterior probability is maximized, i.e.,\narg max\nj\nˆπjϕσ (yi −x⊺\ni ˆβj)\n∑\nˆ\nK\nk=1 ˆπkϕσ (yi −x⊺\ni ˆβk)\n.\n(36)\nWe report the average adjusted Rand index (Hubert and Arabie, 1985), the proportion\nof samples that were correctly identified as having three components, and the average\nWasserstein-2 distance (Nguyen, 2011) between the probability measures ˆG produced by\nthe algorithm in question and G∗, i.e.,\nW2(G∗, ˆG) ∶=\n⎡⎢⎢⎢⎢⎣\nmin\nγ∈Γ(π,ˆπ)\nK\n∑\ni=1\nˆ\nK\n∑\nj=1\n∣∣βi −ˆβj∣∣2γij\n⎤⎥⎥⎥⎥⎦\n1/2\n,\n(37)\nwhere γ = (γij)1≤i≤K,1≤j≤ˆ\nK ∈[0,1]K× ˆ\nK, Γ(π, ˆπ) is the set of all matrices γ satisfying ∑K\ni=1 γij =\nˆπj, ∑\nˆ\nK\nj=1 γij = πi, where π = {π1,⋯,πK} are the true weights, and ˆπ = {ˆπ1,⋯, ˆπ ˆ\nK} are the\nestimated weights; and βi’s are the true coefficients and ˆβj’s are the estimated coefficients\n(see Table 1).\nAdditionally, note that some points at the intersections of the linear regression com-\nponents will unavoidably be mislabeled using (36) by any estimators of G∗, which results\nin an adjusted Rand index less than 1. As an oracle reference for our algorithm, we cal-\nculated the average adjusted Rand index when G was specified as G∗, and the posterior\nprobabilities were directly calculated. We also report the bias and standard deviation of the\nestimated coefficients and weights when three components, the true number of components,\nwere estimated.\nEM-NPKMLE:\nFor the results of the EM-NPKMLE algorithm, see Table 1 for infor-\nmation on the proportion of simulations in which the true number of components were\n20\n\ndetected, and Table 2 for bias and standard deviation of the errors. Standard deviations\nare also reported for all values (in parentheses), where applicable.\nTable 1: Results of Simulation 1 using EM-NPKMLE algorithm.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\nAvg W-2 Distance\n500\n0.415 (0.119)\n0.665 (0.033)\n0.540\n0.818 (0.137)\n1000\n0.567 (0.034)\n0.663 (0.023)\n0.990\n0.562 (0.097)\n5000\n0.645 (0.011)\n0.664 (0.010)\n0.995\n0.343 (0.080)\n10000\n0.651 (0.008)\n0.663 (0.008)\n0.995\n0.288 (0.081)\nTable 2: Results of Simulation 1 using EM-NPKMLE algorithm.\nSample Size\nEstimate\n1st Comp\n2nd Comp\n3rd Comp\n500\nβ: bias ± SD\n(-0.360, 0.227)±(0.053, 0.074)\n(-0.565, -0.363)±(0.234, 0.138)\n(0.460, 0.230)±(0.119, 0.077)\nπ: bias ± SD\n-0.005±0.024\n-0.026±0.059\n0.031±0.056\n1000\nβ: bias ± SD\n(-0.291, 0.157)±(0.037, 0.042)\n(-0.397, -0.164)±(0.112, 0.062)\n(0.297, 0.143)±(0.048, 0.046)\nπ: bias ± SD\n0.002±0.019\n-0.003±0.017\n0.001±0.019\n5000\nβ: bias ± SD\n(-0.155, 0.033)±(0.014, 0.016)\n(-0.157, 0.003)±(0.032, 0.018)\n(0.149, 0.038)±(0.019, 0.014)\nπ: bias ± SD\n-0.001±0.008\n0±0.007\n0.001±0.009\n10000\nβ: bias ± SD\n(-0.112, 0.004)±(0.011, 0.010)\n(-0.115, 0.013)±(0.018, 0.012)\n(0.113, 0.027)±(0.013, 0.010)\nπ: bias ± SD\n-0.001±0.006\n0±0.006\n0.001±0.006\nThe results suggest that our algorithm produces consistent estimates of the β compo-\nnents and weights, as well as a consistent estimator of the number of components. As\nn increases, the values of these estimates approach the true parameter values and true\nnumber of components. As n gets larger, we observe that all of the points that are mis-\nlabeled are at the intersections of the components, and the slightly worse performance of\nthe estimated G from our algorithm than using G∗is due to the bias of the β estimates\ncreating intersection areas that are slightly larger than when G is set to G∗. Since we\nuse the bandwidth based on the oversmoothing principle, we accept slightly larger bias in\nexchange for avoiding detecting spurious modes (components) in the data.\nIn what follows we compare the above results using the EM-NPKMLE algorithm with\nthose using the the EM-NPMLE algorithm with mean shift as a post-processing step as\n21\n\ndescribed in Section 2.3 and the conditional gradient method (CGM) as described in Jiang\nand Guntuboyina (2025). We stepped through the same sequence of sample sizes as in\nTables 1 and 2.\nEM-NPMLE and mean-shift post-processing:\nAs shown in Tables 3 and 4, the\nmean shift algorithm as a post-processing step after EM-NPMLE produced results that\nsuggest consistent estimation of the true β values and weights, as well as the true number\nof components. When using the mean shift for post-processing, we also used the bandwidth\nbased on the maximal smoothing principle as described previously.\nCGM algorithm in Jiang and Guntuboyina (2025):\nAs shown in Tables 5 and\n6, the results of the CGM algorithm do not suggest that this algorithm can consistently\nestimate the number of modes, though the decreasing average W-2 distance with increasing\nsample size supports the idea of convergence in distribution between the true and estimated\ndistributions.\nTable 3: Results of Simulation 1 using EM-NPMLE and mean-shift post-processing.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\nAvg W-2 Distance\n500\n0.620 (0.094)\n0.665 (0.033)\n0.915\n0.612 (0.238)\n1000\n0.657 (0.024)\n0.663 (0.023)\n0.995\n0.465 (0.141)\n5000\n0.663 (0.010)\n0.664 (0.010)\n0.990\n0.311 (0.092)\n10000\n0.662 (0.008)\n0.663 (0.008)\n1\n0.265 (0.089)\nEM-NPKMLE with uniform initialization:\nWe also ran our EM-NPKMLE algo-\nrithm on the same model as in (35) using initial points drawn from a uniform distribution\nover [−4,4]2 to demonstrate that our algorithm does not depend on initial points sampled\nfrom the estimated density g derived from the EM-NPKMLE algorithm. We currently do\nnot have a bandwidth selection strategy for the uniform initialization, and we manually\ntuned the bandwidth by using the maximal smoothing principle multiplied by a constant\n22\n\nTable 4: Results of Simulation 1 using EM-NPMLE and mean-shift post-processing.\nSample Size\nEstimate\n1st Comp\n2nd Comp\n3rd Comp\n500\nβ: bias ± SD\n(-0.012, 0.034)±(0.062, 0.040)\n(-0.173, -0.080)±(0.108, 0.045)\n(0.088, 0.031)±(0.062, 0.042)\nπ: bias ± SD\n-0.001±0.025\n-0.002±0.027\n0.030±0.028\n1000\nβ: bias ± SD\n(-0.002, 0)±(0.049, 0.031)\n(-0.077, -0.021)±(0.061, 0.029)\n(0.041, 0.013)±(0.044, 0.026)\nπ: bias ± SD\n0.002±0.019\n-0.002±0.018\n0±0.019\n5000\nβ: bias ± SD\n(0, 0)±(0.021, 0.015)\n(-0.009, 0.005)±(0.023, 0.012)\n(0.013, -0.003)±(0.018, 0.013)\nπ: bias ± SD\n-0.001±0.008\n0±0.008\n0.001±0.009\n10000\nβ: bias ± SD\n(0, 0)±(0.014, 0.010)\n(-0.007, 0.004)±(0.017, 0.011)\n(0.010, -0.002)±(0.012, 0.008)\nπ: bias ± SD\n-0.001±0.006\n0±0.006\n0±0.006\nTable 5: Results of Simulation 1 using CGM.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\n500\n0.590 (0.066)\n0.665 (0.033)\n0\n1000\n0.581 (0.068)\n0.663 (0.023)\n0\n5000\n0.602 (0.053)\n0.664 (0.010)\n0\n10000\n0.596 (0.057)\n0.663 (0.008)\n0\nTable 6: Results of Simulation 1 using CGM.\nSample Size\nAvg W-2 Distance\nMean No. of Comp\n500\n0.439 (0.109)\n9.295 (2.095)\n1000\n0.373 (0.093)\n8.550 (1.744)\n5000\n0.260 (0.069)\n7.800 (1.504)\n10000\n0.229 (0.064)\n7.570 (1.274)\nc = 1.15. As shown in Tables 7 and 8, the results are comparable to running our EM-\nNPKMLE algorithm using the output of EM-NPMLE for the initialization.\nTable 7: Results of Simulation 1 using EM-NPKMLE Algorithm with uniform initialization.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\nAvg W-2 Dist\n500\n0.146 (0.043)\n0.665 (0.033)\n0.030\n1.130 (0.059)\n1000\n0.283 (0.140)\n0.663 (0.023)\n0.515\n1.020 (0.114)\n5000\n0.624 (0.012)\n0.664 (0.010)\n1\n0.422 (0.060)\n10000\n0.642 (0.008)\n0.663 (0.008)\n1\n0.320 (0.052)\n23\n\nTable 8: Results of Simulation 1 using EM-NPKMLE Algorithm with uniform initialization.\nSample Size\nEstimate\n1st Comp\n2nd Comp\n3rd Comp\n500\nβ: bias ± SD\n(-0.453, 0.273)±(0.035, 0.072)\n(-0.888, -0.504)±(0.259, 0.149)\n(0.643, 0.325)±(0.136, 0.070)\nπ: bias ± SD\n0.029±0.017\n-0.039±0.211\n0.010±0.218\n1000\nβ: bias ± SD\n(-0.430, 0.333)±(0.039, 0.048)\n(-0.717, -0.364)±(0.211, 0.094)\n(0.556, 0.299)±(0.094, 0.075)\nπ: bias ± SD\n0.049±0.018\n-0.119±0.066\n0.070±0.063\n5000\nβ: bias ± SD\n(-0.224, 0.101)±(0.013, 0.018)\n(-0.234, -0.029)±(0.036, 0.020)\n(0.222, 0.073)±(0.020, 0.016)\nπ: bias ± SD\n0.009±0.007\n-0.018±0.006\n0.009±0.007\n10000\nβ: bias ± SD\n(-0.164, 0.047)±(0.010, 0.011)\n(-0.164, 0.001)±(0.019, 0.013)\n(0.164, 0.043)±(0.014, 0.011)\nπ: bias ± SD\n0±0.005\n-0.008±0.004\n0.008±0.005\nGEM-NPKMLE:\nAs discussed at the end of Section 2.2, GEM-NPKMLE is a variant\nof EM-NPKMLE, where we run a single iteration in the inner loop instead of running it to\nconvergence. This approach saves computation time. For example, for the simulation with\nn=5000 and the EM-NPMLE initialization, the EM-NPKLME took an average of 115.888\n(40.759) minutes, while the GEM-NPKMLE took an average of 15.448 (2.194) minutes. It\nalso produced similar results compared to Tables 1, 2, 3, and 4 for larger sample sizes, as\ncan be seen in Tables 9 and 10. This suggests that the GEM-NPKMLE approach is viable\nwhen saving computation time is important and there is a sufficient sample size. The initial\npoints we used for GEM-NPKMLE were sampled from the output of EM-NPMLE, which\nwas the same strategy that was used in the first part of our simulations. However, in this\ncase, we multiplied the initial bandwidth by a constant c = 1.2 to improve the algorithm’s\nperformance.\nTable 9: Results of Simulation 1 using GEM-NPKMLE algorithm.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\nAvg W-2 Dist\n500\n0.133 (0.034)\n0.665 (0.033)\n0.235\n1.070 (0.062)\n1000\n0.213 (0.106)\n0.663 (0.023)\n0.195\n0.993 (0.059)\n5000\n0.614 (0.012)\n0.664 (0.010)\n0.840\n0.401 (0.066)\n10000\n0.637 (0.008)\n0.663 (0.008)\n0.930\n0.323 (0.070)\n24\n\nTable 10: Results of Simulation 1 using GEM-NPKMLE algorithm.\nSample Size\nEstimate\n1st Comp\n2nd Comp\n3rd Comp\n500\nβ: bias ± SD\n(-0.527,0.360)±(0.067, 0.083)\n(-0.983,-0.642)±(0.135,0.071)\n(0.839,0.387)±(0.071,0.067)\nπ: bias ± SD\n0.005±(0.027)\n-0.298±(0.002)\n0.292±(0.027)\n1000\nβ: bias ± SD\n(-0.414,0.308)±(0.052,0.059)\n(-1.052,-0.551)±(0.175,0.079)\n(0.757,0.402)±(0.092,0.075)\nπ: bias ± SD\n0.007±0.019\n-0.291±0.048\n0.284±0.042\n5000\nβ: bias ± SD\n(-0.241,0.104)±(0.014,0.018)\n(-0.282,-0.069)±(0.041,0.023)\n(0.231,0.088)±(0.020,0.016)\nπ: bias ± SD\n-0.001±0.008\n0±0.007\n0.001±0.009\n10000\nβ: bias ± SD\n(-0.184,0.056)±(0.010,0.011)\n(-0.196,-0.009)±(0.022,0.013)\n(0.172,0.053)±(0.014,0.011)\nπ: bias ± SD\n-0.001±(0.006)\n0±(0.006)\n0.001±(0.006)\nCross-validation for unknown σ:\nIf σ is unknown, we can implement a cross-validation\n(CV) approach found in Jiang and Guntuboyina (2025) to estimate the parameter σ. First\nwe divide the dataset T = {(xi,yi) ∶\ni = 1,...,n} into C folds: T1,...,TC.\nFor each\nc ∈{1,2,...C}, one fold Tc is left out as test data and the remaining folds are used to obtain\nan estimator ˆG−c for G∗. Let\nf\nˆG−c\nxi (yi) = ∫ϕσ (yi −x⊺\ni β)d ˆG−c(β), i = 1,⋯,n.\nThe σ value that minimizes the following objective function is selected and denoted by ˆσ:\nCV (σ) = −\nC\n∑\nc=1\n∑\n(xi,yi)∈Tc\nlog(f\nˆG−c\nxi (yi)).\nIn our simulation, we used a random sample from the output of the EM-NPMLE al-\ngorithm as the initialization for our GEM-NPKMLE algorithm (with the oversmoothing\nbandwidth) to calculate ˆG−c. We tested this method on the model in (35) with C = 5 to get\nˆσ, which was further plugged into the EM-NPKMLE algorithm. The results are provided\nin Tables 11 and 12, which are comparable to those in Tables 1 and 2.\n3.1.2\nSimulation 2\nUnlike the EM-NPMLE algorithm, our EM-NPKMLE algorithm does not assume the\nknowledge of a particular structure in the true distribution, such as a low-dimensional\n25\n\nTable 11: Results of Simulation 1 using EM-NPKMLE algorithm and cross-validation to\nestimate σ.\nSample Size\nAvg Adj RI\nAvg Adj RI G*\nProp of Sim with 3 Comp\nAvg W-2 Dist\n500\n0.302 (0.204)\n0.665 (0.033)\n0.390\n0.891 (0.192)\n1000\n0.574 (0.045)\n0.663 (0.023)\n0.975\n0.557 (0.118)\n5000\n0.645 (0.011)\n0.664 (0.010)\n0.995\n0.343 (0.080)\n10000\n0.651 (0.008)\n0.663 (0.008)\n0.995\n0.288 (0.081)\nTable 12: Results of Simulation 1 using EM-NPKMLE algorithm and cross-validation to\nestimate σ.\nSample Size\nEstimate\n1st Comp\n2nd Comp\n3rd Comp\n500\nβ: bias ± SD\n(-0.328,0.201)±(0.080,0.090)\n(-0.425,-0.281)±(0.236,0.177)\n(0.375,0.193)±(0.162,0.091)\nπ: bias ± SD\n-0.010±0.025\n-0.028±0.071\n0.039±0.066\n1000\nβ: bias ± SD\n(-0.273, 0.143)±(0.048, 0.047)\n(-0.355,-0.151)±(0.137,0.068)\n(0.271,0.136)±(0.073,0.047)\nπ: bias ± SD\n-0.003±0.018\n-0.002±0.018\n0.005±0.019\n5000\nβ: bias ± SD\n(-0.155, 0.033)±(0.0143, 0.016)\n(-0.157,0.003)±(0.032,0.018)\n(0.149,0.038)±(0.019,0.014)\nπ: bias ± SD\n-0.001±0.008\n0±0.007\n0.001±0.009\n10000\nβ: bias ± SD\n(-0.112, 0.004)±(0.011, 0.010)\n(-0.115, 0.013)±(0.018, 0.012)\n(0.113, 0.027)±(0.013, 0.010)\nπ: bias ± SD\n-0.001±0.006\n0±0.006\n0.001±0.006\nmanifold as the support. To further demonstrate this, we use a continuous distribution as\nthe model, which consists of a bivariate distribution represented by a mixture of uniform\ndistributions over two concentric circles, each with center at the origin (see Figure 4). The\nmodel is given in (1), where σ = .2 is assumed to be known, and the true distribution G∗is\n1\n2 × Uniform{B(1)} + 1\n2 × Uniform{B(2)},\nwhere B(r) = {β ∈R2 ∶∣∣β∣∣= r}. This model has also been used in (Jiang and Guntuboyina,\n2025).\nWe uniformly sampled 10000 design points xi’s from [−1,3]. For our EM-NPKMLE\nalgorithm, we used the output of the EM-NPMLE algorithm as the initial β(0) points and\nused the oversmoothing bandwidth. We got similar results when we used points sampled\nfrom a uniform distribution on [−4,4]2 as the initial β(0) and applied a constant to the\n26\n\nFigure 4: Left panel: the support of G∗in Simulation 2; Right panel: the sample used in\nSimulation 2.\noversmoothing bandwidth.\nIn this simulation we also tested the EM-NPMLE algorithm with and without SCMS\nas a post-processing step, as described in Section 2.3. The EM-NPMLE algorithm was\ninitialized by a uniform density over [−4,4]2. Then the SCMS algorithm was applied to\n10000 β points sampled from the estimated density produced by the EM-NPMLE algorithm.\nThe bandwidth used for the SCMS algorithm was selected by the biased cross-validation\nmethod aiming to minimize an asymptotic mean integrated squared error (AMISE) estimate\nbetween the estimated and true density gradients (Sain et al., 1994). This was implemented\nby the ‘Hbcv.diag’ function in the R package ks (Duong, 2024), and we used the mean of\nthe diagonal entries in the selected diagonal bandwidth matrix.\nAs shown in Figure 5 for a single random sample, the points produced by the EM-\nNPKMLE algorithm are roughly evenly distributed near the support of G∗, while the\nsupport does not seem to be recovered well by the CGM method. The distribution produced\nby the EM-NPMLE algorithm has a visible concentration near the support of G∗, and\n27\n\napplying the SCMS algorithm as a post-processing step further extracted the 1-dimensional\nstructure.\nIn Table 13 we show the results of applying the Wasserstein-2 distance between the\nprobability measures G∗and ˆG from the applicable algorithms. Using SCMS algorithm for\npost-processing has the best performance, followed by the EM-NPMLE algorithm without\npost-processing. As two algorithms without knowing the prior distribution is continuous\nor requiring a post-processing step, the EM-NPKMLE is an improvement over the CGM.\nTable 13: W-2 Distance between G∗and ˆG for the simulation depicted in Figure 5\nMethod\nW-2 Distance\nEM-NPKMLE\n0.328\nCGM\n0.394\nEM-NPMLE\n0.190\nEM-NPMLE + SCMS\n0.157\n28\n\n(a)\n(b)\n(c)\n(d)\nFigure 5: Comparison of the true mixture distribution and the estimated distributions of\nβ for different algorithms in Simulation 2, where the red circles are the support of G∗, and\npoint sizes are proportional to weights. (a) EM-NPKMLE algorithm: 52 support points,\nweights ranging from 0.0002 to 0.085. (b) CGM method: 34 support points, weights ranging\nfrom 0.011 to 0.054. (c) 104 points sampled from the output of the EM-NPMLE algorithm,\nalso used as the input for the SCMS algorithm. (d) post-processing result after using the\nSCMS algorithm, where all points have weight= 10−4.\n29\n\n3.2\nApplications\nApplication:\nCO2-GDP\nWe tested our EM-NPKMLE algorithm on data of 159 countries’ CO2 emissions and\nGDP per capita in 2015 taken from Bolt et al. (2018); Friedlingstein et al. (2020); Ritchie\net al. (2020); and (UN, 2022). This dataset was also analyzed in Jiang and Guntuboyina\n(2025). Identifying distinct components among the relationship between CO2 emissions\nand economic output could reveal countries that have high economic output and low CO2\nemissions, potentially revealing favorable policies to pursue. Other papers have also looked\ninto this relationship from a mixture of regression approach, such as Huang and Yao (2012)\nand Hurn et al. (2003), though these papers use different data sources. As in Jiang and\nGuntuboyina (2025), we used data on CO2 emission in per capita in 10-ton increments and\nGDP per capita in 10 thousand U.S. dollars.\nWe used a 10-fold cross-validation method as described in Section 3.1 for the CO2-GDP\ndataset, and used the same bandwidth selection method as described at the beginning of\nSection 3. Our estimate of σ from this CV method is 0.160. The results of our algorithm\nwith this estimated σ suggest that there are two growth paths with larger estimated prior\nweights, with possible deviations as evidenced by the other components with smaller weights\n(see Table 14 and Figure 6). This result differs from Jiang and Guntuboyina (2025), who\nidentified ten components before applying the BIC trimming, and five components after\nBIC trimming. We note that the two components with largest weights detected from our\nalgorithm ((0.022, 0.179) and (-0.070, 0.343)) are similar to the two components with the\nlargest weights from Jiang and Guntuboyina (2025) before and after their BIC trimming\nmethod was applied, that is, (0.030, 0.170) and (-0.010, 0.230), with pre-BIC trimming\nweights of 0.290 and 0.180, and post-BIC trimming weights of 0.270 and 0.280, respectively.\nApplication:\nMusic Tone Perception\nWe also tested our EM-NPKMLE algorithm on another dataset analyzed in Jiang and\n30\n\nFigure 6: CO2-GDP Relationship: The points represent CO2 emissions in per capita in\n10-ton increments and GDP per capita in 10 thousand U.S. dollars for individual countries.\nThe lines represent the estimated component regression lines, whose estimated coefficients\ncan be found in Table 14. The thicker lines are the components with larger weights.\nTable 14: Results from CO2/GDP Application\n1st Comp\n2nd Comp\n3rd Comp\n4th Comp\n5th Comp\nβ Estimates\n(0.022, 0.179)\n(-0.070, 0.343)\n(-0.101, 0.628)\n(0.105, 0.120)\n(0.839, -0.051)\nEstimated Weights\n0.484\n0.358\n0.088\n0.057\n0.013\nGuntuboyina (2025). The data was initially collected by Cohen (1980) in an experiment\nexamining how trained musicians perceive tone. See Cohen (1980) for further information\non the experiment. This dataset, which can be found in the R packages mixtools and fpc,\n31\n\ncontains one musician’s tuning that was observed 150 times (Benaglia et al., 2009; Hennig,\n2024). See Figure 7, where the y-axis displays the musician’s assessed tone ratio and the x-\naxis depicts the true tone ratio (Viele and Tong, 2002; Yao and Song, 2015). Other studies\nof this dataset were carried out in De Veaux (1989); Viele and Tong (2002); and Yao and\nSong (2015); but in these studies the number of components was pre-specified at two. We\napplied our EM algorithm that does not require prespecifying the number of components.\nWe used the same cross-validation method as in the CO2-GDP application to estimate\nσ, and our estimate is 0.210. We also used the same bandwidth selection method as in the\nCO2-GDP application, that is, maximal smoothing bandwidth. Our results revealed two\nmain components (see Figure 7 and Table 15). In terms of the number of components, the\nresults are consistent with the music perception theory when the experiment took place,\nwith one component as y = x and another as y = 2 (Jiang and Guntuboyina, 2025). Our\nresults (see Table 15) are comparable to the two largest components computed using the\nCGM algorithm (both pre- and post BIC trimming) in Jiang and Guntuboyina (2025),\nthat is, (1.898, 0.054), (0, 0.989), with weights 0.540 and 0.260 (pre-BIC trimming), and\n0.700 and 0.300 (post-BIC trimming), respectively. However, the CGM method detected\nsix components before the BIC trimming method was applied.\nTable 15: Results from Music Tone Application\n1st Comp\n2nd Comp\nβ Estimates\n(0.012, 0.967)\n(1.292, 0.361)\nEstimated Weights\n0.427\n0.573\n4\nDiscussion\nIn this paper we first introduce an EM algorithm to approximate NPMLE in the setting of\nmixture of linear regressions when the prior distribution for the coefficients has a density.\n32\n\nFigure 7: Music Tone Perception: The blue points represent a musician’s assessed tone\nratio (y) and the true tone ratio (x) pairs. The estimated regression lines (see Table 15)\nare also depicted for the two components. The thicker line is the component with the larger\nweight.\nWe then develop another EM algorithm to approximate NPKMLE that can automatically\ndetect the number of components for a discrete prior distribution, and is able to uncover\na continuous prior distribution. We provide some theoretical support for our algorithms\nas well as demonstrate their performance in simulations and applications. In particular,\nour simulation results suggest that our EM-NPKMLE algorithm can consistently estimate\nthe true number of components and the β coefficients and their associated weights when\nthe prior distribution is discrete.\nWhile our EM-NPKMLE algorithm does have useful\n33\n\nqualities, it requires the selection of a bandwidth, which can be challenging. We selected\nthe oversmoothing bandwidth based on its ability to avoid the appearance of spurious\nfeatures that may not belong to the true prior distribution, but further investigation into\nthe bandwidth selection could yield better methods, e.g., adaptive bandwidths for different\niterations of the algorithm, or introducing unique bandwidths for each dimension of the\ndata. While we demonstrate the algorithm’s ability to uncover both discrete and continuous\ndistributions, investigating its theoretical properties is an area of further research. It is clear\nthat our methods can be extended to the setting of mixture of distributions, and we will\ninvestigate this in a separate article.\nA\nAppendix\nA.1\nProof of Proposition 1\nProof. Let the Kullback-Leibler (KL) divergence between distributions P and Q with den-\nsities p and q be\nDKL(P∥Q) = ∫p(x)log p(x)\nq(x)dx.\n(38)\nWe can write\nEβ∣x,y,G(t)L(g∣x,y,β)\n(39)\n= −\nn\n∑\ni=1 ∫f (t+1)\ni\n(β)log f (t+1)\ni\n(β)\ng(β)\ndβ +\nn\n∑\ni=1 ∫log [ϕσ (yi −x⊺\ni β)f (t+1)\ni\n(β)]f (t+1)\ni\n(β)dβ\n(40)\n= −\nn\n∑\ni=1\nDKL(F (t+1)\ni\n∥G) +\nn\n∑\ni=1 ∫log [ϕσ (yi −x⊺\ni β)f (t+1)\ni\n(β)]f (t+1)\ni\n(β)dβ,\n(41)\nwhere F (t+1)\ni\nis the distribution associated with f (t+1)\ni\n. Hence, maximizing Eβ∣x,y,G(t)L(g∣x,y,β)\nover g is equivalent to minimizing\n1\nn ∑n\ni=1 DKL(F (t+1)\ni\n∥G) over G ∈Gden.\nLet g(t+1) =\n1\nn ∑n\ni=1 f (t+1)\ni\n. Notice that\n1\nn\nn\n∑\ni=1\nδDKL(F (t+1)\ni\n∥G)\nδg\n∣\ng=g(t+1)\n= −1\nn\nn\n∑\ni=1\nf (t+1)\ni\ng(t+1) = −1.\n(42)\n34\n\nUsing Theorem 2 in (Nishiyama, 2020), we conclude that g(t+1) is the unique minimizer of\n1\nn ∑n\ni=1 DKL(F (t+1)\ni\n∥G), and hence the unique maximizer of Eβ∣x,y,G(t)L(g∣x,y,β,G(t)).\nA.2\nProof of Theorem 2.2\nProof. Using the subgradient inequality for differentiable convex functions, we have\nlog[\nn\n∑\nℓ=1\nv(yℓ)] −log[\nn\n∑\nℓ=1\nv(xℓ)] ≥∑n\nℓ=1 v′(xℓ)(yℓ−xℓ)\n∑n\nℓ=1 v(xℓ)\n,\n∀xℓ,yℓ∈[0,∞).\nAlso, recall that w = −v′. Then\nQ(Gν(r+1);G(t)) −Q(Gν(r);G(t))\n(43)\n=\nn\n∑\ni=1\n∫Rd log [ϕσ (yi −x⊺\ni β)\n1\nnhd ∑n\nℓ=1 vh(∥β −ν(r+1)\nℓ\n∥2)]Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(44)\n−\nn\n∑\ni=1\n∫Rd log [ϕσ (yi −x⊺\ni β)\n1\nnhd ∑n\nℓ=1 vh(∥β −ν(r)\nℓ∥2)]Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(45)\n≥\nn\n∑\ni=1\n∫Rd\n∑n\nℓ=1 wh(∥β−ν(r)\nℓ\n∥2)[∥β−ν(r)\nℓ\n∥2−∥β−ν(r+1)\nℓ\n∥2]\n∑n\nj=1 vh(∥β−ν(r)\nj\n∥2)\nSi(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(46)\n=\nn\n∑\ni=1\n∫Rd\n∑n\nℓ=1 wh(∥β−ν(r)\nℓ\n∥2)[2ν(r+1)⊺\nℓ\nβ−2ν(r)⊺\nℓ\nβ−∥ν(r+1)\nℓ\n∥2+∥ν(r)\nℓ\n∥2]\n∑n\nj=1 vh(∥β−ν(r+1)\nj\n∥2)\nSi(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(47)\n=\nn\n∑\nℓ=1\n2ν(r+1)⊺\nℓ\nn\n∑\ni=1\n∫Rd\nwh(∥β−ν(r)\nℓ\n∥2)β\n∑n\nj=1 vh(∥β−ν(r)\nj\n∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(48)\n−\nn\n∑\nℓ=1\n2ν(r)⊺\nℓ\nn\n∑\ni=1\n∫Rd\nwh(∥β−ν(r)\nℓ\n∥2)β\n∑n\nj=1 vh(∥β−ν(r)\nj\n∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(49)\n−\nn\n∑\nℓ=1\n∥ν(r+1)\nℓ\n∥2\nn\n∑\ni=1\n∫Rd\nwh(∥β−ν(r)\nℓ\n∥2)\n∑n\nj=1 vh(∥β−ν(r)\nj\n∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(50)\n+\nn\n∑\nℓ=1\n∥ν(r)\nℓ∥2\nn\n∑\ni=1\n∫Rd\nwh(∥β−ν(r)\nℓ\n∥2)\n∑n\nj=1 vh(∥β−ν(r)\nj\n∥2)Si(β,β(t),x,y)dβ\n∫Rd Si(β,β(t),x,y)dβ\n(51)\n=\nn\n∑\nℓ=1\n[2ν(r+1)⊺\nℓ\nA(ν(r)\nℓ;β(t),x,y) −2ν(r)⊺\nℓ\nA(ν(r)\nℓ;β(t),x,y)\n(52)\n−∥ν(r+1)\nℓ\n∥2C(ν(r)\nℓ,β(t),x,y) + ∥ν(r)\nℓ∥2C(ν(r)\nℓ,β(t),x,y)].\n(53)\n35\n\nIt then follows from (30),(31), and (32) that\nQ(Gν(r+1);G(t)) −Q(Gν(r);G(t))\n(54)\n≥\nn\n∑\nℓ=1\n−∥ν(r+1)\nℓ\n∥2C(ν(r)\nℓ,β(t),x,y) + 2∥ν(r+1)\nℓ\n∥2C(ν(r)\nℓ;β(t),x,y)\n(55)\n−2ν(r)⊺\nℓ\nν(r+1)\nℓ\nC(ν(r)\nℓ;β(t),x,y) + ∥ν(r)\nℓ∥2C(ν(r)\nℓ,β(t),x,y)]\n(56)\n=\nn\n∑\nℓ=1\nC(ν(r)\nℓ,β(t),x,y)∥ν(r+1)\nℓ\n−ν(r)\nℓ∥2\n(57)\n≥0,\n(58)\nwhere the last inequality holds because C(ν(r)\nℓ,β(t),x,y) > 0 by noticing that w is always\npositive.\nNext we show that Q(Gν;G(t)) has an upper bound. Notice that\nsup\nβ\n∣ϕσ (yi −x⊺\ni β) 1\nnhd\nn\n∑\nℓ=1\nvh(∥β −νℓ∥2)∣≤\n∥v∥∞\n√\n2πσhd =∶B,\n(59)\nwhere ∥v∥∞= supt ∣v(t)∣. It then follows from (27) that\nsup\nν Q(Gν;G(t)) ≤nlog B.\n(60)\nHence the sequence Q(Gν(r);G(t)) converges.\nA.3\nProof of Theorem 2.3\nProof. For a distribution G on Rd with density function g, let\nP(β,G) =\nn\n∑\ni=1\nlog(fi(βi ∣xi,yi,g)),\n(61)\nwhere fi(βi ∣xi,yi,g) is defined in (12). We can write\nP(β,G) =\nn\n∑\ni=1\nlog [\nϕσ (yi −x⊺\ni βi)g(βi)\n∫Rd ϕσ (yi −x⊺\ni β)g(β)dβ ]\n=\nn\n∑\ni=1\nlog [ϕσ (yi −x⊺\ni βi)g(βi)] −\nn\n∑\ni=1\nlog [∫Rd ϕσ (yi −x⊺\ni β)g(β)dβ].\n36\n\nUsing the definitions in (6) and (11),\nL(G) = L(G∣x,y,β) −P(β,G).\nTaking the expectation of each side with respect to β∣x,y,G(t), we have\nL(G) = Eβ∣x,y,G(t)(L(G)) = Eβ∣x,y,G(t)(L(G∣x,y,β)) −Eβ∣x,y,G(t)(P(β,G))\n=∶Q(G;G(t)) −H(G;G(t)).\nNotice that\nL(G(t+1)) −L(G(t)) = [Q(G(t+1);G(t)) −Q(G(t);G(t))] −[H(G(t+1);G(t)) −H(G(t);G(t))].\nWe have already shown in Theorem 2.2 that Q(G(t+1);G(t)) ≥Q(G(t);G(t)).\nTo show\nL(G(t+1)) −L(G(t)) ≥0, it remains to prove that\nH(G(t+1);G(t)) −H(G(t);G(t)) ≤0.\nThis is true because for any arbitrary distribution G with density function g,\nH(G;G(t)) −H(G(t),G(t))\n=Eβ∣x,y,G(t) [\nn\n∑\ni=1\nlog(fi(βi ∣xi,yi,g))] −Eβ∣x,y,G(t) [\nn\n∑\ni=1\nlog(fi(βi ∣xi,yi,g(t)))]\n=Eβ∣x,y,G(t) {\nn\n∑\ni=1\nlog [ fi(βi ∣xi,yi,g)\nfi(βi ∣xi,yi,g(t))]}\n=\nn\n∑\ni=1\nEβ∣x,y,G(t) {log [ fi(βi ∣xi,yi,g)\nfi(βi ∣xi,yi,g(t))]}\n≤\nn\n∑\ni=1\nlog {Eβ∣x,y,G(t) [ fi(βi ∣xi,yi,g)\nfi(βi ∣xi,yi,g(t))]}\n(by Jensen’s inequality and the concavity of the log function)\n=\nn\n∑\ni=1\nlog [∫\nfi(β ∣xi,yi,g)\nfi(β ∣xi,yi,g(t))fi(β ∣xi,yi,g(t))dβ]\n=\nn\n∑\ni=1\nlog [∫fi(β ∣xi,yi,g)dβ]\n=0.\n37\n\nWe can also conclude from Theorem 1 in (Jiang and Guntuboyina, 2025) that the incom-\nplete log likelihood L(G) is bounded from above because of the existence of a maximizer\nˆG. By the monotone convergence theorem, L(G(t)) converges.\nAcknowledgments\nThis work was partially supported by resources provided by the Office of Research Com-\nputing at George Mason University (URL: https://orc.gmu.edu).\nData Availability Statement\nThe music tone data that supports the findings of this study is openly available in the\nR packages mixtools (https://cran.r-project.org/web/packages/mixtools/index.html), fpc\n(https://cran.r-project.org/web//packages/fpc/index.html). The CO2-GDP data can be\nfound at https://github.com/hanshengjiang/npmle git.\nReferences\nBenaglia, T., Chauveau, D., Hunter, D. R., and Young, D. (2009). mixtools: An R package\nfor analyzing finite mixture models. Journal of Statistical Software, 32(6):1–29.\nBolt, J., Inklaar, R., de Jong, H., and Van Zanden, J. L. (2018). Rebasing ‘maddison’: new\nincome comparisons and the shape of long-run economic development. 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(2000). Market segmentation: Conceptual and method-\nological foundations. Springer Science & Business Media.\nYao, W. and Song, W. (2015). Mixtures of linear regression with measurement errors.\nCommunications in Statistics-Theory and Methods, 44(8):1602–1614.\n43\n"},"cleaned":{"kind":"string","value":"EM Approaches to Nonparametric Estimation for Mixture of Linear Regressions Andrew Welbaum 17, 2025 Abstract In a mixture of linear regression model, the regression coefficients are treated as random vectors that may follow either a continuous or discrete distribution. We propose two Expectation-Maximization (EM) algorithms to estimate this prior distribution. The first algorithm solves a kernelized version of the nonparametric maximum likelihood estimation (NPMLE). This method not only recovers continuous prior distributions but also accurately estimates the number of clusters when the prior is discrete. The second algorithm, designed to approximate the NPMLE, targets prior distributions with a density. It also performs well for discrete priors when combined with a post-processing step. We study the convergence properties of both algorithms and demonstrate their effectiveness through simulations and applications to real datasets. Keywords: Mixture of linear regressions, EM algorithm, nonparametric maximum likelihood estimation, gradient ascent, mixture models, clustering 1 16 Oct 2025 1 Introduction The mixture of linear regression model extends the multiple linear regression by allowing the coefficient vector to have a continuous or discrete prior distribution, and has a wide number of applications when the response and covariates can have individualized or clustered linear relation, including market segmentation (Wedel and Kamakura, 2000), medical studies (Schlattmann, 2009), educational research (Ding, 2006), and various industry and economic studies, such as housing construction (Quandt, 1972), wage prediction (Quandt and Ramsey, 1978), and climate-economic growth relationships (Yao and Song, 2015). The model we consider in this article is as follows: There are n independent observations (x1,y1),⋯,(xn,yn) ∈Rd × R generated from the model yi = x⊺ i βi + σzi, (1) where β1,⋯,βn,z1,⋯,zn are independent of β1,⋯,βn iid∼G∗and z1,⋯,zn iid∼N(0,1), and σ > 0. Here, G∗is a probability distribution on Rd, which may be discrete or continuous. We will assume that σ is known, but G∗, the true distribution, is not known and needs to be estimated. In practice, an unknown σ can be estimated using cross-validation. The estimation of G∗can be further used to make statistical inference for individual βi through a plug-in approach. The posterior distribution of βi given (xi,yi) with respect to a prior distribution G (which is G∗if the prior is correctly identified) is P(βi ∈A ∣xi,yi,G) = ∫A φσ (yi -x⊺ i β)dG(β) ∫Rd φσ (yi -x⊺ i β)dG(β), (2) for any measurable set A ⊂Rd, where φ is the standard normal density, and φσ(⋅) = (1/σ)φ(⋅/σ). The posterior mean of βi given (xi,yi) is E(βi ∣xi,yi,G) = ∫Rd φσ (yi -x⊺ i β)βdG(β) ∫Rd φσ (yi -x⊺ i β)dG(β) . (3) We can then obtain a point estimate of βi if we replace G by an estimator of G∗in E(βi ∣xi,yi,G). 2 When G∗is discrete, an additional interesting task is to cluster (xi,yi)'s based on this estimation. Suppose G∗(β) = ∑K j=1 πjδβj(β), where K is unknown, πj represents the weight of the component j such that ∑K k=1 πk = 1 and πk > 0, and δβj is the Dirac delta with point mass at βj. The posterior distribution in (2) when G = G∗takes the form of K ∑ j=1 [ πjφσ (yi -x⊺ i βj) ∑K k=1 πkφσ (yi -x⊺ i βk)]δβj(β). (4) In an oracle setting, if we knew the true prior distribution G∗= G, we could use this to identify cluster membership of the points (xi,yi)'s by assigning (xi,yi) to the component j with the highest posterior probability, i.e., arg max j∈{1,...,K} πjφσ (yi -x⊺ i βj) ∑K k=1 πkφσ (yi -x⊺ i βk). (5) If the estimator of G∗is in the form of ∑ ˆ K j=1 ˆπjδˆβj, the above clustering rule is updated by replacing {(πk,βk) ∶k = 1,...,K} with {(ˆπk, ˆβk) ∶k = 1,..., ˆK}. A nonparametric approach to estimating G∗is nonparametric maximum likelihood estimation (NPMLE), which does not assume a parametric form of G∗. The NPMLE for mixture models was first described by Kiefer and Wolfowitz (1956). Some recent NPMLErelated works include Jiang and Zhang (2009); Koenker and Mizera (2014); Dicker and Zhao (2016); Jagabathula et al. (2020); Polyanskiy and Wu (2020); Saha and Guntuboyina (2020); Deb et al. (2021); Gu and Koenker (2022); and Jiang and Guntuboyina (2025). Of note, Jiang and Zhang (2009) introduce a general maximum likelihood empirical Bayesian method for estimatng a vector of means in a mixture of Gaussians problem, assuming independent and identically distributed errors. Their method aims to find the MLE of the distribution of the unknown parameters with a discrete set of support points using an EM algorithm. Koenker and Mizera (2014) introduce an interior point method that estimates the nonparametric MLE of the Gaussian mixture problem. Also, Jiang and Guntuboyina (2025) describe an NPMLE approach to the mixture of linear regression problem using a method called Conditional Gradient Method (CGM). 3 We now describe the NPMLE to estimate G∗. The log-likelihood function for model (1) with a prior distribution G given (x1,y1),⋯,(xn,yn) is G ↦L(G) ∶= n ∑ i=1 log f G xi(yi), (6) where f G xi(yi) = ∫φσ (yi -x⊺ i β)dG(β), i = 1,⋯,n. (7) We seek a maximizer of L over the set G of probability distributions G supported on X ⊂Rd. That is, the NPMLE problem can be stated as ˆG ∈arg max G∈G L(G). (8) If X is compact or satisfies a regularity condition met by Rd, (Jiang and Guntuboyina, 2025) establishes the existence of an NPMLE ˆG that is a probability measure supported on at most n points in X and is a consistent estimator of G∗in terms of the L ́evy-Prokhorov metric. If G is the class of mixture of K probability distributions where K is known, that is, G = GK ∶= {G = K ∑ j=1 πjδβj ∶ K ∑ j=1 πj = 1, πj > 0, βj ∈Rd}, then the model becomes a finite mixture of linear regressions, and this problem can be solved by the regular EM algorithm for a known number of components (Leisch, 2004). The modern EM algorithm was formalized in Dempster et al. (1977). DeSarbo and Cron (1988) introduced an EM algorithm for the cluster of linear regression problem with K components, which extended previous work on estimating two-component models such as in Quandt (1972); Hosmer (1974); and Quandt and Ramsey (1978). In the classic EM algorithm, the number of components, if it is unknown, must be estimated prior to initiating the algorithm. Some approaches to estimating this number include methods based on the incomplete and complete log-likelihoods, Fischer information matrix, Bayesian criteria, and information theory (e.g., AIC) (Hawkins et al., 2001; 4 Melnykov and Melnykov, 2012). On the other hand, methods based on NPMLE such as the approach in Jiang and Guntuboyina (2025) do not appear to be able to accurately detect the true number of components in the mixture model. Although the consistency for NPMLE has been established in Jiang and Guntuboyina (2025), in practice, a true cluster may correspond to a few small clusters detected by NPMLE, where the sum of the estimated weights approximate the true weight of that single cluster (see Figure 1). Figure 1: Estimated regression lines for a mixture of linear regression model with three true components: y = 3 -x, y = 1 + 1.5x, and y = -1 + .5x with true weights (.3, .3, and .4), respectively, and noise with σ = .5. Gray points are (xi,yi)'s of size n = 5000. Left panel: NPMLE solution estimated using the CGM method in Jiang and Guntuboyina (2025). There are 10 estimated regression lines with weights ranging from 0.012 to 0.343, although three clusters appear to be formed with aggregated weights 0.310, 0.296, and 0.393. Right panel: NPKMLE solution estimated by an EM algorithm developed in this paper. There are exactly 3 estimated regression lines with weights 0.314, 0.299 and 0.386, respectively. In our approach, instead of maximizing (8) without imposing a form restriction on G, we restrict the set G to Gkde, which is the set of kernel density estimators with a given kernel function and a bandwidth based on n points in Rd and solve ˆGkde ∈arg max G∈Gkde L(G). (9) 5 We call this the nonparametric kernel maximum likelihood estimator (NPKMLE). The optimization is reduced to finding the n points that define the KDE solution since the kernel function and the bandwidth are given. We develop an EM algorithm that aims to find the solution to the above optimization problem. The algorithm is able to estimate a mixture of linear regression model with a true prior distribution that is discrete or continuous, as opposed to the classical EM-algorithm's restriction to estimating the parameters of a mixture of regression model with a finite number of components. In the case of a discrete prior distribution, our algorithm does not require a prior estimate of the number of components, and is able to determine the true number of components automatically (see Figure 1). Hence the output of our algorithm can be used for clustering in the finite mixture of linear regression model. If the prior distribution G∗is known to have a density, we also develop another EM algorithm that converges to NPMLE. This EM algorithm falls short when the density does not exist. However, specific post-processing steps tailored for the underlying structure of G∗can remediate the issue, for example, when G∗is supported on a finite set or has a low dimensional manifold as it support. This article is structured as follows: In Section 2, we present the mixture of linear regression problem and discuss our approaches to solving it. We introduce an EM algorithm that can be used when the true distribution G∗has a density, as well as a different EM algorithm when we drop this assumption. We also discuss the algorithms' properties. In Section 3, we demonstrate the performance of our algorithms using simulations and show their utility in real-world applications. All the proofs are provided in the Appendix. 2 Methods In the discussion below, we distinguish two cases between G∗has a density and G∗does not necessarily have a density (e.g., when G∗is discrete with an unknown number of 6 components), and describe an EM algorithm for each case. Our starting point is the NPMLE. Note that L(G) in (6) is the \"incomplete-data\" log-likelihood function (meaning that the information of βi for each pair (xi,yi) is not available). If G has a density g, then, abusing the notation L, which was used for L(G), the log-likelihood can be written as g ↦L(g) ∶= n ∑ i=1 log [∫φσ (yi -x⊺ i β)g(β)dβ]. (10) The corresponding \"complete-data\" log-likelihood function is L(G∣x,y,β) = n ∑ i=1 log [φσ (yi -x⊺ i βi)g(βi)], (11) where x = (x1,⋯,xn), y = (y1,⋯,yn), and β = {β1,⋯,βn}. Here g should be understood as the density of G if it exists; If G is discrete such that G = ∑K j=1 πjδβj, then g(β) = ∑K j=1 πj1βj(β), where 1βj is the indicator function at βj. 2.1 EM-NPMLE algorithm: when G∗has a density function We first assume G∗is a continuous distribution for which a density function g∗exists. In this case, the posterior density of βi given (xi,yi) with respect to g∗exists and is fi(β ∣xi,yi,g∗) = φσ (yi -x⊺ i β)g∗(β) ∫Rd φσ (yi -x⊺ i β)g∗(β)dβ . (12) We provide an iterative algorithm to approximate g∗in what follows. With an initial estimate of g∗, we can obtain the posterior density of βi given (xi,yi) with respect to this initial estimate. Then it is a natural idea to use the average of the posterior densities across all βi to update the estimate of g∗, and the process is continued iteratively until convergence. More formally, with an initialization density g(0) for βi's, for t = 0,1,2,⋯, the algorithm iterates between f (t+1) i (β) ≡fi(β ∣xi,yi,g(t)) = φσ (yi -x⊺ i β)g(t)(β) ∫Rd φσ (yi -x⊺ i β)g(t)(β)dβ , i = 1,⋯,n, (13) 7 and g(t+1) = 1 n n ∑ i=1 f (t+1) i . (14) It turns out that this simple but elegant algorithm can be interpreted as an EM algorithm, and it is inspired by a similar algorithm in the setting of mixture of distributions (Vardi et al., 1985; Laird and Louis, 1991; Vardi and Lee, 1993; Chung and Lindsay, 2015; Chae et al., 2018). We will call this an EM-NPMLE algorithm. To see this is an EM algorithm, taking the expectation of L(g∣x,y,β) (we again abuse notation L) with respect to the posterior density of β given x,y,g(t), we get Eβ∣x,y,g(t)L(g∣x,y,β) = n ∑ i=1 ∫Rd log [φσ (yi -x⊺ i β)g(β)]φσ (yi -x⊺ i β)g(t)(β)dβ ∫Rd φσ (yi -x⊺ i β)g(t)(β)dβ . (15) This is the E-step in an EM algorithm. Let Gden be the space of distributions on Rd for which their density functions exist. We would like to find a maximizer of (15) in Gden. Note that ∫Rd g(u)du = 1 for G ∈Gden such that g is the density function of G, which is a restriction in the optimization. Define the Lagrangian function for the optimization as F(g;g(t)) ∶= Eβ∣x,y,g(t)L(g∣x,y,β) -n[∫Rd g(u)du -1], (16) where n is the Lagrange multiplier. Notice that arg max G∈Gden F(g;g(t)) = arg max G∈Gden Eβ∣x,y,g(t)L(g∣x,y,β). (17) To find the solution to (17), we can first take the derivative of F(g;g(t)) with respect to g, which is a functional derivative. In general, the functional derivative of a functional D(f) is defined as δD/δf such that lim ε→0 [D[f + εη] -D[f] ε ] = ∫ δD δf (x)η(x)dx, (18) where η is an arbitrary function and ε is a scalar value (Parr and Weitao, 1994). 8 Taking the functional derivative of F(g;g(t)) with respect to g, we obtain δF(g;g(t)) δg = n ∑ i=1 φσ (yi -x⊺ i β)g(t)(β) g(β)∫Rd φσ (yi -x⊺ i β)g(t)(β)dβ -n. (19) By letting δF(g;g(t)) δg = 0 and setting the solution as g(t+1), we recover the algorithm in (14): g(t+1)(β) = 1 n n ∑ i=1 φσ (yi -x⊺ i β)g(t)(β) ∫Rd φσ (yi -x⊺ i β)g(t)(β)dβ . (20) The following proposition states that (14) provides the solution to the M-step. Proposition 1. In the above setting, g(t+1) = arg max G∈Gden Eβ∣x,y,g(t)L(g∣x,y,β), t = 0,1,2,⋯, (21) that is, g(t+1) is the unique maximizer of the expectation of the complete log-likelihood given g(t). The output of this algorithm converges to NPMLE under mild assumptions, and the convergence limit g(∞) can then be used as a final estimate for g∗. The following theorem is similar to Proposition 2 in Chae et al. (2018) and its proof is omitted. Recall that X is the support of G∗. Theorem 2.1. Suppose for every i = 1,⋯,n, that β →φσ (yi -x⊺ i β) is a continuous and strictly positive map on X. In addition, assume that for each ε > 0 there exists a compact X0 ⊂X where sup β∈X ∁ 0 φσ (yi -x⊺ i β) 0 and a 10 kernel function V based on n points. The optimization problem becomes NPKMLE in (9). The kernel function V is a probability density function on Rd and we specifically require that it is differentiable and spherically symmetric such that V (x) = v(∥x∥2) for some v, where v ∶R≥0 →R≥0 is called the profile of V (Comaniciu and Meer, 2002). An example of V is the density function of the standard normal distribution on Rd. We can write Gkde ≡Gkde(v,h,n) = { 1 nhd n ∑ l=1 v(∥⋅- ̃βl∥2 h2 ) ∶ ̃β1,..., ̃βn ∈Rd}. (22) Here we fix v and h so that each element in Gkde is determined by ̃β = { ̃β1,..., ̃βn} ⊂Rd, and the corresponding distribution is denoted Gβ. The optimization in (9) is reduced to finding the ̃β associated with the NPKMLE, with the solution denoted by ˆβ. The empirical distribution based on ˆβ (which is discrete) is used to estimate G∗and cluster all (xi,yi)'s using a plug-in approach as described in Section 1. The above approach works even when the true distribution is discrete, because we can always get a kernel density estimate using a sample, even from a discrete distribution. To solve NPKMLE, we will again apply the EM idea. Recall that given a current estimate G(t) of G∗with density g(t), the expectation of \"complete-data\" log-likelihood in (11) with respect to the posterior distribution is given in (15). Since we require g(t) to be a KDE in Gkde, G(t) is characterized by a set of n points in Rd denoted by β(t) = {β(t) l ∈ Rd ∶l= 1,⋯,n}, such that we can write G(t) = Gβ(t). This is the E-step of our algorithm. For the M-step, we seek a g(t+1) such that g(t+1) ∈arg max g∈Gkde Q(G;G(t)), (23) where Q(G;G(t)) = Eβ∣x,y,G(t)L(G∣x,y,β). (24) In other words, we would like to maximize the function in (15) with respect to g but require g to take the form of 11 g(β) = g(t+1)(β) = 1 nhd n ∑ l=1 v(∥β -β(t+1) l ∥2 h2 ), (25) where {β(t+1) l ∈Rd ∶l= 1,⋯,n} =∶β(t+1) is a new set of points to be determined, which are viewed as an update from β(t). In what follows we also use β(t) as a vector by stacking all its elements together. Since any g ∈Gkde is only determined by n (unspecified) points in Rd, the optimization problem in (23) is reduced to finding these n points. There is no closed form solution to this optimization problem in general, and therefore a gradient-ascent type algorithm is proposed, which features adaptive step sizes that do not need to be chosen. The gradient-ascent algorithm has no guarantee to converge to a global maximum, and hence the optimization in (23) is understood to find a local maximum. This leads to an EM algorithm involving two loops, as given in Algorithm 1 below, which we call EM-NPKMLE. Algorithm 1 EM-NPKMLE algorithm Input: x,y, β(0) = {β(0) 1 ,⋯,β(0) n } For t = 0,1,2,... ν(0) = β(t) For r = 0,1,2,... ν(r+1) ←Ð ξ(ν(r);β(t),x,y), where ξ is given in (32) End β(t+1) ←Ð ν(∞) End Output: empirical distribution of {β(∞) j ∶j = 1,⋯,n} The key step in the algorithm is the update from ν(r) to ν(r+1), which can be indeed understood as a gradient ascent method because it can be expressed in the following form: 12 ν(r+1) = ξ(ν(r);β(t),x,y) ∶= ν(r) + 1 C(ν(r),β(t),x,y)∇Q(Gν(r);G(t)), (26) where ν(r) is the current update approaching to β(t+1), ∇Q is the gradient of Q(Gν;G(t)) with respect to ν, and 1/C(ν(r),β(t),x,y) is an adaptive step size-see (31) below for the definition of C. We now derive the result in (26). Denote vh(∥x∥2) ∶= v(∥x∥2/h2) and v′ h(∥x∥2) ∶= v′(∥x∥2/h2). Also, let w = -v′ and wh = -v′ h. Plugging (25) into (15), we get Q(Gν;G(t)) =Eβ∣x,y,G(t)L(Gν∣x,y,β) = n ∑ i=1 ∫Rd log [φσ (yi -x⊺ i β) 1 nhd ∑n l=1 vh(∥β -νl∥2)]Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ , (27) where Si(β,β(t),x,y) = φσ (yi -x⊺ i β) n ∑ l=1 vh(∥β -β(t) l∥2). (28) Taking the derivative of Q with respect to νl, l= 1,⋯,n, where νlis the lth entry of ν, we have ζ(νl;β(t),x,y) ∶= ∂Q(Gν;G(t)) ∂νl = n ∑ i=1 ∫Rd wh(∥β-νl∥2)(β-νl) ∑n m=1 vh(∥β-νm∥2) Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ , l= 1,⋯,n. (29) Setting ζ(νl;β(t),x,y) = 0 we can write n ∑ i=1 ∫Rd wh(∥β-νl∥2)β ∑n m=1 vh(∥β-νm∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ ́11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 ̧11111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111¶ =A(νl;β(t),x,y) (30) =νl n ∑ i=1 ∫Rd wh(∥β-νl∥2) ∑n m=1 vh(∥β-νm∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ , l= 1,⋯,n ́111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111 ̧1111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111111¶ =C(νl,β(t),x,y) . (31) 13 Or equivalently, ξ(νl;β(t),x,y) ∶= A(νl;β(t),x,y) C(νl,β(t),x,y) = νl, l= 1,⋯,n. (32) Hence, the gradient of Q(Gν;G(t)) with respect to ν is ∇Q(Gν;G(t)) (33) =( ζ(ν1;β(t),x,y),⋯,ζ(νn;β(t),x,y) ) ⊺ =( C(ν1,β(t),x,y)[ξ(ν1;β(t),x,y) -ν1],⋯,C(νn,β(t),x,y)[ξ(νn;β(t),x,y) -νn] ) ⊺ , which justifies (26). We now show that Algorithm 1 generates an increasing sequence of (27) as a function of ν(r). Here we require that the profile v is monotonically decreasing and function log(∑n i=1 v(xi)) is convex, which is satisfied when V is the Gaussian kernel since the LogSumExp function is convex.1 Theorem 2.2. In the current setting, the expectation of the complete log-likelihood function Q in (27) as a function of ν(r) is monotonically non-decreasing, i.e., for r = 0,1,⋯, Q(Gν(r+1);G(t)) ≥Q(Gν(r);G(t)), and Q(Gν(r);G(t)) converges as r →∞, where Gν(r+1), Gν(r), and G(t) are the distributions of the kernel density estimations of G using ν(r+1), ν(r), and β(t), respectively. The convergence of ν(r) is not implied by the above result but will be assumed in what follows. Now we show the monotonicity of the incomplete likelihood function in (6) as a function of the sequence G(t). Theorem 2.3. The incomplete likelihood function L(G) in (6) is bounded from above, and is a monotonically non-decreasing function of the sequence G(t), that is, L(G(t+1)) ≥ L(G(t)), and hence L(G(t)) converges as t →∞. 1See https://www.math.uwaterloo.ca/~hwolkowi/henry/teaching/w10/367.w10/367miscfiles/ pages48to60.pdf. 14 In the EM-NPKMLE algorithm, we take the sequence νr in our Algorithm to convergence, which can be viewed as a complete M-step to locally maximize (23), and can be computationally expensive sometimes. As a lightweight variant, we may just run the gradient ascent step in the inner loop of the algorithm one time instead of taking it to convergence, which is a generalized (or gradient) EM (GEM) algorithm, and is called GEM-NPKMLE. Similar to the result in Theorem 2.3, the incomplete likelihood function L evaluated at the distribution sequence G produced by GEM-NPKMLE algorithm is also non-decreasing and convergent. 2.3 Post-processing EM-NPMLE: when G∗possibly does not have a density Although the EM-NPMLE algorithm presented in Section 2.1 to approximate NPMLE is not directly suitable for the case where the true prior distribution is discrete, if a clustering algorithm is applied as a post-processing step to a random sample generated from the density g(t) for t sufficiently large produced in (14), we can still obtain a good result to estimate the components in the finite mixture of the linear regression model. The key observation is that as t grows to infinity, g(t) gets closer to the NPMLE and the true prior distribution, formed by peaks near the true support points (see Figure 2). The clustering algorithm we choose to use in this postprocessing step is the mean shift algorithm, which does not require the specification of the number of clusters, and naturally uses local modes (peaks in density) as the central idea of clustering. The mean shift algorithm is a densitybased clustering algorithm that iteratively calculates a local weighted mean starting at a sample point. A kernel function acts as a weight function for this local weighted mean, and a bandwidth determines the level of smoothing. A sequence of local weighted means is iteratively calculated until convergence, which can be shown to approximate the gradient flow of the kernel density estimator using the given sample. All sample points that converge 15 to the same local mode are considered part of the same cluster (Comaniciu et al., 2003). The post-processing idea can be applied not only when we know the true prior distribution is discrete, but can also be extended to some other settings when we have some specific information about the distribution. For example, an interesting scenario occurs when the true prior distribution is supported on a low-dimensional submanifold in Rd. In this case, the distribution does not have a density with respect to the Lebesgue measure, and hence does not satisfy the assumptions when we apply the EM-NPMLE algorithm in Section 2.1. However, we can use a random sample from the output of this EM-NPMLE algorithm as initialization and apply the subspace constrained mean shift (SCMS) algorithm (Ozertem and Erdogmus, 2011; Qiao and Polonik, 2021), which extends the mean shift algorithm to extract low-dimensional features called ridges from data-in fact, we can view a set of local modes as a 0-dimensional submanifold for which the mean shift algorithm is applicable. Like the mean shift algorithm, the SCMS algorithm also requires a kernel (weight) function and a smoothing bandwidth. However, the SCMS projects the gradient of the kernel density estimator onto a subspace spanned by a subset of the eigenvectors of the Hessian matrix using the kernel density estimator. We demonstrate this approach in a simulation study in Section 3.1.2. We emphasize that the specific post-processing procedure depends on the further knowledge or assumption about the true prior distribution. Jiang and Guntuboyina (2025) develop an algorithm to approximate NPMLE, and in order to improve the estimation of the clusters when the prior distribution is discrete, they add a BIC trimming step, which has a similar purpose as the mean shift post-processing step using the output of EM-NPMLE algorithm in Section 2.1. In practice, it can be difficult to decide whether the prior distribution is continuous or discrete (see the CO2 vs GDP real data example in Section 3.2), which can create challenges of determining the best post-processing steps to use. As a comparison, the EM-NPKMLE 16 algorithm we present in Section 2.2 can be used to handle both discrete and continuous prior distributions without requiring a post-processing step. 3 Numerical Results For the EM-NPMLE algorithm introduced in Section 2.1, it requires an initial density g(0), which can be chosen as a (non-informative) uniform distribution over a bounded subset in Rd. For the EM-NPKMLE algorithm developed in Section 2.2, we need a set of initial points β(0), which can be chosen by sampling from the same uniform distribution. Alternatively, one can also use a random sample from the estimated density after running the EM-NPMLE algorithm. This creates initial data points that reflect the underlying distribution G∗more accurately than assuming no prior knowledge of the distribution G∗and using samples from an arbitrary distribution (e.g., a uniform distribution) as the starting values. Our simulation results below indicate that the choice of the initial points β(0) does not have a strong influence on the results, as the performance of the algorithm is good for either choice described above. We used the Gaussian kernel as the kernel function V for EM-NPKMLE in all the simulations below. We propose using a bandwidth based on the maximal smoothing principle as described in Terrell (1990). Based on this principle, the bandwidth is chosen based on the maximum smoothing consistent with the density's estimated dispersion. We find this approach advantageous because by using such a bandwidth, we guard against choosing a smaller bandwidth that may introduce spurious features, such as additional local modes in the estimated distribution that do not exist in G∗. The bandwidth is defined as follows: hOS = U × [(d + 8) d+6 2 πd/2R(V ) 16nΓ(d+8 2 )d(d + 2) ] 1 d+4 , (34) where R(V ) = ∫V 2, which is equal to 1/(4π) when V is a Gaussian kernel (Terrell, 1990; Davies, 2013). For U, we calculate the Gaussian-scaled interquartile range (IQR) (i.e., 17 IQR/1.34, a robust estimator of the standard deviation of a normal density) of the values sampled from the estimated density in each dimension in our data and then take the mean, as proposed in Davies and Hazelton (2010). We emphasize that this bandwidth selection strategy is used only when the initialization is based on sampling from the EM-NPMLE output. Otherwise an adjustment may be needed for a different initilization. The numerical study includes simulations where the true distribution is discrete and continuous. For the simulation using a discrete distribution, when we examined the performance of the EM-NPKMLE, we tested two different initialization procedures. We also ran the GEM-NPKMLE with a single iteration in the inner loop, as discussed at the end of Section 2.2. We compared the EM-NPKMLE performance with the mean-shift algorithm as a post-processing step after sampling from the output density based on the EM-NPMLE algorithm, and with the CGM developed in Jiang and Guntuboyina (2025). In addition, we included a study verifying that cross-validation can effectively estimate σ that was assumed to be known. For the continuous distribution case, we generated one simulation and compared the results visually using the EM-NPMKLE, the EM-NPMLE with and without a post-processing step, and the CGM. We also implemented our EM algorithms in two real applications. We used R version 4.1.2 (R Core Team, 2023) for all the numerical results except for the CGM method in Jiang and Guntuboyina (2025), where we used the Python code referenced in their paper. 3.1 Simulations 3.1.1 Simulation 1 We used the following model to test the algorithm, which has also been used in Jiang and Guntuboyina (2025). The mixture of linear regression model has three components with 18 prior weights .3, .3, and .4, respectively: y = 3 -x + ε, y = 1 + 1.5x + ε, (35) y = -1 + .5x + ε, where the noise ε has a N(0,σ2) distribution with σ = .5. The x values were generated from a uniform distribution on [-1,3]. We ran our algorithm on 200 simulated datasets (see Figure 3 for an example). We assume that σ is known until we present the result using cross-validation. Figure 3: An example of a simulated dataset from the model in (35) for σ = .5 and n=500 (left panel) and the corresponding true components in the mixture of linear regression model represented by different colors (right panel). In our simulation experiment, we used the uniform distribution over [-4,4]2 as the initialization for the EM-NPMLE algorithm, to iteratively calculate g(t+1) using the expression in (14) until the L2 distance between g(t) and g(t+1) fell below a predefined small threshold. The final estimated density was used to sample n points as the starting points for the EM-NPKMLE algorithm. We stepped through sample sizes starting from 500 until 10,000. The EM-NPKMLE algorithm produced a set of points that defines a KDE solution 19 to (23). Suppose its empirical distribution can be represented by ˆG = ∑ ˆ K j=1 ˆπjδˆβj where the ˆβj's are all distinct, that is, we aggregate the solution points according to their unique values. For a mixture of linear regression model that has K components with K unknown, the true prior distribution G∗= ∑K j=1 πjδβj is estimated by ˆG, where each βj is estimated by the closest ˆβj, and each πj by the associated ˆπj. To cluster a pair of points (xi,yi), it is assigned to a class j for which the posterior probability is maximized, i.e., arg max j ˆπjφσ (yi -x⊺ i ˆβj) ∑ ˆ K k=1 ˆπkφσ (yi -x⊺ i ˆβk) . (36) We report the average adjusted Rand index (Hubert and Arabie, 1985), the proportion of samples that were correctly identified as having three components, and the average Wasserstein-2 distance (Nguyen, 2011) between the probability measures ˆG produced by the algorithm in question and G∗, i.e., W2(G∗, ˆG) ∶= ⎡⎢⎢⎢⎢⎣ min γ∈Γ(π,ˆπ) K ∑ i=1 ˆ K ∑ j=1 ∣∣βi -ˆβj∣∣2γij ⎤⎥⎥⎥⎥⎦ 1/2 , (37) where γ = (γij)1≤i≤K,1≤j≤ˆ K ∈[0,1]K× ˆ K, Γ(π, ˆπ) is the set of all matrices γ satisfying ∑K i=1 γij = ˆπj, ∑ ˆ K j=1 γij = πi, where π = {π1,⋯,πK} are the true weights, and ˆπ = {ˆπ1,⋯, ˆπ ˆ K} are the estimated weights; and βi's are the true coefficients and ˆβj's are the estimated coefficients (see Table 1). Additionally, note that some points at the intersections of the linear regression components will unavoidably be mislabeled using (36) by any estimators of G∗, which results in an adjusted Rand index less than 1. As an oracle reference for our algorithm, we calculated the average adjusted Rand index when G was specified as G∗, and the posterior probabilities were directly calculated. We also report the bias and standard deviation of the estimated coefficients and weights when three components, the true number of components, were estimated. EM-NPKMLE: For the results of the EM-NPKMLE algorithm, see Table 1 for information on the proportion of simulations in which the true number of components were 20 detected, and Table 2 for bias and standard deviation of the errors. Standard deviations are also reported for all values (in parentheses), where applicable. Table 1: Results of Simulation 1 using EM-NPKMLE algorithm. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp Avg W-2 Distance 500 0.415 (0.119) 0.665 (0.033) 0.540 0.818 (0.137) 1000 0.567 (0.034) 0.663 (0.023) 0.990 0.562 (0.097) 5000 0.645 (0.011) 0.664 (0.010) 0.995 0.343 (0.080) 10000 0.651 (0.008) 0.663 (0.008) 0.995 0.288 (0.081) Table 2: Results of Simulation 1 using EM-NPKMLE algorithm. Sample Size Estimate 1st Comp 2nd Comp 3rd Comp 500 β: bias ± SD (-0.360, 0.227)±(0.053, 0.074) (-0.565, -0.363)±(0.234, 0.138) (0.460, 0.230)±(0.119, 0.077) π: bias ± SD -0.005±0.024 -0.026±0.059 0.031±0.056 1000 β: bias ± SD (-0.291, 0.157)±(0.037, 0.042) (-0.397, -0.164)±(0.112, 0.062) (0.297, 0.143)±(0.048, 0.046) π: bias ± SD 0.002±0.019 -0.003±0.017 0.001±0.019 5000 β: bias ± SD (-0.155, 0.033)±(0.014, 0.016) (-0.157, 0.003)±(0.032, 0.018) (0.149, 0.038)±(0.019, 0.014) π: bias ± SD -0.001±0.008 0±0.007 0.001±0.009 10000 β: bias ± SD (-0.112, 0.004)±(0.011, 0.010) (-0.115, 0.013)±(0.018, 0.012) (0.113, 0.027)±(0.013, 0.010) π: bias ± SD -0.001±0.006 0±0.006 0.001±0.006 The results suggest that our algorithm produces consistent estimates of the β components and weights, as well as a consistent estimator of the number of components. As n increases, the values of these estimates approach the true parameter values and true number of components. As n gets larger, we observe that all of the points that are mislabeled are at the intersections of the components, and the slightly worse performance of the estimated G from our algorithm than using G∗is due to the bias of the β estimates creating intersection areas that are slightly larger than when G is set to G∗. Since we use the bandwidth based on the oversmoothing principle, we accept slightly larger bias in exchange for avoiding detecting spurious modes (components) in the data. In what follows we compare the above results using the EM-NPKMLE algorithm with those using the the EM-NPMLE algorithm with mean shift as a post-processing step as 21 described in Section 2.3 and the conditional gradient method (CGM) as described in Jiang and Guntuboyina (2025). We stepped through the same sequence of sample sizes as in Tables 1 and 2. EM-NPMLE and mean-shift post-processing: As shown in Tables 3 and 4, the mean shift algorithm as a post-processing step after EM-NPMLE produced results that suggest consistent estimation of the true β values and weights, as well as the true number of components. When using the mean shift for post-processing, we also used the bandwidth based on the maximal smoothing principle as described previously. CGM algorithm in Jiang and Guntuboyina (2025): As shown in Tables 5 and 6, the results of the CGM algorithm do not suggest that this algorithm can consistently estimate the number of modes, though the decreasing average W-2 distance with increasing sample size supports the idea of convergence in distribution between the true and estimated distributions. Table 3: Results of Simulation 1 using EM-NPMLE and mean-shift post-processing. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp Avg W-2 Distance 500 0.620 (0.094) 0.665 (0.033) 0.915 0.612 (0.238) 1000 0.657 (0.024) 0.663 (0.023) 0.995 0.465 (0.141) 5000 0.663 (0.010) 0.664 (0.010) 0.990 0.311 (0.092) 10000 0.662 (0.008) 0.663 (0.008) 1 0.265 (0.089) EM-NPKMLE with uniform initialization: We also ran our EM-NPKMLE algorithm on the same model as in (35) using initial points drawn from a uniform distribution over [-4,4]2 to demonstrate that our algorithm does not depend on initial points sampled from the estimated density g derived from the EM-NPKMLE algorithm. We currently do not have a bandwidth selection strategy for the uniform initialization, and we manually tuned the bandwidth by using the maximal smoothing principle multiplied by a constant 22 Table 4: Results of Simulation 1 using EM-NPMLE and mean-shift post-processing. Sample Size Estimate 1st Comp 2nd Comp 3rd Comp 500 β: bias ± SD (-0.012, 0.034)±(0.062, 0.040) (-0.173, -0.080)±(0.108, 0.045) (0.088, 0.031)±(0.062, 0.042) π: bias ± SD -0.001±0.025 -0.002±0.027 0.030±0.028 1000 β: bias ± SD (-0.002, 0)±(0.049, 0.031) (-0.077, -0.021)±(0.061, 0.029) (0.041, 0.013)±(0.044, 0.026) π: bias ± SD 0.002±0.019 -0.002±0.018 0±0.019 5000 β: bias ± SD (0, 0)±(0.021, 0.015) (-0.009, 0.005)±(0.023, 0.012) (0.013, -0.003)±(0.018, 0.013) π: bias ± SD -0.001±0.008 0±0.008 0.001±0.009 10000 β: bias ± SD (0, 0)±(0.014, 0.010) (-0.007, 0.004)±(0.017, 0.011) (0.010, -0.002)±(0.012, 0.008) π: bias ± SD -0.001±0.006 0±0.006 0±0.006 Table 5: Results of Simulation 1 using CGM. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp 500 0.590 (0.066) 0.665 (0.033) 0 1000 0.581 (0.068) 0.663 (0.023) 0 5000 0.602 (0.053) 0.664 (0.010) 0 10000 0.596 (0.057) 0.663 (0.008) 0 Table 6: Results of Simulation 1 using CGM. Sample Size Avg W-2 Distance Mean No. of Comp 500 0.439 (0.109) 9.295 (2.095) 1000 0.373 (0.093) 8.550 (1.744) 5000 0.260 (0.069) 7.800 (1.504) 10000 0.229 (0.064) 7.570 (1.274) c = 1.15. As shown in Tables 7 and 8, the results are comparable to running our EMNPKMLE algorithm using the output of EM-NPMLE for the initialization. Table 7: Results of Simulation 1 using EM-NPKMLE Algorithm with uniform initialization. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp Avg W-2 Dist 500 0.146 (0.043) 0.665 (0.033) 0.030 1.130 (0.059) 1000 0.283 (0.140) 0.663 (0.023) 0.515 1.020 (0.114) 5000 0.624 (0.012) 0.664 (0.010) 1 0.422 (0.060) 10000 0.642 (0.008) 0.663 (0.008) 1 0.320 (0.052) 23 Table 8: Results of Simulation 1 using EM-NPKMLE Algorithm with uniform initialization. Sample Size Estimate 1st Comp 2nd Comp 3rd Comp 500 β: bias ± SD (-0.453, 0.273)±(0.035, 0.072) (-0.888, -0.504)±(0.259, 0.149) (0.643, 0.325)±(0.136, 0.070) π: bias ± SD 0.029±0.017 -0.039±0.211 0.010±0.218 1000 β: bias ± SD (-0.430, 0.333)±(0.039, 0.048) (-0.717, -0.364)±(0.211, 0.094) (0.556, 0.299)±(0.094, 0.075) π: bias ± SD 0.049±0.018 -0.119±0.066 0.070±0.063 5000 β: bias ± SD (-0.224, 0.101)±(0.013, 0.018) (-0.234, -0.029)±(0.036, 0.020) (0.222, 0.073)±(0.020, 0.016) π: bias ± SD 0.009±0.007 -0.018±0.006 0.009±0.007 10000 β: bias ± SD (-0.164, 0.047)±(0.010, 0.011) (-0.164, 0.001)±(0.019, 0.013) (0.164, 0.043)±(0.014, 0.011) π: bias ± SD 0±0.005 -0.008±0.004 0.008±0.005 GEM-NPKMLE: As discussed at the end of Section 2.2, GEM-NPKMLE is a variant of EM-NPKMLE, where we run a single iteration in the inner loop instead of running it to convergence. This approach saves computation time. For example, for the simulation with n=5000 and the EM-NPMLE initialization, the EM-NPKLME took an average of 115.888 (40.759) minutes, while the GEM-NPKMLE took an average of 15.448 (2.194) minutes. It also produced similar results compared to Tables 1, 2, 3, and 4 for larger sample sizes, as can be seen in Tables 9 and 10. This suggests that the GEM-NPKMLE approach is viable when saving computation time is important and there is a sufficient sample size. The initial points we used for GEM-NPKMLE were sampled from the output of EM-NPMLE, which was the same strategy that was used in the first part of our simulations. However, in this case, we multiplied the initial bandwidth by a constant c = 1.2 to improve the algorithm's performance. Table 9: Results of Simulation 1 using GEM-NPKMLE algorithm. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp Avg W-2 Dist 500 0.133 (0.034) 0.665 (0.033) 0.235 1.070 (0.062) 1000 0.213 (0.106) 0.663 (0.023) 0.195 0.993 (0.059) 5000 0.614 (0.012) 0.664 (0.010) 0.840 0.401 (0.066) 10000 0.637 (0.008) 0.663 (0.008) 0.930 0.323 (0.070) 24 Table 10: Results of Simulation 1 using GEM-NPKMLE algorithm. Sample Size Estimate 1st Comp 2nd Comp 3rd Comp 500 β: bias ± SD (-0.527,0.360)±(0.067, 0.083) (-0.983,-0.642)±(0.135,0.071) (0.839,0.387)±(0.071,0.067) π: bias ± SD 0.005±(0.027) -0.298±(0.002) 0.292±(0.027) 1000 β: bias ± SD (-0.414,0.308)±(0.052,0.059) (-1.052,-0.551)±(0.175,0.079) (0.757,0.402)±(0.092,0.075) π: bias ± SD 0.007±0.019 -0.291±0.048 0.284±0.042 5000 β: bias ± SD (-0.241,0.104)±(0.014,0.018) (-0.282,-0.069)±(0.041,0.023) (0.231,0.088)±(0.020,0.016) π: bias ± SD -0.001±0.008 0±0.007 0.001±0.009 10000 β: bias ± SD (-0.184,0.056)±(0.010,0.011) (-0.196,-0.009)±(0.022,0.013) (0.172,0.053)±(0.014,0.011) π: bias ± SD -0.001±(0.006) 0±(0.006) 0.001±(0.006) Cross-validation for unknown σ: If σ is unknown, we can implement a cross-validation (CV) approach found in Jiang and Guntuboyina (2025) to estimate the parameter σ. First we divide the dataset T = {(xi,yi) ∶ i = 1,...,n} into C folds: T1,...,TC. For each c ∈{1,2,...C}, one fold Tc is left out as test data and the remaining folds are used to obtain an estimator ˆG-c for G∗. Let f ˆG-c xi (yi) = ∫φσ (yi -x⊺ i β)d ˆG-c(β), i = 1,⋯,n. The σ value that minimizes the following objective function is selected and denoted by ˆσ: CV (σ) = - C ∑ c=1 ∑ (xi,yi)∈Tc log(f ˆG-c xi (yi)). In our simulation, we used a random sample from the output of the EM-NPMLE algorithm as the initialization for our GEM-NPKMLE algorithm (with the oversmoothing bandwidth) to calculate ˆG-c. We tested this method on the model in (35) with C = 5 to get ˆσ, which was further plugged into the EM-NPKMLE algorithm. The results are provided in Tables 11 and 12, which are comparable to those in Tables 1 and 2. 3.1.2 Simulation 2 Unlike the EM-NPMLE algorithm, our EM-NPKMLE algorithm does not assume the knowledge of a particular structure in the true distribution, such as a low-dimensional 25 Table 11: Results of Simulation 1 using EM-NPKMLE algorithm and cross-validation to estimate σ. Sample Size Avg Adj RI Avg Adj RI G* Prop of Sim with 3 Comp Avg W-2 Dist 500 0.302 (0.204) 0.665 (0.033) 0.390 0.891 (0.192) 1000 0.574 (0.045) 0.663 (0.023) 0.975 0.557 (0.118) 5000 0.645 (0.011) 0.664 (0.010) 0.995 0.343 (0.080) 10000 0.651 (0.008) 0.663 (0.008) 0.995 0.288 (0.081) Table 12: Results of Simulation 1 using EM-NPKMLE algorithm and cross-validation to estimate σ. Sample Size Estimate 1st Comp 2nd Comp 3rd Comp 500 β: bias ± SD (-0.328,0.201)±(0.080,0.090) (-0.425,-0.281)±(0.236,0.177) (0.375,0.193)±(0.162,0.091) π: bias ± SD -0.010±0.025 -0.028±0.071 0.039±0.066 1000 β: bias ± SD (-0.273, 0.143)±(0.048, 0.047) (-0.355,-0.151)±(0.137,0.068) (0.271,0.136)±(0.073,0.047) π: bias ± SD -0.003±0.018 -0.002±0.018 0.005±0.019 5000 β: bias ± SD (-0.155, 0.033)±(0.0143, 0.016) (-0.157,0.003)±(0.032,0.018) (0.149,0.038)±(0.019,0.014) π: bias ± SD -0.001±0.008 0±0.007 0.001±0.009 10000 β: bias ± SD (-0.112, 0.004)±(0.011, 0.010) (-0.115, 0.013)±(0.018, 0.012) (0.113, 0.027)±(0.013, 0.010) π: bias ± SD -0.001±0.006 0±0.006 0.001±0.006 manifold as the support. To further demonstrate this, we use a continuous distribution as the model, which consists of a bivariate distribution represented by a mixture of uniform distributions over two concentric circles, each with center at the origin (see Figure 4). The model is given in (1), where σ = .2 is assumed to be known, and the true distribution G∗is 1 2 × Uniform{B(1)} + 1 2 × Uniform{B(2)}, where B(r) = {β ∈R2 ∶∣∣β∣∣= r}. This model has also been used in (Jiang and Guntuboyina, 2025). We uniformly sampled 10000 design points xi's from [-1,3]. For our EM-NPKMLE algorithm, we used the output of the EM-NPMLE algorithm as the initial β(0) points and used the oversmoothing bandwidth. We got similar results when we used points sampled from a uniform distribution on [-4,4]2 as the initial β(0) and applied a constant to the 26 Figure 4: Left panel: the support of G∗in Simulation 2; Right panel: the sample used in Simulation 2. oversmoothing bandwidth. In this simulation we also tested the EM-NPMLE algorithm with and without SCMS as a post-processing step, as described in Section 2.3. The EM-NPMLE algorithm was initialized by a uniform density over [-4,4]2. Then the SCMS algorithm was applied to 10000 β points sampled from the estimated density produced by the EM-NPMLE algorithm. The bandwidth used for the SCMS algorithm was selected by the biased cross-validation method aiming to minimize an asymptotic mean integrated squared error (AMISE) estimate between the estimated and true density gradients (Sain et al., 1994). This was implemented by the 'Hbcv.diag' function in the R package ks (Duong, 2024), and we used the mean of the diagonal entries in the selected diagonal bandwidth matrix. As shown in Figure 5 for a single random sample, the points produced by the EMNPKMLE algorithm are roughly evenly distributed near the support of G∗, while the support does not seem to be recovered well by the CGM method. The distribution produced by the EM-NPMLE algorithm has a visible concentration near the support of G∗, and 27 applying the SCMS algorithm as a post-processing step further extracted the 1-dimensional structure. In Table 13 we show the results of applying the Wasserstein-2 distance between the probability measures G∗and ˆG from the applicable algorithms. Using SCMS algorithm for post-processing has the best performance, followed by the EM-NPMLE algorithm without post-processing. As two algorithms without knowing the prior distribution is continuous or requiring a post-processing step, the EM-NPKMLE is an improvement over the CGM. Table 13: W-2 Distance between G∗and ˆG for the simulation depicted in Figure 5 Method W-2 Distance EM-NPKMLE 0.328 CGM 0.394 EM-NPMLE 0.190 EM-NPMLE + SCMS 0.157 28 (a) (b) (c) (d) Figure 5: Comparison of the true mixture distribution and the estimated distributions of β for different algorithms in Simulation 2, where the red circles are the support of G∗, and point sizes are proportional to weights. (a) EM-NPKMLE algorithm: 52 support points, weights ranging from 0.0002 to 0.085. (b) CGM method: 34 support points, weights ranging from 0.011 to 0.054. (c) 104 points sampled from the output of the EM-NPMLE algorithm, also used as the input for the SCMS algorithm. (d) post-processing result after using the SCMS algorithm, where all points have weight= 10-4. 29 3.2 Applications Application: CO2-GDP We tested our EM-NPKMLE algorithm on data of 159 countries' CO2 emissions and GDP per capita in 2015 taken from Bolt et al. (2018); Friedlingstein et al. (2020); Ritchie et al. (2020); and (UN, 2022). This dataset was also analyzed in Jiang and Guntuboyina (2025). Identifying distinct components among the relationship between CO2 emissions and economic output could reveal countries that have high economic output and low CO2 emissions, potentially revealing favorable policies to pursue. Other papers have also looked into this relationship from a mixture of regression approach, such as Huang and Yao (2012) and Hurn et al. (2003), though these papers use different data sources. As in Jiang and Guntuboyina (2025), we used data on CO2 emission in per capita in 10-ton increments and GDP per capita in 10 thousand U.S. dollars. We used a 10-fold cross-validation method as described in Section 3.1 for the CO2-GDP dataset, and used the same bandwidth selection method as described at the beginning of Section 3. Our estimate of σ from this CV method is 0.160. The results of our algorithm with this estimated σ suggest that there are two growth paths with larger estimated prior weights, with possible deviations as evidenced by the other components with smaller weights (see Table 14 and Figure 6). This result differs from Jiang and Guntuboyina (2025), who identified ten components before applying the BIC trimming, and five components after BIC trimming. We note that the two components with largest weights detected from our algorithm ((0.022, 0.179) and (-0.070, 0.343)) are similar to the two components with the largest weights from Jiang and Guntuboyina (2025) before and after their BIC trimming method was applied, that is, (0.030, 0.170) and (-0.010, 0.230), with pre-BIC trimming weights of 0.290 and 0.180, and post-BIC trimming weights of 0.270 and 0.280, respectively. Application: Music Tone Perception We also tested our EM-NPKMLE algorithm on another dataset analyzed in Jiang and 30 Figure 6: CO2-GDP Relationship: The points represent CO2 emissions in per capita in 10-ton increments and GDP per capita in 10 thousand U.S. dollars for individual countries. The lines represent the estimated component regression lines, whose estimated coefficients can be found in Table 14. The thicker lines are the components with larger weights. Table 14: Results from CO2/GDP Application 1st Comp 2nd Comp 3rd Comp 4th Comp 5th Comp β Estimates (0.022, 0.179) (-0.070, 0.343) (-0.101, 0.628) (0.105, 0.120) (0.839, -0.051) Estimated Weights 0.484 0.358 0.088 0.057 0.013 Guntuboyina (2025). The data was initially collected by Cohen (1980) in an experiment examining how trained musicians perceive tone. See Cohen (1980) for further information on the experiment. This dataset, which can be found in the R packages mixtools and fpc, 31 contains one musician's tuning that was observed 150 times (Benaglia et al., 2009; Hennig, 2024). See Figure 7, where the y-axis displays the musician's assessed tone ratio and the xaxis depicts the true tone ratio (Viele and Tong, 2002; Yao and Song, 2015). Other studies of this dataset were carried out in De Veaux (1989); Viele and Tong (2002); and Yao and Song (2015); but in these studies the number of components was pre-specified at two. We applied our EM algorithm that does not require prespecifying the number of components. We used the same cross-validation method as in the CO2-GDP application to estimate σ, and our estimate is 0.210. We also used the same bandwidth selection method as in the CO2-GDP application, that is, maximal smoothing bandwidth. Our results revealed two main components (see Figure 7 and Table 15). In terms of the number of components, the results are consistent with the music perception theory when the experiment took place, with one component as y = x and another as y = 2 (Jiang and Guntuboyina, 2025). Our results (see Table 15) are comparable to the two largest components computed using the CGM algorithm (both pre- and post BIC trimming) in Jiang and Guntuboyina (2025), that is, (1.898, 0.054), (0, 0.989), with weights 0.540 and 0.260 (pre-BIC trimming), and 0.700 and 0.300 (post-BIC trimming), respectively. However, the CGM method detected six components before the BIC trimming method was applied. Table 15: Results from Music Tone Application 1st Comp 2nd Comp β Estimates (0.012, 0.967) (1.292, 0.361) Estimated Weights 0.427 0.573 4 Discussion In this paper we first introduce an EM algorithm to approximate NPMLE in the setting of mixture of linear regressions when the prior distribution for the coefficients has a density. 32 Figure 7: Music Tone Perception: The blue points represent a musician's assessed tone ratio (y) and the true tone ratio (x) pairs. The estimated regression lines (see Table 15) are also depicted for the two components. The thicker line is the component with the larger weight. We then develop another EM algorithm to approximate NPKMLE that can automatically detect the number of components for a discrete prior distribution, and is able to uncover a continuous prior distribution. We provide some theoretical support for our algorithms as well as demonstrate their performance in simulations and applications. In particular, our simulation results suggest that our EM-NPKMLE algorithm can consistently estimate the true number of components and the β coefficients and their associated weights when the prior distribution is discrete. While our EM-NPKMLE algorithm does have useful 33 qualities, it requires the selection of a bandwidth, which can be challenging. We selected the oversmoothing bandwidth based on its ability to avoid the appearance of spurious features that may not belong to the true prior distribution, but further investigation into the bandwidth selection could yield better methods, e.g., adaptive bandwidths for different iterations of the algorithm, or introducing unique bandwidths for each dimension of the data. While we demonstrate the algorithm's ability to uncover both discrete and continuous distributions, investigating its theoretical properties is an area of further research. It is clear that our methods can be extended to the setting of mixture of distributions, and we will investigate this in a separate article. A Appendix A.1 Proof of Proposition 1 Proof. Let the Kullback-Leibler (KL) divergence between distributions P and Q with densities p and q be DKL(P∥Q) = ∫p(x)log p(x) q(x)dx. (38) We can write Eβ∣x,y,G(t)L(g∣x,y,β) (39) = - n ∑ i=1 ∫f (t+1) i (β)log f (t+1) i (β) g(β) dβ + n ∑ i=1 ∫log [φσ (yi -x⊺ i β)f (t+1) i (β)]f (t+1) i (β)dβ (40) = - n ∑ i=1 DKL(F (t+1) i ∥G) + n ∑ i=1 ∫log [φσ (yi -x⊺ i β)f (t+1) i (β)]f (t+1) i (β)dβ, (41) where F (t+1) i is the distribution associated with f (t+1) i . Hence, maximizing Eβ∣x,y,G(t)L(g∣x,y,β) over g is equivalent to minimizing 1 n ∑n i=1 DKL(F (t+1) i ∥G) over G ∈Gden. Let g(t+1) = 1 n ∑n i=1 f (t+1) i . Notice that 1 n n ∑ i=1 δDKL(F (t+1) i ∥G) δg ∣ g=g(t+1) = -1 n n ∑ i=1 f (t+1) i g(t+1) = -1. (42) 34 Using Theorem 2 in (Nishiyama, 2020), we conclude that g(t+1) is the unique minimizer of 1 n ∑n i=1 DKL(F (t+1) i ∥G), and hence the unique maximizer of Eβ∣x,y,G(t)L(g∣x,y,β,G(t)). A.2 Proof of Theorem 2.2 Proof. Using the subgradient inequality for differentiable convex functions, we have log[ n ∑ l=1 v(yl)] -log[ n ∑ l=1 v(xl)] ≥∑n l=1 v′(xl)(yl-xl) ∑n l=1 v(xl) , ∀xl,yl∈[0,∞). Also, recall that w = -v′. Then Q(Gν(r+1);G(t)) -Q(Gν(r);G(t)) (43) = n ∑ i=1 ∫Rd log [φσ (yi -x⊺ i β) 1 nhd ∑n l=1 vh(∥β -ν(r+1) l ∥2)]Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (44) - n ∑ i=1 ∫Rd log [φσ (yi -x⊺ i β) 1 nhd ∑n l=1 vh(∥β -ν(r) l∥2)]Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (45) ≥ n ∑ i=1 ∫Rd ∑n l=1 wh(∥β-ν(r) l ∥2)[∥β-ν(r) l ∥2-∥β-ν(r+1) l ∥2] ∑n j=1 vh(∥β-ν(r) j ∥2) Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (46) = n ∑ i=1 ∫Rd ∑n l=1 wh(∥β-ν(r) l ∥2)[2ν(r+1)⊺ l β-2ν(r)⊺ l β-∥ν(r+1) l ∥2+∥ν(r) l ∥2] ∑n j=1 vh(∥β-ν(r+1) j ∥2) Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (47) = n ∑ l=1 2ν(r+1)⊺ l n ∑ i=1 ∫Rd wh(∥β-ν(r) l ∥2)β ∑n j=1 vh(∥β-ν(r) j ∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (48) - n ∑ l=1 2ν(r)⊺ l n ∑ i=1 ∫Rd wh(∥β-ν(r) l ∥2)β ∑n j=1 vh(∥β-ν(r) j ∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (49) - n ∑ l=1 ∥ν(r+1) l ∥2 n ∑ i=1 ∫Rd wh(∥β-ν(r) l ∥2) ∑n j=1 vh(∥β-ν(r) j ∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (50) + n ∑ l=1 ∥ν(r) l∥2 n ∑ i=1 ∫Rd wh(∥β-ν(r) l ∥2) ∑n j=1 vh(∥β-ν(r) j ∥2)Si(β,β(t),x,y)dβ ∫Rd Si(β,β(t),x,y)dβ (51) = n ∑ l=1 [2ν(r+1)⊺ l A(ν(r) l;β(t),x,y) -2ν(r)⊺ l A(ν(r) l;β(t),x,y) (52) -∥ν(r+1) l ∥2C(ν(r) l,β(t),x,y) + ∥ν(r) l∥2C(ν(r) l,β(t),x,y)]. (53) 35 It then follows from (30),(31), and (32) that Q(Gν(r+1);G(t)) -Q(Gν(r);G(t)) (54) ≥ n ∑ l=1 -∥ν(r+1) l ∥2C(ν(r) l,β(t),x,y) + 2∥ν(r+1) l ∥2C(ν(r) l;β(t),x,y) (55) -2ν(r)⊺ l ν(r+1) l C(ν(r) l;β(t),x,y) + ∥ν(r) l∥2C(ν(r) l,β(t),x,y)] (56) = n ∑ l=1 C(ν(r) l,β(t),x,y)∥ν(r+1) l -ν(r) l∥2 (57) ≥0, (58) where the last inequality holds because C(ν(r) l,β(t),x,y) > 0 by noticing that w is always positive. Next we show that Q(Gν;G(t)) has an upper bound. Notice that sup β ∣φσ (yi -x⊺ i β) 1 nhd n ∑ l=1 vh(∥β -νl∥2)∣≤ ∥v∥∞ √ 2πσhd =∶B, (59) where ∥v∥∞= supt ∣v(t)∣. It then follows from (27) that sup ν Q(Gν;G(t)) ≤nlog B. (60) Hence the sequence Q(Gν(r);G(t)) converges. A.3 Proof of Theorem 2.3 Proof. For a distribution G on Rd with density function g, let P(β,G) = n ∑ i=1 log(fi(βi ∣xi,yi,g)), (61) where fi(βi ∣xi,yi,g) is defined in (12). We can write P(β,G) = n ∑ i=1 log [ φσ (yi -x⊺ i βi)g(βi) ∫Rd φσ (yi -x⊺ i β)g(β)dβ ] = n ∑ i=1 log [φσ (yi -x⊺ i βi)g(βi)] - n ∑ i=1 log [∫Rd φσ (yi -x⊺ i β)g(β)dβ]. 36 Using the definitions in (6) and (11), L(G) = L(G∣x,y,β) -P(β,G). Taking the expectation of each side with respect to β∣x,y,G(t), we have L(G) = Eβ∣x,y,G(t)(L(G)) = Eβ∣x,y,G(t)(L(G∣x,y,β)) -Eβ∣x,y,G(t)(P(β,G)) =∶Q(G;G(t)) -H(G;G(t)). Notice that L(G(t+1)) -L(G(t)) = [Q(G(t+1);G(t)) -Q(G(t);G(t))] -[H(G(t+1);G(t)) -H(G(t);G(t))]. We have already shown in Theorem 2.2 that Q(G(t+1);G(t)) ≥Q(G(t);G(t)). To show L(G(t+1)) -L(G(t)) ≥0, it remains to prove that H(G(t+1);G(t)) -H(G(t);G(t)) ≤0. This is true because for any arbitrary distribution G with density function g, H(G;G(t)) -H(G(t),G(t)) =Eβ∣x,y,G(t) [ n ∑ i=1 log(fi(βi ∣xi,yi,g))] -Eβ∣x,y,G(t) [ n ∑ i=1 log(fi(βi ∣xi,yi,g(t)))] =Eβ∣x,y,G(t) { n ∑ i=1 log [ fi(βi ∣xi,yi,g) fi(βi ∣xi,yi,g(t))]} = n ∑ i=1 Eβ∣x,y,G(t) {log [ fi(βi ∣xi,yi,g) fi(βi ∣xi,yi,g(t))]} ≤ n ∑ i=1 log {Eβ∣x,y,G(t) [ fi(βi ∣xi,yi,g) fi(βi ∣xi,yi,g(t))]} (by Jensen's inequality and the concavity of the log function) = n ∑ i=1 log [∫ fi(β ∣xi,yi,g) fi(β ∣xi,yi,g(t))fi(β ∣xi,yi,g(t))dβ] = n ∑ i=1 log [∫fi(β ∣xi,yi,g)dβ] =0. 37 We can also conclude from Theorem 1 in (Jiang and Guntuboyina, 2025) that the incomplete log likelihood L(G) is bounded from above because of the existence of a maximizer ˆG. By the monotone convergence theorem, L(G(t)) converges. Acknowledgments This work was partially supported by resources provided by the Office of Research Computing at George Mason University (URL: https://orc.gmu.edu). 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Communications in Statistics-Theory and Methods, 44(8):1602-1614. 43"}}},{"rowIdx":96,"cells":{"id":{"kind":"string","value":"2510.14887"},"raw":{"kind":"string","value":"Prediction-Specific Design of Learning-Augmented Algorithms\nSizhe Li*1, Nicolas Christianson†2, and Tongxin Li‡1\n1School of Data Science, The Chinese University of Hong Kong, Shenzhen\n2Department of Management Science and Engineering, Stanford University\nAbstract\nAlgorithms with predictions has emerged as a powerful framework to combine the robustness of traditional\nonline algorithms with the data-driven performance benefits of machine-learned (ML) predictions. However,\nmost existing approaches in this paradigm are overly conservative, as they do not leverage problem structure\nto optimize performance in a prediction-specific manner. In this paper, we show that such prediction-specific\nperformance criteria can enable significant performance improvements over the coarser notions of consistency\nand robustness considered in prior work. Specifically, we propose a notion of strongly-optimal algorithms\nwith predictions, which obtain Pareto optimality not just in the worst-case tradeoff between robustness and\nconsistency, but also in the prediction-specific tradeoff between these metrics. We develop a general bi-level\noptimization framework that enables systematically designing strongly-optimal algorithms in a wide variety\nof problem settings, and we propose explicit strongly-optimal algorithms for several classic online problems:\ndeterministic and randomized ski rental, and one-max search. Our analysis reveals new structural insights\ninto how predictions can be optimally integrated into online algorithms by leveraging a prediction-specific\ndesign. To validate the benefits of our proposed framework, we empirically evaluate our algorithms in case\nstudies on problems including dynamic power management and volatility-based index trading. Our results\ndemonstrate that prediction-specific, strongly-optimal algorithms can significantly improve performance\nacross a variety of online decision-making settings.\n1\nIntroduction\nOnline algorithms operate in environments where decisions must be made sequentially without full knowl-\nedge of future inputs. Traditionally, these algorithms are designed to guarantee robust performance on adversarial\nproblem instances, providing competitive ratio bounds that hold under worst-case inputs [1]. While theoretically\nrobust, this adversarial perspective often yields overly pessimistic strategies that can underperform in the real\nworld, where worst-case instances are uncommon. Recent work on algorithms with predictions addresses this lim-\nitation by integrating machine-learned (ML) predictions into classical online decision-making frameworks [2, 3].\nThis learning-augmented algorithm design paradigm has been applied to a variety of online problems including\nski rental, caching, and metrical task systems [2–4] and applications including GPU power management [5],\nbattery energy storage system control [6], carbon-aware workload management [7, 8], and electric vehicle (EV)\ncharging [9, 10]—with algorithms that achieve near-optimal performance when predictions are accurate, while\nmaintaining robust worst-case guarantees when they are not.\nExisting approaches to learning-augmented algorithm design typically evaluate performance through a\nso-called consistency-robustness tradeoff. In particular, consistency measures worst-case algorithm performance\nwhen the prediction is perfectly accurate, while robustness measures the worst-case competitive ratio over all\npossible predictions and instances.\n*Email: lisizhe@link.cuhk.edu.cn\n†Email: christianson@stanford.edu\n‡Email: litongxin@cuhk.edu.cn\n1\narXiv:2510.14887v1 [cs.DS] 16 Oct 2025\n\nTable 1: Comparison of our technical results on algorithm (weak and strong) optimality.\nGeneral\nDSR (LARGE b)\nRSR (LARGE b)\nOMS\nϵ-OMS\nWEAK\nAlgorithm 1\nKD Kumar et al.\nKR Kumar et al.\nSun et al.\n\\\nSTRONG\nAlgorithm 1\nPDSR\nPRSR\nPST\nϵ-Tolerant PST\nProposition 1\nTheorem 4.2\nTheorem 5.3\nTheorem 6.2\nTheorem 7.3\nNote: Algorithm 1, PDSR, PRSR, PST, and ϵ-Tolerant PST are results from this work.\nDSR/RSR=Deterministic/Randomized Ski Rental, OMS=One-Max Search, ϵ-OMS=Error Tolerant OMS\n0\n50\n100\n150\n200\nPrediction y\n1.0\n1.5\n2.0\n2.5\n3.0\nConsistency \ny\nKD ( = 0.5) \ny\nKD ( = 0.5) \ny\nPDSR (Ours) \ny\nPDSR (Ours) \ny\n1.0\n1.5\n2.0\n2.5\n3.0\nRobustness \ny\n0\n50\n100\n150\n200\nPrediction y\n1.0\n1.5\n2.0\n2.5\nConsistency \ny\nKR ( = 0.5) \ny\nKR ( = 0.5) \ny\nPRSR (Ours) \ny\nPRSR (Ours) \ny\n1.0\n1.5\n2.0\n2.5\nRobustness \ny\n10\n12\n14\n16\n18\n20\nPrediction y\n1.50\n1.75\n2.00\ny ×\ny\nSun et al. ( = 0.5)\nPST (Ours)\nFigure 1: Prediction-specific consistency βy and robustness γy under different predictions y for DSR with\nb = 100 (LEFT), RSR with b = 100 (MIDDLE), and OMS with L = 10, U = 20 (RIGHT).\nImportantly, both metrics are inherently worst-case in nature: consistency reflects the algorithm’s perfor-\nmance under the least favorable instance with the least favorable, yet accurate prediction, and robustness measures\nperformance under the least favorable (inaccurate) prediction and instance. Because they are worst-case, neither\nof these metrics measures whether algorithms can achieve better performance for specific predictions.\nAs such, while much prior work has sought to design algorithms that obtain the optimal tradeoff between\nconsistency and robustness (e.g., [2, 11]), and some existing algorithms have sought to improve performance by\noptimizing decisions in a prediction-specific manner (e.g., [8, 12]), no prior work has considered the question of\nhow to design online algorithms with prediction-specific guarantees on optimality.\nMotivated by this gap and the potential to improve the performance of algorithms with predictions, our work\nexplores the following question:\nHow can we design algorithms that achieve Pareto-optimal tradeoffs between consistency and robustness that\nare tailored to specific prediction values?\nTo this end, we introduce a prediction-specific framework for algorithm design, enabling the development of\nexplicit and tractable online algorithms that adapt to each prediction’s characteristics to ensure Pareto-optimal\nrobustness and consistency for each individual prediction value. Instantiating this framework, we design explicit\nprediction-specific algorithms for the problems of deterministic and randomized ski rental and one-max search,\ntwo classic online problems with connections to real-world applications including TCP acknowledgment [13],\ncloud cost management [14], dynamic power management [5], and energy market operations [15, 16].\n1.1\nContributions\nThe main contributions of this paper are as follows.\nFramework and Theoretical Results. We introduce a novel prediction-specific framework for the design and\nanalysis of learning-augmented algorithms. Specifically, Definition 2 extends the classic notions of consistency\nand robustness by defining the prediction-specific consistency βy and robustness γy for each possible prediction\nvalue y. Here, βy evaluates the algorithm’s performance on the worst-case instance which is consistent with\nthe prediction y, while γy measures its worst-case performance under prediction y. Furthermore, Definition 4\nintroduces the notion of strong optimality: an algorithm is strongly optimal if it is Pareto-optimal not only\n2\n\nDSR\nRSR\nOMS\nExisting Results\nKD ([2], Algorithm 2)\nKR ([2], Algorithm 3)\nSun et al. [11]\nThis Work\nPDSR (Algorithm 2)\nPRSR (Algorithm 9)\nPST (Algorithm 4)\nImprovement Ratio\n\u00001 + 1\nλ\n\u0001\n/2\n( e−1\ne ) · (1 −e−λ)−1\n√\n(1−λ)2+4λθ−(1−λ)\n2λ\n√\nθ\nTable 2: Comparison of proposed algorithms in this work with existing results. The Improvement Ratio\nquantifies the maximum ratio between the consistency-robustness score (βy · γy) of our algorithm over the best\nexisting score for all y.\nin the classic sense (referred to as weak optimality, see Definition 3), but also in the (βy, γy) plane for every\nprediction value y in the prediction space F. This reframes the evaluation of algorithm performance from a\nsingle worst-case trade-off to a richer, per-prediction perspective, thereby enabling a more fine-grained analysis\nof algorithm behavior across different prediction values.\nUnder this new framework, we show that the existing weakly-optimal algorithms for several canonical\nonline problems—deterministic ski rental, randomized ski rental, and one-max search (see Table 3)—are not\nstrongly-optimal (see Theorems 4.1, 5.1, and 6.1). As such, we propose new algorithms for these problems\n(Algorithms 2, 9, and 4) and prove their strong optimality (see Theorems 4.2, 5.3, and 6.2). Notably, our strongly-\noptimal algorithms can obtain significant performance improvements over prior weakly-optimal algorithms for\ncertain prediction values; Table 2 summarizes the best improvement ratio of the consistency–robustness score\n(βy · γy) across all y for each problem, reflecting the relative benefit of our algorithms over existing ones under\nthe most favorable y. The parameter λ represents some confidence governing the tradeoff between robustness\nand consistency used in [2, 11]. Note that for both deterministic and randomized ski rental, the maximum\nimprovement ratio is obtained as λ ↓0, where it diverges to infinity with asymptotic order O(λ−1). In contrast,\nfor one-max search, the maximum improvement ratio is\n√\nθ, which is also attained as λ ↓0. Consequently, our\nmethods can yield substantial performance improvements for certain predictions and choices of the parameter λ.\nWe also show that there exist problems for which existing weakly-optimal algorithms in the literature\nare already strongly-optimal: in particular, we establish prediction-specific bounds in Theorem 2.1 for the\nweakly-optimal algorithm of Wei and Zhang [17] for non-clairvoyant job scheduling when the job size n = 2,\nand prove its strong optimality in this case.\nNovel Techniques and Methodology. We propose a bi-level optimization methodology (see Problems 1\nand 2 in Section 3.1) for systematically deriving strongly-optimal algorithms. Given an initial robustness target\nγ, Problem 1 finds the best prediction-specific consistency βy, and Problem 2 then finds a decision with the\nbest prediction-specific robustness γy given the fixed βy. The bi-level optimization pipeline naturally forms\na meta-algorithm (Algorithm 1), which we prove in Proposition 1 yields a strategy on the prediction-specific\nPareto front for each y, guaranteeing strong optimality. This approach offers two key benefits. (1) Generality\n– the bi-level optimization framework is broadly applicable: it can be used to design algorithms for online\nproblems beyond those described in Table 3. (2) Flexibility – this approach exhibits two kinds of flexibility.\nFirst, the consistency–robustness trade-off in the meta-algorithm (Algorithm 1) can be tuned by adjusting the\nrobustness target γ. Second, the bi-level optimization problem can be adapted to different kinds of predictions\nand performance objectives. For example, the objectives and constraints in Problems 1 (P1) and 2 (P2) can be\nextended to incorporate error tolerance to enable the design of algorithms which perform well even if predictions\nare erroneous, thus alleviating, to some extent, the issues of brittleness and smooth performance degradation\nwhich have been considered in a number of recent works [18–20]. More specifically, in Section 7, we introduce\nthe notion of ϵ-consistency to formalize the goal of preserving consistency (and avoiding algorithm brittleness)\nwhen faced with small prediction error ϵ, and we show that it is possible to obtain both classic Pareto optimality\nbetween ϵ-consistency and robustness, as well as a corresponding version of strong optimality in this case (see\nTheorem 7.3).\nIn addition, our analysis of strong optimality in the randomized ski rental problem (see Appendix E) employs\n3\n\na novel prediction-specific primal–dual technique. Specifically, we first use a perturbation-based analysis to\nderive structural properties of the optimal randomized distribution, and then formulate and analyze a primal–dual\noptimization problem for each prediction. This provides a structured and potentially generalizable approach to\nestablish Pareto optimality with respect to a given prediction.\nExperimental Results and Real-World Implications. We evaluate our algorithms through both synthetic\nsimulations and real-world case studies, spanning a range of online decision-making problems. Overall,\nour methods consistently outperform state-of-the-art learning-augmented and classical baseline algorithms,\nconfirming their theoretical soundness and practical value.\nMore specifically, we apply the deterministic and randomized ski rental algorithms to Dynamic Power\nManagement (DPM) traces [5], an important benchmark for energy-efficient computing, and we apply the\none-max search algorithms to VIX trading, a representative financial market task marked by high volatility and\nuncertainty. In both domains, our methods deliver significant improvements across most prediction regimes,\nillustrating how prediction-specific design can translate improved theoretical guarantees into tangible real-world\ngains.\n1.2\nRelated Work\nAlgorithms with Predictions. Although the study of online algorithms with ML predictions is still relatively\nnew [2, 3], significant research progress has been made in recent years on various problems such as online\noptimization [6], control [21, 22], and reinforcement learning [23, 24]. Similar frameworks have also been\nadopted to design and analyze algorithms for a number of other online problems, such as caching [3, 25–27],\nonline knapsack [28–30], secretary and bipartite matching [31], metrical task systems [4, 32, 33], and convex\nbody/function chasing [32, 34]. Most of these works make no assumptions about prediction quality, seeking to\nbalance consistency (competitive ratio when predictions are accurate) and robustness (worst-case competitive\nratio regardless of prediction quality), though the precise, formal definitions of these metrics vary slightly across\nworks.\nBeyond their theoretical appeal, these frameworks have begun to influence practical systems domains,\nenabling advances in areas such as data-center scheduling [35], energy-aware computing [7, 15], and networked\ncontrol systems [6, 22], where ML-driven forecasts are increasingly available but inherently imperfect.\nNote also that some recent works depart from the standard paradigm of robustness and consistency and\nconsider alternative prediction models and performance measures. Sun et al. [36] proposed online algorithms\nwith uncertainty-quantified (UQ) predictions, which leverage UQ to assess prediction quality. They introduced\nthe distributionally robust competitive ratio (DRCR), which weighs both the setting where the UQ prediction\naccurately describes the instance, and the worst-case adversarial setting, and applied this metric to the problems\nof ski rental and online search. Mahdian et al. [37] proposed a general framework for online optimization\nunder uncertain inputs, where the algorithm has access to an optimistic strategy that performs well when the\nfuture unfolds favorably. They developed a meta-algorithm that balances between this optimistic policy and a\nrobust fallback, achieving a trade-off between worst-case guarantees and performance under accurate predictions\nwithout relying on a formal error model.\nSki Rental and Scheduling with ML Predictions. Regarding online problems that are closely related\nto those specific examples considered in this work, Kumar et al. [2] studied ski rental and non-clairvoyant\nscheduling with ML predictions. Their framework introduces a tunable trade-off between consistency and\nrobustness through a user-specified hyperparameter. Wei and Zhang [17] subsequently gave general lower\nbounds on the trade-off obtainable in these problems, thus proving that the deterministic and randomized\nalgorithms of Kumar et al. [2] for ski rental achieve the Pareto-optimal trade-off between consistency and\nrobustness. Furthermore, they demonstrated that the meta-algorithm proposed by Kumar et al. [2] for non-\nclairvoyant scheduling does not achieve the tight trade-off, and introduced a novel two-stage scheduling strategy\nthat is provably tight for the case of n = 2.\nOne-Max Search with ML Predictions. In the learning-augmented setting of one-max search, where\nalgorithms receive a prediction of the maximum element value, Sun et al. [11] established a fundamental lower\n4\n\nbound on the trade-off between consistency and robustness. They also proposed a threshold-based algorithm and\nshowed that it achieves this lower bound, making it Pareto-optimal in terms of these two performance measures.\nAlgorithm Smoothness. A recent line of work has shown that some existing learning-augmented algorithms\nare brittle, suffering sharp performance drops under small prediction errors. Elenter et al. [18] addressed this by\ndesigning algorithms that follow user-specified error-performance profiles, ensuring controlled degradation in\nperformance for the one-way trading problem. Benomar et al. [19] further proposed smooth one-max search\nalgorithms that are Pareto-optimal and exhibit gradual performance degradation with increasing prediction error,\nachieving a “triple Pareto optimality” among consistency, robustness, and smoothness. Unlike prior work that\nfocuses on structural smoothness of the Pareto frontier, our formulation provides a principled relaxation of\nconsistency itself, leading to algorithms that are both robust and tolerant to small predictive errors.\n2\nProblem Formulation\nWe consider online cost minimization problems over a set of instances I. For each instance I ∈I, let\nALG(A, I) and OPT(I) denote the cost incurred by an online algorithm A and the cost of the offline optimum,\nrespectively. We assume that the algorithms have no prior knowledge of the instance I. Under the competitive\nanalysis framework [1], the goal is to find an online algorithm A that minimizes the worst-case competitive ratio\nα(A), which is defined as1\nα(A) = max\nI∈I\nALG(A, I)\nOPT(I) .\nThis worst-case focus, however, can be overly pessimistic in practical settings. To move beyond the\nlimitations of worst-case algorithms, machine-learned predictions can be integrated into algorithm design\n(see [2, 3]). In this setting, an online algorithm Aω, potentially parameterized by ω ∈Ω, receives not only the\ninstance I ∈I online but also an ML prediction y ∈F concerning some relevant but unknown feature x(I) ∈F\nof I, where F denotes a prediction space. The feature x(I) encapsulates useful information about I (e.g., the\nmaximum element value for online one-max search) or may even fully specify I in some problems (e.g., the\ntotal number of skiing days for discrete-time ski rental). Let ALG(Aω, I, y) denote the (expected) cost incurred\nby algorithm Aω on instance I given prediction y.\nClassic Consistency and Robustness. Since the ML prediction y ∈F may be erroneous, a number of\nalgorithms have been designed (see those summarized in Sections 1.2 and 2.3) to (1) achieve near-optimal\nperformance when the prediction y is accurate (consistency), while (2) simultaneously maintaining a bounded\nperformance guarantee even when the prediction is arbitrarily wrong (robustness). To formalize this, let Iy ⊆I\nrepresent the set of instances for which prediction y is considered “accurate” or “consistent”; the precise\ndefinition depends on the specific problem, the form of prediction, and the prediction quality measure. The\nclassic consistency and robustness metrics are defined as follows.\nDefinition 1 (Classic Metrics). Given an online algorithm Aω that takes predictions, the consistency β(Aω)\nand robustness γ(Aω) are defined as:\nβ(Aω) := sup\ny∈F\nsup\nI∈Iy\nALG(Aω, I, y)\nOPT(I)\n,\nγ(Aω) := sup\ny∈F\nsup\nI∈I\nALG(Aω, I, y)\nOPT(I)\n.\n(1)\nIf an algorithm Aω achieves β(Aω) consistency and γ(Aω) robustness, it is called β(Aω)-consistent and\nγ(Aω)-robust, respectively.\nA typical choice of Iy is Iy := {I ∈I : x(I) = y} (following [2, 3]). The equality constraint in Iy\ncan be generalized in a number of ways—for instance, to a ball centered at x(I), which will emphasize small\nerror tolerance of the algorithm (see Section 7 for more details). Here, the consistency β(Aω) captures the\n1In online profit maximization problems, the competitive ratio is defined as the worst-case ratio between the optimal offline profit and\nthat obtained by the online algorithm, which is the inverse of the minimization setting.\n5\n\nalgorithm’s performance guarantee assuming the prediction is correct, taking the worst case over all possible\ncorrect predictions, and the robustness γ(Aω) represents the worst-case guarantee under all possible instances\nand any prediction y, regardless of accuracy. An ideal algorithm minimizes both β(Aω) (aiming for near 1) and\nγ(Aω).\n2.1\nPrediction-Specific Consistency and Robustness\nThe standard metrics in (1) evaluate performance by taking the worst case over both instances I and\npredictions F. While adopting a worst-case perspective for instances I is a standard and reasonable approach\nin competitive analysis due to future uncertainty, applying this same perspective to the entire prediction space\nF is unnecessarily conservative, as better prediction-specific performance may be possible. This motivates a\nmore nuanced framework to evaluate performance conditioned on the specific prediction value y ∈F that the\nalgorithm receives. We thus introduce the notions of prediction-specific consistency and prediction-specific\nrobustness.\nDefinition 2 (Prediction-Specific Metrics). Given an online algorithm Aω and a specific prediction value y ∈F,\nthe prediction-specific consistency βy(Aω) and prediction-specific robustness γy(Aω) relative to y are defined\nas:\nβy(Aω) := sup\nI∈Iy\nALG(Aω, I, y)\nOPT(I)\n,\nγy(Aω) := sup\nI∈I\nALG(Aω, I, y)\nOPT(I)\n.\n(2)\nIf an algorithm Aω achieves βy(Aω) and γy(Aω) for a given prediction value y, we say Aω is βy(Aω)-consistent\nand γy(Aω)-robust under y.\nThe prediction-specific metrics in Definition 2 enable a finer-grained analysis for algorithms with predictions.\nInstead of considering only the worst-case consistency or robustness over all predictions, we may instead tailor\nan algorithm’s strategy based on the characteristics of each observed prediction value y. This adaptiveness can\nenable better performance compared to standard algorithms, which only optimize for worst-case consistency and\nrobustness (1).\n2.2\nWEAK and STRONG Optimality\nWhile these prediction-specific metrics provide valuable insight into algorithm performance conditioned on\nthe received prediction, the standard metrics of consistency and robustness still remain valuable to characterize\nan algorithm’s overall worst-case guarantees across all instances and predictions.\nTo formally capture the notion of algorithms that perform well both overall and for specific predictions,\nwe introduce two notions of optimality based on these different settings. Our goal is to distinguish algorithms\nwhich both achieve the optimal tradeoff between robustness and consistency for the worst-case prediction, as\nwell as on a per-prediction basis. This leads to the following definitions of weakly-optimal and strongly-optimal\nalgorithms.\nWe begin by defining weak optimality, which characterizes algorithms with the optimal tradeoff between the\nstandard notions of consistency and robustness.\nDefinition 3 (WEAK Optimality). For a fixed online problem, consider an online algorithm A that achieves\nβ consistency and γ robustness. A is WEAKLY-optimal if there does not exist another online algorithm with\nconsistency β′ and robustness γ′ such that β′ ≤β and γ′ ≤γ, with at least one inequality being strict.\nBuilding upon this, we define the stricter notion of strong optimality incorporating prediction-specific\nperformance.\nDefinition 4 (STRONG Optimality). For a fixed online problem, consider an online algorithm A that achieves βy\nprediction-specific consistency and γy prediction-specific robustness under prediction y ∈F. A is STRONGLY-\noptimal if\n6\n\n1. A is weakly-optimal;\n2. for any prediction y ∈F, there does not exist another online algorithm with prediction-specific consistency\nβ′\ny and robustness γ′\ny such that β′\ny ≤βy and γ′\ny ≤γy, with at least one inequality being strict.\nThe notion of strong optimality in Definition 4 extends classic Pareto optimality to the prediction-specific\nsetting. This definition requires Pareto optimality in the two-dimensional consistency-robustness plane (βy, γy)\nfor each specific prediction y ∈F, in addition to the baseline weak optimality. This stricter criterion captures\nwhether an algorithm is unimprovable in its fundamental performance trade-off across the entire prediction space\nF.\n2.3\nOnline Problems Studied\nWe now briefly introduce each of the problems studied in this work; these problems are summarized\nin Table 3.\nTable 3: Instantiation of the general problem components I, x, and y for problems analyzed.\nDiscrete-Time Ski Rental\nOne-Max Search\nScheduling\nInstance I\nNumber of days\nn values in [L, U]\nSet of n jobs\nFeature x\nNumber of days\nMaximum value\nProcessing times (x1, . . . , xn)\nPrediction y\nPredicted number of days\nPredicted maximum value\nPredicted times (y1, . . . , yn)\nDiscrete-Time Ski Rental. The discrete-time ski rental problem is a classic online decision-making\nproblem, wherein a decision-maker decides each day whether to rent skis for 1 unit of cost per day or purchase\nthem outright for a fixed cost b ∈N+, without prior knowledge of the total number of skiing days x. The\noptimal cost is given by min{b, x}. In the standard competitive analysis framework, where no predictions are\navailable, the optimal deterministic algorithm employs a simple rent-or-buy strategy, purchasing the skis on day\nM = b; this strategy achieves a competitive ratio of 2 −1/b. The optimal randomized algorithm is known as\nKarlin’s algorithm [38], which strategically balances renting and buying through a carefully-designed probability\ndistribution, achieving a competitive ratio of approximately e/(e −1) ≈1.582.\nFor the ski rental problem, Kumar et al. [2] proposed algorithms that trade off consistency (performance\nunder accurate predictions) and robustness (worst-case performance). They presented a deterministic algorithm\nachieving (1 + λ) consistency and (1 + 1/λ) robustness (for λ ∈(0, 1)), and a randomized algorithm with\n(\nλ\n1−e−λ ) consistency and (\n1+1/b\n1−e−(λ−1/b) ) robustness (for λ ∈(1/b, 1)). Subsequently, Wei and Zhang [17]\nestablished fundamental lower bounds on this trade-off, showing that as b →∞, for deterministic algorithms,\nany (1+λ)-consistent algorithm must have a robustness of at least (1+1/λ) for λ ∈(0, 1), while for randomized\nalgorithms, any γ-robust algorithm must have a consistency of at least γ log(1 + 1/(γ −1)). These results show\nthat the algorithms in [2] achieve weak optimality in the limit b →∞(see Definition 3).\nIn this work, we provide a deeper analysis of the algorithms proposed by Kumar et al. [2], examining their\nprediction-specific consistency and robustness. To maintain consistency and comparability with [2, 17], we also\nconsider the asymptotic regime b →∞. Our findings indicate that neither the deterministic nor randomized\nalgorithms in [2] are strongly-optimal within this framework (see Theorems 4.1 and 5.1). Consequently, we\npropose novel algorithms (see Algorithms 2 and 9) that achieve strong optimality (see Theorems 4.2 and 5.3),\nthereby improving upon existing approaches for this problem setting with ML predictions.\nOne-Max Search. The one-max search problem considers a sequence of n elements with values in the range\n[L, U], where L and U are known positive constants. At each step, an element is observed, and the algorithm\nmust decide whether to accept it immediately or irrevocably discard it. The objective is to select the element with\nthe maximum value. The instance’s difficulty is characterized by the ratio θ = U/L. In classical competitive\nanalysis, the optimal competitive ratio is\n√\nθ, achieved by an algorithm using the fixed threshold\n√\nLU [39].\nNote that this is a reward maximization problem, rather than a loss minimization; thus, for this problem, we will\n7\n\nconsider definitions of classic (1) and prediction-specific (2) consistency and robustness with the ratio between\nALG and OPT inverted.\nFor learning-augmented one-max search, where algorithms receive a prediction of the maximum element\nvalue, Sun et al. [11] established a fundamental lower bound on the consistency-robustness trade-off: any\nγ-robust algorithm must have a consistency of at least θ/γ. They further proposed a threshold-based algorithm\nthat achieves this lower bound, implying its weak optimality.\nIn this work, we provide a deeper analysis of the algorithm proposed by Sun et al. [11] within this prediction-\nspecific framework. Our analysis reveals that their algorithm is not strongly-optimal (see Theorem 6.1).\nConsequently, we propose a novel algorithm (see Algorithm 4) that is strongly-optimal (see Theorem 6.2) and\noffers improved performance.\nNon-Clairvoyant Scheduling. The non-clairvoyant scheduling problem concerns scheduling n jobs on\na single machine without prior knowledge of their processing times. We focus on the preemptive setting,\nwhere jobs can be interrupted and resumed without cost. In the standard competitive analysis framework [1],\nRound-Robin (RR) achieves an optimal competitive ratio of\n2n\nn+1 [40].\nExtending this, Wei and Zhang [17] established a fundamental lower bound on the consistency-robustness\ntrade-off, showing that any algorithm must have robustness of at least\nn+n(n−1)λ\n1+λ(n+1)(n+2)/2 if it is (1 + λ)-consistent.\nThey also propose a two-stage schedule algorithm (see Algorithm 6 in Section A.1) and show it achieves the\ntight tradeoff for n = 2 jobs in this learning-augmented setting. In the next section, as a warm-up, we provide a\ndeeper analysis of this two-stage algorithm’s prediction-specific consistency and robustness, establishing that it\nis strongly-optimal.\n2.4\nWarm-Up: An Existing Algorithm that is Strongly-Optimal\nWhile our proposed notion of strong optimality is much stricter than weak optimality, some existing\nalgorithms known only to be weakly-optimal can be shown to satisfy strong optimality. In this section, we\nconsider the problem of non-clairvoyant scheduling with n = 2 jobs (with length predictions y = (y1, y2)\nwith y1 ≤y2). It is well known that the two-stage scheduling algorithm proposed by Wei and Zhang [17] (see\nAlgorithm 6 in Section A.1) achieves the optimal trade-off between classic consistency and robustness and is thus\nweakly-optimal. Notably, we can show that this algorithm also satisfies prediction-specific Pareto optimality:\nTheorem 2.1. The two-stage algorithm in [17] with n = 2 is 1 + min{y1/(2y1 + y2), λ}-consistent and\n1 + max{1/3, y1/(y1 + 2λ(2y1 + y2))}-robust under y = (y1, y2). Moreover, it is strongly-optimal.\nWe prove this theorem in Section A.2. Despite the fact that this algorithm is strongly optimal for non-\nclairvoyant scheduling, for many other canonical online problems, including the ski rental and one-max search\nproblems, we shall soon see that existing weakly-optimal algorithms (such as those in [2, 11]) fail to achieve\nstrong optimality (see Theorems 4.1, 5.1, and 6.1). As such, the rest of this paper will consider the development\nof new algorithms that can achieve strong optimality in these settings.\n3\nOptimization-Based Meta-Algorithm\nIn this section, we introduce a general optimization-based approach to systematically identify strongly-\noptimal algorithms for online problems. Our goal is to obtain a general meta-algorithm that, given a prediction\ny ∈F and a target robustness upper bound γ, returns an algorithm Aω which is strongly-optimal.\n3.1\nBi-Level Optimization Formulation\nWe begin by considering the following question: given some target upper bound γ on robustness and a\nprediction y ∈F, how can we obtain an algorithm that both satisfies the target robustness bound, and obtains a\nprediction-specific optimal tradeoff between consistency and robustness? Here, γ ∈Λγ := [α∗, +∞) can be any\npossible robustness level, where α∗denotes the optimal competitive ratio achievable for the problem.\n8\n\nAlgorithm 1 BI-LEVEL OPTIMIZATION-BASED META-ALGORITHM\n1: Input: Desired robustness upper bound γ ∈Λγ := [α∗, +∞)\n2: Receive a prediction y;\n3: Compute {β∗\ny, ω} by solving P1(γ, y) (Problem 1);\n4: Obtain {γ∗\ny, ω∗} by solving P2(β∗\ny, y) (Problem 2) ;\n5: Deploy the online decision rule induced by Aω∗;\nTo achieve this goal, we specify a bi-level optimization framework comprising Problems 1 and 2. Recall\nthat a prediction y determines a consistent subset of instances Iy ⊆I, as described in Section 2, and that\nALG(Aω, I, y) and OPT(I) denote, respectively, the costs of the algorithm Aω augmented by the prediction\ny and optimal offline algorithm under instance I ∈I. Our first optimization problem (Problem 1), referred\nto as P1(γ, y), determines the best achievable prediction-specific consistency of any γ-robust algorithm under\nthe prediction y ∈F. Let {β∗\ny, ω} denote its optimal solution. We then solve a second optimization problem\n(Problem 2), referred to as P2(β∗\ny, y), to find the most robust algorithm among those achieving β∗\ny consistency\nunder the prediction y. Let {γ∗\ny, ω∗} denote its optimal solution. The resulting algorithm Aω∗with the optimal\nsolution ω∗is then implemented for the prediction y.\nProblem 1 (P1): Minimize Consistency\nP1(γ, y) := min\nβy,ω∈Ω\nβy\ns.t.\n(3a)\nALG(Aω, I, y) ≤γ OPT(I), ∀I ∈I,\n(3b)\nALG(Aω, I, y) ≤βyOPT(I), ∀I ∈Iy.\n(3c)\nConstraint (3b) enforces the initial robustness level γ.\nProblem 2 (P2): Minimize Robustness\nP2(β∗\ny, y) := min\nγy,ω∈Ω\nγy\ns.t.\n(4a)\nALG(Aω, I, y) ≤γyOPT(I), ∀I ∈I,\n(4b)\nALG(Aω, I, y) ≤β∗\nyOPT(I), ∀I ∈Iy.\n(4c)\nConstraint (4c) enforces the optimal prediction-\nspecific consistency β∗\ny found in Problem 1.\n3.2\nMeta-Algorithm Formulation\nThe preceding approach (Problems 1 and 2) focuses on deriving an instance-specific solution given a fixed\nprediction y. To operationalize this idea in the online setting, where predictions may vary, we introduce a meta-\nalgorithm (Algorithm 1) that invokes this two-stage optimization procedure for any realized prediction. This\nenables prediction-specific robust and consistent decision-making across varying inputs, with a user-specified\nrobustness bound γ.\nIn the following proposition, which we prove in Section B, we show that this meta-algorithm yields decisions\non the prediction-specific Pareto frontier and satisfying the desired robustness bound. Moreover, the condition\nthat γ is on the (non-prediction-specific) Pareto frontier guarantees the weak optimality of Algorithm 1; this\nholds if tight Pareto fronts are available off-the-shelf for the problem at hand, such as the tight tradeoffs between\nclassical consistency and robustness available for ski rental and one-max search [2, 11, 17].\nProposition 1. Suppose there exists a weakly-optimal algorithm with robustness γ. With ω∗being an optimal\nsolution of Problem 2 in line 4 of Algorithm 1, Aω∗is β∗\ny-consistent and γ∗\ny-robust with respect to the prediction\ny, with γ∗\ny ≤γ. Furthermore, Algorithm 1 is strongly-optimal.\nNotably, the meta-algorithm (Algorithm 1) offers a systematic pipeline to achieve Pareto optimality for each\nprediction y ∈F while also guaranteeing weak optimality. While the tractability of solving the constituent\nbi-level optimization problems P1(γ, y) and P2(β∗\ny, y) depends on the structure of the specific online problem,\nthis framework provides a foundation for deriving explicit and tractable strongly-optimal algorithms for the\nproblems we discuss in the remainder of this paper.\n9\n\n3.3\nGeneralizations of this framework\nWe briefly note that the bi-level optimization framework we have proposed is quite flexible, and could be\ngeneralized in a number of ways to enable its application to different problem settings or objectives. For example,\nProblems 1 and 2 could be augmented to include practical considerations such as risk- or uncertainty-sensitive\nconstraints and objectives [36, 41, 42] or tolerance to erroneous predictions. In particular, in this work, we\ndevelop an error-tolerant variant of this framework, motivated by the continuous nature of certain problems\n(e.g., prices in the one-max search problem) and the fact that prediction errors are often unavoidable in practice.\nInstead of analyzing the tradeoff between consistency and robustness under perfectly accurate predictions, we\ncan instead study the tradeoff between ϵ-consistency and robustness, where ϵ-consistency denotes the worst-case\nguarantee when the prediction error is bounded by a chosen constant ϵ. We further describe this generalized\nframework, and how to design algorithms in this setting, in Section 7.\nWhile our approach could, in principle, be extended to more general and complex dynamic multi-round\nproblems, we anticipate that such an extension would likely require nontrivial technical developments. In\nparticular, such an extension would require a number of additional structural assumptions, e.g., that the overall\nprediction space F remains fixed and independent of the algorithm’s actions, and that the algorithm’s actions\nexert negligible influence on future prediction values. Moreover, depending on the problem, the resulting\noptimization problems may be considerably harder to analyze. For complex problems like metrical task systems\nand non-clairvoyant scheduling with n jobs, achieving even weak optimality remains an open question [17, 33];\nas such, designing strongly-optimal algorithms is inherently challenging. Thus, while our methodology in its\ncurrent form may not be directly applicable to these settings, we posit that it may still serve as a useful conceptual\nframework for designing prediction-specific algorithms for these problems. In the rest of this paper, we will\nconsider problem settings where this methodology can be tractably instantiated.\n4\nDeterministic Ski Rental\nWe begin our investigation of strongly-optimal algorithm design in specific problem settings by considering\ndeterministic algorithms for the discrete-time ski rental problem described in Section 2.3. If a deterministic\ndecision-maker buys skis at the start of day M ∈N (that may depend on the predicted day y), then the induced\ncost is\nALGDSR(M, x, y) := x · 1M(y)>x + (b + M(y) −1) · 1M(y)≤x,\nwhere b ∈N denotes the price of the skis. Then, for this problem, the general definitions of the prediction-specific\nconsistency and robustness in Equation (2) are instantiated as:\nβDSR\ny\n(AM(y)) := ALGDSR(M, y, y)\nmin{b, y}\n,\nγDSR\ny\n(AM(y)) := sup\nx∈N\nALGDSR(M, x, y)\nmin{b, x}\n.\n(5)\n4.1\nThe Deterministic Algorithm of Kumar et al. is Not Strongly-Optimal\nWe first provide a deeper analysis of the deterministic algorithm proposed by Kumar et al. [2], examining its\nprediction-specific consistency and robustness.\nThe deterministic algorithm of Kumar et al. [2, Algorithm 2], which we denote by KD, purchases at the\nbeginning of day ⌈λb⌉if y ≥b, and at the beginning of day ⌈b/λ⌉otherwise. It achieves (1+λ)-consistency and\n(1 + 1/λ)-robustness, where λ ∈(0, 1) is a tunable hyper-parameter. By the lower bound of Wei and Zhang [17],\nKD is weakly-optimal as the price b →+∞. However, as we show in the following theorem (which is proved\nin Section C.1), the KD algorithm is not strongly-optimal.\nTheorem 4.1. KD is 1-consistent and (1+1/λ)-robust when y < b, and (1+λ)-consistent and (1+1/λ)-robust\nwhen y ≥b. Furthermore, KD is not strongly-optimal even for b →+∞.\n10\n\n4.2\nA Strongly-Optimal Algorithm for Deterministic Ski Rental\nWe now turn to the design of an algorithm that is strongly-optimal in the asymptotic regime b →∞. Suppose\nthe decision maker buys at the beginning of day M. The objective is to determine M for each y ∈F such\nthat the resulting prediction-specific consistency-robustness pair (βy(M), γy(M)) is not dominated by that\nof any alternative decision M′ ̸= M, while remaining within the universal consistency–robustness bound\n(1 + λ, 1 + 1/λ). We consider the following two cases.\nCase I: y < b.\nGiven the decision M, the prediction-specific consistency and robustness are\nβy(M) =\n(\nM−1+b\ny\nif M ≤y,\n1\nif M > y.\nγy(M) =\n(\nM−1+b\nM\nif M ≤b,\nM−1+b\nb\nif M > b.\nIt’s clear that the decision M = b simultaneously achieves the optimal consistency 1 and the optimal\nrobustness 2 −1\nb with respect to y, thus dominating any other decisions.\nCase II: y ≥b.\nGiven the decision M, the prediction-specific consistency and robustness are\nβy(M) =\n(\nM−1+b\nb\nif M ≤y,\ny\nb\nif M > y.\nγy(M) =\n(\nM−1+b\nM\nif M ≤b,\nM−1+b\nb\nif M > b.\nAs shown in Figure 2 and 3, we observe that choosing M = y + 1 dominates all options with M > y + 1,\nas it offers better robustness without compromising consistency, making M = y + 1 a strong choice. Note\nthat βy(y + 1) = y/b and γy(y + 1) = (y + b)/b. Therefore, two critical decision boundaries within (0, b) are\nM1 = y + 1 −b and M2 = (b2 −b)/y.2 Specifically, when M1 < M2, M = y + 1 dominates any choice\nwithin (M1, M2). Our strategy is to set M = ⌈λb⌉as the primary decision, and to take M = y + 1 whenever\nβy(M)\n1\nb\nO\ny\nM\nβy(M) = M −1 + b\nb\nβy(M) = y\nb\ny + 1 −b\ny + 1\ny/b\nFigure 2: Prediction-specific consistency βy(M) ver-\nsus decision M given y ≥b.\nO\nγy(M) = M −1 + b\nM\nb\ny + 1\nγy(M) = M −1 + b\nb\n(y + b)/b\n(b2 −b)/y\nM\nγy(M)\nFigure 3: Prediction-specific robustness γy(M) versus\ndecision M given y ≥b.\nthe primary choice is dominated. Given λ ∈(0, 1), the condition that y + 1 dominates ⌈λb⌉in the asymptotic\nregime b →∞requires that y simultaneously satisfies the following constraints: (i) λb ≥y + 1 −b, (ii)\nλb ≤(b2 −b)/y, which together imply y ∈[b, min {b(λ + 1) −1, (b −1)/λ}].\nTherefore, when y ∈[b, min {b(λ + 1) −1, (b −1)/λ}], ⌈λb⌉is dominated by y + 1, thereby making\ny + 1 a better choice. On the other hand, when y > min {b(λ + 1) −1, (b −1)/λ}, ⌈λb⌉remains on the Pareto\nfront.\nAll together, these cases motivate the design of our deterministic algorithm, PDSR (Algorithm 2). In the\nfollowing theorem, we characterize the prediction-specific consistency and robustness of PDSR, establishing its\nstrong optimality in the asymptotic regime b →+∞; the full proof is deferred to Section C.2.\n2Note that b2−b\ny\nmay not be an integer.\n11\n\nAlgorithm 2 PDSR: PREDICTION-SPECIFIC DETERMINISTIC SKI RENTAL\n1: Input: λ ∈(0, 1)\n2: If y < b then determine M = b;\n3: Else if y ∈[b, min{b(λ + 1) −1, b−1\nλ }] then determine M = y + 1;\n4: Else if y > min{b(λ + 1) −1, b−1\nλ } then determine M = ⌈λb⌉;\n5: Buy skis on day M.\nTheorem 4.2. PDSR presented in Algorithm 2 is strongly-optimal when b →+∞, and is\n\n\n\n\n\n\n\n1-consistent and (2 −1\nb)-robust\nif y < b;\ny\nb-consistent and\n\u00001 + y\nb\n\u0001\n-robust\nif y ∈\n\u0002\nb, min\n\b\nb(λ + 1) −1, b−1\nλ\n\t\u0003\n;\n(1 + λ)-consistent and (1 + 1\nλ)-robust\nif y > min\n\b\nb(λ + 1) −1, b−1\nλ\n\t\n.\n5\nRandomized Ski Rental\nWe now turn to considering the design of randomized algorithms for the discrete-time ski rental problem.\nConsider a randomized algorithm π = (πi)i∈N+ (that may depend on the predicted day y) that chooses to buy\nskis at the start of day i ∈N+ with some probability πi. The algorithm’s average cost is\nALGRSR(π, x, y) :=\n+∞\nX\ni=1\nπi(y) · [x · 1i>x + (b + i −1) · 1i≤x],\nwhere b ∈N denotes the price of the skis. For this problem, the prediction-specific consistency and robustness\nin Equation (2) are instantiated as:\nβRSR\ny\n(Aπ(y)) := ALGRSR(π, y, y)\nmin{b, y}\n,\nγRSR\ny\n(Aπ(y)) := sup\nx∈N\nALGRSR(π, x, y)\nmin{b, x}\n.\n(6)\n5.1\nThe Randomized Algorithm of Kumar et al. is Not Strongly-Optimal\nKumar et al. [2] propose a randomized ski rental algorithm, which we denote by KR, which is a variant\nof Karlin’s classical randomized strategy [38]. Given a prediction y and a hyper-parameter λ ∈(1/b, 1), KR\nchooses the purchase day i according to the distribution\nπi =\n\n\n\n\u0000 b−1\nb\n\u0001m−i ·\n1\nb\n\u0010\n1−\n\u0010\n1−1\nb\n\u0011m\u0011,\nif i ≤m,\n0,\nif i > m,\n(7)\nwhere\nm :=\n(\n⌈b/λ⌉,\nif y < b,\n⌊λb⌋,\nif y ≥b.\nUnder this strategy, KR achieves consistency λ/(1 −e−λ) and robustness 1/(1 −e−(λ−1/b)) [2]. It is known\nthat KR is weakly-optimal as b →+∞[17]; however, as we show in the following theorem, which is proved in\nSection E.3, KR is not strongly-optimal.\nTheorem 5.1. KR is not strongly-optimal, even for b →+∞.\n12\n\nAlgorithm 3 Variant of Algorithm 1 for randomized ski rental\n1: Input: γ ∈Λγ := [eb/(eb −1), ∞);\n2: Compute {β∗\ny, π} by solving PRSR\n1\n(γ, y) (Problem 8);\n3: Obtain {γ∗\ny, π∗} by solving PRSR\n2\n(β∗\ny, y) (Problem 9);\n4: Choose i randomly according to the distribution π∗;\n5: Buy the skis at the start of day i.\n5.2\nA Strongly-Optimal Algorithms for Randomized Ski Rental\nIn the analysis that follows, we focus on designing a strongly-optimal algorithm for randomized ski rental.\nWe use R(π, x) to denote the expected ratio between the cost achieved by a randomized algorithm that uses a\ndistribution π over purchase days and the offline solution when the actual ski season lasts x days, i.e.,\nR(π, x) := ALGRSR(π, x, y)/min{b, x}.\nLet βy(π) and γy(π) denote the prediction-specific consistency and robustness of π; thus,\nβy(π) = R(π, y)\nγy(π) = max\nx∈N+ R(π, x).\nAn Optimization-Based Algorithm. While the bi-level optimization-based meta-algorithm (Algorithm 1)\nprovides a general method for computing strongly-optimal strategies, solving the underlying optimization\nproblems can be complicated in general—in particular, since they require optimizing over all possible algorithms.\nFortunately, for randomized ski rental, it is possible to restrict the support of the randomized strategy to a finite\nset U(b, y) := [b]∪{y +1} without loss of optimality, thus enabling a computationally tractable solution to these\nproblems (see Lemma 1 in Section E.1). Given a target robustness level γ ∈Λγ := [eb/(eb −1), +∞), where\neb := (1 +\n1\nb−1)b, this structural result allows us to specialize the bi-level optimization framework introduced\nin Section 3.1 as follows:\nProblem 1 (PRSR\n1\n): Minimize Consistency\nmin\nπ,βy\nβy\n(8a)\ns.t.\nR(π, x) ≤γ, ∀x ∈U(b, y)\n(8b)\nR(π, y) ≤βy\n(8c)\nLet {π, β∗} denote the optimal solution.\nProblem 2 (PRSR\n2\n): Minimize Robustness\nmin\nπ,γy\nγy\n(9a)\ns.t.\nR(π, x) ≤γy, ∀x ∈U(b, y)\n(9b)\nR(π, y) ≤β∗\ny\n(9c)\nLet {π∗, γ∗\ny} denote the optimal solution.\nNote that both of these problems are linear programs, and thus the corresponding meta-algorithm (Algo-\nrithm 3) is tractable to implement in this case. As such, we obtain as a consequence of Proposition 1 the\nfollowing strong optimality result (see Section E.4 for a full proof).\nTheorem 5.2. Algorithm 3 is strongly-optimal when b →∞.\nExplicit Algorithm Design. While the optimization-based algorithm described in Algorithm 3 is strongly-\noptimal, it does not provide much insight into the analytic structure of the resulting probability distribution\nover purchase days. Complementing this result, we can in fact leverage the problem structure to derive\na novel and explicit strongly-optimal randomized algorithm PRSR (Algorithm 9), whose construction we\ndetail in Appendix D. The algorithm builds on two transformation procedures, OPERATION A (Algorithm 7)\nand OPERATION B (Algorithm 8), which systematically adjust the “equalizing distributions” (Theorem D.1,\nEquation 18) by reallocating probability mass to trace the prediction-specific Pareto frontier for each y. As we\nshow in the following theorem, PRSR, like the optimization-based approach, achieves strong optimality.\nTheorem 5.3. The algorithm PRSR (Algorithm 9) is strongly-optimal when b →+∞.\n13\n\nAlgorithm 4 PST: PREDICTION-SPECIFIC THRESHOLDING\n1: Input: λ ∈[0, 1]\n2: Determine M = λL + (1 −λ)\n√\nLU;\n3: If y ∈[L, M] then set Φ =\n√\nLU;\n4: Else if y ∈(M,\n√\nLU] then set Φ = y;\n5: Else if y ∈(\n√\nLU, U] then determine µ =\n(1−λ)\n√\nθ\n(1−λ)\n√\nθ+λ and set Φ = µ\n√\nLU + (1 −µ)y;\n6: Perform OTA with threshold Φ.\nThe proof of Theorem 5.3 is detailed in Section E.5. We begin with a perturbation-based analysis that char-\nacterizes the structure of the optimal randomized distribution, and then establish prediction-specific optimality\nvia a primal–dual formulation.\n6\nOne-Max Search\nIn this section, we shift our focus to the one-max search problem described in Section 2.3 with predicted\nmaximum price. Let x denote the true highest price and let y denote the predicted highest price. If the decision-\nmaker decides to set their purchase threshold to Φ ∈[L, U]—i.e., they purchase at the first price exceeding Φ,\nwhich may depend on the prediction y—then their reward is\nALGOMS(Φ, x, y) := Φ(y) · 1Φ(y)≤x + L · 1Φ(y)>x.\nFor this problem, the prediction-specific consistency and robustness are:\nβOMS\ny\n(AΦ(y)) :=\ny\nALGOMS(Φ, y, y),\nγOMS\ny\n(AΦ(y)) :=\nsup\nx∈[L,U]\nx\nALGOMS(Φ, x, y).\n(10)\n6.1\nThe Algorithm of Sun et al. is Not Strongly-Optimal\nSun et al. [11] proposed a β-consistent and γ-robust Online Threshold-based Algorithm (OTA) using the\nthreshold\nΦ =\n\n\n\n\n\nLβ,\nif y ∈[L, Lβ);\nλLγ + (1 −λ)y/β\nif y ∈[Lβ, Lγ);\nLγ\nif y ∈[Lγ, U],\nwhere β = 2λθ/[\np\n(1 −λ)2 + 4λθ −(1 −λ)], γ = θ/β, and λ ∈[0, 1]. Furthermore, they show that any\nγ-robust deterministic algorithm for one-max search must have consistency β ≥θ/γ, implying their algorithm\nis weakly-optimal. However, as we establish in the following theorem (which is proved in Section F.1), their\nalgorithm is not strongly-optimal.\nTheorem 6.1. Sun’s algorithm is not strongly-optimal.\n6.2\nA Strongly-Optimal Algorithm for One-Max Search\nWe propose in Algorithm 4 a new approach, PST, which, by more carefully selecting the purchase threshold,\nachieves strong optimality. In particular, it achieves the prediction-specific consistency and robustness values\nestablished in the following theorem.\nTheorem 6.2. PST (Algorithm 4) is strongly-optimal, and is\n\n\n\n\n\n\n\n(y/L)-consistent and\n√\nθ-robust\nif y ∈[L, λL + (1 −λ)\n√\nLU);\n1-consistent and (U/y)-robust\nif y ∈[λL + (1 −λ)\n√\nLU,\n√\nLU];\n\u0010\n(1−λ)\n√\nθy+λy\n(1−λ)U+λy\n\u0011\n-consistent and\n\u0010\n(1−λ)U+λy\n(1−λ)\n√\nLU+λL\n\u0011\n-robust\nif y ∈(\n√\nLU, U].\n14\n\nAlgorithm 5 ϵ-Tolerant PST\n1: Input: λ ∈[0, 1], ϵ > 0\n2: Determine M = λ(L + 3ϵ) + (1 −λ)(\n√\nLU −ϵ);\n3: If y ∈[L, M −2ϵ] then set Φ =\n√\nLU;\n4: Else if y ∈(M −2ϵ, M) set Φ = M −ϵ;\n5: Else if y ∈[M,\n√\nLU + ϵ] then set Φ = y −ϵ;\n6: Else if y ∈(\n√\nLU + ϵ, U −ϵ) then set µ = (U−2ϵ)−LU/(M−ϵ)\n(U−2ϵ)−\n√\nLU\n, Φ = µ\n√\nLU + (1 −µ)(y −ϵ);\n7: Else if y ∈[U −ϵ, U] then set Φ = LU/(M −ϵ);\n8: Perform OTA with threshold Φ.\nWe prove Theorem 6.2 in Section F.2; the proof identifies prediction-specific optimal thresholds (Φ) by\npartitioning the prediction space and deriving each threshold (which is a convex combination involving µ) that\nensures Pareto-optimality for each segment of predictions.\n7\nError-Tolerant Algorithms\nPareto-optimal algorithms can exhibit brittleness, a vulnerability noted by [18, 19], where the competitive\nratio degrades sharply toward the worst-case robustness bound even with small prediction errors. This issue\nstems from the standard definition of consistency (see Definition 1), which assumes strictly perfect predictions\n(x(I) = y) and thus fails to consider performance under erroneous predictions. To address this, we use the\none-max search problem as an example to demonstrate how explicit and tractable Pareto-optimal algorithms\nincorporating a “generalized consistency” can be constructed in our prediction-specific framework, offering\ngood performance tradeoffs when faced with minor prediction errors. It is worth emphasizing that the notion of\n“error tolerance” considered in this section differs from the concept of “smoothness” discussed in [18–20]. The\nformer concerns the tradeoff between generalized consistency (allowing small prediction errors) and robustness\n(with arbitrarily large errors), whereas the latter requires that the algorithm’s performance degrade smoothly as\nthe prediction error increases. Nevertheless, we view the two notions as related, both contributing to alleviating\n\"brittleness\".\nTo account for small prediction errors, we specify a desired error tolerance ϵ and define a relaxed consistent\nset Iϵ\ny := {I ∈I : L(x(I), y) ≤ϵ} , where L : F × F →R≥0 is a chosen loss function that measures\nprediction error. When substituting Iy with Iϵ\ny, we refer to the corresponding consistency metrics in Equation (1)\nand Equation (2) as ϵ-consistency and prediction-specific ϵ-consistency, denoted by βϵ and βϵ\ny, respectively.\nSubstituting β, βy and Iy with βϵ, βϵ\ny and Iϵ\ny in Section 3, we can generalize our meta-algorithm to incorporate\nerror tolerance.\nFor the one-max search problem, its inherent continuity renders prediction errors unavoidable in practice,\nunderscoring the necessity of incorporating error tolerance. We fix L(x, y) = |x −y| and assume that ϵ is small\nrelative to the scale of the problem. We propose an error-tolerant algorithm for this problem, ϵ-tolerant PST,\nin Algorithm 5. To conclude the section, we present three theorems that characterize the performance of this\nalgorithm. In particular, Theorem 7.1 establishes the tradeoffs between ϵ-consistency and robustness achieved by\nϵ-tolerant PST, together with their prediction-specific counterparts, while Theorem 7.2 provides a lower bound\non the global ϵ-consistency–robustness tradeoff. Finally, Theorem 7.3 shows that the ϵ-tolerant PST algorithm is\nstrongly optimal. The proofs of all three results are given in Section G.\nTheorem 7.1. ϵ-Tolerant PST achieves [(M −ϵ)/L] ϵ-consistency and U/(M −ϵ) robustness. Specifically,\n15\n\nϵ-Tolerant PST achieves\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n\n[(y + ϵ)/L] ϵ-consistency and\n√\nθ robustness\nif y ∈[L, M −2ϵ];\n(M −ϵ)/L ϵ-consistency and U/(M −ϵ) robustness\nif y ∈(M −2ϵ, M);\n(y + ϵ)/(y −ϵ) ϵ-consistency and U/(y −ϵ) robustness\nif y ∈[M,\n√\nLU + ϵ];\n(y+ϵ)\nµ\n√\nLU+(1−µ)(y−ϵ) ϵ-consistency and µ\n√\nLU+(1−µ)(y−ϵ)\nL\nrobustness\nif y ∈(\n√\nLU + ϵ, U −ϵ);\n(M −ϵ)/L ϵ-consistency and U/(M −ϵ) robustness\nif y ∈[U −ϵ, U];\nwhere M = λ(L + 3ϵ) + (1 −λ)(\n√\nLU −ϵ).\nTheorem 7.2. Any γ-robust algorithm has at least (θ/γ) ϵ-consistency, and any algorithm that achieves βϵ\nϵ-consistency must be at least (θ/βϵ)-robust.\nTheorem 7.3. Assume ϵ ≤(\n√\nLU −L)/4. The ϵ-consistency and robustness of ϵ-Tolerant PST (Algorithm 5)\nare jointly Pareto optimal. Moreover, for every prediction y ∈F = [L, U], ϵ-Tolerant PST achieves prediction-\nspecific ϵ-consistency βϵ\ny and robustness γy that are jointly Pareto optimal.\n8\nNumerical Experiments\nIn this section, we evaluate the performance of our proposed algorithms in three different case studies\nspanning synthetic and real-world settings.3\n8.1\nCase Study 1: Synthetic Data Experiments for Ski Rental\nWe begin by testing the performance of our algorithms for the ski rental problem via simulations on synthetic\ninstances. We let the actual number of skiing days x be a uniformly random integer drawn from [1, 10b], where\nb = 100 is buying costs of skis. The prediction y is generated with accuracy p: with probability p, the prediction\nis accurate (i.e., y = x), and with probability (1 −p), the prediction y ∼N(x, σ2), where σ = 500 and the\noutput is rounded and made positive.\nFor the deterministic setting, we compare PDSR (Algorithm 2) with KD ([2], Algorithm 2) and the classic\ncompetitive algorithm (which always buys on day b), using the same parameter λ = 0.5 for both PDSR and\nKD. For the randomized setting, we compare PRSR (Algorithm 9) with KR ([2], Algorithm 3) and Karlin’s\nalgorithm [38], using γ = 3 for PRSR and λ = ln(3/2) for KR, ensuring PRSR and KR have the same\nrobustness 3. Each setup is evaluated over 10000 independent trials. Figures 4 and 5 present the empirical results\nof average competitive ratio versus accuracy p. We observe that our proposed algorithms, PDSR and PRSR,\nconsistently outperform both classic online algorithms and existing learning-augmented algorithms across both\nsettings.\n8.2\nCase Study 2: Ski Rental on Dynamic Power Management\nWe next evaluate our ski rental algorithms on real-world traces for a Dynamic Power Management (DPM)\nproblem, where we control the idle and active periods of a computing device. Modern processors typically\nsupport multiple power states: deeper states disable more components, leading to lower operating cost/energy\nbut higher wake-up penalties/overhead. During each idle interval, a DPM controller must decide whether to stay\nactive or transition into a deeper sleep state without knowing the remaining idle duration.\n3Our code is publicly available at https://github.com/Bill-SizheLi/Prediction_Specific_Design_of_\nLearning-Augmented_Algorithms\n16\n\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nAccuracy of ML Predictor\n1.5\n1.6\n1.7\n1.8\n1.9\nEmpirical Competitive Ratio\nPDSR (Ours)\nKD\nCompetitive\nFigure 4: Empirical competitive ratios versus accuracy\np in deterministic setting.\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nAccuracy of ML Predictor\n1.20\n1.25\n1.30\n1.35\n1.40\n1.45\n1.50\n1.55\nEmpirical Competitive Ratio\nPRSR (Ours)\nKR\nCompetitive\nFigure 5: Empirical competitive ratios versus accuracy\np in randomized setting.\nlog126_copyDtoH\nlog176_Booting\nlog225_MusicFaceBook\nlog225_MusicTwitter\nlog245_MusicFaceBook\nlog126_copyDtoH\nlog176_Booting\nlog225_MusicFaceBook\nlog225_MusicTwitter\nlog245_MusicFaceBook\n1.0\n1.5\n2.0\n2.5\n3.0\n3.5\nEmpirical ratio\nGood predictions\nBad predictions\n(\ne\ne\n1)-competitive\nBlindly Trust\nKR\nPRSR(Ours)\nFigure 6: Empirical competitive ratios on DPM traces for good and bad predictions.\nThe two-state DPM system (one active and one sleep state with zero operating cost) is equivalent to the ski\nrental problem, where remaining active corresponds to renting and transitioning to the sleep state corresponds to\nbuying. Moreover, Antoniadis et al. [5] demonstrated that randomized ski rental algorithms can be converted to\nmulti-state DPM algorithms.\nSetup. We consider a DPM problem with 4 states. Specifically, we use the same problem setting as\nAntoniadis et al. [5], employing I/O traces4 collected from a Nexus 5 smartphone [43], from which idle intervals\nbetween consecutive requests are extracted. We adopt the IBM mobile hard-drive power states reported in [44],\nconsistent with the setup in [5]. The idle periods are scaled in the same way as in [5].\nWe use the five largest traces for evaluation. Since the main goal of this section is to probe the algorithms’\nperformance under the two extremes of very good and very bad predictions, we consider the following method\nfor generating predictions: \"good predictions\" and \"bad predictions\" are obtained by perturbing the ground truth\nwith N(0, σ2\ngood) and N(0, σ2\nbad) noises, respectively. In this experiment, we set σgood = 0.02 and σbad = 20.\nWe compare four algorithms: the classic (\ne\ne−1)-competitive algorithm, Blindly Trust (which treats the prediction\nas if it is correct and optimizes accordingly), the randomized algorithm of Kumar et al. (KR), and our randomized\nalgorithm (PRSR). For the learning-augmented algorithms, we use the same parameter values for λ and γ as in\nCase Study 1.\nResults. Figure 6 reports the empirical competitive ratios on the real DPM traces. We observe that\nour strongly-optimal algorithm PRSR consistently achieves the lowest competitive ratios, except for when\n4The traces are available at http://iotta.snia.org/traces/block-io.\n17\n\nthe prediction quality is very good (in which case, the non-robust Blindly Trust algorithm outperforms it.\nThis validates our algorithm’s ability to exploit specific predictions to enable good performance regardless of\nprediction value or quality.\n8.3\nCase Study 3: One-Max Search on VIX Trading\nFigure 7: VIX Closing Price from January 2020\nto December 2024. The VIX index soared from a\npre-pandemic level of 12-15 to 83 in March 2020.\nThe VIX, often referred to as the fear index, exhibits\nsharp volatility spikes that make it a natural benchmark for\nevaluating online search algorithms. Its uncertainty and\nheavy-tailed dynamics closely mirror the one-max search\nsetting, where the core challenge lies in optimally timing\na single exit. This characterization is vividly illustrated\nby the extreme volatility in early 2020: within just a few\nweeks at the onset of the COVID-19 market shock, the VIX\nsurged from below $15 to over $80 (shown in Figure 7).\nSetup. We evaluate our one-max search algorithms\nin a case study using the daily closing prices of the VIX\nindex from January 2020 to December 2024 (shown in Fig-\nure 7), which consist of publicly available values obtained\nfrom the Cboe Options Exchange. We assume that at the\nbeginning of each month, an agent holds one unit of VIX\nand must choose a single day within that month to sell it. Over the course of five years, there are 60 trading\nrounds (one per month), each offering approximately 20 to 21 trading opportunities, as the VIX is only traded on\nweekdays. We set L and U as the historical minimum and maximum prices over the entire 5 years.5\nBaselines. We compare our proposed methods, PST(Algorithm 4) and ϵ-Tolerant PST(Algorithm 5), to\nthree baseline algorithms: (i) blindly trusting the prediction, (ii) the classical online algorithm of El-Yaniv [39],\nand (iii) prior learning-augmented algorithms (Sun’s [11] and Benomar’s [19]).\nExperiment 1. In this experiment, we consider a naive prediction strategy that simply uses the highest\nobserved VIX price from the previous month. As the evaluation metric, we use the empirical ratio6, defined as\nthe cumulative online outcome up to the current round divided by the cumulative offline optimum. This metric\nreflects how well an algorithm performs in practice relative to the hindsight-optimal strategy, averaged over time.\nWe run the algorithms over the 60-month horizon using historical VIX data, and report the empirical ratios at\neach round to visualize both long-term trends and the stability of performance across different market periods in\nFigure 8.\nFor our proposed algorithms, we fix the trade-off parameter λ = 0.3 in PST, and use λ = 0.3 and ϵ = 1.8\nin ϵ-Tolerant PST. For baseline algorithms with tunable parameters, including those from Sun et al. [11] and\nBenomar et al. [19], we find that setting λ = 1.0 yields the best cumulative empirical ratio over the full 60-month\nhorizon. However, to better illustrate performance variation across different regimes, we also include their results\nunder λ = 0.3 and λ = 0.6.\nExperiment 2. In practical settings, machine-learned predictions are often more accurate than the naive\npredictor used in Experiment 1, though they remain imperfect due to model limitations and data noise. The degree\nof prediction accuracy varies with the capability and training of the underlying ML model. To systematically\nevaluate algorithmic performance under varying prediction quality, we introduce the notion of an error level –\na scalar value between 0 and 1 that quantifies the deviation from perfect information. For each trading round,\nthe prediction is constructed via linear interpolation between the previous month’s maximum (naive prediction)\n5The focus of this paper is not on the impact of L and U; therefore, we simply set them to historical values. In practical trading\nscenarios, L and U can be viewed as predetermined parameters representing the stop-loss and take-profit thresholds in the exit strategy\nof the trading process.\n6Note that the empirical ratio here is the inverse of that used in the theoretical analysis, so as to better reflect the proportion of the\nhindsight optimum that the online or learning-augmented algorithm can achieve (or recover).\n18\n\n0\n10\n20\n30\n40\n50\n60\nRound\n0.60\n0.65\n0.70\n0.75\n0.80\n0.85\n0.90\n0.95\nEmpirical Ratio\nBlindly Trust\nEl-Yaniv's (Sun's ( =1) & Benomar's( =1))\nSun's ( =0.6)\nSun's ( =0.3)\nBenomar's ( =0.6)\nBenomar's ( =0.3)\nPST (Ours)\n-Tolerant PST (Ours)\nFigure 8: Empirical competitive ratios over 60 trading\nrounds.\n0.0\n0.2\n0.4\n0.6\n0.8\n1.0\nError Level\n0.800\n0.825\n0.850\n0.875\n0.900\n0.925\n0.950\n0.975\n1.000\nCumulative Empirical Ratio\nBlindly Trust\nEl-Yaniv's\nSun's\nBenomar's\nPST (Ours)\n-Tolerant PST (Ours)\nFigure 9: Cumulative empirical competitive ratio with\nvarying prediction error levels.\nand the current month’s actual maximum (perfect prediction), where the error level determines the interpolation\nweight. An error level of 1.0 corresponds to the naive prediction, while 0.0 yields the perfect prediction.\nWe assess the cumulative empirical ratio of each algorithm over all 60 trading rounds under varying error\nlevels from 0 to 1, and report the result in Figure 9. To ensure a fair comparison across prediction regimes, we fix\nthe trade-off parameter λ = 0.5 for all tunable methods, including those of Sun, Benomar, PST, and ϵ-Tolerant\nPST. For ϵ-Tolerant PST, we additionally set ϵ = 0.5 to account for moderate tolerance to prediction error.\nResults. The results for Experiment 1 are shown in Figure 8. Our methods consistently outperform all\nbaselines across the decision horizon, with final empirical competitive ratios of 87.2% (ϵ-Tolerant PST) and\n85.7% (PST), compared to 82.4%–84.4% for all baselines. The results for Experiment 2 are shown in Figure 9;\nϵ-Tolerant PST remains consistently superior across nearly all error levels.\n9\nConcluding Remarks\nIn this work, we introduce a prediction-specific analysis framework and a finer-grained notion of strong\noptimality for online algorithms with predictions. We further provide a systematic approach to designing\nPareto-optimal online algorithms with better prediction-specific performance than prior algorithms, and we\nshow how this methodology can yield significant performance improvements for the problems of ski rental\n(deterministic and randomized) and one-max search.\nFuture Directions. In contrast to the ski rental and one-max search settings, the existing weakly-optimal\nnon-clairvoyant scheduling algorithm of Wei and Zhang [17] is strongly-optimal when n = 2. Thus, designing a\nstrongly-optimal algorithm for n-job non-clairvoyant scheduling remains an open question. Similarly, as we\ndid for one-max search, developing explicit error-tolerant strongly-optimal algorithms for both deterministic\nand randomized ski rental is also an interesting future direction. In addition, exploring whether the bi-level\noptimization in the meta-algorithm (Algorithm 1) in Section 3 can be tractably solved for more complex,\nmulti-stage problems represents a challenging but potentially impactful direction for future study.\nReferences\n[1] Allan Borodin and Ran El-Yaniv. Online Computation and Competitive Analysis. Cambridge University\nPress, 2005. 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As detailed in Algorithm 6 in\nSection A.1, the algorithm proceeds in two phases based on predicted job lengths y = (y1, y2), where y1 ≤y2.\nIn Theorem 2.1, we show that this algorithm also satisfies prediction-specific Pareto optimality under n = 2.\nA.1\nWei et al.’s Two-Stage Schedule\nWei and Zhang [17] propose an algorithm called the two-stage schedule (see Algorithm 6) that achieves a\nconsistency of (1 + λ) and a robustness of (1 +\n1\n1+6λ) under n = 2. By [17] and Definition 3, this two-stage\nschedule algorithm is weakly-optimal for n = 2.\nAlgorithm 6 Two-Stage Schedule\n1: At any point, if a job finishes with processing time less or more than its prediction, use round robin forever.\n2: Stage 1: Round robin for at most λn · OPTy/\n\u0000n\n2\n\u0001\nunits of time.\n3: Stage 2: Process jobs in predicted order\n(starting from the unfinished job with the least predicted time).\nA.2\nProof of Theorem 2.1\nIn this subsection, we further analyze the prediction-specific consistency and robustness and prove that\ntwo-stage schedule is strongly-optimal under n = 2.\nProof of Theorem 2.1. Note that the jobs are ranked based on their predicted lengths; thus we have y1 ≤y2. We\nfirst prove the prediction-specific consistency and robustness under a specific set of predictions y = (y1, y2).\nWe first consider λ ≤\ny1\n2y1+y2 , i.e. λ(2y1 + y2) ≤y1. Regarding the consistency, assume that x1 = y1, x2 =\ny2. In stage 1, the algorithm runs round-robin for 2λ · (2y1 + y2) time. Since λ · (2y1 + y2) ≤y1, job 1 cannot\nfinish in stage 1. Therefore, the completion time of job 1 is\n2λ(2y1 + y2) + y1 −λ(2y1 + y2) = y1 + λ(2y1 + y2),\nand that of job 2 is y1 + λ(2y1 + y2) + y2 −λ(2y1 + y2) = y1 + y2, thus, ALG = 2y1 + y2 + λ(2y1 + y2),\nand OPT = 2y1 + y2, yeilding a consistency of 1 + λ.\nRegarding the robustness, we consider an adversarial attack x1, x2. Let δ denote an infinitesimal quantity.\nCase I: If x1 ≤λ(2y1 + y2) or x2 ≤λ(2y1 + y2) , i.e. some incorrect prediction is found in stage 1. In this\ncase, the algorithm runs round-robin from beginning to end, resulting in a robustness of at most 4/3.\nCase II: If λ(2y1 + y2) < x1 ≤y1 , i.e. job 1 finishes no later than its prediction. In this case, the algorithm\nruns round-robin for 2λ(2y1 + y2) time and processes job 1 until completion, and then turns to job 2. Thus,\nALG = 2\n\u0010\n2λ(2y1 + y2) +\n\u0000x1 −λ(2y1 + y2)\n\u0001\u0011\n+ (x2 −λ(2y1 + y2))\n= 2x1 + x2 + λ(2y1 + y2).\nCase II(a): λ(2y1 + y2) < x2 ≤x1 In this case, OPT = 2x2 + x1. This yields a robustness of 1 +\ny1\ny1+2λ(2y1+y2) ≥4/3, which is attained when x1 = y1 and x2 = λ(2y1 + y2) + δ.\n24\n\nCase II(b): x2 > x1 In this case, OPT = 2x1 + x2. This results in a robustness of 1 + λ(2y1+y2)\n3y1\n≤4/3,\nwhich is achieved when x1 = y1 and x2 = y1 + δ.\nCase III: If x1 > y1 , i.e. job 1 finishes later than its prediction. In this case, the algorithm first runs round-robin\nfor 2λ · (2y1 + y2) time, then processes job 1 for y1 −λ(2y1 + y2) time, and finally runs round-robin till the end.\nCase III(a) λ(2y1 + y2) < x2 < x1 + λ(2y1 + y2) −y1 , i.e. job 1 finishes later than job 2.\nIn this case, ALG = x1 + 3x2 −λ(2y1 + y2) + y1 and OPT = 2x2 + x1. This yields a robustness of\n1 +\ny1\ny1+2λ(2y1+y2) ≥4/3, which is achieved when x1 = y1 + δ and x2 = λ(2y1 + y2) + δ.\nCase III(b) x2 ≥x1 + λ(2y1 + y2) −y1 , i.e. job 1 finishes no later than job 2.\nNote that in this case, we have ALG = 3x1 + x2 + λ(2y1 + y2) −y1.\nFirst, if x1 < x2, OPT = 2x1 + x2. This generates a robustness of 1 + λ(2y1+y2)\n3y1\n≤4/3, which is achieved\nwhen x1 = y1 + δ and x2 = y1 + 2δ.\nOtherwise, if x1 + λ(2y1 + y2) −y1 < x2 < x1, OPT = 2x2 + x1. This yields a robustness of 1 +\ny1\ny1+2λ(2y1+y2) ≥4/3, which is achieved when x1 = y1 + δ and x2 = λ(2y1 + y2) + 2δ.\nTo sum up, if λ ≤y1/(2y1 + y2), the algorithm is (1 + λ)-consistent and (1 +\ny1\ny1+2λ(2y1+y2))-robust.\nOtherwise, if λ > y1/(2y1 + y2), the algorithm runs round-robin forever, and is (1 +\ny1\n2y1+y2 )-consistent and\n(4/3)-robust.\nIn conclusion, given prediction y = (y1, y2), the prediction-specific consistency and robustness are\nβy = 1 + min{\ny1\n2y1 + y2\n, λ},\nγy = 1 + max{1/3,\ny1\ny1 + 2λ(2y1 + y2)}.\nSince the two-stage schedule is already proven to be weakly-optimal, we then prove that it is strongly-optimal\nby demonstrating the Pareto optimality of their prediction-specific consistency and robustness.\nWhen λ ≤y1/(2y1 + y2), the algorithm is (1 + λ)-consistent and\n\u0010\n1 +\ny1\ny1+2λ(2y1+y2)\n\u0011\n-robust. Consider\nalgorithm Aω that completes r > λ(2y1 + y2) amount of work for job 2 when it finishes job 1 in the case where\npredictions are accurate. Then, it follows that\nALG = 2 · (y1 + r) + y2 −r = 2y1 + y2 + r\nOPT = 2y1 + y2,\nwhich yields a competitive ratio of 1 + r/(2y1 + y2) > 1 + λ. Therefore, any (1 + λ)-consistent algorithm\nprocesses at most λ(2y1 +y2) amount of work for job 2 when it finishes job 1 or finds any incorrect prediction of\njob 1. Then, we consider an incorrect prediction x1 = y1, x2 = λ(2y1 + y2) + δ. Consider a (1 + λ)-consistent\nalgorithm B that completes r ≤λ(2y1 + y2) amount of work when it finishes job 1. Then, upon x1 = y1 and\nx2 = r + δ, ALG = 2(y1 + r) + (r + δ −r) = 2y1 + 2r + δ and OPT = 2r + y1. This leads to a robustness\nno better than\nmin\nr≤λ(2y1+y2)\n2y1 + 2r\ny1 + 2r = 1 +\ny1\ny1 + 2λ(2y1 + y2)\nfor all r ≤λ(2y1 + y2).\nWhen λ > y1/(2y1 + y2), the algorithm, equivalent to round-robin (RR), is (1 + y1/(2y1 + y2))-consistent\nand 4/3-robust. Note that RR is the only algorithm that achieves 4/3 competitive ratio. Therefore, under\nprediction y = (y1, y2), no other algorithm achieves a robustness equal to or less than 4/3.\nBy Definition 4, the two-stage schedule is strongly-optimal.\n25\n\nB\nAdditional Details for Section 3\nIn this section, we provide a proof of Proposition 1 and discuss the scenario in which a weakly-optimal\nalgorithm with robustness γ is unavailable.\nB.1\nProof of Proposition 1\nProof of Proposition 1. Let {β∗\ny, ω} and {γ∗\ny, ω∗} denote the optimal solution to Problem 1 and Problem 2,\nrespectively. Since γ ≥α∗, {γ, ω} is always a feasible solution to Problem 2. Thus, γ∗\ny ≤γ, where γ∗\ny is the\noptimal objective value to Problem 2. Since {γ∗\ny, ω∗} is a feasible solution to Problem 2, we have\nALG(Aω∗, I, y) ≤γ∗\ny · OPT(I), ∀I ∈I,\nALG(Aω∗, I, y) ≤β∗\ny · OPT(I), ∀I ∈Iy.\nBy Definition 2, Aω∗is β∗\ny-consistent and γ∗\ny-robust with respect to y.\nConsider ω′ ∈Ω. Let β′\ny and γ′\ny denote the prediction-specific consistency and robustness of Aω′ with\nrespect to y. We consider two cases.\nCase I: β′\ny > β∗\ny If β′\ny > β∗\ny, then (β′\ny, γ′\ny) produces no Pareto improvement over (β∗\ny, γ∗\ny).\nCase II: β′\ny ≤β∗\ny . If β′\ny ≤β∗\ny, since Aω′ is β′\ny-consistent and γ′\ny-robust with respect to y, by Definition 2,\nALG(Aω′, I, y) ≤β′\ny · OPT(I), ∀I ∈Iy,\n(11a)\nALG(Aω′, I, y) ≤γ′\ny · OPT(I), ∀I ∈I.\n(11b)\nSince β′\ny ≤β∗\ny, by Inequality (11a), we further have\nALG(Aω′, I, y) ≤β∗\ny · OPT(I), ∀I ∈Iy.\n(12)\nBy Inequalities (11b) and (12), {γ′\ny, ω′} is a feasible solution to Problem 2, thus we have γ∗\ny ≤γ′\ny, where γ∗\ny is\nthe optimal objective value to Problem 2.\nCase II (a): γ′\ny > γ∗\ny If γ′\ny > γ∗\ny, then (β′\ny, γ′\ny) produces no Pareto improvement over (β∗\ny, γ∗\ny).\nCase II (b): γ′\ny = γ∗\ny If γ′\ny = γ∗\ny, since γ∗\ny ≤γ, we have γ′\ny ≤γ. By Inequality (11b), we further have\nALG(Aω′, I, y) ≤γ · OPT(I), ∀I ∈I.\n(13)\nBy Inequalities (11a) and (13), {β′\ny, ω′} is a feasible solution to Problem 1, thus we have β∗\ny ≤β′\ny, where β∗\ny\nis the optimal objective value to problem 1. Because β′\ny ≤β∗\ny and β∗\ny ≤β′\ny, we have β′\ny = β∗\ny. Since γ′\ny = γ∗\ny\nand β′\ny = β∗\ny, (β′\ny, γ′\ny) produces no Pareto improvement over (β∗\ny, γ∗\ny).\nConsequently, (β∗\ny, γ∗\ny) are jointly Pareto optimal in terms of the prediction-specific consistency and robust-\nness with respect to y. Since γ∗\ny ≤γ for all y ∈F, we have supy∈F γ∗\ny ≤γ. By Equation (1) and Equation (2),\nAlgorithm 1 is γ-robust.\nNow, there exists a weakly-optimal algorithm Aω with robustness γ, we assume the consistency of Aω is β\nand the prediction-specific consistency of Aω under prediction y is βy. Since Aω is βy-consistent with respect to\ny, by Definition 2,\nALG(Aω, I, y) ≤βy · OPT(I), ∀I ∈Iy.\n(14)\nSince Aω is γ-robust, it is also γ-robust with respect to y, thus we have\nALG(Aω, I, y) ≤γ · OPT(I), ∀I ∈Iy.\n(15)\n26\n\nBy Inequalities (14) and (15), {βy, ω} is a feasible solution to Problem 1. Therefore, β∗\ny ≤βy, where β∗\ny is\nthe optimal objective value to Problem 1. Consequently,\nsup\ny∈F\nβ∗\ny ≤sup\ny∈F\nβy = β.\nBy Equation (1) and Equation (2), Algorithm 1 is β-consistent.\nBy weak optimality of Aω, any γ-robust algorithm is at least β-consistent, and any β-consistent algorithm is\nat least γ-robust. Since Algorithm 1 is β-consistent and γ-robust, by Definition 3, it is weakly-optimal.\nMoreover, ∀y ∈F, (β∗\ny, γ∗\ny) are jointly Pareto optimal in terms of the prediction-specific consistency and\nrobustness with respect to y. Thus, Algorithm 1 is strongly-optimal by Definition 4.\nB.2\nAddressing the Absence of a Weakly-Optimal Algorithm with γ Robustness.\nIn general, even if (β′, γ) is not on the Pareto front for any β′ ≥1, a process that first determines a tight\nconsistency bound β = supy′∈F P1(γ, y′), then determines a tight robustness bound γ = supy′∈F P2(β, y′)\ncan generate a tight Pareto-optimal consistency-robustness tradeoff (β, γ) so that γ becomes a valid input of\nAlgorithm 1.\nLet β, γ denote supy′∈F P1(γ, y′), supy′∈F P2(β, y′), respectively. Define y1 := arg maxy′∈F P1(γ, y′),\ny2 := arg maxy′∈F P2(β, y′).\nSince β = supy′∈F P1(γ, y′), we have ∀y′ ∈F, P1(γ, y′) ≤β. Therefore,\n∀y′ ∈F, ∃ω′ ∈Ω, s.t. {γ, ω′} is a feasible solution to P2(β, y′),\ni.e., ∀y′ ∈F, γ ≥P2(β, y′), where P2(β, y′) is the optimal objective value. Consequently,\nγ ≥sup\ny′∈F\nP2(β, y′) = γ.\n(16)\nWe can use similar techniques to prove\nβ ≥sup\ny′∈F\nP1(γ, y′).\nSince β = supy′∈F P1(γ, y′), any γ-robust algorithm is at least β-consistent. We prove this by contradiction,\nassuming there exists a γ-robust algorithm A that has consistency βA < β. Then, under the prediction y1, A\nachieves a prediction-specific consistency\nβA\ny1 ≤βA < β = P1(γ, y1).\nMoreover, A is γ-robust under y1, thus satisfying Constraint 3b. Therefore, P1(γ, y1) is not the optimal\nobjective value, since βA\ny1 < P1(γ, y1), yielding a contradiction.\nNote that by Inequality (16), γ ≥γ; thus we can further conclude that any γ-robust algorithm is at least\nβ-consistent.\nSimilarly, since γ = supy′∈F P2(β, y′), any β-consistent algorithm is at least γ-robust. Therefore, (β, γ) is\na Pareto-optimal consistency-robustness tradeoff.\n27\n\nC\nProofs for Section 4\nWe provide the proofs of Theorem 4.1 and 4.2 in the following.\nC.1\nProof of Theorem 4.1\nProof of Theorem 4.1. We begin by analyzing the prediction-specific consistency and robustness of KD, consid-\nering two distinct cases: y < b and y ≥b.\nCase I: If y < b , then the algorithm buys on day ⌈b/λ⌉.\nTo obtain the prediction-specific consistency, we assume that the prediction is accurate, i.e. x = y. Since\nthe algorithm postpones the purchase beyond the predicted day, ALG = OPT = y, thus βy = 1. To analyze\nthe prediction-specific robustness, we consider incorrect predictions. Since the worst-case attack arises when\nx = ⌈b/λ⌉, we have ALG = ⌈b/λ⌉−1 + b and OPT = b. Thus, γy = ALG/OPT < (b/λ + b)/b = 1 + 1/λ.\nCase II: If y ≥b , then the algorithm buys on day ⌈λb⌉.\nTo obtain the prediction-specific consistency, we assume x = y. ALG = ⌈λb⌉−1 + b and OPT = b,\nyielding βy = ALG/OPT < 1 + λ. To obtain the prediction-specific robustness, we consider incorrect\npredictions. Observe that the worst-case attack occurs when x = ⌈λb⌉, we have ALG = ⌈λb⌉−1 + b, and\nOPT = ⌈λb⌉. Hence, γy = ALG/OPT < (λb + b)/⌈λb⌉≤(λb + b)/(λb) = 1 + 1/λ.\nWe now prove that KD is not strongly-optimal. Consider a simple algorithm that always buys on day b. It is\nstraightforward to verify that buy-to-rent algorithm is 1-consistent and (2 −1/b)-robust under y < b. Recall that\nunder y < b, KD’s prediction-specific consistency and robustness are 1 and ⌈b/λ⌉−1+b\nb\n, respectively. Since\n⌈b/λ⌉−1 + b\nb\n> b −1 + b\nb\n= 2 −1/b\nfor all λ ∈(0, 1), by Definition 4, KD is not strong-optimal.\nC.2\nProof of Theorem 4.2\nProof of Theorem 4.2. Denote the prediction-specific consistency and robustness of Algorithm 2 with respect to\ny as βy and γy.\nWe start by analyzing the prediction-specific consistency and robustness of Algorithm 2 by considering the\nfollowing three different cases.\nCase I: y < b . In this case, Algorithm 2 purchases on day b. It is straightforward that the algorithm is\n1-consistent and (2 −1/b)-robust with respect to y.\nCase II: y ∈[b, min{b(λ + 1) −1, (b −1)/λ}] . In this case, Algorithm 2 purchases on day y+1. To prove the\nprediction-specific consistency, we assume x = y. ALG = y and OPT = b, yielding βy = ALG/OPT = y/b.\nTo prove the prediction-specific robustness, we consider inaccurate predictions. Observe that x = y + 1 brings\nthe best attack to the algorithm, in which case ALG = y +b, OPT = b, leading to γy = ALG/OPT = 1+y/b.\nCase III: y > min{b(λ + 1) −1, (b −1)/λ} . In this case, Algorithm 2 buys on day ⌈λb⌉. To obtain the\nprediction-specific consistency, we assume x = y. ALG = ⌈λb⌉−1 + b, OPT = b, βy = ALG/OPT < 1 + λ.\nTo get the prediction-specific robustness, we consider incorrect predictions. Observe that the worst-case attack\noccurs at x = ⌈λb⌉, in which case ALG = ⌈λb⌉−1 + b, OPT = ⌈λb⌉. Thus, γy = ALG/OPT < 1 + 1/λ.\nNow, we prove that Algorithm 2 is strongly-optimal.\nBy considering the worst-case prediction y over the βy and γy in Theorem 4.2, Algorithm 2 is (1 + λ)-\nconsistent and (1+1/λ)-robust, where the worst-case prediction occurs when y > min{b(λ+1)−1, (b−1)/λ}.\nBased on the lower bound by Wei and Zhang [17], Algorithm 2 is weakly-optimal, as b →∞.\n28\n\nWe consider the following three cases to prove the Pareto optimality of (βy, γy) when b →∞.\nCase I: y < b . Algorithm 2 is 1-consistent and (2 −1/b)-robust, which is already optimal. This is because\nconsistency can be no less than 1, and (2−1/b) is the best competitive ratio α achievable in competitive analysis.\nCase II: y ∈[b, min{b(λ + 1) −1, (b −1)/λ}] . Algorithm 2, which purchases on day y + 1, is (y/b)-\nconsistent and (1 + y/b)-robust with respect to y. Since b ≤y ≤min{b(λ + 1) −1, (b −1)/λ}, we have (i)\nλb ≥y + 1 −b, (ii) λb ≤b2−b\ny , which together implies\n(b2 −b)/y ≥y + 1 −b.\n(17)\nConsider another algorithm P that buys on day p (̸= y + 1) with prediction-specific consistency and robustness\nβP\ny and γP\ny .\nCase II (a): p < (b2 −b)/y . In this case, it holds that\nγP\ny = p −1 + b\np\n> [(b2 −b)/y] −1 + b\n[(b2 −b)/y]\n= 1 + y/b = γy.\nCase II (b) : p = (b2 −b)/y (This case applies only if b2−b\ny\n∈N+). In this case, γP\ny = p−1+b\np\n= 1+y/b =\nγy. By Inequality (17), we have the following holds:\nβP\ny = p −1 + b\nb\n= [(b2 −b)/y] + (b −1)\nb\n≥[y + 1 −b] + (b −1)\nb\n= y/b = βy.\nCase II (c) : (b2 −b)/y < p < y + 1 . In this case, by Inequality (17), it follows that\nβP\ny = p −1 + b\nb\n> [(b2 −b)/y] −1 + b\nb\n≥[y + 1 −b] −1 + b\nb\n= y/b = βy.\nCase II (d) : p > y + 1 . In this case, we see that\nγP\ny = p −1 + b\nb\n> (y + 1) −1 + b\nb\n= 1 + y/b = γy.\nIn either case, (βP\ny , γP\ny ) provides no Pareto improvement over (βy, γy).\nCase III: y > min{b(λ + 1) −1, (b −1)/λ} . Algorithm 2, purchasing on day ⌈λb⌉, achieves a consistency\nof ⌈λb⌉−1+b\nb\nand a robustness of ⌈λb⌉−1+b\n⌈λb⌉\n. Consider another algorithm Q that buys on day q (̸= ⌈λb⌉) with\nprediction-specific consistency and robustness βQ\ny and γQ\ny .\nCase III (a): q < ⌈λb⌉. In this case, γQ\ny = q−1+b\nq\n> ⌈λb⌉−1+b\n⌈λb⌉\n= γy.\nCase III (b): ⌈λb⌉< q ≤y . In this case, βQ\ny = q−1+b\nb\n> ⌈λb⌉−1+b\nb\n= βy.\nCase III (c): q > y . If y > b(λ+1)−1, we have λb < y +1−b. Note that limb→+∞\n⌈λb⌉\nb\n= λ. Therefore,\nβy = ⌈λb⌉−1 + b\nb\n< (y + 1 −b) −1 + b\nb\n= y\nb = βQ\ny\nas b →∞. If y > (b −1)/λ, we have ⌈λb⌉≥λb > b2−b\ny . Therefore,\nγy = ⌈λb⌉−1 + b\n⌈λb⌉\n< [(b2 −b)/y] −1 + b\n[(b2 −b)/y]\n= y + b\nb\n≤γQ\ny .\nIn either case, (βP\ny , γP\ny ) yields no Pareto improvement over (βy, γy). By Definition 4, Algorithm 2 is\nstrongly-optimal as b →∞.\n29\n\nD\nAdditional Details for Section 5\nIn this section, we provide some supplementary details for the construction of explicit algorithms for\nrandomized ski rental. The design of the explicit strongly-optimal algorithm is largely inspired by an equalizing\nproperty, which plays a central role in the classic randomized ski rental problem [38].\nD.1\nEqualizing Distributions\nWe formally define equalizing distributions as follows.\nDefinition 5. Given integers m and n with 1 ≤m ≤n ≤b, a distribution πeq[m, n] is an equalizing\ndistribution in [m, n] if it is only supported on {m, m + 1, . . . , n}, and R(πeq[m, n], x) = α(πeq[m, n]) for all\nx ∈{m, m + 1, . . . , n}, where α(π) is the competitive ratio of distribution π.\nFurthermore, the following theorem gives the explicit form of the equalizing distribution.\nTheorem D.1. Let πeq[m, n] be an equalizing distribution in [m, n]. Then it satisfies the following recursive\nformula:\nπeq\ni [m, n] =\n\n\n\n\n\n\n\n\n\n\u0010\n1 + m+b−1\nm\n·\n\u0010\n(\nb\nb−1)n−m −1\n\u0011\u0011−1\nfor i = m;\nπeq\nm[m, n] · m+b−1\nm(b−1) ·\n\u0010\nb\nb−1\n\u0011i−m−1\n,\nfor i ∈{m + 1, . . . , n};\n0,\notherwise.\n(18)\nNote that πeq[1, b] is exactly the same as Karlin’s distribution [38], while πeq[1, ⌊λb⌋] coincides with the\ndistribution used in Kumar’s randomized algorithm [2] when y ≥b.\nTheorem D.2. For any y < b, the equalizing distribution on [y + 1, b], denoted πeq[y + 1, b], is 1-consistent and\nminimizes robustness among all 1-consistent distributions under y.\nThe proof of Theorem D.1 and D.2 is detailed in Appendix E.2.\nD.2\nOPERATION A and OPERATION B\nWe now propose two types of operations (OPERATION A and OPERATION B) that transform an equalizing\ndistribution into a distribution with the desired robustness level on the Pareto frontier.\nOPERATION A: Consistency Boosting. OPERATION A (see Algorithm 7) is employed when y ≥b. It\nstarts with an equalizing distribution πeq[1, n], where the precise choice of n depends on the desired robustness\nlevel γ. The core idea of OPERATION A is to enhance the prediction-specific consistency βy of the distribution\nwhile not sacrificing the prediction-specific robustness γy.\nRemark 1. Reaching r = 1 in Step 0 only occurs when y = b; otherwise, the condition r ≤y + 1 −b\nwill be satisfied earlier. In the special case where y = b, any two-point distribution over {1, b + 1} is trivially\n1-consistent. In this case, OPERATION A returns the distribution among them that achieves the lowest possible\nrobustness.\nRemark 2. When y ≥b, the prediction-specific consistency is βy =\nPy\ni=1 πi·(b+i−1)\nb\n+\nP∞\ni=y+1 πi·y\nb\n. As a result,\nshifting probability mass from πr to πy+1 can potentially improve prediction-specific consistency only in cases\nwhere b + r −1 > y. This explains why, when r ≤y + 1 −b, we immediately terminate the shifting process in\nStep 0.\n30\n\nAlgorithm 7 OPERATION A: CONSISTENCY BOOSTING\n1: Input: Initial distribution πeq[1, n] and prediction y ≥b;\n2: Initialization: π ←πeq[1, n], iterative index r ←n, initial robustness γ ←γy(π);\n3: Step 0:\n4:\n// Check whether the current distribution is a two-point distribution.\n5:\nIf r = 1, then update π1 ←1/b and πb+1 ←(b −1)/b, go to Step 4;\n6:\n// Check if further shifting enhances consistency.\n7:\nIf r ≤y + 1 −b, then go to Step 4;\n8: Step 1:\n9:\n// Shifting probability mass from πr to πy+1.\n10:\nUpdate πy+1 ←πy+1 + πr and πr ←0;\n11: Step 2:\n12:\n// Determine if more shifting is necessary.\n13:\nIf R(π, y + 1) < γ, then set r ←r −1, go to Step 0;\n14: Step 3:\n15:\n//Determine the maximum amount of probability mass that can be shifted.\n16:\nSet π′ ←(π1, . . . , πr−1, π′\nr, 0, . . . , 0, π′\ny+1);\n17:\nSolve π′\nr, π′\ny+1 through R(π′, y + 1) = γ and π′\nr + π′\ny+1 = 1 −Pr−1\ni=1 πi;\n18:\nUpdate πr, πy+1 ←π′\nr, π′\ny+1;\n19: Step 4: Return π.\nOPERATION B: Robustness Seeking. OPERATION B (see Algorithm 8) is employed when y < b.\nOPERATION B (see Algorithm 8) initializes from the most robust 1-consistent distribution πeq[y + 1, b] given\nthe prediction y, and incrementally sacrifices consistency to gain robustness. This process continues until a\ndistribution is obtained that achieves Pareto-optimality under the consistency-robustness trade-off, subject to a\ndesired robustness level.\nAlgorithm 8 OPERATION B: ROBUSTNESS SEEKING\n1: Input: Initial distribution πeq[y + 1, b](y < b) and desired robustness γ;\n2: Initialization: π ←πeq[y + 1, b], iterative index r ←1;\n3: Step 1:\n4:\n// Establish an equalized distribution over {1, . . . , r, y + 1, . . . , b}.\n5:\nSet π′ ←(π′\n1, . . . , π′\nr, 0, . . . , 0, π′\ny+1, . . . , π′\nb);\n6:\nSolve (π′, γ′) through Pb\ni=1 π′\ni = 1 and R(π′, x) = γ′, x ∈{1, . . . , r, y + 1, . . . , b};\n7: Step 2:\n8:\n// Check if the robustness of the newly constructed distribution meets the desired level.\n9:\nIf γ′ > γ, then set r ←r + 1, and go to Step 1;\n10: Step 3:\n11:\n// After determining the appropriate value of r ,\n12:\n// Prioritize assigning probability mass to {1, . . . , r −1, y + 1, . . . , b}.\n13:\nSolve π′ through Pb\ni=1 π′\ni = 1 and R(π′, x) = γ, x ∈{1, . . . , r −1, y + 1, . . . , b};\n14:\nUpdate π ←π′;\n15: Step 4: Return π.\n31\n\nD.3\nPRSR: Prediction-specific Randomized Ski Rental\nWith these foundations in place, we proceed to introduce an algorithm for randomized ski rental, referred as\nPRSR(see Algorithm 9).\nAlgorithm 9 PRSR: PREDICTION-SPECIFIC RANDOMIZED SKI RENTAL\n1: Input: γ ∈[eb/(eb −1), b −2)\n2: // Determine the smallest n such that πeq[1, n] is γ-robust.\n3: Determine n ←⌈logb/(b−1)(1 + 1/(γ −1))⌉;\n4: // Determine the adjusted robustness γ′, defined as that of πeq[1, n].\n5: Determine γ′ ←[(\nb\nb−1)n −1]−1 + 1;\n6: If y ≥b then\n7:\nπ ←OPERATION A(πeq[1, n], y); (see Algorithm 7)\n8: Else if y < b then\n9:\n// Ensure that the adjusted robustness does not exceed the upper bound γν := γ(πeq[y + 1, b]).\n10:\nDetermine γν ←γ(πeq[y + 1, b]) and determine γ′′ ←min{γν, γ′};\n11:\nπ ←OPERATION B(πeq[y + 1, b], γ′′); (see Algorithm 8)\n12: Choose i randomly according to the distribution π;\n13: Buy the skis at the start of day i.\nRemark 3. Consider the equalizing distribution πeq[1, n] with n ≤b. Let ˆγ denote the robustness of πeq[1, n].\nNote that\nˆγ = R(πeq[1, n], 1) = (b −1) · πeq\n1 [1, n] + 1.\n(19)\nMoreover, by Theorem D.1,\nπeq\n1 [1, n] ·\n \n1 −(\nb\nb−1)n\n1 −(\nb\nb−1)\n!\n= 1.\n(20)\nBy Equation (19) and Equation (20),\nn = logb/(b−1)(1 + 1/(ˆγ −1)).\nTherefore, choosing n = ⌈logb/(b−1)(1 + 1/(1 −γ))⌉ensures that πeq[1, n] is γ-robust.\nRemark 4. The requirement that γ < b −2 is primarily enforced to guarantee n > 1.\nIn Section E.5, we formally verify the prediction-specific Pareto optimality and strong optimality of PRSR.\n32\n\nE\nProofs for Section 5 and Appendix D\nThis section provides proofs for Section 5 and Appendix D.\nE.1\nProofs of Useful Lemmas\nLemma 1. Consider the ski rental randomized problem with b > 1, for any distribution π supported on N+,\nthere exists another distribution π′ supported on a finite set [b] such that α(π′) ≤α(π). Specifically, when the\nbest attack for π only occurs at x > b, we have α(π′) < α(π). Furthermore, in the learning-augmented setting\nwith prediction y, for any distribution π supported on N+, there exists another distribution π′ supported on the\nfinite set [b] ∪{y + 1} such that βy(π′) ≤βy(π) and γy(π′) ≤γy(π).\nProof. We first consider the traditional competitive analysis setting.\nLet π be a distribution with P∞\ni=b+1 πi ̸= 0. Consider another distribution π′ with π′\ni = πi for all i ∈[b −1],\nand π′\nb = P∞\ni=b πi. Let r be the maximal index on which π has probability mass, i.e. r = max{i ∈N | πi ̸= 0}.\nLet α(π), α(π′) denote the competitive ratio of a randomized algorithm that uses π and π′, respectively.\nCase I: x < b . By the definition of R(π, x) in Section 5, R(π, x) = R(π′, x).\nCase II: x ≥b . In this range, the worst-case attack against π occurs at x = r, while the worst-case attack\nagainst π′ occurs at x = b. According to the definition of R(π, x), we have R(π, r) > R(π′, b).\nNote that α(π) = supx∈N+ R(π, x) and α(π′) = supx∈N+ R(π′, x). We have α(π) ≥α(π′). Specifically,\nwhen the best attack for π occurs at x > b, we have α(π) > α(π′).\nIn the learning-augmented setting, we divide our discussion into two parts: y < b and y ≥b.\nCase I: y < b . In this case, [b] ∪{y + 1} = [b]. Let π′ be the distribution after transferring all probability\nmass beyond b of π to b (i.e., π′\ni = πi, ∀i ∈[b −1] and π′\nb = P∞\ni=b πi). By Lemma 1, γy(π′) ≤γy(π). By\nEquation (6), we have βy(π′) = βy(π).\nCase II: y ≥b . Construct another distribution π′ by transferring all the probability mass of π on the {b +\n1, . . . , y} to b, and all the mass beyond y + 1 to y + 1 (i.e., π′\ni = πi, ∀i ∈[b −1]; π′\nb = Py\ni=b πi and\nπ′\ny+1 = P∞\ni=y+1 πi). By a simple verification, we obtain that βy(π′) ≤βy(π) and γ(π′) ≤γ(π).\nLemma 2. Consider a randomized ski rental problem with b > 1. Consider a probability distribution π over\nN+. If R(π, x) = α(π) for all x ∈{m, m + 1, . . . , n}, then we have\nπi+1 =\nb\nb −1 · πi,\n∀i ∈{m + 1, . . . , n −1}.\nProof. By the definition of R(π, x) in Section 5, when x ≤b:\nR(π, x) =\nPx\ni=1 πi(x −1 + b) + P∞\ni=x+1 πi · x\nx\n.\nConsider the following equations:\nR(π, i) = α(π),\nm ≤i ≤m + 2\n(21)\nSubtracting both sides of (m + 1)×Equation (21) with i = m + 1 by m×Equation (21) with i = m,\nb · πm+1 +\n∞\nX\ni=m+2\nπi = α(π).\n(22)\n33\n\nSubtracting both sides of (m + 2)× Equation (21) with i = m + 2 by (m + 1)× Equation (21) with i = m + 1,\nb · πm+2 +\n∞\nX\ni=m+3\nπi = α(π).\n(23)\nFinally, subtracting both sides of Equation (23) by Equation (22) yields πm+2 =\nb\nb−1πm+1. Similarly, we can\nshow that\nπi+1 =\nb\nb −1πi,\n∀i ∈{m + 2, . . . , n −1}.\nE.2\nProof of Theorem D.1 and D.2\nProof of Theorem D.1. By Definition 5, πeq[m, n] only has positive support over {m, . . . , n}.\nConsider x = m and x = m + 1,\n(m −1 + b) · πeq\nm[m, n]\nm\n+ (1 −πeq\nm[m, n]) = α(π).\n(24)\n(m −1 + b)πeq\nm[m, n] + (m + b)πeq\nm+1[m, n]]\n(m + 1)\n+ (1 −πeq\nm[m, n] −πeq\nm+1[m, n]) = α(π).\n(25)\nEquating the left-hand sides of Equation (24) and Equation (25) yields:\nπeq\nm+1[m, n] = m(b −1)\nm + b −1πeq\nm[m, n].\n(26)\nBy Lemma 2, we have\nπeq\ni+1[m, n] =\nb\nb −1 · πeq\ni [m, n],\n∀i ∈{m + 1, . . . , n −1}.\n(27)\nEquation (26), Equation (27), together with the constraint Pn\ni=m πi = 1, imply that\nπeq\ni [m, n] =\n\n\n\n\n\n\n\n\n\n\u0010\n1 + m+(b−1)\nm\n·\n\u0010\n(\nb\nb−1)n−m −1\n\u0011\u0011−1\nfor i = m;\nπeq\nm[m, n] · m+(b−1)\nm(b−1) ·\n\u0010\nb\nb−1\n\u0011i−m−1\n,\nfor i ∈{m + 1, . . . , n};\n0,\notherwise.\nProof of Theorem D.2. Applying Lemma 1, we first reduce the support under consideration to [b]. Furthermore,\nthe condition of 1 consistency requires πi = 0 for all i ∈[y]. We consider the following optimization problem.\nmin\nπy+1,...,πb,γy\nγy\n(Primal Problem)\ns.t.\ni\nX\nj=y+1\nπj · (b + j −1) +\nb\nX\nj=i+1\nπj · i ≤γy · i,\n∀i ∈{y + 1, . . . , b},\nb\nX\ni=y+1\nπi = 1,\nπi ≥0,\n∀i ∈{y + 1, . . . , b}.\nγy ≥0\n34\n\nLet π∗\ny+1, . . . , π∗\nb, γ∗\ny denote the optimal solution to Primal Problem. It’s clear that\nγ∗\ny =\nmin\nπ∗\ny+1,...,π∗\nb\nmax\ni∈{y+1,...,b} R(π∗, i).\n(28)\nWe claim that π∗\ny+1 ̸= 0. We prove this by contradiction, assuming π∗\ny+1 = 0. Let r be the minimal index\nwith non-zero probability mass, i.e. r := min{i | π∗\ni ̸= 0}. Consider another distribution π′ with π′\ny+1 = ϵ,\nπ′\nr = π∗\nr −ϵ and π′\ni = π∗\ni for all i ∈Z+ \\ {y + 1, r}, where we denote\nϵ := min\n\u001a\nπ∗\nr, y + 1\nb −1 · (γ∗\ny −1\n2\n)\n\u001b\n.\nNote that\nR(π′, y + 1) = (b + y) · π′\ny+1 + (y + 1) · (1 −πy+1)\ny + 1\n≤γ∗\ny −1\n2\n+ 1 = γ∗\ny + 1\n2\n< γ∗\ny.\nMoreover, it follows that\nR(π′, i) < R(π∗, i),\n∀i ∈{y + 2, . . . , b}.\nTherefore,\nmax\ni∈{y+1,...,b} R(π′, i) <\nmax\ni∈{y+1,...,b} R(π∗, i) = γ∗\ny,\nwhich contradicts with Equation (28).\nWe then claim that π∗\nb ̸= 0. We prove this by contradiction, assuming π∗\nb = 0. Consider another distribution\nπ′ with π′\ny+1 = π∗\ny+1 −ϵ, π′\nb = ϵ, and π′\ni = π∗\ni for all i ∈Z+ \\ {y + 1, r}, where ϵ = min{π∗\ny+1, γ∗\ny/(2b −1)}.\nWe verify that\nR(π′, i) < R(π∗, i),\n∀i ∈{y + 1, . . . , b −1}.\nFurthermore,\nR(π′, b) <\nPb−1\ni=y+1(i −1 + b) · π∗\ni + (2b −1) · ϵ\nb\n≤(b −1) · γ∗\ny + (2b −1) · ϵ\nb\n≤γ∗\ny.\nConsequently,\nmax\ni∈{y+1,...,b} R(π′, i) <\nmax\ni∈{y+1,...,b} R(π∗, i) = γ∗\ny,\nwhich contradicts with Equation (28).\nWe further claim that π∗\ni ̸= 0, for all i ∈{y + 2, . . . , b −1}. We prove this by contradiction, assuming that\nthere exists some q ∈{y + 2, . . . , b −1}, such that π∗\nq = 0. Let r = min{i > q | πi ̸= 0}. Since π∗\nb ̸= 0, such\nr is guaranteed to exist. Consider another distribution π′ with π′\ny+1 = π∗\ny+1 −ϵ1, π′\nq = ϵ1 + ϵ2, πr = π∗\nr −ϵ2,\nand π′\ni = π∗\ni for all i ∈Z+ \\ {y + 1, q, r}, where we denote\nϵ1 := min\n\u001a\nπ∗\ny+1,\nr −q\n2(q −y −1) · ϵ2\n\u001b\n,\nϵ2 := min\n\u001a\nπ∗\nr,\n2(q −y −1)\n[(r −q) + 2(q −y −1)](q −1 + b) · γ∗\ny\n\u001b\n.\nSimilarly, we verify that\nR(π′, i) < R(π∗, i),\n∀i ∈{y + 1, . . . , q −1}.\nNote that ϵ1 + ϵ2 ≤\nγ∗\ny\nq−1+b. We have\nR(π′, q) < (q −1) · γ∗\ny + (q −1 + b) · (ϵ1 + ϵ2)\nq\n≤γ∗\ny.\n35\n\nSince (q −y −1)ϵ1 < 2(q −y −1)ϵ1 ≤(r −q)ϵ2, we have\nR(π′, i) < R(π∗, i),\n∀i ∈{q + 1, . . . , b}.\nTherefore,\nmax\ni∈{y+1,...,b} R(π′, i) <\nmax\ni∈{y+1,...,b} R(π∗, i) = γ∗\ny,\nwhich contradicts with Equation (28).\nAs a result, we conclude that π∗\ni ̸= 0, ∀i ∈{y + 1, . . . , b}.\nNow, let the dual variables be λy+1, . . . , λb, λ. The dual problem can be formulated as following:\nmax\nλy+1,...,λb,λ\nλ\n(Dual Problem)\ns.t.\nx−1\nX\ni=y+1\ni · λi + (b + x −1) ·\nb\nX\ni=x\nλi + λ ≤0,\n∀x ∈{y + 1, . . . , b},\nb\nX\ni=y+1\n−i · λi ≤1,\nλi ≤0,\n∀i ∈{y + 1, . . . , b},\nλ is free.\nLet λ∗\ny+1, . . . , λ∗\nb, λ∗denote the optimal solution to Dual Problem.\nBy the complementary slackness,\nπ∗\nx ·\n\n\nx−1\nX\ni=y+1\ni · λ∗\ni + (b + x −1) ·\nb\nX\ni=x\nλ∗\ni + λ∗\n\n= 0,\n∀x ∈{y + 1, . . . , b};\nγ∗\ny ·\n\n1 +\nb\nX\ni=y+1\ni · λ∗\ni\n\n= 0.\nSince π∗\ni ̸= 0, ∀i ∈{y + 1, . . . , b}, we have\nx−1\nX\ni=y+1\ni · λ∗\ni + (b + x −1) ·\nb\nX\ni=x\nλ∗\ni + λ∗= 0,\n∀x ∈{y + 1, . . . , b}.\nBecause γ∗\ny ̸= 0, we have Pb\ni=y+1 −i · λ∗\ni = 1.\nLet Λi := −i · λi for all i ∈{y + 1, . . . , b}. Then, the dual problem can be transformed into:\nmax\nΛy+1,...,Λb,λ\nλ\ns.t.\nx−1\nX\ni=y+1\nΛi +\nb\nX\ni=x\n(b + x −1)\ni\n· Λi = λ,\n∀x ∈{y + 1, . . . , b},\nb\nX\ni=y+1\nΛi = 1,\nΛi ≥0,\n∀i ∈{y + 1, . . . , b},\nλ is free.\n36\n\nLet Λ∗\ny+1, . . . , Λ∗\nb, λ∗denote the optimization problem.\nWe claim that Λ∗\ny+1 ̸= 0. Consider x = y + 1 and x = y + 2, the optimal solution should satisfy that\n( b+y\ny+1 · Λ∗\ny+1 +\nb+y\ny+2 · Λ∗\ny+2 + · · · +\nb+y\nb\n· Λ∗\nb = λ∗\nΛ∗\ny+1 + b+y+1\ny+2\n· Λ∗\ny+2 + · · · + b+y+1\nb\n· Λ∗\nb = λ∗\nIf Λ∗\ny+1 = 0, since\nb + y\ny + 2 < b + y + 1\ny + 2\n, . . . , b + y\nb\n< b + y + 1\nb\n,\nthen we have\nΛ∗\ny+2 = · · · = Λ∗\nb = 0,\nwhich contradicts with Pb\ni=y+1 Λ∗\ni = 1. We then claim that Λ∗\ni ̸= 0, for all i ∈{y + 2, . . . , b −1}. Consider\nx = i and x = i + 1, the optimal solution should satisfy that\n(\nΛ∗\ny+1 + · · · + Λ∗\ni−1 + b+i−1\ni\n· Λ∗\ni + b+i−1\ni+1\n· Λ∗\ni+1 + · · · + b+i−1\nb\n· Λ∗\nb = λ∗\nΛ∗\ny+1 + · · · + Λ∗\ni−1 +\nΛ∗\ni +\nb+i\ni+1 · Λ∗\ni+1 + · · · +\nb+i\nb · Λ∗\nb = λ∗\nAssume Λ∗\ni = 0. Since\nb + i −1\ni + 1\n< b + i\ni + 1, . . . , b + i −1\nb\n< b + i\nb\n,\nwe have\nΛ∗\ni+1 = Λ∗\ni+2 = · · · = Λ∗\nb = 0.\nSimilarly, consider x = i −1 and x = i, we have\n(\nΛ∗\ny+1 + · · · + Λ∗\ni−2 + b+i−2\ni−1\n· Λ∗\ni−1 + b+i−2\ni\n· Λ∗\ni + · · · + b+i−2\nb\n· Λ∗\nb = λ∗\nΛ∗\ny+1 + · · · + Λ∗\ni−2 +\nΛ∗\ni−1 + b+i−1\ni\n· Λ∗\ni + · · · + b+i−1\nb\n· Λ∗\nb = λ∗\nWe further have Λ∗\ni−1 = 0.\nRepeating this process, we obtain Λ∗\ny+1 = 0, which leads to a contradiction.\nWe finally claim that Λ∗\nb ̸= 0. Consider x = b −1 and x = b,\n(\nΛ∗\ny+1 + · · · + Λ∗\nb−2 + 2b−2\nb−1 · Λ∗\nb−1 + 2b−2\nb\n· Λ∗\nb = λ∗\nΛ∗\ny+1 + · · · + Λ∗\nb−2 +\nΛ∗\nb−1 + 2b−1\nb\n· Λ∗\nb = λ∗\nIf Λ∗\nb = 0, we can deduce that Λ∗\nb−1 = 0, which contradicts our previous results.\nTherefore, we have\nΛ∗\ni ̸= 0,\n∀i ∈{y + 1, . . . , b},\nwhich is equivalent to\nλ∗\ni ̸= 0,\n∀i ∈{y + 1, . . . , b}.\nFurthermore, by complementary slackness,\nλ∗\ni ·\n\n\ni\nX\nj=y+1\nπj · (b + j −1) +\nb\nX\nj=i+1\nπj · i −γy · i\n\n= 0,\n∀i ∈{y + 1, . . . , b}\nIt follows that\ni\nX\nj=y+1\nπj · (b + j −1) +\nb\nX\nj=i+1\nπj · i = γy · i,\n∀i ∈{y + 1, . . . , b}\nBy Definition 5, we conclude that πeq[y + 1, b] is the most robust 1-consistent distribution.\n37\n\nE.3\nProof of Theorem 5.1\nProof of Theorem 5.1. Consider y < b. Kumar’s algorithm uses distribution in (7), where m = ⌈b/λ⌉> b. We\ndenote this distribution as π, and consider another distribution π′ that transfer all probability masses beyond b\nto b (i.e. π′\ni = πi, ∀i ∈[b −1] and π′\nb = P∞\ni=b πi). Recall that βy(π), γy(π), βy(π′), γy(π′) are the consistency\nand robustness of π and π′ under prediction y, respectively.\nSince y < b, by Equation (6), we have βy(π′) = βy(π). Observe that the worst-case attack against π only\noccurs at x > b, by Lemma 1, we have γy(π′) < γy(π). By Definition 4, KR is not strongly-optimal.\nE.4\nProof of Theorem 5.2\nProof of Theorem 5.2. We start by finding a weakly-optimal algorithm with robustness γ for any given γ ∈Λγ.\nIf γ = eb/(eb −1), Karlin’s algorithm [38] has exactly γ robustness. Since it is the only eb/(eb −1)-\ncompetitive algorithm, it is weakly-optimal.\nAs b →∞, for any γ ∈(eb/(eb −1), ∞), there exists λ ∈(1/b, 1), such that KR’s robustness is exactly γ.\nFurthermore, KR is weakly-optimal as b →∞.\nTherefore, we find a weakly-optimal algorithm with robustness γ for any γ ∈Λγ.\nWe now conclude with the strong optimality of Algorithm 3: by Lemma 1, the bi-level optimization problems\ncan be reduced to Problem 8 and Problem 9. By immediate consequence of Proposition 1, Algorithm 3 is\nstrongly-optimal.\nE.5\nProof of Theorem 5.3\nBefore proving Theorem 5.3, we first prove the Pareto optimality of PRSR(see Algorithm 9)’s prediction-\nspecific consistency and robustness. Our analysis is divided into two cases: y < b and y ≥b.\nFor the case where y < b, we begin by considering the following optimization problem, referred to\nas Problem A, which plays a key role in the subsequent analysis.\nmin\nπ1,π2,...,πb\nPy\ni=1 πi · (b + i −1) + Pb\ni=y+1 πi · y\ny\n(Problem A)\ns.t.\nPx\ni=1 πi · (b + i −1) + Pb\ni=x+1 πi · x\nx\n≤γy,\n∀x ∈{1, 2, . . . , b},\nb\nX\ni=1\nπi = 1,\nπi ≥0,\n∀i ∈{1, 2, . . . , b}.\nLet πA = (πA\n1 , πA\n2 , . . . , πA\nb ) denote the optimal solution to Problem A.\nLemma 3. Assume that γy ∈[γξ, γν]. For Problem A, πA\ni ̸= 0, for all i ∈{y + 1, y + 2, . . . , b}.\nProof. For any γy ∈[γξ, γν], πeq[1, b] is always a feasible solution to Problem A. This guarantees the existence\nof a optimal solution πA.\nWe prove by contradiction. Suppose there exists r ∈{y + 1, y + 2, . . . , b} such that πA\nr = 0. We analyze\nthe following cases.\nCase I: Py\ni=1 πA\ni = 0 . In this case, πA\n1 = · · · = πA\ny = πA\nr = 0. When γy = γν, πeq[y + 1, r] is the\nonly solution to Problem A if we requires πA\n1 = · · · = πA\ny = 0. If we further require πA\nr = 0 for some\nr ∈{y + 1, y + 2, . . . , b}, there is no feasible solution to Problem A for any γy ∈[γξ, γν].\n38\n\nCase II(a): Py\ni=1 πA\ni ̸= 0 and Pb\ni=r+1 πA\ni ̸= 0 . Let p be the largest index in [y] that has non-zero proba-\nbility mass, i.e. p := max{i ∈[y] | πA\ni ̸= 0}. Let q be the smallest index in [b]\\[r] that has non-zero probability\nmass, i.e.\nq := min\n\b\ni ∈[b] \\ [r] | πA\ni ̸= 0\n\t\n.\nSince Py\ni=1 πA\ni ̸= 0 and Pb\ni=r+1 πA\ni ̸= 0, such p and q are guaranteed to exist. Based on this, we can always\nconstruct another solution\nπ′ =\n\u0000πA\n1 , πA\n2 , . . . , πA\np −ϵ1, . . . , πr−1, ϵ1 + ϵ2, . . . , πA\nq −ϵ2, . . . , πA\nb\n\u0001\n,\nwhere ϵ2 = min\nn\n(Pr\ni=1 πA\ni )b\n(q−r+b−1)(r−1), πA\nq\no\n> 0, ϵ1 = min\nn\n( q−r\nr−p)ϵ2, πA\np\no\n> 0.\nWe first prove that π′ is still a feasible solution to Problem A.\nRecall that when x ≤b (see the definition of R(π, x) in Section 5)\nR(π, x) =\nPx\ni=1 πi(i −1 + b) +\n\u0010Pb\ni=x+1 πi\n\u0011\nx\nx\n.\n(1) Consider x < p. R(π′, x) = R(πA, x) ≤γy.\n(2) Consider p ≤x < r. Since π′\ni = πA\ni for all i ∈[x] \\ {p}, π′\np = πA\np −ϵ1, and\n\u0010Pb\ni=x+1 π′\ni\n\u0011\n=\n\u0010Pb\ni=x+1 πA\ni\n\u0011\n+ ϵ1, we have\nxR(π′, x) = xR(πA, x) −ϵ1(p −1 + b) + ϵ1x < xR(πA, x).\nTherefore, we have\nR(π′, x) < R(πA, x) ≤γy.\n(3) Consider x = r. Note that\nrR(π′, r) =\nr\nX\ni=1\nπ′\ni(b + i −1) +\n \n1 −\nr\nX\ni=1\nπ′\ni\n!\nr,\n(29)\nrR(πA, r) =\nr\nX\ni=1\nπA\ni (b + i −1) +\n \n1 −\nr\nX\ni=1\nπA\ni\n!\nr.\n(30)\nSince π′\ni = πA\ni for all i ∈[r] \\ {p, r}, π′\np = πA\np −ϵ1, π′\nr = πA\nr + (ϵ1 + ϵ2), and (Pb\ni=r+1 π′\ni) =\n(Pb\ni=r+1 πA\ni ) −ϵ2, we have\nrR(π′, r) = rR(πA, r) −ϵ1(b + p −1) + (ϵ1 + ϵ2)(b + r −1) −ϵ2r.\nBy arranging terms, we have\nrR(π′, r) = rR(πA, r) + ϵ1(r −p) + ϵ2(b −1).\n(31)\nNote that r −p > 0, b −1 > 0, and ϵ1 ≤( q−r\nr−p)ϵ2, ϵ2 ≤\n(Pr\ni=1 πA\ni )b\n(q−r+b−1)(r−1). We have\nϵ1(r −p) + ϵ2(b −1) ≤(Pr\ni=1 πA\ni )b\nr −1\n≤\nPr\ni=1 πA\ni (b + i −1)\nr −1\n.\n(32)\n39\n\nBy (30), (31) and (32),\nrR(π′, r) ≤r Pr\ni=1 πA\ni (b + i −1) + r(r −1)(1 −Pr\ni=1 πA\ni )\nr −1\n.\n(33)\nSince πA\nr = 0, (33) implies\nrR(π′, r) ≤r Pr−1\ni=1 πA\ni (b + i −1) + r(r −1)(1 −Pr−1\ni=1 πA\ni )\nr −1\n.\nTherefore,\nR(π′, r) ≤\nPr−1\ni=1 πA\ni (b + i −1) + (1 −Pr−1\ni=1 πA\ni )(r −1)\nr −1\n= R(πA, r −1) ≤γy.\n(34)\n(4) Consider r < x < q. Note that q = min{i ∈[b] \\ [r] | πA\ni ̸= 0}. Thus πA\nr+1 = · · · = πA\nq−1 = 0. Hence,\nπ′\nr+1 = · · · = π′\nq−1 = 0. It follows that\nxR(π′, x) = rR(π′, r).\n(35)\nBy (34) and (35), we have\nR(π′, x) = r\nx · R(π′, r) < R(π′, r) ≤γy.\n(5) Consider q ≤x ≤b. Since π′\ni = πA\ni for all i ∈[b] \\ {p, r, q}, π′\np = πA\np −ϵ1, π′\nr = πA\nr + (ϵ1 + ϵ2), and\nπ′\nq = πA\nq −ϵ2, we have\nxR(π′, x) = xR(πA, x) −ϵ1(b + p −1) + (ϵ1 + ϵ2)(b + r −1) −ϵ2(b + q −1).\nBy arrange terms, we have\nxR(π′, x) = xR(πA, x) + ϵ1(r −p) −ϵ2(q −r).\n(36)\nNote that ϵ1 ≤( q−r\nr−p)ϵ2, and r −p > 0, q −r > 0. (36) further indicates that\nxR(π′, x) ≤xR(πA, x).\nTherefore,\nR(π′, x) ≤R(πA, x) ≤γy.\nConsequently, R(π′, x) ≤γy for all x ∈[b]. Note that Pb\ni=1 π′\ni = 1 and π′\ni ≥0 for all i ∈[b]. Therefore,\nπ′ is a feasible solution to Problem A.\nWe then compare the objective value of Problem A at π′ and πA.\nNote that p ≤y < r. Consequently, as established in Case II(a)(2),\nyR(π′, y) = yR(πA, y) −ϵ1(b + p −1) + ϵ1y < yR(πA, y).\nThis leads to\nR(π′, y) < R(πA, y).\nThis makes πA impossible to be the optimal solution to Problem A under the assumptions of Case II (a),\nsince there is always another feasible solution π′ that achieves a smaller objective value.\nCase II(b): Py\ni=1 πA\ni ̸= 0 and Pb\ni=r+1 πA\ni = 0 .\n40\n\nLet p be the largest index in [y] that has non-zero probability mass, i.e. p := max{i ∈[y] | πA\ni ̸= 0}. We\ncan always construct another solution\nπ′ = (πA\n1 , . . . , πA\np−1, πA\np −ϵ, . . . , πA\nr−1, ϵ, . . . , πA\nb ),\nwhere ϵ = min{πA\np ,\nb\n(r−1)·(r−p)} > 0.\nSimilarly we first investigate the feasibility of π′.\n(1) Consider x < p. It is clear that\nR(π′, x) = R(πA, x) ≤γy.\n(2) Consider p ≤x < r. Note that\nxR(π′, x) = xR(πA, x) −ϵ(b + p −1) + ϵx < xR(πA, x).\nTherefore,\nR(π′, x) < R(πA, x) ≤γy.\n(3) Consider x = r. Since Pb\ni=r+1 πA\ni = 0, we have πA\ni = 0, ∀i ∈{r + 1, . . . , b}. Note that\nrR(π′, r) = rR(πA, r) −ϵ(b + p −1) + ϵ(b + r −1).\nBy arranging terms,\nrR(π′, r) = rR(πA, r) + ϵ(r −p).\n(37)\nGiven that ϵ ≤\nb\n(r−1)(r−p),\nrR(πA, r) + ϵ(r −p) ≤rR(πA, r) +\nb\nr −1.\n(38)\nSince Pb\ni=r+1 πA\ni = 0, we have Pr\ni=1 πA\ni = 1. Therefore,\nrR(πA, r) +\nb\nr −1 ≤rR(πA, r) +\nPr\ni=1 πA\ni (b + i −1)\nr −1\n.\n(39)\nBy definition,\nR(πA, r) =\nPr\ni=1 πA\ni (b + i −1) + Pb\ni=r+1 πA\ni r\nr\n.\nNote that Pb\ni=r+1 πA\ni = 0. We further conclude\nrR(πA, r) +\nPr\ni=1 πA\ni (b + i −1)\nr −1\n= r Pr\ni=1 πA\ni (b + i −1)\nr −1\n.\n(40)\nBy (37), (38), (39) and (40),\nrR(π′, r) ≤r Pr\ni=1 πA\ni (b + i −1)\nr −1\n.\nNote that πA\nr = 0. We have\nR(π′, r) ≤\nPr−1\ni=1 πA\ni (b + i −1)\nr −1\n= R(πA, r −1) ≤γy.\n(4) Consider r < x ≤b. Consider r < x ≤b. Since π′\ni = πA\ni = 0 for all i ≥r + 1, it is clear that\nR(π′, x) < R(π′, r) ≤γy.\n41\n\nConsequently, R(π′, x) ≤γy for all x ∈[b]. This implies that π′ is also a feasible solution to Problem A.\nHowever,\nyR(π′, y) = yR(πA, y) −ϵ(b + p −1) + ϵy < yR(πA, y).\nThis implies R(π′, y) < R(πA, y), which contradicts with the optimality of πA.\nIn conclusion, πA\ni ̸= 0, ∀i ∈{y + 1, y + 2, . . . , b}.\nLemma 4. For any γy ∈[γξ, γν] in Problem A, at the optimal solution πA, the (y+1)-th through b-th constraints\nare binding, i.e., they are satisfied as equalities.\nProof. The dual problem to Problem A (referred to as Problem B) can be formulated as\nmax\nλ1,λ2,...,λb+1\nγy ·\n b\nX\ni=1\nλi\n!\n+ λb+1\n(Problem B)\ns.t.\nx−1\nX\ni=1\nλi +\nb\nX\ni=x\nb + x −1\ni\n· λi + λb+1 ≤b + x −1\ny\n,\n∀x ∈{1, 2, . . . , y},\nx−1\nX\ni=1\nλi +\nb\nX\ni=x\nb + x −1\ni\n· λi + λb+1 ≤1,\n∀x ∈{y + 1, . . . , b},\nλi ≤0,\n∀i ∈{1, 2, . . . , b},\nλb+1 is free.\nLet λB = (λB\n1 , λB\n2 , . . . , λB\nb+1) be the optimal solution to Problem B.\nBy the complementary slackness,\n\"x−1\nX\ni=1\nλB\ni +\nb\nX\ni=x\nb + x −1\ni\n· λB\ni + λB\nb+1 −1\n#\n· πA\nx = 0,\n∀x ∈{y + 1, . . . , b}.\nBy Lemma 3, πA\nx ̸= 0 for all x ∈{y + 1, . . . , b}, we have\nx−1\nX\ni=1\nλB\ni +\nb\nX\ni=x\nb + x −1\ni\n· λB\ni + λB\nb+1 = 1,\n∀x ∈{y + 1, . . . . , b},\n(41)\nwhere we later refer to them as Eq.(y + 1) to Eq.(b).\nWe consider the following three cases.\nCase I: γy = γν . Note that πeq[y + 1, b] is a feasible solution to Problem A when γy = γν. Moreover, by\nTheorem D.2, it’s the only distribution that achieves R(π, y) = 1. Therefore, πeq[y +1, b] is the optimal solution\nto Problem A when γy = γν. By Definition 5, (y + 1)-th to b-th constraints in Problem A are binding at\nπA = πeq[y + 1, b].\nCase II: γy ∈(γξ, γν) . Let p∗denote the optimal objective value in Problem A (the primal problem) and let d∗\ndenote the optimal value in Problem B (the dual problem). It is clear that p∗> 1 when γy ∈(γξ, γν).\nSince Problem A is a convex optimization problem, and πeq[1, b] (i.e. Karlin’s distribution [38]) is always an\ninterior point, by the Slater’s condition, d∗= p∗> 1.\nWe then show that in Problem B, λB\ni ̸= 0 for all i ∈{y+1, . . . , b}. We prove this by contradiction, assuming\nthat there exists r ∈{y + 1, . . . , b} such that λB\nr = 0.\n42\n\nCase II(a): r = y + 1 . Consider Eq.(y + 1) and Eq.(y + 2):\n(\nλ1 + · · · + λy + b+y\ny+1 · λy+1 +\nb+y\ny+2 · λy+2 + · · · +\nb+y\nb\n· λb + λb+1 = 1\nEq.(y + 1)\nλ1 + · · · + λy +\nλy+1 + b+y+1\ny+2\n· λy+2 + · · · + b+y+1\nb\n· λb + λb+1 = 1\nEq.(y + 2)\nNote that b+y\ny+2 < b+y+1\ny+2 , . . . , b+y\nb\n< b+y+1\nb\n. If λB\ny+1 = 0, then (Py\ni=1 λB\ni ) + λB\nb+1 = 1.\nCase II(b): y + 1 < r < b . Consider Eq.(r) and Eq.(r + 1):\n(\nλ1 + · · · + λr−1 + b+r−1\nr\n· λr + b+r−1\nr+1 · λr+1 + · · · + b+r−1\nb\n· λb + λb+1 = 1\nEq.(r)\nλ1 + · · · + λr−1 +\nλr +\nb+r\nr+1 · λr+1 + · · · +\nb+r\nb\n· λb + λb+1 = 1\nEq.(r + 1)\nNote that b+r−1\nr+1\n< b+r\nr+1, . . . , b+r−1\nb\n< b+r\nb . If λB\nr = 0, then\n\u0010Pr−1\ni=1 λB\ni\n\u0011\n+ λB\nb+1 = 1.\nBased on this, consider Eq.(r −1) and Eq.(r):\n(\nλ1 + · · · + λr−2 + b+r−2\nr−1\n· λr−1 + b+r−2\nr\n· λr + · · · + b+r−2\nb\n· λb + λb+1 = 1\nEq.(r −1)\nλ1 + · · · + λr−2 +\nλr−1 + b+r−1\nr\n· λr + · · · + b+r−1\nb\n· λb + λb+1 = 1\nEq.(r)\nWe can deduce λB\nr−1 = 0. Thus,\n\u0010Pr−2\ni=1 λB\ni\n\u0011\n+ λB\nb+1 = 1.\nRepeating this process, we obtain\n\u0000Py\ni=1 λB\ni\n\u0001\n+ λB\nb+1 = 1.\nCase II(c): r = b . Consider Eq.(b −1) and Eq.(b):\n(\nλ1 + · · · + λb−2 + 2b−2\nb−1 · λb−1 + 2b−2\nb\n· λb = 1\nEq.(b −1)\nλ1 + · · · + λb−2 +\nλb−1 + 2b−1\nb\n· λb = 1\nEq.(b)\nIf λB\nb = 0, then λB\nb−1 = 0. This implies that there exists r′ ∈(y + 1, b) such that λB\nr′ = 0. Based on the analysis\nin (b), we have (Py\ni=1 λB\ni ) + λB\nb+1 = 1.\nNote that in Problem Problem B, we have λi ≤0 for all i ∈[b], λb+1 is a free variable, and γy > 1.\nTherefore, the optimal solution to Problem B is λB\ni = 0, ∀i ∈[b] and λB\nb+1 = 1, achieving an objective value of\nd∗= 1. This leads to a contradiction. It thus follows that λB\ni ̸= 0 for all i ∈{y + 1, . . . , b}.\nNow, by the complementary slackness,\nλB\nx ·\n\"Px\ni=1 πA\ni (b + i −1) + Pb\ni=y+1 πA\ni x\nx\n−γy\n#\n= 0,\n∀x ∈{y + 1, . . . , b}.\nSince λB\ni ̸= 0 for all i ∈{y + 1, . . . , b}, we have\nPx\ni=1 πA\ni (b + i −1) + Pb\ni=y+1 πA\ni x\nx\n= γy,\n∀x ∈{y + 1, . . . , b}.\nTherefore, the (y + 1)-th to b-th constraints in Problem A are binding at πA.\nCase III: γy = γξ . Note that πeq[1, b] (i.e. Karlin’s distribution [38]) is the only probability distribution\nthat is γξ-competitive. We can verify that the (y + 1)-th to b-th constraints in Problem A are binding at\nπA = πeq[1, b].\nLemma 5. Consider y < b. Let γ ∈[γξ, γν], where γξ = γ(πeq[1, b]) =\neb\neb−1 and γν = γ(πeq[y + 1, b]). Let\nthe consistency and robustness of OPERATION B(πeq[y + 1, b], γ) (see Algorithm 8) under prediction y be βy\nand γy, respectively. Then, any γy-robust algorithm’s consistency under prediction y is at least βy.\n43\n\nProof. Let ∆denote the set of probability distributions on N+, i.e. ∆:= {π | P∞\ni=1 πi = 1}. Let ∆b be the set\nof probability distributions on [b], i.e. ∆b := {π | Pb\ni=1 πi = 1}. Let ∆′\nb be the set of probability distributions\non [b] that achieve equalizing ratios on {y + 1, . . . , b}, i.e.\n∆′\nb :=\n(\nπ |\nb\nX\ni=1\nπi = 1; R(π, x) = α(π), ∀x ∈{y + 1, . . . , b}\n)\n.\nIt is straightforward to see that ∆′\nb ⊆∆b ⊆∆.\nLet ∆γy denote the set of probability distributions on N+ that is γy-robust under prediction y, i.e.\n∆γy :=\n(\nπ |\n∞\nX\ni=1\nπi = 1; γy(π) = γy\n)\n.\nIt is straightforward to see that ∆γy ⊆∆.\nTo prove that any γy-robust algorithm’s consistency under prediction y is at least βy, it suffices to show\nmin\nπ∈∆γy\nβy(π) = βy.\nConsider π ∈∆\\ ∆b. Let π′ be the distribution after transferring all probability mass at i > b to b. It is clear\nthat βy(π′) = βy(π), ∀y < b. By Lemma 1, γy(π′) ≤γy(π). Therefore,\nmin\nπ∈∆γy\nβy(π) =\nmin\nπ∈∆γy ∩∆b βy(π).\n(42)\nNote that minπ∈∆γy ∩∆b βy(π) is the optimal objective value of Problem A.\nBy Lemma 4, Problem A reduces to the following problem (referred to as Problem C):\nmin\nπ1,π2,...,πb\nPy\ni=1 πi · (b + i −1) + Pb\ni=y+1 πi · y\ny\n(Problem C)\ns.t.\nPx\ni=1 πi · (b + i −1) + Pb\ni=x+1 πi · x\nx\n≤γy,\n∀x ∈{1, 2, . . . , y},\nPx\ni=1 πi · (b + i −1) + Pb\ni=x+1 πi · x\nx\n= γy,\n∀x ∈{y + 1, . . . , b},\nb\nX\ni=1\nπi = 1,\nπi ≥0,\n∀i ∈{1, 2, . . . , b}.\nIn other words, we have\nmin\nπ∈∆γy ∩∆b βy(π) =\nmin\nπ∈∆γy ∩∆′\nb\nβy(π).\n(43)\nLet π∗denote OPERATION B(πeq[y+1, b], γ). Suppose π∗has positive support over {1, . . . , k, y+1, . . . , b}.\nFrom the construction of OPERATION B (see Algorithm 8), it follows that\nR(π∗, x) = γ,\n∀x ∈{1, . . . , k −1, y + 1, . . . , b}\n(44a)\nR(π∗, x) ≤γ,\n∀x ∈{k, . . . , y}\n(44b)\nEquation (44a) and (44b) together imply γ = γy(π∗) = γy. Since Problem C requires R(π, x) = γy for all\nx ∈{y + 1, . . . , b}, by Lemma 2, a necessary condition for the optimal solution is\nπi+1 =\nb\nb −1 · πi,\n∀i ∈{y + 2, . . . , b −1}.\n(45)\n44\n\nConsider π ∈(∆γy ∩∆′\nb) \\ {π∗}. Let r denote the minimal index such that π∗\nr ̸= πr, i.e. r = min{i | π∗\ni ̸=\nπi}. Obviously, r < b. We consider the following cases.\nCase I: y < r < b . In this case, πi = π∗\ni , for all i ∈[y]. Since πy+1 ̸= π∗\ny+1 leads to R(π, y + 1) ̸=\nR(π∗, y + 1) = γy, we have r ̸= y + 1. Thus, πi = π∗\ni , for all i ∈[y + 1]. Based on this, we have\nPb\ni=y+2 πi = Pb\ni=y+2 π∗\ni . If there exists r ∈{y + 2, . . . , b}, such that πr ̸= π∗\nr, then π violates Condition 45.\nTherefore, π is not the optimal solution to Problem C.\nCase II: r ≤y .\nCase II(a): πr > π∗\nr . Note that R(π∗, x) = γy for all x ∈[k −1]. This makes r > k −1. Therefore, we\nhave\nπi = π∗\ni ,\n∀i ∈[k −1]\nπk ≥π∗\nk and πi ≥π∗\ni = 0,\n∀i ∈{k + 1, . . . , y}\nwith some r ∈{k, . . . , y} such that πr > π∗\nr. Thus, Pb\ni=y+1 πi < Pb\ni=y+1 π∗\ni . Combing them together,\nβy(π) = R(π, y) > R(π∗, y) = βy(π∗).\nTherefore, π cannot be an optimal solution to Problem C.\nCase II(b): πr < π∗\nr . Note that π∗\ni = 0 for all i ∈{k + 1, . . . , y}. Therefore, r ≤k. Since πi = π∗\ni\nfor all i ∈[r −1] and πr < π∗\nr, we have Py\ni=r+1 πi ̸= 0; otherwise γy(π) > γy(π∗) = γy. Let r′ denote\nthe minimal index in {r + 1, . . . , y} such that πr′ ̸= 0, i.e. r′ = min {i ∈{r + 1, . . . , y} | πi ̸= 0}. Consider\nπ′ = (π1, . . . , πr + ϵ, . . . , πr′ −ϵ, . . .), where ϵ = min{π∗\nr −πr, πr′}. It is straightforward to verify that\nγy(π′) = γy(π),\nβy(π′) < βy(π).\n(46)\nNote that\nβy(π′) ≥min\nπ∈∆γy\nβy(π) ≥\nmin\nπ∈∆γy ∩∆′\nb\nβy(π).\nIf βy(π) = minπ∈∆γy ∩∆′\nb βy(π), then we have\nβy(π′) ≥βy(π).\n(47)\nHowever, (47) Contradict with (46). Therefore, βy(π) ̸= minπ∈∆γy ∩∆′\nb, which makes π suboptimal to Problem\nC.\nIn conclusion, considering (a) and (b), ∀π ∈(∆γy ∩∆′\nb) \\ {π∗}, π is not an optimal solution to Problem C.\nIn other words, π∗is the optimal solution to Problem C and thus\nβy = βy(π∗) =\nmin\nπ∈∆γy ∩∆′\nb\nβy(π).\n(48)\nBy Equation (42), Equation (43) and Equation (48), we have\nmin\nπ∈∆γy\nβy(π) = βy.\nTherefore, any γy-robust algorithm’s consistency under prediction y is at least βy.\nLemma 6. Consider y < b. Let γ ∈[γξ, γν], where γξ = γ(πeq[1, b]) =\neb\neb−1 and γν = γ(πeq[y + 1, b]). Let\nβy(γ) and γy(γ) denote the consistency and robustness of OPERATION B(πeq[y + 1, b], γ) (see Algorithm 8)\nwith respect to prediction y, respectively. βy(γ) and γy(γ) are jointly Pareto optimal.\n45\n\nProof. Consider OPERATION B (πeq[y + 1, b], γ) and let βy(γ) and γy(γ) denote its prediction-specific consis-\ntency and robustness. Note that βy(γ) is a strictly decreasing function of γ, and γy(γ) = γ. By Lemma 5, for\nany γ ∈[γξ, γν], any γy(γ)-robust algorithm is at least βy(γ)-consistent under prediction y. Then, the only way\nto disprove the Pareto optimality of (βy(γ), γy(γ)) is to find another βy(γ)-consistent and γQ\ny -robust algorithm\nQ, where γQ\ny < γy(γ).\nAssume such algorithm Q exists. Since γQ\ny < γy(γ) and βy(γ) is a strictly decreasing function of γ, we\nhave\nβy(γQ\ny ) > βy(γy(γ)) = βy(γ).\n(49)\nNext, we invoke Lemma 6 once again, any γQ\ny -robust algorithm is at least βy(γQ\ny )-consistent. Note that\nγQ\ny -robust algorithm Q is βy(γ)-consistent. Thus\nβy(γQ\ny ) ≤βy(γ).\n(50)\nEquation (49) and Equation (50) lead to a Contradiction. Therefore, for any γ ∈[γξ, γν], βy(γ) and γy(γ)\nare jointly Pareto optimal.\nConsider OPERATION A(πeq[1, n], y) and denote its consistency and robustness under prediction y by βy\nand γy, respectively. For the case when y ≥b, the following result holds.\nLemma 7. Suppose y ≥b and 1 < n ≤b. The consistency and robustness of OPERATION A(πeq[1, n], y) (see\nAlgorithm 7) under prediction y, denoted by βy and γy are jointly Pareto optimal.\nProof. Let π∗denote OPERATION A(πeq[1, n], y). Our proof is divided into two cases.\nCase I: π∗\n2 = 0 . Note that n > 1, from the construction of Algorithm 7, π∗\n2 = 0 could only happen when\ny = b. In this case, we have π∗\n1 = 1/b, π∗\nb+1 = (b −1)/b, and π∗\ni = 0 for all i ∈N+ \\ {1, b + 1}. Note that\nits prediction-specific consistency and robustness are 1 and 2 −(1/b), respectively, i.e. βy = βy(π∗) = 1 and\nγy = γy(π∗) = 2 −(1/b).\nNote that all 1-consistent algorithm under y = b can only have probability mass on {1, b + 1}. Let π1 and\n1 −π1 represent the probability mass on 1 and b + 1, respectively. then\nγy(π) = max\nπ1∈[0,1]{bπ1 + (1 −π1), π1 + 2(1 −π1)} ≥2 −(1/b).\nTherefore, βy and γy are jointly Pareto optimal.\nCase II: π∗\n2 ̸= 0 . Let k := max{i ≤b | π∗\ni ̸= 0}. Since π∗\n2 ̸= 0, we have k ≥2. According to the structure of\nAlgorithm 7, we have that at least one of the following holds:\nR(π∗, y + 1) = γy,\n(51)\ny −b + 1 = k.\n(52)\nConsider π ̸= π∗. Let r be the minimal index such that πi ̸= π∗\ni , i.e. r := min{i | πi ̸= π∗\ni }.\nCase II(a): πr > π∗\nr . In this case, we must have r < y + 1. If r ≤k −1, based on the design of πeq[1, n]\nand OPERATION A (see Algorithm 7), we have R(π∗, x) = γy for all x ∈[k −1], therefore, πr > π∗\nr will lead\nto γy(π) > γy(π∗) = γy.\nIf k ≤r < y + 1, then we must have π∗\ny+1 ̸= 0, and thus k −1 + b ≥y. It’s clear that βy(π∗) ≤βy(π).\nNote that γy(π) ≥R(π, 1) = R(π∗, 1) = γy(π∗) = γy. Therefore, π cannot achieve strictly better consistency\nor robustness with respect to y.\n46\n\nCase II(b): πr < π∗\nr . A necessary condition for π to dominate π∗is that π must be γy-robust. In the\nsubsequent analysis, we proceed under this assumption. Note that it suffices to restrict the support of the\ndistribution to [y + 1], since assigning probability mass beyond y+1 yields no improvement in consistency while\npotentially worsening robustness. Then, we have r < y + 1. Since π∗\ni = 0 for all k < i < y + 1, we further\nhave r ≤k.\nLet v := min{i > r | πi ̸= 0}. Consider\nπ′ = {π1, . . . , πr−1, πr + ϵ, . . . , πv −ϵ, . . .},\nwhere ϵ = {πv, π∗\nr −πr}. It is straightforward to verify γy(π′) ≤γy.\nIf b+k−1 < y, by construction of OPERATION A (see Algorithm 7), this can only happen when b+n−1 < y\nand the operation terminates upon encountering the first occurrence of Step 0. Since r < v ≤k < y, it’s clear\nthat βy(π′) < βy(π).\nIf b + k −1 = y, since r ≤k, we have r −1 + b ≤y, thus\nbβy(π′) = bβy(π) + ϵ · [1{v ≤y}(r −v) + 1{v > y}(r −1 + b −y)] ≤bβy(π).\nEquivalently,\nβy(π′) ≤βy(π).\nAttaining equality requires r −1 + b = y and r = k.\nIf b + k −1 > y, then Condition (52) does not hold; thus, Condition (51) must hold, i.e., we have\nR(π∗, y + 1) = γy. If v = y + 1, then πi = π∗\ni for all i ∈[r −1], πr < π∗\nr, πi = 0 for all r < i < y + 1. It’s\nclear that γ(π) > γy. This makes v < y + 1. Therefore,\nbβy(π′) = bβy(π) + ϵ(r −v) < bβy(π).\nEquivalently,\nβy(π′) < βy(π).\nTherefore, for all π ̸= π∗, π is not the most consistent γy-robust distribution except that r −1 + b = y and\nr = k hold simultaneously.\nWhen r = k ≥2 and r −1 + b = y hold simultaneously, we have π1 = π∗\n1. Thus, γy(π) ≥R(π, 1) =\nR(π∗, 1) = γy(π∗) = γy. Since π is γy-robust, γy(π) ≤γy. Thus, γy(π) = γy. Note that\nβy(π) = R(π, y) =\nPy\ni=1(i −1 + b)πi + πy+1y\nb\n,\nand i −1 + b ≥r −1 + b = y for all k ≤i ≤y. This makes βy(π) ≥βy(π∗).\nTherefore, any π ̸= π∗makes no Pareto improvement over π∗.\nIn conclusion, βy and γy are jointly Pareto optimal.\nWith those foundational lemmas, we prove Theorem 5.3.\nProof of Theorem 5.3. Assume that the user-specified parameter is γ. Let n = ⌈logb/(b−1) 1 + 1/(γ + 1)⌉. Let\nγ′ = γ(πeq[1, n]) = 1 + [(\nb\nb−1)n −1]−1. Let π denote the final distribution obtained after applying either\nOPERATION A or OPERATION B.\nCase I: y ≥b . Let πA denote OPERATION A(πeq[1, n], y).\nCase I(a):\nπA is a two-point distribution . We have βy(πA) = 1 and γy(πA) =\n2b−1\nb . By design,\nOPERATION A never increases the robustness of the distribution at any point in time. Therefore, the a condition\nfor forming a two-point should be γ′ ≥γy(πA). In this case, it’s easy to verify that\n47\n\nβy(πA) ≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\n(53)\nCase I(b): πA is not a two-point distribution . In this case, we have\nπA\n1 = πeq\n1 [1, n] = [(b −1)((\nb\nb −1)n −1)]−1.\nThus, we have\nγy(πA) = R(πA, 1) = πA\n1 · b + (1 −πA\n1 ) = 1 + [(\nb\nb −1)n −1]−1 = γ′.\nNote that as b →∞, Kumar’s algorithm is\nλ\n1−e−λ -consistent and\n1\n1−e−λ -robust. (see [2]) Take λ =\n−log(1 −γ′−1). In this case, Kumar’s algorithm is γ′ log(1 +\n1\nγ′−1)-consistent and γ′-robust. Since γ < b −2\nand n = ⌈logb/(b−1) 1 + 1/(γ + 1)⌉, we have n > 1. By Lemma 7, (βy(πA), γy(πA)) is Pareto optimal.\nTherefore, we have\nβy(πA) ≤γy(πA) log\n\u0012\n1 +\n1\nγy(πA) −1\n\u0013\n.\nSince γ′ = γy(πA), we have for any y ≥b,\nβy(πA) ≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\n(54)\nCase II: y < b . Let γ′′ = min{γν, γ′}. Recall that γν = γ(πeq[y+1, b]). Let πB denote OPERATION B(πeq[y+\n1, b], γ′′). Note that\nγν = γ(πeq[y + 1, b]) > γ(πeq[1, b]) = γξ,\nand\n1 + [(\nb\nb −1)n −1]−1 = γ(πeq[1, n]) ≥γ(πeq[1, b]) = γξ.\nTherefore, γ′′ = min{γν, γ′} ≥γξ, and γ′′ = min{γν, γ′} ≤γν. Thus, we have\nγξ ≤γ′′ ≤γν.\nTherefore, γy(πB) = γ′′. We consider the following two cases.\nCase II(a): γ′′ = γ′ . In this case, we consider λ = −log(1−γ−1\ny ), Kumar’s algorithm is γy log(1+\n1\nγy−1)-\nconsistent and γy-robust. Since γξ ≤γ′′ ≤γν, by Lemma 5,\nβy(πB) ≤γy(πB) log\n\u0012\n1 +\n1\nγy(πB) −1\n\u0013\n.\nSince γ′ = γ′′ = γy(πB), we have for any y < b,\nβy(πB) ≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\n(55)\nCase II(b): γ′′ = γν < γ′ . In this case, πB = πeq[y + 1, b], and βy(πB) = 1. We can verify that\nβy(πB) ≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\n(56)\n48\n\nBy (53), (54), (55) and (56), we conclude that for any y ∈N+,\nβy(πB) ≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\nTherefore, by Definition 1 and Definition 2,\nβ(π) = sup\ny∈N+\nβy(π) = max\n(\nsup\ny≥b\nβy(πA), sup\ny<b\n(βy(πB))\n)\n≤γ′ log\n\u0012\n1 +\n1\nγ′ −1\n\u0013\n.\nAcross all of the cases discussed above, it holds that γy(π) ≤γ′. Similarly, by Definition 1 and Definition 2,\nγ(π) = supy∈N+ γy(π) ≤γ′.\nSince F(x) := x log(1 +\n1\nx−1) is decreasing on x ∈(1, +∞), we further have\nβ(π) ≤γ(π) log\n\u0012\n1 +\n1\nγ(π) −1\n\u0013\n.\nWei’s lower bound [17] and Definition 3, PRSRis weakly-optimal.\nBy Lemma 6 and Lemma 7, PRSR’s prediction-specific consistency and robustness are Pareto optimal.\nAccording to Definition 4, PRSR is strongly-optimal.\n49\n\nF\nProofs for Section 6\nIn this section, we prove Theorem 6.1 and 6.2.\nF.1\nProof of Theorem 6.1\nProof of Theorem 6.1. Let βS\ny and γS\ny denote the prediction-specific consistency and robustness of Sun’s algo-\nrithm with respect to prediction y. Let βS and γS denote the consistency and robustness of Sun’s algorithm.\nConsider y ∈[L, LβS). In this case, Sun’s algorithm sets thresholds to Φ = LβS. To analyze the prediction-\nspecific consistency, we assume that the maximum price is exactly y. Since y < LβS, we have ALG = L and\nOPT = y. By Definition 2, βS\ny = y/L. To obtain the robustness under prediction y, we consider incorrect\npredictions. Observe that in the worst case, ALG = LβS, OPT = U. Therefore, γS\ny = U/(LβS) = θ/βS.\nConsider the canonical competitive algorithm that adopts a fixed threshold policy with Φ =\n√\nLU, which has\nprediction-specific consistency βC\ny = y/L and robustness γC\ny =\n√\nθ. Note that ∀λ ∈[0, 1), γS >\n√\nθ, which\nmakes\nβS\ny = βC\ny ,\nγS\ny > γC\ny .\nBy Definition 4, Sun’s algorithm is not strongly-optimal.\nF.2\nProof of Theorem 6.2\nProof of Theorem 6.2. We first prove the prediction-specific consistency and robustness of PST. We consider\nthe following three cases.\nCase I: y ∈[L, λL + (1 −λ)\n√\nLU) . In this case, PST sets the threshold at Φ =\n√\nLU. Since λL + (1 −\nλ)\n√\nLU ≤\n√\nLU, we have y <\n√\nLU, therefore βy = y/L. Since Φ =\n√\nLU, γy = max{Φ/L, U/Φ} =\n√\nθ.\nCase II: y ∈[λL + (1 −λ)\n√\nLU,\n√\nLU] . In this case, PST sets the threshold at Φ = y. Obviously, βy = 1.\nSince y ≤\n√\nLU, we have γy = max{Φ/L, U/Φ} = U/y.\nCase III: y ∈(\n√\nLU, U] . In this case, PST sets the threshold at Φ = µ\n√\nLU + (1 −µ)y, where\nµ =\n(1 −λ)\n√\nθ\n(1 −λ)\n√\nθ + λ\n.\nSince\n√\nLU < y, Φ = µ\n√\nLU + (1 −µ)y ≤y. Under worst-case construction, we conclude\nβy = y\nΦ =\ny\nµ\n√\nLU + (1 −µ)y\n= (1 −λ)\n√\nθy + λy\n(1 −λ)U + λy ,\nγy = max{Φ/L, U/Φ} = Φ\nL =\n(1 −λ)U + λy\n(1 −λ)\n√\nLU + λL\n.\nWe then prove the STRONG optimality of PST. As in worst-case instances any deterministic algorithm\nperforms equivalently to a threshold algorithm, it is not restrictive to only consider OTAs.\nBy considering the worst-case predictions for both prediction-specific consistency and robustness, we can\nconclude that PST is λ + (1 −λ)\n√\nθ-consistent and\nθ\nλ+(1−λ)\n√\nθ-robust. Building on the lower bound established\nby Sun et al. [11], PST is weakly-optimal.\nConsider Φ′ ̸= Φ. Let (β′\ny, γ′\ny) denote the consistency and robustness of OTA that uses threshold Φ′ with\nrespect to y.\n50\n\nAssume y ∈[L, λL + (1 −λ)\n√\nLU). Since Φ′ ̸= Φ =\n√\nLU, γ′\ny = max{Φ′/L, U/Φ′} >\n√\nLU = γy.\nTherefore, (βy, γy) is Pareto optimal.\nAssume y ∈[λL+(1−λ)\n√\nLU,\n√\nLU]. Since Φ′ ̸= Φ = y, βy > 1 = βy. Thus, (βy, γy) is Pareto optimal.\nAssume y ∈(\n√\nLU, U]. If Φ′ < Φ, then β′\ny = y/Φ′ > y/Φ = βy. If Φ′ > Φ, since Φ′ > Φ ≥\n√\nLU, we\nhave γy = Φ/\n√\nLU and γ′\ny = Φ′/\n√\nLU, thus γ′\ny > γy. Therefore, (βy, γy) is Pareto optimal.\nBy Definition 4, PST is strongly-optimal.\n51\n\nG\nProofs for Section 7\nIn this section, we prove Theorem 7.1, 7.2 and 7.3.\nG.1\nProof of Theorem 7.1\nProof of Theorem 7.1. We consider the following five cases to investigate the prediction-specific ϵ-consistency\nand robustness.\nCase I: y ∈[L, M −2ϵ] . In this case, ϵ-Tolerant PST decides to set the threshold to Φ =\n√\nLU and the price\nrange relevant to ϵ-consistency is restricted to [max{L, y −ϵ}, y +ϵ]. To determine the ϵ-consistency, we assume\nthat the highest price x ∈[max{L, y −ϵ}, y + ϵ]. Note that y + ϵ ≤(M −2ϵ) + ϵ <\n√\nLU = Φ. We have\nβϵ\ny = (y + ϵ)/L. To obtain the robustness, we consider incorrect predictions. γy = max{Φ/L, U/Φ} =\n√\nθ.\nCase II: y ∈(M −2ϵ, M) . In this case, ϵ-Tolerant PST decides to set the threshold to Φ = M −ϵ and the\nprice range relevant to ϵ-consistency is restricted to [y −ϵ, y + ϵ]. For ϵ-consistency, we observe the best attack\ncan potentially occur at x = M −ϵ −δ or x = y + ϵ, where δ is the infinitesimal quantity. The corresponding\nratios are r1(M) := (M −ϵ)/L and r2(M) := (M + ϵ)/(M −ϵ). Note that r1(M) is an increasing function\nof M and r2(M) is a decreasing function of M. Since\nr1(L + 3ϵ) = L + 2ϵ\nL\n≥L + 4ϵ\nL + 2ϵ = r2(L + 3ϵ),\nwe can conclude r1(M) ≥r2(M), for all M ∈[L + 3ϵ,\n√\nLU −ϵ]. thus βϵ\ny = (M −ϵ)/L. To obtain the\nrobustness, we consider incorrect predictions. Since M −ϵ <\n√\nLU, γy = U/(M −ϵ).\nCase III: y ∈[M,\n√\nLU + ϵ] . In this case, ϵ-Tolerant PST decides to set the threshold to Φ = y −ϵ and the\nprice range relevant to ϵ-consistency is restricted to [y −ϵ, y + ϵ]. To determine the ϵ-consistency, we assume\nthat x ∈[y −ϵ, y + ϵ]. Observe that the worst case occurs at x = y + ϵ, we have βϵ\ny = (y + ϵ)/(y −ϵ). To\nobtain the robustness, we consider incorrect predictions. Note that\ny −ϵ ≤(\n√\nLU + ϵ) −ϵ =\n√\nLU.\nWe have γy = U/Φ = U/(y −ϵ).\nCase IV: y ∈(\n√\nLU + ϵ, U −ϵ) . In this case, ϵ-Tolerant PST decides to set the threshold to Φ = µ\n√\nLU+(1−\nµ)(y −ϵ) and the price range relevant to ϵ-consistency is restricted to [y −ϵ, y +ϵ]. Since µ = (U−2ϵ)−LU/(M−ϵ)\n(U−2ϵ)−\n√\nLU\n,\nby noting that\n(U −2ϵ) −\nLU\n(L + 3ϵ −ϵ) > 0,\n(U −2ϵ) −\nLU\n(\n√\nLU −ϵ −ϵ)\n< (U −2ϵ) −\n√\nLU,\nwe have µ ∈(0, 1). To determine the ϵ-consistency, we assume that x ∈[y −ϵ, y + ϵ]. Note that Φ ≤y −ϵ. We\nhave\nβϵ\ny = y + ϵ\nΦ\n=\ny + ϵ\nµ\n√\nLU + (1 −µ)(y −ϵ)\n.\nTo obtain the robustness, we consider incorrect predictions. Note that Φ ≥\n√\nLU. We have γy = Φ/L =\n[µ\n√\nLU + (1 −µ)(y −ϵ)]/L.\n52\n\nCase V: y ∈[U −ϵ, U] . In this case, ϵ-Tolerant PST decides to set the threshold to Φ = LU/(M −ϵ) and the\nprice range relevant to ϵ-consistency is restricted to [y −ϵ, U]. Since ϵ ≤(\n√\nLU −L)/4 < (U −L)/2, we have\nΦ =\nLU\nM −ϵ ≤\nLU\nL + 2ϵ ≤U −2ϵ ≤y −ϵ,\nthus, βϵ\ny = U\nΦ = M−ϵ\nL . Furthermore, γy = Φ\nL =\nU\nM−ϵ.\nThen, we focus on the ϵ-consistency and robustness of ϵ-Tolerant PST.\nIn Case III, we have\nβϵ\ny = (y + ϵ)\n(y −ϵ) < L + 2ϵ\nL\n≤M −ϵ\nL\n.\nIn Case IV, since µ = (U−2ϵ)−LU/(M−ϵ)\n(U−2ϵ)−\n√\nLU\n, we get\nβϵ\ny ≤\nU\nµ\n√\nLU + (1 −µ)(U −2ϵ)\n= M −ϵ\nL\n.\nSimilarly, we obtain\nγy = µ\n√\nLU + (1 −µ)(y −ϵ)\nL\n≤µ\n√\nLU + (1 −µ)(U −2ϵ)\nL\n=\nU\nM −ϵ.\nBy considering the worst-case y over F = [L, U], we conclude that ϵ-Tolerant PST’s ϵ-consistency and\nrobustness is βϵ = (M −ϵ)/L and γ = U/(M −ϵ), respectively.\nG.2\nProof of Theorem 7.2\nProof of Theorem 7.2. By the lower bound provided by Sun et al. [11], any γ-robust algorithm must be at least\n(θ/γ)-consistent. By Definition 1 and the definition of ϵ-consistency, we have βϵ ≥β for any ϵ > 0. Therefore,\nany γ-robust algorithm has at least (θ/γ) ϵ-consistency.\nSimilarly, given ϵ > 0, any algorithm that achieves βϵ ϵ-consistency is βϵ-consistent. Based on the lower\nbound by Sun et al. [11], it must be at least (θ/βϵ)-robust.\nG.3\nProof of Theorem 7.3\nProof of Theorem 7.3. By Theorem 7.1 and Theorem 7.2, we conclude that ϵ-Tolerant PST’s ϵ-consistency and\nrobustness are jointly Pareto optimal.\nTo prove the Pareto optimality of prediction-specific ϵ-consistency and robustness, we consider Φ′ ̸= Φ,\nwhich achieves ϵ-consistency βϵ\ny\n′ and robustness γy′ with respect to y.\nCase I: y ∈[L, M −2ϵ] . In this case, γy =\n√\nθ. Note that Φ =\n√\nLU is the only threshold that achieves\nrobustness\n√\nθ. We can conclude that (βϵ\ny, γy) is Pareto optimal.\nCase II: y ∈(M −2ϵ, M) . In this case, ϵ-Tolerant PST sets the threshold to Φ = M −ϵ, achieving\nβϵ\ny = (M −ϵ)/L and γy = U/(M −ϵ). If Φ′ < Φ, then γ′\ny > γy. If Φ′ > Φ, then consider the attack\nx = min{Φ′ −δ, y + ϵ} for threshold Φ′, where δ is the infinitesimal quantity. This makes\nβϵ\ny\n′ ≥min{Φ′, y + ϵ}/L > (M −ϵ)/L = βϵ\ny.\nTherefore, (βϵ\ny, γy) is Pareto optimal.\n53\n\nCase III: y ∈[M,\n√\nLU + ϵ] . In this case, ϵ-Tolerant PST sets the threshold to Φ = y −ϵ, achieving\nβϵ\ny = (y + ϵ)/(y −ϵ) and γy = U/(y −ϵ). Note that Φ = y −ϵ <\n√\nLU. If Φ′ < Φ, then γ′\ny > γy. If Φ′ > Φ,\nconsider x = min{Φ′ −δ, y + ϵ} for threshold Φ′, where δ is the infinitesimal quantity. This guarantees the\nfollowing inequality:\nβϵ\ny\n′ ≥min{Φ′, y + ϵ}/L > (y −ϵ)/L.\nNote that r′(y) := (y −ϵ)/L is an increasing function of y, and r(y) := (y + ϵ)/(y −ϵ) is a decreasing function\nof y. Since y ≥M ≥L + 3ϵ and\nr(L + 3ϵ) = (L + 4ϵ)/(L + 2ϵ) < (L + 2ϵ)/L = r′(L + 3ϵ),\nwe have r(y) < r′(y), ∀y ∈[M,\n√\nLU + ϵ]. This gives βϵ\ny < βϵ\ny\n′. Therefore, (βϵ\ny, γy) is Pareto optimal.\nCase IV: y ∈(\n√\nLU + ϵ, U −ϵ) . In this case, ϵ-Tolerant PST sets the threshold to Φ = µ\n√\nLU+(1−µ)(y−ϵ),\nachieving βϵ\ny = (y + ϵ)/Φ and γy = Φ/L. If Φ′ < Φ, it follows that\nβϵ\ny\n′ ≥(y + ϵ)/Φ′ > (y + ϵ)/Φ = βϵ\ny.\nIf Φ′ > Φ, since Φ >\n√\nLU + ϵ >\n√\nLU, γ′\ny > γy. Therefore, (βϵ\ny, γy) is Pareto optimal.\nCase V: y ∈[U −ϵ, U] . In this case, ϵ-Tolerant PST sets the threshold to Φ = LU/(M −ϵ), achieving\nβϵ\ny = (M −ϵ)/L and γy = U/(M −ϵ). If Φ′ > Φ, then γ′\ny > γy. If Φ′ < Φ, consider highest price x = U\nwith |x −y| ≤ϵ, βϵ\ny\n′ ≥U/Φ′ > U/Φ = βϵ\ny. Therefore, (βϵ\ny, γy) is Pareto optimal.\n54\n"},"cleaned":{"kind":"string","value":"Prediction-Specific Design of Learning-Augmented Algorithms Sizhe Li*1, Nicolas Christianson†2, and Tongxin Li‡1 1 2 -driven performance benefits of machine-learned (ML) predictions. However, most existing approaches in this paradigm are overly conservative, as they do not leverage problem structure to optimize performance in a prediction-specific manner. In this paper, we show that such prediction-specific performance criteria can enable significant performance improvements over the coarser notions of consistency and robustness considered in prior work. Specifically, we propose a notion of strongly-optimal algorithms with predictions, which obtain Pareto optimality not just in the worst-case tradeoff between robustness and consistency, but also in the prediction-specific tradeoff between these metrics. We develop a general bi-level optimization framework that enables systematically designing strongly-optimal algorithms in a wide variety of problem settings, and we propose explicit strongly-optimal algorithms for several classic online problems: deterministic and randomized ski rental, and one-max search. Our analysis reveals new structural insights into how predictions can be optimally integrated into online algorithms by leveraging a prediction-specific design. To validate the benefits of our proposed framework, we empirically evaluate our algorithms in case studies on problems including dynamic power management and volatility-based index trading. Our results demonstrate that prediction-specific, strongly-optimal algorithms can significantly improve performance across a variety of online decision-making settings. 1 Introduction Online algorithms operate in environments where decisions must be made sequentially without full knowledge of future inputs. Traditionally, these algorithms are designed to guarantee robust performance on adversarial problem instances, providing competitive ratio bounds that hold under worst-case inputs [1]. While theoretically robust, this adversarial perspective often yields overly pessimistic strategies that can underperform in the real world, where worst-case instances are uncommon. Recent work on algorithms with predictions addresses this limitation by integrating machine-learned (ML) predictions into classical online decision-making frameworks [2, 3]. This learning-augmented algorithm design paradigm has been applied to a variety of online problems including ski rental, caching, and metrical task systems [2-4] and applications including GPU power management [5], battery energy storage system control [6], carbon-aware workload management [7, 8], and electric vehicle (EV) charging [9, 10]-with algorithms that achieve near-optimal performance when predictions are accurate, while maintaining robust worst-case guarantees when they are not. Existing approaches to learning-augmented algorithm design typically evaluate performance through a so-called consistency-robustness tradeoff. In particular, consistency measures worst-case algorithm performance when the prediction is perfectly accurate, while robustness measures the worst-case competitive ratio over all possible predictions and instances. *Email: †Email: ‡Email: 1 16 Oct 2025 Table 1: Comparison of our technical results on algorithm (weak and strong) optimality. General DSR (LARGE b) RSR (LARGE b) OMS ε-OMS WEAK Algorithm 1 KD Kumar et al. KR Kumar et al. Sun et al. \\ STRONG Algorithm 1 PDSR PRSR PST ε-Tolerant PST Proposition 1 Theorem 4.2 Theorem 5.3 Theorem 6.2 Theorem 7.3 Note: Algorithm 1, PDSR, PRSR, PST, and ε-Tolerant PST are results from this work. DSR/RSR=Deterministic/Randomized Ski Rental, OMS=One-Max Search, ε-OMS=Error Tolerant OMS 0 50 100 150 200 Prediction y 1.0 1.5 2.0 2.5 3.0 Consistency y KD ( = 0.5) y KD ( = 0.5) y PDSR (Ours) y PDSR (Ours) y 1.0 1.5 2.0 2.5 3.0 Robustness y 0 50 100 150 200 Prediction y 1.0 1.5 2.0 2.5 Consistency y KR ( = 0.5) y KR ( = 0.5) y PRSR (Ours) y PRSR (Ours) y 1.0 1.5 2.0 2.5 Robustness y 10 12 14 16 18 20 Prediction y 1.50 1.75 2.00 y × y Sun et al. ( = 0.5) PST (Ours) Figure 1: Prediction-specific consistency βy and robustness γy under different predictions y for DSR with b = 100 (LEFT), RSR with b = 100 (MIDDLE), and OMS with L = 10, U = 20 (RIGHT). Importantly, both metrics are inherently worst-case in nature: consistency reflects the algorithm's performance under the least favorable instance with the least favorable, yet accurate prediction, and robustness measures performance under the least favorable (inaccurate) prediction and instance. Because they are worst-case, neither of these metrics measures whether algorithms can achieve better performance for specific predictions. As such, while much prior work has sought to design algorithms that obtain the optimal tradeoff between consistency and robustness (e.g., [2, 11]), and some existing algorithms have sought to improve performance by optimizing decisions in a prediction-specific manner (e.g., [8, 12]), no prior work has considered the question of how to design online algorithms with prediction-specific guarantees on optimality. Motivated by this gap and the potential to improve the performance of algorithms with predictions, our work explores the following question: How can we design algorithms that achieve Pareto-optimal tradeoffs between consistency and robustness that are tailored to specific prediction values? To this end, we introduce a prediction-specific framework for algorithm design, enabling the development of explicit and tractable online algorithms that adapt to each prediction's characteristics to ensure Pareto-optimal robustness and consistency for each individual prediction value. Instantiating this framework, we design explicit prediction-specific algorithms for the problems of deterministic and randomized ski rental and one-max search, two classic online problems with connections to real-world applications including TCP acknowledgment [13], cloud cost management [14], dynamic power management [5], and energy market operations [15, 16]. 1.1 Contributions The main contributions of this paper are as follows. Framework and Theoretical Results. We introduce a novel prediction-specific framework for the design and analysis of learning-augmented algorithms. Specifically, Definition 2 extends the classic notions of consistency and robustness by defining the prediction-specific consistency βy and robustness γy for each possible prediction value y. Here, βy evaluates the algorithm's performance on the worst-case instance which is consistent with the prediction y, while γy measures its worst-case performance under prediction y. Furthermore, Definition 4 introduces the notion of strong optimality: an algorithm is strongly optimal if it is Pareto-optimal not only 2 DSR RSR OMS Existing Results KD ([2], Algorithm 2) KR ([2], Algorithm 3) Sun et al. [11] This Work PDSR (Algorithm 2) PRSR (Algorithm 9) PST (Algorithm 4) Improvement Ratio 1 + 1 λ /2 ( e-1 e ) · (1 -e-λ)-1 √ (1-λ)2+4λθ-(1-λ) 2λ √ θ Table 2: Comparison of proposed algorithms in this work with existing results. The Improvement Ratio quantifies the maximum ratio between the consistency-robustness score (βy · γy) of our algorithm over the best existing score for all y. in the classic sense (referred to as weak optimality, see Definition 3), but also in the (βy, γy) plane for every prediction value y in the prediction space F. This reframes the evaluation of algorithm performance from a single worst-case trade-off to a richer, per-prediction perspective, thereby enabling a more fine-grained analysis of algorithm behavior across different prediction values. Under this new framework, we show that the existing weakly-optimal algorithms for several canonical online problems-deterministic ski rental, randomized ski rental, and one-max search (see Table 3)-are not strongly-optimal (see Theorems 4.1, 5.1, and 6.1). As such, we propose new algorithms for these problems (Algorithms 2, 9, and 4) and prove their strong optimality (see Theorems 4.2, 5.3, and 6.2). Notably, our stronglyoptimal algorithms can obtain significant performance improvements over prior weakly-optimal algorithms for certain prediction values; Table 2 summarizes the best improvement ratio of the consistency-robustness score (βy · γy) across all y for each problem, reflecting the relative benefit of our algorithms over existing ones under the most favorable y. The parameter λ represents some confidence governing the tradeoff between robustness and consistency used in [2, 11]. Note that for both deterministic and randomized ski rental, the maximum improvement ratio is obtained as λ ↓0, where it diverges to infinity with asymptotic order O(λ-1). In contrast, for one-max search, the maximum improvement ratio is √ θ, which is also attained as λ ↓0. Consequently, our methods can yield substantial performance improvements for certain predictions and choices of the parameter λ. We also show that there exist problems for which existing weakly-optimal algorithms in the literature are already strongly-optimal: in particular, we establish prediction-specific bounds in Theorem 2.1 for the weakly-optimal algorithm of Wei and Zhang [17] for non-clairvoyant job scheduling when the job size n = 2, and prove its strong optimality in this case. Novel Techniques and Methodology. We propose a bi-level optimization methodology (see Problems 1 and 2 in Section 3.1) for systematically deriving strongly-optimal algorithms. Given an initial robustness target γ, Problem 1 finds the best prediction-specific consistency βy, and Problem 2 then finds a decision with the best prediction-specific robustness γy given the fixed βy. The bi-level optimization pipeline naturally forms a meta-algorithm (Algorithm 1), which we prove in Proposition 1 yields a strategy on the prediction-specific Pareto front for each y, guaranteeing strong optimality. This approach offers two key benefits. (1) Generality - the bi-level optimization framework is broadly applicable: it can be used to design algorithms for online problems beyond those described in Table 3. (2) Flexibility - this approach exhibits two kinds of flexibility. First, the consistency-robustness trade-off in the meta-algorithm (Algorithm 1) can be tuned by adjusting the robustness target γ. Second, the bi-level optimization problem can be adapted to different kinds of predictions and performance objectives. For example, the objectives and constraints in Problems 1 (P1) and 2 (P2) can be extended to incorporate error tolerance to enable the design of algorithms which perform well even if predictions are erroneous, thus alleviating, to some extent, the issues of brittleness and smooth performance degradation which have been considered in a number of recent works [18-20]. More specifically, in Section 7, we introduce the notion of ε-consistency to formalize the goal of preserving consistency (and avoiding algorithm brittleness) when faced with small prediction error ε, and we show that it is possible to obtain both classic Pareto optimality between ε-consistency and robustness, as well as a corresponding version of strong optimality in this case (see Theorem 7.3). In addition, our analysis of strong optimality in the randomized ski rental problem (see Appendix E) employs 3 a novel prediction-specific primal-dual technique. Specifically, we first use a perturbation-based analysis to derive structural properties of the optimal randomized distribution, and then formulate and analyze a primal-dual optimization problem for each prediction. This provides a structured and potentially generalizable approach to establish Pareto optimality with respect to a given prediction. Experimental Results and Real-World Implications. We evaluate our algorithms through both synthetic simulations and real-world case studies, spanning a range of online decision-making problems. Overall, our methods consistently outperform state-of-the-art learning-augmented and classical baseline algorithms, confirming their theoretical soundness and practical value. More specifically, we apply the deterministic and randomized ski rental algorithms to Dynamic Power Management (DPM) traces [5], an important benchmark for energy-efficient computing, and we apply the one-max search algorithms to VIX trading, a representative financial market task marked by high volatility and uncertainty. In both domains, our methods deliver significant improvements across most prediction regimes, illustrating how prediction-specific design can translate improved theoretical guarantees into tangible real-world gains. 1.2 Related Work Algorithms with Predictions. Although the study of online algorithms with ML predictions is still relatively new [2, 3], significant research progress has been made in recent years on various problems such as online optimization [6], control [21, 22], and reinforcement learning [23, 24]. Similar frameworks have also been adopted to design and analyze algorithms for a number of other online problems, such as caching [3, 25-27], online knapsack [28-30], secretary and bipartite matching [31], metrical task systems [4, 32, 33], and convex body/function chasing [32, 34]. Most of these works make no assumptions about prediction quality, seeking to balance consistency (competitive ratio when predictions are accurate) and robustness (worst-case competitive ratio regardless of prediction quality), though the precise, formal definitions of these metrics vary slightly across works. Beyond their theoretical appeal, these frameworks have begun to influence practical systems domains, enabling advances in areas such as data-center scheduling [35], energy-aware computing [7, 15], and networked control systems [6, 22], where ML-driven forecasts are increasingly available but inherently imperfect. Note also that some recent works depart from the standard paradigm of robustness and consistency and consider alternative prediction models and performance measures. Sun et al. [36] proposed online algorithms with uncertainty-quantified (UQ) predictions, which leverage UQ to assess prediction quality. They introduced the distributionally robust competitive ratio (DRCR), which weighs both the setting where the UQ prediction accurately describes the instance, and the worst-case adversarial setting, and applied this metric to the problems of ski rental and online search. Mahdian et al. [37] proposed a general framework for online optimization under uncertain inputs, where the algorithm has access to an optimistic strategy that performs well when the future unfolds favorably. They developed a meta-algorithm that balances between this optimistic policy and a robust fallback, achieving a trade-off between worst-case guarantees and performance under accurate predictions without relying on a formal error model. Ski Rental and Scheduling with ML Predictions. Regarding online problems that are closely related to those specific examples considered in this work, Kumar et al. [2] studied ski rental and non-clairvoyant scheduling with ML predictions. Their framework introduces a tunable trade-off between consistency and robustness through a user-specified hyperparameter. Wei and Zhang [17] subsequently gave general lower bounds on the trade-off obtainable in these problems, thus proving that the deterministic and randomized algorithms of Kumar et al. [2] for ski rental achieve the Pareto-optimal trade-off between consistency and robustness. Furthermore, they demonstrated that the meta-algorithm proposed by Kumar et al. [2] for nonclairvoyant scheduling does not achieve the tight trade-off, and introduced a novel two-stage scheduling strategy that is provably tight for the case of n = 2. One-Max Search with ML Predictions. In the learning-augmented setting of one-max search, where algorithms receive a prediction of the maximum element value, Sun et al. [11] established a fundamental lower 4 bound on the trade-off between consistency and robustness. They also proposed a threshold-based algorithm and showed that it achieves this lower bound, making it Pareto-optimal in terms of these two performance measures. Algorithm Smoothness. A recent line of work has shown that some existing learning-augmented algorithms are brittle, suffering sharp performance drops under small prediction errors. Elenter et al. [18] addressed this by designing algorithms that follow user-specified error-performance profiles, ensuring controlled degradation in performance for the one-way trading problem. Benomar et al. [19] further proposed smooth one-max search algorithms that are Pareto-optimal and exhibit gradual performance degradation with increasing prediction error, achieving a \"triple Pareto optimality\" among consistency, robustness, and smoothness. Unlike prior work that focuses on structural smoothness of the Pareto frontier, our formulation provides a principled relaxation of consistency itself, leading to algorithms that are both robust and tolerant to small predictive errors. 2 Problem Formulation We consider online cost minimization problems over a set of instances I. For each instance I ∈I, let ALG(A, I) and OPT(I) denote the cost incurred by an online algorithm A and the cost of the offline optimum, respectively. We assume that the algorithms have no prior knowledge of the instance I. Under the competitive analysis framework [1], the goal is to find an online algorithm A that minimizes the worst-case competitive ratio α(A), which is defined as1 α(A) = max I∈I ALG(A, I) OPT(I) . This worst-case focus, however, can be overly pessimistic in practical settings. To move beyond the limitations of worst-case algorithms, machine-learned predictions can be integrated into algorithm design (see [2, 3]). In this setting, an online algorithm Aω, potentially parameterized by ω ∈Ω, receives not only the instance I ∈I online but also an ML prediction y ∈F concerning some relevant but unknown feature x(I) ∈F of I, where F denotes a prediction space. The feature x(I) encapsulates useful information about I (e.g., the maximum element value for online one-max search) or may even fully specify I in some problems (e.g., the total number of skiing days for discrete-time ski rental). Let ALG(Aω, I, y) denote the (expected) cost incurred by algorithm Aω on instance I given prediction y. Classic Consistency and Robustness. Since the ML prediction y ∈F may be erroneous, a number of algorithms have been designed (see those summarized in Sections 1.2 and 2.3) to (1) achieve near-optimal performance when the prediction y is accurate (consistency), while (2) simultaneously maintaining a bounded performance guarantee even when the prediction is arbitrarily wrong (robustness). To formalize this, let Iy ⊆I represent the set of instances for which prediction y is considered \"accurate\" or \"consistent\"; the precise definition depends on the specific problem, the form of prediction, and the prediction quality measure. The classic consistency and robustness metrics are defined as follows. Definition 1 (Classic Metrics). Given an online algorithm Aω that takes predictions, the consistency β(Aω) and robustness γ(Aω) are defined as: β(Aω) := sup y∈F sup I∈Iy ALG(Aω, I, y) OPT(I) , γ(Aω) := sup y∈F sup I∈I ALG(Aω, I, y) OPT(I) . (1) If an algorithm Aω achieves β(Aω) consistency and γ(Aω) robustness, it is called β(Aω)-consistent and γ(Aω)-robust, respectively. A typical choice of Iy is Iy := {I ∈I : x(I) = y} (following [2, 3]). The equality constraint in Iy can be generalized in a number of ways-for instance, to a ball centered at x(I), which will emphasize small error tolerance of the algorithm (see Section 7 for more details). Here, the consistency β(Aω) captures the 1In online profit maximization problems, the competitive ratio is defined as the worst-case ratio between the optimal offline profit and that obtained by the online algorithm, which is the inverse of the minimization setting. 5 algorithm's performance guarantee assuming the prediction is correct, taking the worst case over all possible correct predictions, and the robustness γ(Aω) represents the worst-case guarantee under all possible instances and any prediction y, regardless of accuracy. An ideal algorithm minimizes both β(Aω) (aiming for near 1) and γ(Aω). 2.1 Prediction-Specific Consistency and Robustness The standard metrics in (1) evaluate performance by taking the worst case over both instances I and predictions F. While adopting a worst-case perspective for instances I is a standard and reasonable approach in competitive analysis due to future uncertainty, applying this same perspective to the entire prediction space F is unnecessarily conservative, as better prediction-specific performance may be possible. This motivates a more nuanced framework to evaluate performance conditioned on the specific prediction value y ∈F that the algorithm receives. We thus introduce the notions of prediction-specific consistency and prediction-specific robustness. Definition 2 (Prediction-Specific Metrics). Given an online algorithm Aω and a specific prediction value y ∈F, the prediction-specific consistency βy(Aω) and prediction-specific robustness γy(Aω) relative to y are defined as: βy(Aω) := sup I∈Iy ALG(Aω, I, y) OPT(I) , γy(Aω) := sup I∈I ALG(Aω, I, y) OPT(I) . (2) If an algorithm Aω achieves βy(Aω) and γy(Aω) for a given prediction value y, we say Aω is βy(Aω)-consistent and γy(Aω)-robust under y. The prediction-specific metrics in Definition 2 enable a finer-grained analysis for algorithms with predictions. Instead of considering only the worst-case consistency or robustness over all predictions, we may instead tailor an algorithm's strategy based on the characteristics of each observed prediction value y. This adaptiveness can enable better performance compared to standard algorithms, which only optimize for worst-case consistency and robustness (1). 2.2 WEAK and STRONG Optimality While these prediction-specific metrics provide valuable insight into algorithm performance conditioned on the received prediction, the standard metrics of consistency and robustness still remain valuable to characterize an algorithm's overall worst-case guarantees across all instances and predictions. To formally capture the notion of algorithms that perform well both overall and for specific predictions, we introduce two notions of optimality based on these different settings. Our goal is to distinguish algorithms which both achieve the optimal tradeoff between robustness and consistency for the worst-case prediction, as well as on a per-prediction basis. This leads to the following definitions of weakly-optimal and strongly-optimal algorithms. We begin by defining weak optimality, which characterizes algorithms with the optimal tradeoff between the standard notions of consistency and robustness. Definition 3 (WEAK Optimality). For a fixed online problem, consider an online algorithm A that achieves β consistency and γ robustness. A is WEAKLY-optimal if there does not exist another online algorithm with consistency β′ and robustness γ′ such that β′ ≤β and γ′ ≤γ, with at least one inequality being strict. Building upon this, we define the stricter notion of strong optimality incorporating prediction-specific performance. Definition 4 (STRONG Optimality). For a fixed online problem, consider an online algorithm A that achieves βy prediction-specific consistency and γy prediction-specific robustness under prediction y ∈F. A is STRONGLYoptimal if 6 1. A is weakly-optimal; 2. for any prediction y ∈F, there does not exist another online algorithm with prediction-specific consistency β′ y and robustness γ′ y such that β′ y ≤βy and γ′ y ≤γy, with at least one inequality being strict. The notion of strong optimality in Definition 4 extends classic Pareto optimality to the prediction-specific setting. This definition requires Pareto optimality in the two-dimensional consistency-robustness plane (βy, γy) for each specific prediction y ∈F, in addition to the baseline weak optimality. This stricter criterion captures whether an algorithm is unimprovable in its fundamental performance trade-off across the entire prediction space F. 2.3 Online Problems Studied We now briefly introduce each of the problems studied in this work; these problems are summarized in Table 3. Table 3: Instantiation of the general problem components I, x, and y for problems analyzed. Discrete-Time Ski Rental One-Max Search Scheduling Instance I Number of days n values in [L, U] Set of n jobs Feature x Number of days Maximum value Processing times (x1, . . . , xn) Prediction y Predicted number of days Predicted maximum value Predicted times (y1, . . . , yn) Discrete-Time Ski Rental. The discrete-time ski rental problem is a classic online decision-making problem, wherein a decision-maker decides each day whether to rent skis for 1 unit of cost per day or purchase them outright for a fixed cost b ∈N+, without prior knowledge of the total number of skiing days x. The optimal cost is given by min{b, x}. In the standard competitive analysis framework, where no predictions are available, the optimal deterministic algorithm employs a simple rent-or-buy strategy, purchasing the skis on day M = b; this strategy achieves a competitive ratio of 2 -1/b. The optimal randomized algorithm is known as Karlin's algorithm [38], which strategically balances renting and buying through a carefully-designed probability distribution, achieving a competitive ratio of approximately e/(e -1) ≈1.582. For the ski rental problem, Kumar et al. [2] proposed algorithms that trade off consistency (performance under accurate predictions) and robustness (worst-case performance). They presented a deterministic algorithm achieving (1 + λ) consistency and (1 + 1/λ) robustness (for λ ∈(0, 1)), and a randomized algorithm with ( λ 1-e-λ ) consistency and ( 1+1/b 1-e-(λ-1/b) ) robustness (for λ ∈(1/b, 1)). Subsequently, Wei and Zhang [17] established fundamental lower bounds on this trade-off, showing that as b →∞, for deterministic algorithms, any (1+λ)-consistent algorithm must have a robustness of at least (1+1/λ) for λ ∈(0, 1), while for randomized algorithms, any γ-robust algorithm must have a consistency of at least γ log(1 + 1/(γ -1)). These results show that the algorithms in [2] achieve weak optimality in the limit b →∞(see Definition 3). In this work, we provide a deeper analysis of the algorithms proposed by Kumar et al. [2], examining their prediction-specific consistency and robustness. To maintain consistency and comparability with [2, 17], we also consider the asymptotic regime b →∞. Our findings indicate that neither the deterministic nor randomized algorithms in [2] are strongly-optimal within this framework (see Theorems 4.1 and 5.1). Consequently, we propose novel algorithms (see Algorithms 2 and 9) that achieve strong optimality (see Theorems 4.2 and 5.3), thereby improving upon existing approaches for this problem setting with ML predictions. One-Max Search. The one-max search problem considers a sequence of n elements with values in the range [L, U], where L and U are known positive constants. At each step, an element is observed, and the algorithm must decide whether to accept it immediately or irrevocably discard it. The objective is to select the element with the maximum value. The instance's difficulty is characterized by the ratio θ = U/L. In classical competitive analysis, the optimal competitive ratio is √ θ, achieved by an algorithm using the fixed threshold √ LU [39]. Note that this is a reward maximization problem, rather than a loss minimization; thus, for this problem, we will 7 consider definitions of classic (1) and prediction-specific (2) consistency and robustness with the ratio between ALG and OPT inverted. For learning-augmented one-max search, where algorithms receive a prediction of the maximum element value, Sun et al. [11] established a fundamental lower bound on the consistency-robustness trade-off: any γ-robust algorithm must have a consistency of at least θ/γ. They further proposed a threshold-based algorithm that achieves this lower bound, implying its weak optimality. In this work, we provide a deeper analysis of the algorithm proposed by Sun et al. [11] within this predictionspecific framework. Our analysis reveals that their algorithm is not strongly-optimal (see Theorem 6.1). Consequently, we propose a novel algorithm (see Algorithm 4) that is strongly-optimal (see Theorem 6.2) and offers improved performance. Non-Clairvoyant Scheduling. The non-clairvoyant scheduling problem concerns scheduling n jobs on a single machine without prior knowledge of their processing times. We focus on the preemptive setting, where jobs can be interrupted and resumed without cost. In the standard competitive analysis framework [1], Round-Robin (RR) achieves an optimal competitive ratio of 2n n+1 [40]. Extending this, Wei and Zhang [17] established a fundamental lower bound on the consistency-robustness trade-off, showing that any algorithm must have robustness of at least n+n(n-1)λ 1+λ(n+1)(n+2)/2 if it is (1 + λ)-consistent. They also propose a two-stage schedule algorithm (see Algorithm 6 in Section A.1) and show it achieves the tight tradeoff for n = 2 jobs in this learning-augmented setting. In the next section, as a warm-up, we provide a deeper analysis of this two-stage algorithm's prediction-specific consistency and robustness, establishing that it is strongly-optimal. 2.4 Warm-Up: An Existing Algorithm that is Strongly-Optimal While our proposed notion of strong optimality is much stricter than weak optimality, some existing algorithms known only to be weakly-optimal can be shown to satisfy strong optimality. In this section, we consider the problem of non-clairvoyant scheduling with n = 2 jobs (with length predictions y = (y1, y2) with y1 ≤y2). It is well known that the two-stage scheduling algorithm proposed by Wei and Zhang [17] (see Algorithm 6 in Section A.1) achieves the optimal trade-off between classic consistency and robustness and is thus weakly-optimal. Notably, we can show that this algorithm also satisfies prediction-specific Pareto optimality: Theorem 2.1. The two-stage algorithm in [17] with n = 2 is 1 + min{y1/(2y1 + y2), λ}-consistent and 1 + max{1/3, y1/(y1 + 2λ(2y1 + y2))}-robust under y = (y1, y2). Moreover, it is strongly-optimal. We prove this theorem in Section A.2. Despite the fact that this algorithm is strongly optimal for nonclairvoyant scheduling, for many other canonical online problems, including the ski rental and one-max search problems, we shall soon see that existing weakly-optimal algorithms (such as those in [2, 11]) fail to achieve strong optimality (see Theorems 4.1, 5.1, and 6.1). As such, the rest of this paper will consider the development of new algorithms that can achieve strong optimality in these settings. 3 Optimization-Based Meta-Algorithm In this section, we introduce a general optimization-based approach to systematically identify stronglyoptimal algorithms for online problems. Our goal is to obtain a general meta-algorithm that, given a prediction y ∈F and a target robustness upper bound γ, returns an algorithm Aω which is strongly-optimal. 3.1 Bi-Level Optimization Formulation We begin by considering the following question: given some target upper bound γ on robustness and a prediction y ∈F, how can we obtain an algorithm that both satisfies the target robustness bound, and obtains a prediction-specific optimal tradeoff between consistency and robustness? Here, γ ∈Λγ := [α∗, +∞) can be any possible robustness level, where α∗denotes the optimal competitive ratio achievable for the problem. 8 Algorithm 1 BI-LEVEL OPTIMIZATION-BASED META-ALGORITHM 1: Input: Desired robustness upper bound γ ∈Λγ := [α∗, +∞) 2: Receive a prediction y; 3: Compute {β∗ y, ω} by solving P1(γ, y) (Problem 1); 4: Obtain {γ∗ y, ω∗} by solving P2(β∗ y, y) (Problem 2) ; 5: Deploy the online decision rule induced by Aω∗; To achieve this goal, we specify a bi-level optimization framework comprising Problems 1 and 2. Recall that a prediction y determines a consistent subset of instances Iy ⊆I, as described in Section 2, and that ALG(Aω, I, y) and OPT(I) denote, respectively, the costs of the algorithm Aω augmented by the prediction y and optimal offline algorithm under instance I ∈I. Our first optimization problem (Problem 1), referred to as P1(γ, y), determines the best achievable prediction-specific consistency of any γ-robust algorithm under the prediction y ∈F. Let {β∗ y, ω} denote its optimal solution. We then solve a second optimization problem (Problem 2), referred to as P2(β∗ y, y), to find the most robust algorithm among those achieving β∗ y consistency under the prediction y. Let {γ∗ y, ω∗} denote its optimal solution. The resulting algorithm Aω∗with the optimal solution ω∗is then implemented for the prediction y. Problem 1 (P1): Minimize Consistency P1(γ, y) := min βy,ω∈Ω βy s.t. (3a) ALG(Aω, I, y) ≤γ OPT(I), ∀I ∈I, (3b) ALG(Aω, I, y) ≤βyOPT(I), ∀I ∈Iy. (3c) Constraint (3b) enforces the initial robustness level γ. Problem 2 (P2): Minimize Robustness P2(β∗ y, y) := min γy,ω∈Ω γy s.t. (4a) ALG(Aω, I, y) ≤γyOPT(I), ∀I ∈I, (4b) ALG(Aω, I, y) ≤β∗ yOPT(I), ∀I ∈Iy. (4c) Constraint (4c) enforces the optimal predictionspecific consistency β∗ y found in Problem 1. 3.2 Meta-Algorithm Formulation The preceding approach (Problems 1 and 2) focuses on deriving an instance-specific solution given a fixed prediction y. To operationalize this idea in the online setting, where predictions may vary, we introduce a metaalgorithm (Algorithm 1) that invokes this two-stage optimization procedure for any realized prediction. This enables prediction-specific robust and consistent decision-making across varying inputs, with a user-specified robustness bound γ. In the following proposition, which we prove in Section B, we show that this meta-algorithm yields decisions on the prediction-specific Pareto frontier and satisfying the desired robustness bound. Moreover, the condition that γ is on the (non-prediction-specific) Pareto frontier guarantees the weak optimality of Algorithm 1; this holds if tight Pareto fronts are available off-the-shelf for the problem at hand, such as the tight tradeoffs between classical consistency and robustness available for ski rental and one-max search [2, 11, 17]. Proposition 1. Suppose there exists a weakly-optimal algorithm with robustness γ. With ω∗being an optimal solution of Problem 2 in line 4 of Algorithm 1, Aω∗is β∗ y-consistent and γ∗ y-robust with respect to the prediction y, with γ∗ y ≤γ. Furthermore, Algorithm 1 is strongly-optimal. Notably, the meta-algorithm (Algorithm 1) offers a systematic pipeline to achieve Pareto optimality for each prediction y ∈F while also guaranteeing weak optimality. While the tractability of solving the constituent bi-level optimization problems P1(γ, y) and P2(β∗ y, y) depends on the structure of the specific online problem, this framework provides a foundation for deriving explicit and tractable strongly-optimal algorithms for the problems we discuss in the remainder of this paper. 9 3.3 Generalizations of this framework We briefly note that the bi-level optimization framework we have proposed is quite flexible, and could be generalized in a number of ways to enable its application to different problem settings or objectives. For example, Problems 1 and 2 could be augmented to include practical considerations such as risk- or uncertainty-sensitive constraints and objectives [36, 41, 42] or tolerance to erroneous predictions. In particular, in this work, we develop an error-tolerant variant of this framework, motivated by the continuous nature of certain problems (e.g., prices in the one-max search problem) and the fact that prediction errors are often unavoidable in practice. Instead of analyzing the tradeoff between consistency and robustness under perfectly accurate predictions, we can instead study the tradeoff between ε-consistency and robustness, where ε-consistency denotes the worst-case guarantee when the prediction error is bounded by a chosen constant ε. We further describe this generalized framework, and how to design algorithms in this setting, in Section 7. While our approach could, in principle, be extended to more general and complex dynamic multi-round problems, we anticipate that such an extension would likely require nontrivial technical developments. In particular, such an extension would require a number of additional structural assumptions, e.g., that the overall prediction space F remains fixed and independent of the algorithm's actions, and that the algorithm's actions exert negligible influence on future prediction values. Moreover, depending on the problem, the resulting optimization problems may be considerably harder to analyze. For complex problems like metrical task systems and non-clairvoyant scheduling with n jobs, achieving even weak optimality remains an open question [17, 33]; as such, designing strongly-optimal algorithms is inherently challenging. Thus, while our methodology in its current form may not be directly applicable to these settings, we posit that it may still serve as a useful conceptual framework for designing prediction-specific algorithms for these problems. In the rest of this paper, we will consider problem settings where this methodology can be tractably instantiated. 4 Deterministic Ski Rental We begin our investigation of strongly-optimal algorithm design in specific problem settings by considering deterministic algorithms for the discrete-time ski rental problem described in Section 2.3. If a deterministic decision-maker buys skis at the start of day M ∈N (that may depend on the predicted day y), then the induced cost is ALGDSR(M, x, y) := x · 1M(y)>x + (b + M(y) -1) · 1M(y)≤x, where b ∈N denotes the price of the skis. Then, for this problem, the general definitions of the prediction-specific consistency and robustness in Equation (2) are instantiated as: βDSR y (AM(y)) := ALGDSR(M, y, y) min{b, y} , γDSR y (AM(y)) := sup x∈N ALGDSR(M, x, y) min{b, x} . (5) 4.1 The Deterministic Algorithm of Kumar et al. is Not Strongly-Optimal We first provide a deeper analysis of the deterministic algorithm proposed by Kumar et al. [2], examining its prediction-specific consistency and robustness. The deterministic algorithm of Kumar et al. [2, Algorithm 2], which we denote by KD, purchases at the beginning of day ⌈λb⌉if y ≥b, and at the beginning of day ⌈b/λ⌉otherwise. It achieves (1+λ)-consistency and (1 + 1/λ)-robustness, where λ ∈(0, 1) is a tunable hyper-parameter. By the lower bound of Wei and Zhang [17], KD is weakly-optimal as the price b →+∞. However, as we show in the following theorem (which is proved in Section C.1), the KD algorithm is not strongly-optimal. Theorem 4.1. KD is 1-consistent and (1+1/λ)-robust when y y. γy(M) = ( M-1+b M if M ≤b, M-1+b b if M > b. It's clear that the decision M = b simultaneously achieves the optimal consistency 1 and the optimal robustness 2 -1 b with respect to y, thus dominating any other decisions. Case II: y ≥b. Given the decision M, the prediction-specific consistency and robustness are βy(M) = ( M-1+b b if M ≤y, y b if M > y. γy(M) = ( M-1+b M if M ≤b, M-1+b b if M > b. As shown in Figure 2 and 3, we observe that choosing M = y + 1 dominates all options with M > y + 1, as it offers better robustness without compromising consistency, making M = y + 1 a strong choice. Note that βy(y + 1) = y/b and γy(y + 1) = (y + b)/b. Therefore, two critical decision boundaries within (0, b) are M1 = y + 1 -b and M2 = (b2 -b)/y.2 Specifically, when M1 min {b(λ + 1) -1, (b -1)/λ}, ⌈λb⌉remains on the Pareto front. All together, these cases motivate the design of our deterministic algorithm, PDSR (Algorithm 2). In the following theorem, we characterize the prediction-specific consistency and robustness of PDSR, establishing its strong optimality in the asymptotic regime b →+∞; the full proof is deferred to Section C.2. 2Note that b2-b y may not be an integer. 11 Algorithm 2 PDSR: PREDICTION-SPECIFIC DETERMINISTIC SKI RENTAL 1: Input: λ ∈(0, 1) 2: If y min{b(λ + 1) -1, b-1 λ } then determine M = ⌈λb⌉; 5: Buy skis on day M. Theorem 4.2. PDSR presented in Algorithm 2 is strongly-optimal when b →+∞, and is 1-consistent and (2 -1 b)-robust if y min b(λ + 1) -1, b-1 λ . 5 Randomized Ski Rental We now turn to considering the design of randomized algorithms for the discrete-time ski rental problem. Consider a randomized algorithm π = (πi)i∈N+ (that may depend on the predicted day y) that chooses to buy skis at the start of day i ∈N+ with some probability πi. The algorithm's average cost is ALGRSR(π, x, y) := +∞ X i=1 πi(y) · [x · 1i>x + (b + i -1) · 1i≤x], where b ∈N denotes the price of the skis. For this problem, the prediction-specific consistency and robustness in Equation (2) are instantiated as: βRSR y (Aπ(y)) := ALGRSR(π, y, y) min{b, y} , γRSR y (Aπ(y)) := sup x∈N ALGRSR(π, x, y) min{b, x} . (6) 5.1 The Randomized Algorithm of Kumar et al. is Not Strongly-Optimal Kumar et al. [2] propose a randomized ski rental algorithm, which we denote by KR, which is a variant of Karlin's classical randomized strategy [38]. Given a prediction y and a hyper-parameter λ ∈(1/b, 1), KR chooses the purchase day i according to the distribution πi = b-1 b m-i · 1 b 1- 1-1 b m , if i ≤m, 0, if i > m, (7) where m := ( ⌈b/λ⌉, if y x. For this problem, the prediction-specific consistency and robustness are: βOMS y (AΦ(y)) := y ALGOMS(Φ, y, y), γOMS y (AΦ(y)) := sup x∈[L,U] x ALGOMS(Φ, x, y). (10) 6.1 The Algorithm of Sun et al. is Not Strongly-Optimal Sun et al. [11] proposed a β-consistent and γ-robust Online Threshold-based Algorithm (OTA) using the threshold Φ = Lβ, if y ∈[L, Lβ); λLγ + (1 -λ)y/β if y ∈[Lβ, Lγ); Lγ if y ∈[Lγ, U], where β = 2λθ/[ p (1 -λ)2 + 4λθ -(1 -λ)], γ = θ/β, and λ ∈[0, 1]. Furthermore, they show that any γ-robust deterministic algorithm for one-max search must have consistency β ≥θ/γ, implying their algorithm is weakly-optimal. However, as we establish in the following theorem (which is proved in Section F.1), their algorithm is not strongly-optimal. Theorem 6.1. Sun's algorithm is not strongly-optimal. 6.2 A Strongly-Optimal Algorithm for One-Max Search We propose in Algorithm 4 a new approach, PST, which, by more carefully selecting the purchase threshold, achieves strong optimality. In particular, it achieves the prediction-specific consistency and robustness values established in the following theorem. Theorem 6.2. PST (Algorithm 4) is strongly-optimal, and is (y/L)-consistent and √ θ-robust if y ∈[L, λL + (1 -λ) √ LU); 1-consistent and (U/y)-robust if y ∈[λL + (1 -λ) √ LU, √ LU]; (1-λ) √ θy+λy (1-λ)U+λy -consistent and (1-λ)U+λy (1-λ) √ LU+λL -robust if y ∈( √ LU, U]. 14 Algorithm 5 ε-Tolerant PST 1: Input: λ ∈[0, 1], ε > 0 2: Determine M = λ(L + 3ε) + (1 -λ)( √ LU -ε); 3: If y ∈[L, M -2ε] then set Φ = √ LU; 4: Else if y ∈(M -2ε, M) set Φ = M -ε; 5: Else if y ∈[M, √ LU + ε] then set Φ = y -ε; 6: Else if y ∈( √ LU + ε, U -ε) then set μ = (U-2ε)-LU/(M-ε) (U-2ε)- √ LU , Φ = μ √ LU + (1 -μ)(y -ε); 7: Else if y ∈[U -ε, U] then set Φ = LU/(M -ε); 8: Perform OTA with threshold Φ. We prove Theorem 6.2 in Section F.2; the proof identifies prediction-specific optimal thresholds (Φ) by partitioning the prediction space and deriving each threshold (which is a convex combination involving μ) that ensures Pareto-optimality for each segment of predictions. 7 Error-Tolerant Algorithms Pareto-optimal algorithms can exhibit brittleness, a vulnerability noted by [18, 19], where the competitive ratio degrades sharply toward the worst-case robustness bound even with small prediction errors. This issue stems from the standard definition of consistency (see Definition 1), which assumes strictly perfect predictions (x(I) = y) and thus fails to consider performance under erroneous predictions. To address this, we use the one-max search problem as an example to demonstrate how explicit and tractable Pareto-optimal algorithms incorporating a \"generalized consistency\" can be constructed in our prediction-specific framework, offering good performance tradeoffs when faced with minor prediction errors. It is worth emphasizing that the notion of \"error tolerance\" considered in this section differs from the concept of \"smoothness\" discussed in [18-20]. The former concerns the tradeoff between generalized consistency (allowing small prediction errors) and robustness (with arbitrarily large errors), whereas the latter requires that the algorithm's performance degrade smoothly as the prediction error increases. Nevertheless, we view the two notions as related, both contributing to alleviating \"brittleness\". To account for small prediction errors, we specify a desired error tolerance ε and define a relaxed consistent set Iε y := {I ∈I : L(x(I), y) ≤ε} , where L : F × F →R≥0 is a chosen loss function that measures prediction error. When substituting Iy with Iε y, we refer to the corresponding consistency metrics in Equation (1) and Equation (2) as ε-consistency and prediction-specific ε-consistency, denoted by βε and βε y, respectively. Substituting β, βy and Iy with βε, βε y and Iε y in Section 3, we can generalize our meta-algorithm to incorporate error tolerance. For the one-max search problem, its inherent continuity renders prediction errors unavoidable in practice, underscoring the necessity of incorporating error tolerance. We fix L(x, y) = |x -y| and assume that ε is small relative to the scale of the problem. We propose an error-tolerant algorithm for this problem, ε-tolerant PST, in Algorithm 5. To conclude the section, we present three theorems that characterize the performance of this algorithm. In particular, Theorem 7.1 establishes the tradeoffs between ε-consistency and robustness achieved by ε-tolerant PST, together with their prediction-specific counterparts, while Theorem 7.2 provides a lower bound on the global ε-consistency-robustness tradeoff. Finally, Theorem 7.3 shows that the ε-tolerant PST algorithm is strongly optimal. The proofs of all three results are given in Section G. Theorem 7.1. ε-Tolerant PST achieves [(M -ε)/L] ε-consistency and U/(M -ε) robustness. Specifically, 15 ε-Tolerant PST achieves [(y + ε)/L] ε-consistency and √ θ robustness if y ∈[L, M -2ε]; (M -ε)/L ε-consistency and U/(M -ε) robustness if y ∈(M -2ε, M); (y + ε)/(y -ε) ε-consistency and U/(y -ε) robustness if y ∈[M, √ LU + ε]; (y+ε) μ √ LU+(1-μ)(y-ε) ε-consistency and μ √ LU+(1-μ)(y-ε) L robustness if y ∈( √ LU + ε, U -ε); (M -ε)/L ε-consistency and U/(M -ε) robustness if y ∈[U -ε, U]; where M = λ(L + 3ε) + (1 -λ)( √ LU -ε). Theorem 7.2. Any γ-robust algorithm has at least (θ/γ) ε-consistency, and any algorithm that achieves βε ε-consistency must be at least (θ/βε)-robust. Theorem 7.3. Assume ε ≤( √ LU -L)/4. The ε-consistency and robustness of ε-Tolerant PST (Algorithm 5) are jointly Pareto optimal. Moreover, for every prediction y ∈F = [L, U], ε-Tolerant PST achieves predictionspecific ε-consistency βε y and robustness γy that are jointly Pareto optimal. 8 Numerical Experiments In this section, we evaluate the performance of our proposed algorithms in three different case studies spanning synthetic and real-world settings.3 8.1 Case Study 1: Synthetic Data Experiments for Ski Rental We begin by testing the performance of our algorithms for the ski rental problem via simulations on synthetic instances. We let the actual number of skiing days x be a uniformly random integer drawn from [1, 10b], where b = 100 is buying costs of skis. The prediction y is generated with accuracy p: with probability p, the prediction is accurate (i.e., y = x), and with probability (1 -p), the prediction y ∼N(x, σ2), where σ = 500 and the output is rounded and made positive. For the deterministic setting, we compare PDSR (Algorithm 2) with KD ([2], Algorithm 2) and the classic competitive algorithm (which always buys on day b), using the same parameter λ = 0.5 for both PDSR and KD. For the randomized setting, we compare PRSR (Algorithm 9) with KR ([2], Algorithm 3) and Karlin's algorithm [38], using γ = 3 for PRSR and λ = ln(3/2) for KR, ensuring PRSR and KR have the same robustness 3. Each setup is evaluated over 10000 independent trials. Figures 4 and 5 present the empirical results of average competitive ratio versus accuracy p. We observe that our proposed algorithms, PDSR and PRSR, consistently outperform both classic online algorithms and existing learning-augmented algorithms across both settings. 8.2 Case Study 2: Ski Rental on Dynamic Power Management We next evaluate our ski rental algorithms on real-world traces for a Dynamic Power Management (DPM) problem, where we control the idle and active periods of a computing device. Modern processors typically support multiple power states: deeper states disable more components, leading to lower operating cost/energy but higher wake-up penalties/overhead. During each idle interval, a DPM controller must decide whether to stay active or transition into a deeper sleep state without knowing the remaining idle duration. 3Our code is publicly available at https://github.com/Bill-SizheLi/Prediction_Specific_Design_of_ Learning-Augmented_Algorithms 16 0.0 0.2 0.4 0.6 0.8 1.0 Accuracy of ML Predictor 1.5 1.6 1.7 1.8 1.9 Empirical Competitive Ratio PDSR (Ours) KD Competitive Figure 4: Empirical competitive ratios versus accuracy p in deterministic setting. 0.0 0.2 0.4 0.6 0.8 1.0 Accuracy of ML Predictor 1.20 1.25 1.30 1.35 1.40 1.45 1.50 1.55 Empirical Competitive Ratio PRSR (Ours) KR Competitive Figure 5: Empirical competitive ratios versus accuracy p in randomized setting. log126_copyDtoH log176_Booting log225_MusicFaceBook log225_MusicTwitter log245_MusicFaceBook log126_copyDtoH log176_Booting log225_MusicFaceBook log225_MusicTwitter log245_MusicFaceBook 1.0 1.5 2.0 2.5 3.0 3.5 Empirical ratio Good predictions Bad predictions ( e e 1)-competitive Blindly Trust KR PRSR(Ours) Figure 6: Empirical competitive ratios on DPM traces for good and bad predictions. The two-state DPM system (one active and one sleep state with zero operating cost) is equivalent to the ski rental problem, where remaining active corresponds to renting and transitioning to the sleep state corresponds to buying. Moreover, Antoniadis et al. [5] demonstrated that randomized ski rental algorithms can be converted to multi-state DPM algorithms. Setup. We consider a DPM problem with 4 states. Specifically, we use the same problem setting as Antoniadis et al. [5], employing I/O traces4 collected from a Nexus 5 smartphone [43], from which idle intervals between consecutive requests are extracted. We adopt the IBM mobile hard-drive power states reported in [44], consistent with the setup in [5]. The idle periods are scaled in the same way as in [5]. We use the five largest traces for evaluation. Since the main goal of this section is to probe the algorithms' performance under the two extremes of very good and very bad predictions, we consider the following method for generating predictions: \"good predictions\" and \"bad predictions\" are obtained by perturbing the ground truth with N(0, σ2 good) and N(0, σ2 bad) noises, respectively. In this experiment, we set σgood = 0.02 and σbad = 20. We compare four algorithms: the classic ( e e-1)-competitive algorithm, Blindly Trust (which treats the prediction as if it is correct and optimizes accordingly), the randomized algorithm of Kumar et al. (KR), and our randomized algorithm (PRSR). For the learning-augmented algorithms, we use the same parameter values for λ and γ as in Case Study 1. Results. Figure 6 reports the empirical competitive ratios on the real DPM traces. We observe that our strongly-optimal algorithm PRSR consistently achieves the lowest competitive ratios, except for when 4The traces are available at http://iotta.snia.org/traces/block-io. 17 the prediction quality is very good (in which case, the non-robust Blindly Trust algorithm outperforms it. This validates our algorithm's ability to exploit specific predictions to enable good performance regardless of prediction value or quality. 8.3 Case Study 3: One-Max Search on VIX Trading Figure 7: VIX Closing Price from January 2020 to December 2024. The VIX index soared from a pre-pandemic level of 12-15 to 83 in March 2020. The VIX, often referred to as the fear index, exhibits sharp volatility spikes that make it a natural benchmark for evaluating online search algorithms. Its uncertainty and heavy-tailed dynamics closely mirror the one-max search setting, where the core challenge lies in optimally timing a single exit. This characterization is vividly illustrated by the extreme volatility in early 2020: within just a few weeks at the onset of the COVID-19 market shock, the VIX surged from below 80 (shown in Figure 7). Setup. We evaluate our one-max search algorithms in a case study using the daily closing prices of the VIX index from January 2020 to December 2024 (shown in Figure 7), which consist of publicly available values obtained from the Cboe Options Exchange. We assume that at the beginning of each month, an agent holds one unit of VIX and must choose a single day within that month to sell it. Over the course of five years, there are 60 trading rounds (one per month), each offering approximately 20 to 21 trading opportunities, as the VIX is only traded on weekdays. We set L and U as the historical minimum and maximum prices over the entire 5 years.5 Baselines. We compare our proposed methods, PST(Algorithm 4) and ε-Tolerant PST(Algorithm 5), to three baseline algorithms: (i) blindly trusting the prediction, (ii) the classical online algorithm of El-Yaniv [39], and (iii) prior learning-augmented algorithms (Sun's [11] and Benomar's [19]). Experiment 1. In this experiment, we consider a naive prediction strategy that simply uses the highest observed VIX price from the previous month. As the evaluation metric, we use the empirical ratio6, defined as the cumulative online outcome up to the current round divided by the cumulative offline optimum. This metric reflects how well an algorithm performs in practice relative to the hindsight-optimal strategy, averaged over time. We run the algorithms over the 60-month horizon using historical VIX data, and report the empirical ratios at each round to visualize both long-term trends and the stability of performance across different market periods in Figure 8. For our proposed algorithms, we fix the trade-off parameter λ = 0.3 in PST, and use λ = 0.3 and ε = 1.8 in ε-Tolerant PST. For baseline algorithms with tunable parameters, including those from Sun et al. [11] and Benomar et al. [19], we find that setting λ = 1.0 yields the best cumulative empirical ratio over the full 60-month horizon. However, to better illustrate performance variation across different regimes, we also include their results under λ = 0.3 and λ = 0.6. Experiment 2. In practical settings, machine-learned predictions are often more accurate than the naive predictor used in Experiment 1, though they remain imperfect due to model limitations and data noise. The degree of prediction accuracy varies with the capability and training of the underlying ML model. To systematically evaluate algorithmic performance under varying prediction quality, we introduce the notion of an error level - a scalar value between 0 and 1 that quantifies the deviation from perfect information. For each trading round, the prediction is constructed via linear interpolation between the previous month's maximum (naive prediction) 5The focus of this paper is not on the impact of L and U; therefore, we simply set them to historical values. In practical trading scenarios, L and U can be viewed as predetermined parameters representing the stop-loss and take-profit thresholds in the exit strategy of the trading process. 6Note that the empirical ratio here is the inverse of that used in the theoretical analysis, so as to better reflect the proportion of the hindsight optimum that the online or learning-augmented algorithm can achieve (or recover). 18 0 10 20 30 40 50 60 Round 0.60 0.65 0.70 0.75 0.80 0.85 0.90 0.95 Empirical Ratio Blindly Trust El-Yaniv's (Sun's ( =1) & Benomar's( =1)) Sun's ( =0.6) Sun's ( =0.3) Benomar's ( =0.6) Benomar's ( =0.3) PST (Ours) -Tolerant PST (Ours) Figure 8: Empirical competitive ratios over 60 trading rounds. 0.0 0.2 0.4 0.6 0.8 1.0 Error Level 0.800 0.825 0.850 0.875 0.900 0.925 0.950 0.975 1.000 Cumulative Empirical Ratio Blindly Trust El-Yaniv's Sun's Benomar's PST (Ours) -Tolerant PST (Ours) Figure 9: Cumulative empirical competitive ratio with varying prediction error levels. and the current month's actual maximum (perfect prediction), where the error level determines the interpolation weight. An error level of 1.0 corresponds to the naive prediction, while 0.0 yields the perfect prediction. We assess the cumulative empirical ratio of each algorithm over all 60 trading rounds under varying error levels from 0 to 1, and report the result in Figure 9. To ensure a fair comparison across prediction regimes, we fix the trade-off parameter λ = 0.5 for all tunable methods, including those of Sun, Benomar, PST, and ε-Tolerant PST. For ε-Tolerant PST, we additionally set ε = 0.5 to account for moderate tolerance to prediction error. Results. The results for Experiment 1 are shown in Figure 8. Our methods consistently outperform all baselines across the decision horizon, with final empirical competitive ratios of 87.2% (ε-Tolerant PST) and 85.7% (PST), compared to 82.4%-84.4% for all baselines. The results for Experiment 2 are shown in Figure 9; ε-Tolerant PST remains consistently superior across nearly all error levels. 9 Concluding Remarks In this work, we introduce a prediction-specific analysis framework and a finer-grained notion of strong optimality for online algorithms with predictions. We further provide a systematic approach to designing Pareto-optimal online algorithms with better prediction-specific performance than prior algorithms, and we show how this methodology can yield significant performance improvements for the problems of ski rental (deterministic and randomized) and one-max search. Future Directions. In contrast to the ski rental and one-max search settings, the existing weakly-optimal non-clairvoyant scheduling algorithm of Wei and Zhang [17] is strongly-optimal when n = 2. Thus, designing a strongly-optimal algorithm for n-job non-clairvoyant scheduling remains an open question. 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Online strategies for dynamic power management in systems with multiple power-saving states. ACM Transactions on Embedded Computing Systems, 2(3):325-346, 2003. https://dl.acm.org/doi/10.1145/860176.860180. 23 A Additional Details and Proofs for Section 2.4 This section supplements the analysis of the non-clairvoyant scheduling problem for the special case n = 2, as introduced in Section 2.3. We focus on the two-stage scheduling algorithm proposed by Wei and Zhang [17], which is known to achieve the optimal classic trade-off and is weakly-optimal-optimal under Definition 3. As detailed in Algorithm 6 in Section A.1, the algorithm proceeds in two phases based on predicted job lengths y = (y1, y2), where y1 ≤y2. In Theorem 2.1, we show that this algorithm also satisfies prediction-specific Pareto optimality under n = 2. A.1 Wei et al.'s Two-Stage Schedule Wei and Zhang [17] propose an algorithm called the two-stage schedule (see Algorithm 6) that achieves a consistency of (1 + λ) and a robustness of (1 + 1 1+6λ) under n = 2. By [17] and Definition 3, this two-stage schedule algorithm is weakly-optimal for n = 2. Algorithm 6 Two-Stage Schedule 1: At any point, if a job finishes with processing time less or more than its prediction, use round robin forever. 2: Stage 1: Round robin for at most λn · OPTy/ n 2 units of time. 3: Stage 2: Process jobs in predicted order (starting from the unfinished job with the least predicted time). A.2 Proof of Theorem 2.1 In this subsection, we further analyze the prediction-specific consistency and robustness and prove that two-stage schedule is strongly-optimal under n = 2. Proof of Theorem 2.1. Note that the jobs are ranked based on their predicted lengths; thus we have y1 ≤y2. We first prove the prediction-specific consistency and robustness under a specific set of predictions y = (y1, y2). We first consider λ ≤ y1 2y1+y2 , i.e. λ(2y1 + y2) ≤y1. Regarding the consistency, assume that x1 = y1, x2 = y2. In stage 1, the algorithm runs round-robin for 2λ · (2y1 + y2) time. Since λ · (2y1 + y2) ≤y1, job 1 cannot finish in stage 1. Therefore, the completion time of job 1 is 2λ(2y1 + y2) + y1 -λ(2y1 + y2) = y1 + λ(2y1 + y2), and that of job 2 is y1 + λ(2y1 + y2) + y2 -λ(2y1 + y2) = y1 + y2, thus, ALG = 2y1 + y2 + λ(2y1 + y2), and OPT = 2y1 + y2, yeilding a consistency of 1 + λ. Regarding the robustness, we consider an adversarial attack x1, x2. Let δ denote an infinitesimal quantity. Case I: If x1 ≤λ(2y1 + y2) or x2 ≤λ(2y1 + y2) , i.e. some incorrect prediction is found in stage 1. In this case, the algorithm runs round-robin from beginning to end, resulting in a robustness of at most 4/3. Case II: If λ(2y1 + y2) x1 In this case, OPT = 2x1 + x2. This results in a robustness of 1 + λ(2y1+y2) 3y1 ≤4/3, which is achieved when x1 = y1 and x2 = y1 + δ. Case III: If x1 > y1 , i.e. job 1 finishes later than its prediction. In this case, the algorithm first runs round-robin for 2λ · (2y1 + y2) time, then processes job 1 for y1 -λ(2y1 + y2) time, and finally runs round-robin till the end. Case III(a) λ(2y1 + y2) y1/(2y1 + y2), the algorithm runs round-robin forever, and is (1 + y1 2y1+y2 )-consistent and (4/3)-robust. In conclusion, given prediction y = (y1, y2), the prediction-specific consistency and robustness are βy = 1 + min{ y1 2y1 + y2 , λ}, γy = 1 + max{1/3, y1 y1 + 2λ(2y1 + y2)}. Since the two-stage schedule is already proven to be weakly-optimal, we then prove that it is strongly-optimal by demonstrating the Pareto optimality of their prediction-specific consistency and robustness. When λ ≤y1/(2y1 + y2), the algorithm is (1 + λ)-consistent and 1 + y1 y1+2λ(2y1+y2) -robust. Consider algorithm Aω that completes r > λ(2y1 + y2) amount of work for job 2 when it finishes job 1 in the case where predictions are accurate. Then, it follows that ALG = 2 · (y1 + r) + y2 -r = 2y1 + y2 + r OPT = 2y1 + y2, which yields a competitive ratio of 1 + r/(2y1 + y2) > 1 + λ. Therefore, any (1 + λ)-consistent algorithm processes at most λ(2y1 +y2) amount of work for job 2 when it finishes job 1 or finds any incorrect prediction of job 1. Then, we consider an incorrect prediction x1 = y1, x2 = λ(2y1 + y2) + δ. Consider a (1 + λ)-consistent algorithm B that completes r ≤λ(2y1 + y2) amount of work when it finishes job 1. Then, upon x1 = y1 and x2 = r + δ, ALG = 2(y1 + r) + (r + δ -r) = 2y1 + 2r + δ and OPT = 2r + y1. This leads to a robustness no better than min r≤λ(2y1+y2) 2y1 + 2r y1 + 2r = 1 + y1 y1 + 2λ(2y1 + y2) for all r ≤λ(2y1 + y2). When λ > y1/(2y1 + y2), the algorithm, equivalent to round-robin (RR), is (1 + y1/(2y1 + y2))-consistent and 4/3-robust. Note that RR is the only algorithm that achieves 4/3 competitive ratio. Therefore, under prediction y = (y1, y2), no other algorithm achieves a robustness equal to or less than 4/3. By Definition 4, the two-stage schedule is strongly-optimal. 25 B Additional Details for Section 3 In this section, we provide a proof of Proposition 1 and discuss the scenario in which a weakly-optimal algorithm with robustness γ is unavailable. B.1 Proof of Proposition 1 Proof of Proposition 1. Let {β∗ y, ω} and {γ∗ y, ω∗} denote the optimal solution to Problem 1 and Problem 2, respectively. Since γ ≥α∗, {γ, ω} is always a feasible solution to Problem 2. Thus, γ∗ y ≤γ, where γ∗ y is the optimal objective value to Problem 2. Since {γ∗ y, ω∗} is a feasible solution to Problem 2, we have ALG(Aω∗, I, y) ≤γ∗ y · OPT(I), ∀I ∈I, ALG(Aω∗, I, y) ≤β∗ y · OPT(I), ∀I ∈Iy. By Definition 2, Aω∗is β∗ y-consistent and γ∗ y-robust with respect to y. Consider ω′ ∈Ω. Let β′ y and γ′ y denote the prediction-specific consistency and robustness of Aω′ with respect to y. We consider two cases. Case I: β′ y > β∗ y If β′ y > β∗ y, then (β′ y, γ′ y) produces no Pareto improvement over (β∗ y, γ∗ y). Case II: β′ y ≤β∗ y . If β′ y ≤β∗ y, since Aω′ is β′ y-consistent and γ′ y-robust with respect to y, by Definition 2, ALG(Aω′, I, y) ≤β′ y · OPT(I), ∀I ∈Iy, (11a) ALG(Aω′, I, y) ≤γ′ y · OPT(I), ∀I ∈I. (11b) Since β′ y ≤β∗ y, by Inequality (11a), we further have ALG(Aω′, I, y) ≤β∗ y · OPT(I), ∀I ∈Iy. (12) By Inequalities (11b) and (12), {γ′ y, ω′} is a feasible solution to Problem 2, thus we have γ∗ y ≤γ′ y, where γ∗ y is the optimal objective value to Problem 2. Case II (a): γ′ y > γ∗ y If γ′ y > γ∗ y, then (β′ y, γ′ y) produces no Pareto improvement over (β∗ y, γ∗ y). Case II (b): γ′ y = γ∗ y If γ′ y = γ∗ y, since γ∗ y ≤γ, we have γ′ y ≤γ. By Inequality (11b), we further have ALG(Aω′, I, y) ≤γ · OPT(I), ∀I ∈I. (13) By Inequalities (11a) and (13), {β′ y, ω′} is a feasible solution to Problem 1, thus we have β∗ y ≤β′ y, where β∗ y is the optimal objective value to problem 1. Because β′ y ≤β∗ y and β∗ y ≤β′ y, we have β′ y = β∗ y. Since γ′ y = γ∗ y and β′ y = β∗ y, (β′ y, γ′ y) produces no Pareto improvement over (β∗ y, γ∗ y). Consequently, (β∗ y, γ∗ y) are jointly Pareto optimal in terms of the prediction-specific consistency and robustness with respect to y. Since γ∗ y ≤γ for all y ∈F, we have supy∈F γ∗ y ≤γ. By Equation (1) and Equation (2), Algorithm 1 is γ-robust. Now, there exists a weakly-optimal algorithm Aω with robustness γ, we assume the consistency of Aω is β and the prediction-specific consistency of Aω under prediction y is βy. Since Aω is βy-consistent with respect to y, by Definition 2, ALG(Aω, I, y) ≤βy · OPT(I), ∀I ∈Iy. (14) Since Aω is γ-robust, it is also γ-robust with respect to y, thus we have ALG(Aω, I, y) ≤γ · OPT(I), ∀I ∈Iy. (15) 26 By Inequalities (14) and (15), {βy, ω} is a feasible solution to Problem 1. Therefore, β∗ y ≤βy, where β∗ y is the optimal objective value to Problem 1. Consequently, sup y∈F β∗ y ≤sup y∈F βy = β. By Equation (1) and Equation (2), Algorithm 1 is β-consistent. By weak optimality of Aω, any γ-robust algorithm is at least β-consistent, and any β-consistent algorithm is at least γ-robust. Since Algorithm 1 is β-consistent and γ-robust, by Definition 3, it is weakly-optimal. Moreover, ∀y ∈F, (β∗ y, γ∗ y) are jointly Pareto optimal in terms of the prediction-specific consistency and robustness with respect to y. Thus, Algorithm 1 is strongly-optimal by Definition 4. B.2 Addressing the Absence of a Weakly-Optimal Algorithm with γ Robustness. In general, even if (β′, γ) is not on the Pareto front for any β′ ≥1, a process that first determines a tight consistency bound β = supy′∈F P1(γ, y′), then determines a tight robustness bound γ = supy′∈F P2(β, y′) can generate a tight Pareto-optimal consistency-robustness tradeoff (β, γ) so that γ becomes a valid input of Algorithm 1. Let β, γ denote supy′∈F P1(γ, y′), supy′∈F P2(β, y′), respectively. Define y1 := arg maxy′∈F P1(γ, y′), y2 := arg maxy′∈F P2(β, y′). Since β = supy′∈F P1(γ, y′), we have ∀y′ ∈F, P1(γ, y′) ≤β. Therefore, ∀y′ ∈F, ∃ω′ ∈Ω, s.t. {γ, ω′} is a feasible solution to P2(β, y′), i.e., ∀y′ ∈F, γ ≥P2(β, y′), where P2(β, y′) is the optimal objective value. Consequently, γ ≥sup y′∈F P2(β, y′) = γ. (16) We can use similar techniques to prove β ≥sup y′∈F P1(γ, y′). Since β = supy′∈F P1(γ, y′), any γ-robust algorithm is at least β-consistent. We prove this by contradiction, assuming there exists a γ-robust algorithm A that has consistency βA b -1 + b b = 2 -1/b for all λ ∈(0, 1), by Definition 4, KD is not strong-optimal. C.2 Proof of Theorem 4.2 Proof of Theorem 4.2. Denote the prediction-specific consistency and robustness of Algorithm 2 with respect to y as βy and γy. We start by analyzing the prediction-specific consistency and robustness of Algorithm 2 by considering the following three different cases. Case I: y min{b(λ + 1) -1, (b -1)/λ} . In this case, Algorithm 2 buys on day ⌈λb⌉. To obtain the prediction-specific consistency, we assume x = y. ALG = ⌈λb⌉-1 + b, OPT = b, βy = ALG/OPT min{b(λ+1)-1, (b-1)/λ}. Based on the lower bound by Wei and Zhang [17], Algorithm 2 is weakly-optimal, as b →∞. 28 We consider the following three cases to prove the Pareto optimality of (βy, γy) when b →∞. Case I: y [(b2 -b)/y] -1 + b [(b2 -b)/y] = 1 + y/b = γy. Case II (b) : p = (b2 -b)/y (This case applies only if b2-b y ∈N+). In this case, γP y = p-1+b p = 1+y/b = γy. By Inequality (17), we have the following holds: βP y = p -1 + b b = [(b2 -b)/y] + (b -1) b ≥[y + 1 -b] + (b -1) b = y/b = βy. Case II (c) : (b2 -b)/y [(b2 -b)/y] -1 + b b ≥[y + 1 -b] -1 + b b = y/b = βy. Case II (d) : p > y + 1 . In this case, we see that γP y = p -1 + b b > (y + 1) -1 + b b = 1 + y/b = γy. In either case, (βP y , γP y ) provides no Pareto improvement over (βy, γy). Case III: y > min{b(λ + 1) -1, (b -1)/λ} . Algorithm 2, purchasing on day ⌈λb⌉, achieves a consistency of ⌈λb⌉-1+b b and a robustness of ⌈λb⌉-1+b ⌈λb⌉ . Consider another algorithm Q that buys on day q (̸= ⌈λb⌉) with prediction-specific consistency and robustness βQ y and γQ y . Case III (a): q ⌈λb⌉-1+b ⌈λb⌉ = γy. Case III (b): ⌈λb⌉ ⌈λb⌉-1+b b = βy. Case III (c): q > y . If y > b(λ+1)-1, we have λb (b -1)/λ, we have ⌈λb⌉≥λb > b2-b y . Therefore, γy = ⌈λb⌉-1 + b ⌈λb⌉ y. This explains why, when r ≤y + 1 -b, we immediately terminate the shifting process in Step 0. 30 Algorithm 7 OPERATION A: CONSISTENCY BOOSTING 1: Input: Initial distribution πeq[1, n] and prediction y ≥b; 2: Initialization: π ←πeq[1, n], iterative index r ←n, initial robustness γ ←γy(π); 3: Step 0: 4: // Check whether the current distribution is a two-point distribution. 5: If r = 1, then update π1 ←1/b and πb+1 ←(b -1)/b, go to Step 4; 6: // Check if further shifting enhances consistency. 7: If r ≤y + 1 -b, then go to Step 4; 8: Step 1: 9: // Shifting probability mass from πr to πy+1. 10: Update πy+1 ←πy+1 + πr and πr ←0; 11: Step 2: 12: // Determine if more shifting is necessary. 13: If R(π, y + 1) γ, then set r ←r + 1, and go to Step 1; 10: Step 3: 11: // After determining the appropriate value of r , 12: // Prioritize assigning probability mass to {1, . . . , r -1, y + 1, . . . , b}. 13: Solve π′ through Pb i=1 π′ i = 1 and R(π′, x) = γ, x ∈{1, . . . , r -1, y + 1, . . . , b}; 14: Update π ←π′; 15: Step 4: Return π. 31 D.3 PRSR: Prediction-specific Randomized Ski Rental With these foundations in place, we proceed to introduce an algorithm for randomized ski rental, referred as PRSR(see Algorithm 9). Algorithm 9 PRSR: PREDICTION-SPECIFIC RANDOMIZED SKI RENTAL 1: Input: γ ∈[eb/(eb -1), b -2) 2: // Determine the smallest n such that πeq[1, n] is γ-robust. 3: Determine n ←⌈logb/(b-1)(1 + 1/(γ -1))⌉; 4: // Determine the adjusted robustness γ′, defined as that of πeq[1, n]. 5: Determine γ′ ←[( b b-1)n -1]-1 + 1; 6: If y ≥b then 7: π ←OPERATION A(πeq[1, n], y); (see Algorithm 7) 8: Else if y 1. In Section E.5, we formally verify the prediction-specific Pareto optimality and strong optimality of PRSR. 32 E Proofs for Section 5 and Appendix D This section provides proofs for Section 5 and Appendix D. E.1 Proofs of Useful Lemmas Lemma 1. Consider the ski rental randomized problem with b > 1, for any distribution π supported on N+, there exists another distribution π′ supported on a finite set [b] such that α(π′) ≤α(π). Specifically, when the best attack for π only occurs at x > b, we have α(π′) R(π′, b). Note that α(π) = supx∈N+ R(π, x) and α(π′) = supx∈N+ R(π′, x). We have α(π) ≥α(π′). Specifically, when the best attack for π occurs at x > b, we have α(π) > α(π′). In the learning-augmented setting, we divide our discussion into two parts: y 1. Consider a probability distribution π over N+. If R(π, x) = α(π) for all x ∈{m, m + 1, . . . , n}, then we have πi+1 = b b -1 · πi, ∀i ∈{m + 1, . . . , n -1}. Proof. By the definition of R(π, x) in Section 5, when x ≤b: R(π, x) = Px i=1 πi(x -1 + b) + P∞ i=x+1 πi · x x . Consider the following equations: R(π, i) = α(π), m ≤i ≤m + 2 (21) Subtracting both sides of (m + 1)×Equation (21) with i = m + 1 by m×Equation (21) with i = m, b · πm+1 + ∞ X i=m+2 πi = α(π). (22) 33 Subtracting both sides of (m + 2)× Equation (21) with i = m + 2 by (m + 1)× Equation (21) with i = m + 1, b · πm+2 + ∞ X i=m+3 πi = α(π). (23) Finally, subtracting both sides of Equation (23) by Equation (22) yields πm+2 = b b-1πm+1. Similarly, we can show that πi+1 = b b -1πi, ∀i ∈{m + 2, . . . , n -1}. E.2 Proof of Theorem D.1 and D.2 Proof of Theorem D.1. By Definition 5, πeq[m, n] only has positive support over {m, . . . , n}. Consider x = m and x = m + 1, (m -1 + b) · πeq m[m, n] m + (1 -πeq m[m, n]) = α(π). (24) (m -1 + b)πeq m[m, n] + (m + b)πeq m+1[m, n]] (m + 1) + (1 -πeq m[m, n] -πeq m+1[m, n]) = α(π). (25) Equating the left-hand sides of Equation (24) and Equation (25) yields: πeq m+1[m, n] = m(b -1) m + b -1πeq m[m, n]. (26) By Lemma 2, we have πeq i+1[m, n] = b b -1 · πeq i [m, n], ∀i ∈{m + 1, . . . , n -1}. (27) Equation (26), Equation (27), together with the constraint Pn i=m πi = 1, imply that πeq i [m, n] = 1 + m+(b-1) m · ( b b-1)n-m -1 -1 for i = m; πeq m[m, n] · m+(b-1) m(b-1) · b b-1 i-m-1 , for i ∈{m + 1, . . . , n}; 0, otherwise. Proof of Theorem D.2. Applying Lemma 1, we first reduce the support under consideration to [b]. Furthermore, the condition of 1 consistency requires πi = 0 for all i ∈[y]. We consider the following optimization problem. min πy+1,...,πb,γy γy (Primal Problem) s.t. i X j=y+1 πj · (b + j -1) + b X j=i+1 πj · i ≤γy · i, ∀i ∈{y + 1, . . . , b}, b X i=y+1 πi = 1, πi ≥0, ∀i ∈{y + 1, . . . , b}. γy ≥0 34 Let π∗ y+1, . . . , π∗ b, γ∗ y denote the optimal solution to Primal Problem. It's clear that γ∗ y = min π∗ y+1,...,π∗ b max i∈{y+1,...,b} R(π∗, i). (28) We claim that π∗ y+1 ̸= 0. We prove this by contradiction, assuming π∗ y+1 = 0. Let r be the minimal index with non-zero probability mass, i.e. r := min{i | π∗ i ̸= 0}. Consider another distribution π′ with π′ y+1 = ε, π′ r = π∗ r -ε and π′ i = π∗ i for all i ∈Z+ \\ {y + 1, r}, where we denote ε := min π∗ r, y + 1 b -1 · (γ∗ y -1 2 ) . Note that R(π′, y + 1) = (b + y) · π′ y+1 + (y + 1) · (1 -πy+1) y + 1 ≤γ∗ y -1 2 + 1 = γ∗ y + 1 2 q | πi ̸= 0}. Since π∗ b ̸= 0, such r is guaranteed to exist. Consider another distribution π′ with π′ y+1 = π∗ y+1 -ε1, π′ q = ε1 + ε2, πr = π∗ r -ε2, and π′ i = π∗ i for all i ∈Z+ \\ {y + 1, q, r}, where we denote ε1 := min π∗ y+1, r -q 2(q -y -1) · ε2 , ε2 := min π∗ r, 2(q -y -1) [(r -q) + 2(q -y -1)](q -1 + b) · γ∗ y . Similarly, we verify that R(π′, i) b. We denote this distribution as π, and consider another distribution π′ that transfer all probability masses beyond b to b (i.e. π′ i = πi, ∀i ∈[b -1] and π′ b = P∞ i=b πi). Recall that βy(π), γy(π), βy(π′), γy(π′) are the consistency and robustness of π and π′ under prediction y, respectively. Since y b, by Lemma 1, we have γy(π′) 0, ε1 = min n ( q-r r-p)ε2, πA p o > 0. We first prove that π′ is still a feasible solution to Problem A. Recall that when x ≤b (see the definition of R(π, x) in Section 5) R(π, x) = Px i=1 πi(i -1 + b) + Pb i=x+1 πi x x . (1) Consider x 0, b -1 > 0, and ε1 ≤( q-r r-p)ε2, ε2 ≤ (Pr i=1 πA i )b (q-r+b-1)(r-1). We have ε1(r -p) + ε2(b -1) ≤(Pr i=1 πA i )b r -1 ≤ Pr i=1 πA i (b + i -1) r -1 . (32) 39 By (30), (31) and (32), rR(π′, r) ≤r Pr i=1 πA i (b + i -1) + r(r -1)(1 -Pr i=1 πA i ) r -1 . (33) Since πA r = 0, (33) implies rR(π′, r) ≤r Pr-1 i=1 πA i (b + i -1) + r(r -1)(1 -Pr-1 i=1 πA i ) r -1 . Therefore, R(π′, r) ≤ Pr-1 i=1 πA i (b + i -1) + (1 -Pr-1 i=1 πA i )(r -1) r -1 = R(πA, r -1) ≤γy. (34) (4) Consider r 0, q -r > 0. (36) further indicates that xR(π′, x) ≤xR(πA, x). Therefore, R(π′, x) ≤R(πA, x) ≤γy. Consequently, R(π′, x) ≤γy for all x ∈[b]. Note that Pb i=1 π′ i = 1 and π′ i ≥0 for all i ∈[b]. Therefore, π′ is a feasible solution to Problem A. We then compare the objective value of Problem A at π′ and πA. Note that p ≤y 0. Similarly we first investigate the feasibility of π′. (1) Consider x 1 when γy ∈(γξ, γν). Since Problem A is a convex optimization problem, and πeq[1, b] (i.e. Karlin's distribution [38]) is always an interior point, by the Slater's condition, d∗= p∗> 1. We then show that in Problem B, λB i ̸= 0 for all i ∈{y+1, . . . , b}. We prove this by contradiction, assuming that there exists r ∈{y + 1, . . . , b} such that λB r = 0. 42 Case II(a): r = y + 1 . Consider Eq.(y + 1) and Eq.(y + 2): ( λ1 + · · · + λy + b+y y+1 · λy+1 + b+y y+2 · λy+2 + · · · + b+y b · λb + λb+1 = 1 Eq.(y + 1) λ1 + · · · + λy + λy+1 + b+y+1 y+2 · λy+2 + · · · + b+y+1 b · λb + λb+1 = 1 Eq.(y + 2) Note that b+y y+2 1. Therefore, the optimal solution to Problem B is λB i = 0, ∀i ∈[b] and λB b+1 = 1, achieving an objective value of d∗= 1. This leads to a contradiction. It thus follows that λB i ̸= 0 for all i ∈{y + 1, . . . , b}. Now, by the complementary slackness, λB x · \"Px i=1 πA i (b + i -1) + Pb i=y+1 πA i x x -γy # = 0, ∀x ∈{y + 1, . . . , b}. Since λB i ̸= 0 for all i ∈{y + 1, . . . , b}, we have Px i=1 πA i (b + i -1) + Pb i=y+1 πA i x x = γy, ∀x ∈{y + 1, . . . , b}. Therefore, the (y + 1)-th to b-th constraints in Problem A are binding at πA. Case III: γy = γξ . Note that πeq[1, b] (i.e. Karlin's distribution [38]) is the only probability distribution that is γξ-competitive. We can verify that the (y + 1)-th to b-th constraints in Problem A are binding at πA = πeq[1, b]. Lemma 5. Consider y b to b. It is clear that βy(π′) = βy(π), ∀y π∗ r . Note that R(π∗, x) = γy for all x ∈[k -1]. This makes r > k -1. Therefore, we have πi = π∗ i , ∀i ∈[k -1] πk ≥π∗ k and πi ≥π∗ i = 0, ∀i ∈{k + 1, . . . , y} with some r ∈{k, . . . , y} such that πr > π∗ r. Thus, Pb i=y+1 πi R(π∗, y) = βy(π∗). Therefore, π cannot be an optimal solution to Problem C. Case II(b): πr γy(π∗) = γy. Let r′ denote the minimal index in {r + 1, . . . , y} such that πr′ ̸= 0, i.e. r′ = min {i ∈{r + 1, . . . , y} | πi ̸= 0}. Consider π′ = (π1, . . . , πr + ε, . . . , πr′ -ε, . . .), where ε = min{π∗ r -πr, πr′}. It is straightforward to verify that γy(π′) = γy(π), βy(π′) βy(γy(γ)) = βy(γ). (49) Next, we invoke Lemma 6 once again, any γQ y -robust algorithm is at least βy(γQ y )-consistent. Note that γQ y -robust algorithm Q is βy(γ)-consistent. Thus βy(γQ y ) ≤βy(γ). (50) Equation (49) and Equation (50) lead to a Contradiction. Therefore, for any γ ∈[γξ, γν], βy(γ) and γy(γ) are jointly Pareto optimal. Consider OPERATION A(πeq[1, n], y) and denote its consistency and robustness under prediction y by βy and γy, respectively. For the case when y ≥b, the following result holds. Lemma 7. Suppose y ≥b and 1 1, from the construction of Algorithm 7, π∗ 2 = 0 could only happen when y = b. In this case, we have π∗ 1 = 1/b, π∗ b+1 = (b -1)/b, and π∗ i = 0 for all i ∈N+ \\ {1, b + 1}. Note that its prediction-specific consistency and robustness are 1 and 2 -(1/b), respectively, i.e. βy = βy(π∗) = 1 and γy = γy(π∗) = 2 -(1/b). Note that all 1-consistent algorithm under y = b can only have probability mass on {1, b + 1}. Let π1 and 1 -π1 represent the probability mass on 1 and b + 1, respectively. then γy(π) = max π1∈[0,1]{bπ1 + (1 -π1), π1 + 2(1 -π1)} ≥2 -(1/b). Therefore, βy and γy are jointly Pareto optimal. Case II: π∗ 2 ̸= 0 . Let k := max{i ≤b | π∗ i ̸= 0}. Since π∗ 2 ̸= 0, we have k ≥2. According to the structure of Algorithm 7, we have that at least one of the following holds: R(π∗, y + 1) = γy, (51) y -b + 1 = k. (52) Consider π ̸= π∗. Let r be the minimal index such that πi ̸= π∗ i , i.e. r := min{i | πi ̸= π∗ i }. Case II(a): πr > π∗ r . In this case, we must have r π∗ r will lead to γy(π) > γy(π∗) = γy. If k ≤r r | πi ̸= 0}. Consider π′ = {π1, . . . , πr-1, πr + ε, . . . , πv -ε, . . .}, where ε = {πv, π∗ r -πr}. It is straightforward to verify γy(π′) ≤γy. If b+k-1 y}(r -1 + b -y)] ≤bβy(π). Equivalently, βy(π′) ≤βy(π). Attaining equality requires r -1 + b = y and r = k. If b + k -1 > y, then Condition (52) does not hold; thus, Condition (51) must hold, i.e., we have R(π∗, y + 1) = γy. If v = y + 1, then πi = π∗ i for all i ∈[r -1], πr γy. This makes v 1. By Lemma 7, (βy(πA), γy(πA)) is Pareto optimal. Therefore, we have βy(πA) ≤γy(πA) log 1 + 1 γy(πA) -1 . Since γ′ = γy(πA), we have for any y ≥b, βy(πA) ≤γ′ log 1 + 1 γ′ -1 . (54) Case II: y γ(πeq[1, b]) = γξ, and 1 + [( b b -1)n -1]-1 = γ(πeq[1, n]) ≥γ(πeq[1, b]) = γξ. Therefore, γ′′ = min{γν, γ′} ≥γξ, and γ′′ = min{γν, γ′} ≤γν. Thus, we have γξ ≤γ′′ ≤γν. Therefore, γy(πB) = γ′′. We consider the following two cases. Case II(a): γ′′ = γ′ . In this case, we consider λ = -log(1-γ-1 y ), Kumar's algorithm is γy log(1+ 1 γy-1)- consistent and γy-robust. Since γξ ≤γ′′ ≤γν, by Lemma 5, βy(πB) ≤γy(πB) log 1 + 1 γy(πB) -1 . Since γ′ = γ′′ = γy(πB), we have for any y √ θ, which makes βS y = βC y , γS y > γC y . By Definition 4, Sun's algorithm is not strongly-optimal. F.2 Proof of Theorem 6.2 Proof of Theorem 6.2. We first prove the prediction-specific consistency and robustness of PST. We consider the following three cases. Case I: y ∈[L, λL + (1 -λ) √ LU) . In this case, PST sets the threshold at Φ = √ LU. Since λL + (1 - λ) √ LU ≤ √ LU, we have y √ LU = γy. Therefore, (βy, γy) is Pareto optimal. Assume y ∈[λL+(1-λ) √ LU, √ LU]. Since Φ′ ̸= Φ = y, βy > 1 = βy. Thus, (βy, γy) is Pareto optimal. Assume y ∈( √ LU, U]. If Φ′ y/Φ = βy. If Φ′ > Φ, since Φ′ > Φ ≥ √ LU, we have γy = Φ/ √ LU and γ′ y = Φ′/ √ LU, thus γ′ y > γy. Therefore, (βy, γy) is Pareto optimal. By Definition 4, PST is strongly-optimal. 51 G Proofs for Section 7 In this section, we prove Theorem 7.1, 7.2 and 7.3. G.1 Proof of Theorem 7.1 Proof of Theorem 7.1. We consider the following five cases to investigate the prediction-specific ε-consistency and robustness. Case I: y ∈[L, M -2ε] . In this case, ε-Tolerant PST decides to set the threshold to Φ = √ LU and the price range relevant to ε-consistency is restricted to [max{L, y -ε}, y +ε]. To determine the ε-consistency, we assume that the highest price x ∈[max{L, y -ε}, y + ε]. Note that y + ε ≤(M -2ε) + ε 0, (U -2ε) - LU ( √ LU -ε -ε) 0. Therefore, any γ-robust algorithm has at least (θ/γ) ε-consistency. Similarly, given ε > 0, any algorithm that achieves βε ε-consistency is βε-consistent. Based on the lower bound by Sun et al. [11], it must be at least (θ/βε)-robust. G.3 Proof of Theorem 7.3 Proof of Theorem 7.3. By Theorem 7.1 and Theorem 7.2, we conclude that ε-Tolerant PST's ε-consistency and robustness are jointly Pareto optimal. To prove the Pareto optimality of prediction-specific ε-consistency and robustness, we consider Φ′ ̸= Φ, which achieves ε-consistency βε y ′ and robustness γy′ with respect to y. Case I: y ∈[L, M -2ε] . In this case, γy = √ θ. Note that Φ = √ LU is the only threshold that achieves robustness √ θ. We can conclude that (βε y, γy) is Pareto optimal. Case II: y ∈(M -2ε, M) . In this case, ε-Tolerant PST sets the threshold to Φ = M -ε, achieving βε y = (M -ε)/L and γy = U/(M -ε). If Φ′ γy. If Φ′ > Φ, then consider the attack x = min{Φ′ -δ, y + ε} for threshold Φ′, where δ is the infinitesimal quantity. This makes βε y ′ ≥min{Φ′, y + ε}/L > (M -ε)/L = βε y. Therefore, (βε y, γy) is Pareto optimal. 53 Case III: y ∈[M, √ LU + ε] . In this case, ε-Tolerant PST sets the threshold to Φ = y -ε, achieving βε y = (y + ε)/(y -ε) and γy = U/(y -ε). Note that Φ = y -ε γy. If Φ′ > Φ, consider x = min{Φ′ -δ, y + ε} for threshold Φ′, where δ is the infinitesimal quantity. This guarantees the following inequality: βε y ′ ≥min{Φ′, y + ε}/L > (y -ε)/L. Note that r′(y) := (y -ε)/L is an increasing function of y, and r(y) := (y + ε)/(y -ε) is a decreasing function of y. Since y ≥M ≥L + 3ε and r(L + 3ε) = (L + 4ε)/(L + 2ε) (y + ε)/Φ = βε y. If Φ′ > Φ, since Φ > √ LU + ε > √ LU, γ′ y > γy. Therefore, (βε y, γy) is Pareto optimal. Case V: y ∈[U -ε, U] . In this case, ε-Tolerant PST sets the threshold to Φ = LU/(M -ε), achieving βε y = (M -ε)/L and γy = U/(M -ε). If Φ′ > Φ, then γ′ y > γy. If Φ′ U/Φ = βε y. Therefore, (βε y, γy) is Pareto optimal. 54"}}},{"rowIdx":97,"cells":{"id":{"kind":"string","value":"2510.14884"},"raw":{"kind":"string","value":"Learning When Not to Learn:\nRisk-Sensitive Abstention in Bandits with Unbounded Rewards\nSarah Liaw∗\nBenjamin Plaut∗\nHarvard University\nUniversity of California, Berkeley\nAbstract\nIn high-stakes AI applications, even a single\naction can cause irreparable damage. How-\never, nearly all of sequential decision-making\ntheory assumes that all errors are recoverable\n(e.g.,\nby bounding rewards).\nStandard\nbandit algorithms that explore aggressively\nmay cause irreparable damage when this\nassumption fails.\nSome prior work avoids\nirreparable errors by asking for help from\na mentor, but a mentor may not always be\navailable. In this work, we formalize a model\nof learning with unbounded rewards without\na mentor as a two-action contextual bandit\nwith an abstain option: at each round the\nagent observes an input and chooses either to\nabstain (always 0 reward) or to commit (ex-\necute a preexisting task policy). Committing\nyields rewards that are upper-bounded but\ncan be arbitrarily negative, and the commit\nreward is assumed Lipschitz in the input.\nWe propose a caution-based algorithm that\nlearns when not to learn: it chooses a trusted\nregion and commits only where the available\nevidence does not already certify harm.\nUnder these conditions and i.i.d.\ninputs,\nwe establish sublinear regret guarantees,\ntheoretically demonstrating the effectiveness\nof cautious exploration for deploying learning\nagents safely in high-stakes environments.\n1\nINTRODUCTION\nWith AI becoming ubiquitous, many learning systems\nare now deployed in unpredictable, safety-critical\ndomains, such as process control and manufacturing\nrobotics, autonomous driving, and surgical assistance.\nIn these settings, a single ill-chosen action can cause\nirreparable and lasting damage with no opportunity\n∗Equal contribution.\nfor subsequent recovery. For instance, a self-driving car\ncannot compensate for a deadly crash by later driving\nmore safely, nor can a medical robot undo a fatal mis-\ntake during surgery. Following Plaut et al. (2025a,b),\nwe refer to such irreparable errors as catastrophes.\nDespite the risks such deployments pose, there is lim-\nited work (and limited theoretical work in particular)\non how an agent can learn without ever incurring\nan irreparable error. The possibility of catastrophes\nchallenges standard frameworks for sequential decision\nmaking, especially the familiar notion of optimism\nunder uncertainty. Optimism effectively assumes that\nearly mistakes can be offset (or be compensated for)\nby later gains, an assumption that is inappropriate\nwhen errors are irrecoverable. Instead, these settings\ncall for pessimism under uncertainty: when evidence\nis insufficient, prefer inaction to risky action.\nOne approach to mitigate these problems is to let the\nagent ask for help from a mentor in unfamiliar or risky\nsituations. Such human-in-the-loop oversight can block\nunsafe actions and prevent irreparable errors (even if\nordinary, recoverable errors still occur). However, this\napproach depends on the availability of a capable men-\ntor, which can be costly or impractical at scale. This\nmotivates a mentor-free alternative: can an agent\navoid irreparable errors on its own by acting\ncautiously when inputs appear unfamiliar?\nWe propose a model of learning in the presence of\nirreparable costs without a mentor but with an option\nto abstain from action.\nThe key question is when\nto abstain, i.e., when not to learn. To focus on this\nquestion, we assume the agent has previously learned\na baseline policy that works well in-distribution but\nbehaves unpredictably elsewhere. This allows us to\nstreamline the model to two actions:\nabstain (do\nnothing) and commit (follow the baseline policy). Ab-\nstaining yields a deterministic safe reward r(x, 0) = 0,\nwhile committing yields a reward r(x, 1) ∈(−∞, 1]1.\n1The asymmetric bounds on the commit reward reflect\nthat a single action can be catastrophic, whereas it is rare\nfor a single action to yield arbitrarily large benefit.\narXiv:2510.14884v1 [cs.LG] 16 Oct 2025\n\nRisk-Sensitive Abstention in Bandits\nWe treat the origin as fully “in-distribution” and as-\nsume the baseline policy is beneficial there: r(0, 1) > 0.\nWe use the distance from the origin ∥x∥as a measure\nof how out-of-distribution (OOD) an input is. The\ncommit reward is assumed L-Lipschitz, capturing\nthe idea that similar inputs yield similar outcomes.\nFor our main results we focus on a fixed distribution\nν; for our impossibility results we also consider a\nT-dependent distribution νT .\nWe formalize the tension between exploration and\nsafety via two negative results. First, in the worst case,\nany algorithm that begins by always exploring (i.e.,\ncommits on the first round regardless of the input)\ncan suffer infinite expected regret (Thm. 4.1). Second,\nwhen every input lies uniformly far OOD, there is no\nsafe way to explore to identify a beneficial committing\nregion, and sublinear regret is impossible (Thm. 4.2).\nTogether, these results delineate both the necessity\nand the limits of caution.\nMotivated by this perspective, we develop a caution-\nbased algorithm that learns only when it can guarantee\nthat an error is not catastrophic (which essentially\ncorresponds to not-too-OOD inputs). This approach\nyields sublinear expected regret for i.i.d. inputs from\nany fixed distribution, with bounds that also reflect\nhow often the agent encounters far OOD inputs, while\nprioritizing the avoidance of irreparable errors.\nContributions. Our contributions can be summarized\nas follows:\n1. We introduce a formal model of learning with ir-\nreparable costs and no external mentor.\n2. We prove two impossibility results that delineate\nthe necessity and limits of caution.\n3. We develop a caution-based algorithm that achieves\nsublinear regret for any fixed input distribution.\nOrganization. §3 introduces the formal model and no-\ntation. §4 presents the impossibility results (Thms. 4.1\nand 4.2) and their implications for exploration. § 5 de-\nscribes our caution-based learning algorithm and states\nthe main regret bound. § 6 outlines the proof strategy\nand supporting lemmas.\n2\nRELATED WORK\nMost prior work on sequential decision-making and\nsafe exploration focuses on settings where errors are\nultimately recoverable; here we contrast this with our\nsetting where individual actions can cause irreparable\nharm.\n2.1\nSequential decision-making when all\nerrors are recoverable\nThe literature on sequential decision-making is vast,\nspanning bandit problems, reinforcement learning,\nand online learning. See Slivkins et al. (2019), Sutton\net al. (1998), and Cesa-Bianchi and Lugosi (2006) for\nintroductions to these (somewhat overlapping) topics,\nrespectively. However, nearly all of this work assumes\nexplicitly or implicitly that any error can be recovered\nfrom. This assumption enables the agent to ignore risk\nand simply try all possible behaviors, since no matter\nhow badly it performs in the short term, it can always\neventually make up for it. Indeed, most sequential\ndecision-making algorithms with formal regret bounds\nhave this general structure.\nThis assumption can manifest in different ways. In\nbandit settings, it suffices to assume that rewards are\nbounded (or at least have bounded expectation). This\nassumption implies that the expected regret from any\naction on any time step is always bounded, which is\nsufficient for the risk-agnostic exploration mentioned\nabove.\nIn contrast, we allow unbounded negative\nrewards so that actions can be arbitrarily costly.\nIndeed, our first negative result (Thm. 4.1) relies on\nthe expected regret for a single action potentially\nbeing infinite in our model.\nIn Markov Decision Processes (MDPs), the agent’s\nactions determine the next state via a transition func-\ntion, so in addition to bounded rewards, one typically\nassumes that either the environment is reset at the\nstart of each “episode” (e.g., Azar et al., 2017) or that\nany state is reachable from any other (e.g., Jaksch et al.,\n2010). The dependence of standard MDP algorithms\non these assumptions was observed by Moldovan and\nAbbeel (2012a); Cohen et al. (2021), among others.\nRegardless of the specific form of this assumption, it\nclearly does not hold in safety-critical contexts where\na single action can be catastrophic.\n2.2\nSafe exploration\nThese issues have motivated a wide field of safe\nexploration. A full survey is beyond the scope of this\npaper (see Garc´ıa and Fern´andez, 2015; Gu et al., 2024;\nKrasowski et al., 2023; Tan et al., 2022 for surveys), so\nwe cover only the most relevant prior work. Avoiding\nirreparable errors while learning has also been studied\nempirically across multiple domains (e.g., Saunders\net al., 2017; Moldovan and Abbeel, 2012b; Wachi et al.,\n2023; Zhao et al., 2023; Perkins and Barto, 2003),\nbut here we focus on theoretical work, which is most\nrelevant to our setting.\nSafe exploration is modeled in two main ways. The first\n\nSarah Liaw, Benjamin Plaut\napproach is to require the agent to satisfy some sort of\nconstraint in addition to maximizing reward. The con-\nstraint can be entirely separate from reward, as in the\ncase of constrained MDPs (Altman, 1999), or they can\nbe related to the reward (e.g., the agent’s reward must\nalways exceed some baseline). When zero or near-zero\nconstraint violation is required, these formalisms do\ncapture the possibility of irreparable errors. The second\napproach treats reward as the sole objective, with safety\nas a necessary but not sufficient property for maximiz-\ning reward. Here, irreparable errors correspond to ei-\nther unboundedly negative rewards (our work falls into\nthis category) or inescapable “trap” states with poor\nreward. An agent that obtains very negative rewards\nor enters trap states clearly cannot obtain high reward.\nBoth of these models must contend with a fundamental\nobstacle:\nhow does one learn which actions are\ncatastrophic without trying those actions directly?\nThis can be formalized by the so called “Heaven or\nHell problem”. Suppose there are two available actions,\nwhere one has unbounded positive reward and the\nother has unbounded negative reward. In this case,\nthe agent can do no better than simply guessing and\ncan never guarantee good regret. This problem shows\nthat some sort of additional assumption is necessary\nfor any meaningful regret guarantees.\nBelow, we\ncategorize work within safe exploration based on which\nassumption(s) it uses for this purpose.\nFull\nprior\nknowledge.\nPerhaps the simplest\napproach is to assume that the agent knows the precise\nsafety constraint upfront (see Zhao et al., 2023 for\na survey). This immediately resolves the Heaven or\nHell problem; indeed, it eliminates the need for the\nagent to “learn when not to learn” at all. However,\nfull knowledge of the safety constraint may not hold\nin practice. In contrast, we only assume that the (1)\nbaseline policy performs well in-distribution and (2)\nthe agent can always safely abstain.\nLearning constraints using a safe fallback action.\nThere is a growing body of work which shares our\nassumption of a safe fallback action. Liu et al. (2021);\nStradi et al. (2024) use this approach in the constrained\nMDP model, while Wu et al. (2016); Kazerouni et al.\n(2017); Lin et al. (2022); Chen et al. (2022) require the\nreward to exceed a fixed baseline in a bandit model.\nThese papers generally rely on a pair of subtle but\ncrucial assumptions to obtain zero constraint violation:\n(1) the constraint violation on any given time step\nis bounded and (2) the baseline policy satisfies the\nconstraints with a known amount of slack (this is called\nSlater’s gap, although not all of the above papers\nuse this term).\nThis combination of assumptions\nenables the agent to still explore aggressively with\nsome known probability. Furthermore, the resulting\nbounds typically depend inversely on Slater’s gap.\nOur work is complementary to each of these two\nassumptions.\nFirst, rather than assuming global\nboundedness, we assume that rewards decrease at a\nbounded rate, i.e., rewards are Lipschitz continuous.\nSecond, rather than dependence on the reward or cost\nfunction (in the form of Slater’s gap), our bounds\ndepend on the input distribution:\nspecifically, our\nbounds degrade as the agent sees more OOD inputs.\nOur approach may be more or less realistic depending\non the specific context, but it notably diverges from\nthe typical way fallback actions are utilized.\nAsking for help. Perhaps the most common approach\nin this model is relying on external supervision. This\nis a growing body of work which uses limited queries\nto a mentor to prove formal regret guarantees in\nthe presence of irreversible dynamics (Cohen et al.,\n2021; Cohen and Hutter, 2020; Kosoy, 2019; Maillard\net al., 2019; Plaut et al., 2025b,a). However, as the\nnumber of deployed AI systems continues to grow,\nit may be impractical for each one to have a human\nsupervisor.\nEven in cases where external help will\neventually become available, the agent may need to\nbehave safely on its own in the short-term.\nThese\nconsiderations motivate our study of how to learn\nsafely in the absence of external help.\n2.3\nOther related work\nWe briefly discuss some topics that are less directly\nrelevant but still worth mentioning. One is the heavy-\ntailed bandit model (Bubeck et al., 2013; Agrawal\net al., 2021), which studies the case where reward distri-\nbutions are not subgaussian and thus less predictable.\nWhile this model does incorporate elements of safety,\nas long as the expected reward from any action is\nbounded, risk-agnostic exploration remains valid (as\ndiscussed above). Another topic adjacent to our work\nis the standard Lipschitz bandit model with bounded\nrewards and bounded domain (see, e.g., Chapters 4 and\n8 of Slivkins, 2011). This work shares some similarities\nwith ours, like the algorithmic use of discretization.\nHowever, the core of our paper is removing the\nboundedness assumptions, which introduces a host\nof new challenges.\nFinally, there is complementary\nwork on abstention with bounded rewards (Neu and\nZhivotovskiy, 2020; Yang et al., 2024). While this line\nof work also demonstrates the benefits of abstention, it\ndoes not address the possibility of irreparable errors.\n3\nPRELIMINARIES\nWe study a two-action contextual bandit model in\nwhich, on each round, the agent observes an input and\n\nRisk-Sensitive Abstention in Bandits\nchooses either to commit, thus executing a fixed task\npolicy that may yield risky outcomes, or to abstain,\nreceiving a safe default reward of zero. In this section,\nwe introduce the formal notation and assumptions\nused throughout.\nFor k ∈N, let [k] = {1, . . . , k}. Let X = Rn be the\ninput space, T ∈N be the time horizon, and ∥· ∥be\nthe Euclidean norm (though one could also consider a\nmore general metric space). On each time step t ∈[T],\nthe agent observes an input xt ∈X, chooses an action\nyt ∈{0, 1}, and receives a (noisy) scalar reward; the\nprecise noise assumptions are stated below.\nActions and Rewards.\nWe interpret yt = 0 as\n“abstaining”, a safe default which deterministically\nyields r(xt, 0) = 0 for any xt ∈X. We interpret yt = 1\nas “committing”, which executes a preexisting policy\nwhose reward r(xt, 1) may be arbitrarily negative\n(catastrophic) but is assumed to have a constant upper\nbound (rescaled to 1 without loss of generality). This\ncaptures the asymmetry of high-stakes settings where\ncatastrophic losses can be unbounded in magnitude,\nwhereas gains typically saturate.\nInput models. We assume inputs are i.i.d. draws from\nan unknown distribution ν on X, i.e., x1, . . . , xT\ni.i.d.\n∼ν.\nWe typically take ν to be fixed, but in our impossibility\nresults we also consider the case of T-dependent ν\n(denoted νT ).\nWe assume bandit feedback: the agent observes only\nthe realized reward of its chosen action. Abstaining\nprovides no information about the counterfactual com-\nmit reward r(xt, 1), so the agent cannot “learn by\nabstaining”. Formally, at round t the learner observes\nrt = r(xt, yt) + ηt,\nwhere (ηt)T\nt=1 are i.i.d. zero-mean σ-subgaussian noise\nvariables, independent of (xt) and of the learner’s\ninternal randomness (specified formally in Def. 3.1).\nDefinition 3.1 (σ-subgaussian). A random variable\nZ is σ-subgaussian if\nE[exp(λ(Z −E[Z]))] ≤exp\n\u0010\nσ2λ2\n2\n\u0011\nfor all λ ∈R.\nEquivalently, Z −E[Z] has tails that are dominated by\na centered Gaussian with variance proxy σ2.\nRegularity. We make two assumptions on the reward\nfunction: (i) the commit reward r(·, 1) is L-Lipschitz\nin the Euclidean norm, i.e., there exists L > 0 such\nthat for all x, x′ ∈X, |r(x, 1) −r(x′, 1)| ≤L∥x −x′∥.\nThis is a standard smoothness condition in Lipschitz\nbandit models (see, e.g., Slivkins et al., 2019) and\ncaptures the intuition that similar inputs yield similar\ncommit rewards. Since r(x, 0) ≡0, the abstain reward\nis 0-Lipschitz. (ii) The in-distribution baseline input\nyields strictly positive reward when committing,\ni.e.\nr(0, 1) > 0.\nThis guarantees that committing\nis beneficial somewhere (at the origin); without it,\nthe optimal policy would be to always abstain and\ncautious learning would be impossible.\nObjective. The agent’s goal is to minimize its (ex-\npected) regret, which is the difference between its cu-\nmulative reward and the optimal cumulative reward.\nFormally, define\nReg(T) =\nT\nX\nt=1\n\u0010\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, yt)\n\u0011\n.\nWe take the expectations over the input process (in\nthe stochastic model), the observation noise, and the\nlearner’s internal randomness. The goal is to achieve\nsublinear expected regret, i.e., E[Reg(T)] = o(T),\nequivalently E[Reg(T)]/T →0 as T →∞.\n4\nTHE VIRTUES AND LIMITS OF\nCAUTION\nIn this section, we provide two impossibility results\nthat demonstrate the importance and limitations of\ncaution in high-stakes, unbounded reward bandits.\nFirst, caution is necessary: if an agent commits with\nnon-negligible probability on inputs that are far OOD,\ncatastrophic tail losses dominate—indeed, even a single\nrisk-agnostic exploratory commit can incur infinite\nexpected regret. This kind of “incautious exploration”\nis exactly how standard bandit algorithms behave\nwhen they begin by pulling every arm at least once.\nSecond, caution has limits: when the input stream\nis uniformly far OOD, there is no way to explore\ncautiously to identify a beneficial committing region\nwithout risking catastrophe. In such settings, sublinear\nregret is not possible and the optimal strategy is to\nabstain on every time step.\nTheorem 4.1 (The need for caution). Let ν be any dis-\ntribution over X such that Ex∼ν[∥x∥] = ∞and assume\nx1, . . . , xT ∼ν i.i.d. Then there exists a reward func-\ntion r such that any algorithm which always commits\non the first time step satisfies E[Reg(T)] = ∞.\nProof. Define r(x, 1) = 1 −L∥x∥for all x ∈X. Then\nE[Reg(T)] = E\n\" T\nX\nt=1\n\u0012\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, yt)\n\u0013#\n≥E\n\u0014\nmax\ny∗∈{0,1} r(x1, y∗) −r(x1, y1)\n\u0015\n≥E[0 −(1 −L∥x1∥)]\n\nSarah Liaw, Benjamin Plaut\n= L E\nx∼ν[∥x∥] −1\n= ∞\nas required.\nThe proof can easily be modified to handle the\ncases where the first commit is taken with constant\nprobability (rather than probability 1) or where the\nalgorithm abstains for a constant number of initial\nrounds.\nEssentially, this negative result applies to\nany algorithm that is not cautious, i.e., that explores\nwithout considering how OOD xt is.\nHowever, caution can only get us so far. While it pre-\nvents catastrophic first commits, some exploration is\nnecessary to obtain sublinear regret. If all inputs are far\nOOD, then there is no safe way to explore, so the agent\nhas no choice but to always abstain. Equivalently, this\ncan be phrased by considering i.i.d. inputs from a T-\ndependent distribution νT supported on {x : ∥x∥= T}.\nTheorem 4.2 (The limits of caution). Let νT be any\ndistribution supported on {x : ∥x∥= T}, and suppose\nx1, . . . , xT\ni.i.d.\n∼νT . Then no algorithm can guarantee\nE[Reg(T)] ∈o(T).\nProof. Define r−(x, 1) := 1 −L∥x∥and r+(x, 1) := 1,\nwith r±(x, 0) := 0. Since we only care about asymp-\ntotics, we can restrict our attention to T > 1/L. Then\nfor ∥xt∥= T, optimal behavior for r+ is to always\ncommit, while optimal behavior for r−is to always\nabstain. We show maxr∈{r−,r+} E[Reg(T)] ∈Ω(T).\nTo do so, we use a mild version of the probabilistic\nmethod. Let U(r−, r+) be the uniform distribution over\n{r−, r+}. It suffices to show Er∼U E[Reg(T)] ∈Ω(T),\nwhere the second expectation is over x1, . . . , xT and\ny1, . . . , yT .\nLet E be the event that the agent ever\ncommits. If E holds, there exists i ∈[T] with yi = 1.\nSince yi is independent of r,\nE\nr E[Reg(T) | E]\n= E\nr E\n\" T\nX\nt=1\n\u0012\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, yt)\n\u0013\n| E\n#\n≥Pr[r = r−] E\n\u0014\nmax\ny∗∈{0,1} r(xi, y∗) −r(xi, yi) | r = r−\n\u0015\n= LT −1\n2\nOn the other hand, if E does not occur, then\nE\nr E[Reg(T) | ¬E]\n≥Pr[r = r+] E\n\" T\nX\nt=1\n\u0012\nmax\ny∗∈{0,1}r+(xt, y∗)−r+(xt, yt)\n\u0013\n|¬E\n#\n≥T\n2 .\nThen by the law of total expectation,\nE\nr E[Reg(T)]\n= Pr[E] E\nr E[Reg(T) | E] + Pr[¬E] E\nr E[Reg(T) | ¬E]\n≥min\n\u0000Pr[E], Pr[¬E]\n\u0001\nmin\n\u0012LT −1\n2\n, T\n2\n\u0013\n∈Ω(T)\nas required.\n5\nALGORITHM AND MAIN\nRESULT\nFollowing the negative results in § 4, we propose an\nalgorithm (Algorithm 1) that operationalizes cautious\nlearning: only learn in regions that are not too far\nOOD and where the available evidence does not\nalready certify that committing is harmful.\nInformally, we define a trusted region around the origin\nwhose radius grows with the time horizon, reflecting the\nmaximum regret we are willing to tolerate—intuitively,\nthis corresponds to allowing mistakes that are bad\nbut not catastrophic. We then discretize the region\ninto bins to exploit Lipschitz continuity. Within each\nbin, the commit reward cannot vary by more than a\nLipschitz discretization error, so it suffices to estimate\na single per-bin mean.\nThe agent always abstains\noutside the trusted region, and inside it abstains in\nany bin whose pessimistic upper bound on reward is\nnegative; otherwise it commits to gather information.\nMore precisely, the algorithm defines a ball of radius\nm(T) around the origin, treating inputs outside this\nball as too OOD to test.\nThe ball is partitioned\ninto n-dimensional hypercubes (bins) of side length\nw(T). By Lipschitz continuity, the variation of r(·, 1)\nwithin any bin B is at most L√nw(T).\nFor each\nbin B, the algorithm maintains its empirical mean\nˆµB and a confidence radius γ(k) after k commits in\nB. If ˆµB + γ(k) + L√nw(T) < 0, then B is certified\nunsafe and the algorithm abstains there permanently.\nFigure 1 shows a schematic of the algorithm.\nWe saw in § 4 that the problem is impossible when\ninputs are too far OOD. A natural way to quantify this\nis via the amount of probability mass that lies outside a\ngiven radius, captured by the radial survival function:\nDefinition 5.1 (Radial survival function). For any\nradius R ≥0, the radial survival function of ν is\n¯ν(R) := Prx∼ν\n\u0002\n∥x∥≥R\n\u0003\n.\nWe are now ready to state our main result.\n\nRisk-Sensitive Abstention in Bandits\nAlgorithm 1 Risk-Sensitive Abstention Algorithm\nInputs: m : N →R>0, w : N →R>0\nH ←partition of X into n-cubes of side length w(T)\nB ←{B ∈H : ∃x ∈B with ∥x∥≤m(T)}\nσw ←\np\nnL2w(T)2 + σ2\nγ(k) :=\nq\nc−1σ2w ln(2T 4)\nk\nwhere c is the absolute con-\nstant from Lemma A.2\n(kB, ˆµB) = (0, 0) for all B ∈B\nfor t = 1, . . . , T do\nif ∃B ∈B s.t. xt ∈B then\n▷xt is not too OOD: it’s safe to learn\nif ˆµB + γ(kB) + L√nw(T) < 0 then\n▷We already know xt is bad: don’t learn\nAbstain (yt = 0)\nelse\n▷xt might be good: learn\nCommit (yt = 1)\nkB ←kB + 1\nˆµB ←ˆµB + rt−ˆµB\nkB\nelse\n▷xt is far OOD: it’s too risky to learn\nAbstain (yt = 0)\nTheorem 5.2. In the stochastic setting with xt ∼ν\ni.i.d., Algorithm 1 with w(T) = T −1/(n+2) and m(T) =\nln T satisfies\nE[Reg(T)] ∈O\n\u0010\n(L + σ2)T\nn+1\nn+2 (ln T)n+1 + T ¯ν(ln T)\n\u0011\n.\nThe first term is typical for Lipschitz contextual ban-\ndits and reflects the curse of dimensionality (see, e.g.,\nSlivkins et al., 2019, Thms. 4.11–4.12; Plaut et al.,\n2025a, Thm. 10).\nThe T ¯ν(ln T) term is unusual mainly because un-\nbounded domains are unusual: for bounded domains,\n¯ν(ln T) = 0 for all large T. Our analysis deals with\nfar OOD inputs directly and the bound necessarily\ndegrades as such inputs become more frequent. This\ndependence is unavoidable:\nthe construction in\nThm. 4.2 sets xt = T for all t ∈[T], hence ¯ν(ln T) = 1\nand the bound in Thm. 5.2 becomes linear, matching\nthe impossibility result. By contrast, for any fixed\ndistribution ν, we have ¯ν(ln T) →0 as T →∞, so\nT ¯ν(ln T) = o(T) and the overall regret stays sublinear.\nFor example, if ν is subgaussian with ¯ν(r) ≤e−cr2, then\nT ¯ν(ln T) ≤Te−c(ln T )2 = o(1). If ν is subexponential\nwith ¯ν(r) ≤e−cr, then T ¯ν(ln T) ≤T · T −c = T 1−c =\no(T). If ν has polynomial tails with ¯ν(r) ≍r−α for\nα > 0, then T ¯ν(ln T) ≍T/(ln T)α = o(T). If one has\nprior knowledge of ν, the choice of m(T) can be tailored\nmore precisely than our generic setting m(T) = ln T.\nIn particular, for polynomial tails, setting m(T) = T c\n0\nx1\nx2\nw(T)\ngrid (bins)\nbin ∈B (trusted)\ncertified negative\n∥x∥≤m(T)\nFigure 1: Trusted region of radius m(T) around the\norigin, partitioned into bins of side w(T). Any square\nintersecting the ball is shown fully green (bin ∈B).\nCertified negative bins are shown hatched red. The\nagent abstains outside the ball.\nfor small c > 0 improves the bound to O(T 1−cα).\nThus,\nthe\nregret\ndecomposes\ninto\na\ngeomet-\nric/statistical term from discretization and concentra-\ntion inside the trusted region, and a tail term from\nfar OOD inputs; both are sublinear for any fixed ν.\n6\nPROOF SKETCH\nWe now outline the logical structure of the proof\nof Thm. 5.2; we also provide the intuition behind each\nstep. Full technical details and complete proofs of all\nlemmas are deferred to Appendix A.\nLet m(T), w(T) be as in Algorithm 1, and let B be\nthe set of bins intersecting the ball of radius m(T).\nFor any B ∈B, let µB = Ex∼ν[ r(x, 1) | x ∈B ] be its\ntrue mean commit reward, let kB(t) be the number\nof commits taken in B by the end of round t (so\nkB(0) = 0), and let ˆµB(k) be the empirical mean\nin B after k commits (i.e., the running mean from\nAlgorithm 1 indexed by its commit count).\nTo control estimation error we define the confidence\nradius\nγ(k) =\nr\nc−1σ2w ln(2T 4)\nk\n,\nσ2\nw = nL2w(T)2 + σ2,\nwhere c > 0 is the absolute constant from Lemma A.2.\nHere σ2\nw aggregates the combines observation noise\nσ2 with the Lipschitz-induced within-bin variation of\n(L√nw(T))2.\n\nSarah Liaw, Benjamin Plaut\nDefine the good event under which all per-bin estimates\nare uniformly accurate over the realized commit counts:\nG =\nn\n∀B ∈B, ∀t ∈[T] : |ˆµB(kB(t))−µB| ≤γ(kB(t))\no\n.\nOn G, each empirical mean is a reliable proxy for its\nbin’s true mean at every commit count that occurs\nalong the algorithm’s trajectory. The analysis condi-\ntions on G (which holds with high probability by a\nunion bound over realized bin–count pairs), and then\ndecomposes regret into: (i) commits inside the trusted\nregion (handled by certification of negative bins plus a\nmargin term), and (ii) abstentions outside the ball of ra-\ndius m(T) (quantified by the radial survival function).\nLemma 6.1 (Per-bin concentration). For any B ∈B\nand t ∈[T], Pr [|ˆµB(kB(t)) −µB| > γ(kB(t))] ≤T −4.\nProof idea. Fix B and t and condition on the kB(t)\ncommit times t1 < · · · < tkB(t) with xtj ∈B. Decom-\npose\nˆµB(kB(t)) −µB =\n1\nkB(t)\nkB(t)\nX\nj=1\n\u0000r(xtj, 1) −µB\n\u0001\n+\n1\nkB(t)\nkB(t)\nX\nj=1\nηtj.\nBy Lipschitz continuity and the fact that all xtj lie\nin a single cube of side w(T), we have\n\f\fr(xtj, 1) −\nµB\n\f\f ≤L√nw(T) (see Lem. A.3), so the first term\nis bounded and hence subgaussian with variance\nproxy O((L√nw(T))2). The second term is the ob-\nservation noise, which is σ-subgaussian by assump-\ntion.\nStandard subgaussian tail bounds then give\nPr (|ˆµB(kB(t)) −µB| > γ(kB(t))) ≤T −4.\n□\nLem. 6.1 gave concentration for each fixed bin and\ncommit count. To extend this guarantee uniformly,\nwe apply a union bound over the at most T bin–time\npairs actually realized by the algorithm.\nLemma 6.2 (Uniform concentration bound). With\nprobability at least 1 −T −2, the good event G holds.\nEquivalently, Pr(¬G) ≤T −2.\nProof idea. There are at most T 2 relevant pairs (B, t)\nalong the trajectory of the algorithm. Each has failure\nprobability T −4 by Lem. 6.1. A union bound yields\nfailure probability at most T −2.\n□\nRecall that the algorithm abstains permanently in any\nbin once ˆµB(kB(t)) + γ(kB(t)) + L√nw(T) < 0. On\nthe good event G, bins with sufficiently negative mean\nare thus certified unsafe after finitely many commits.\n(For bins near the decision boundary, certification may\nnot occur, but their per-round regret is O(L√nw(T)),\nso their total contribution is small and accounted for\nby the margin term later.)\nWe now compute how\nmany commits are needed to certify a negative bin.\nLemma 6.3 (Samples for negative certification). Con-\nsider any t ∈[T] and B ∈B.\nOn G, if µB <\n−L√nw(T) and kB(t) >\n4c−1σ2\nw ln(2T 4)\n(µB+L√nw(T ))2 , then bin B\nis certified negative at time t.\nProof idea. On the good event G, certification in bin B\noccurs when ˆµB +γ(kB(t))+L√nw(T) < 0. Using the\nworst-case deviation ˆµB = µB + γ(kB(t)), this reduces\nto µB + 2γ(kB(t)) + L√nw(T) < 0. Plugging in the\ndefinition of γ(kB(t)) and solving for k yields the stated\nnumber of commits needed for certification.\n□\nNext, we bound the geometry of the trusted region. By\nconstruction, B consists of all bins intersecting the ball\nof radius m(T). Consequently, their union S\nB∈B B is\ncontained within a slightly larger ball. This enlarged\nregion will be useful both for bounding how negative\nrewards can be (via Lipschitz continuity) and for\ncontrolling the number of bins (via volume packing).\nLet v1 be the volume of the unit ball {x ∈X : ∥x∥≤1}.\nLemma 6.4 (Trusted cover is a slightly larger ball).\nEvery x ∈S\nB∈B B satisfies ∥x∥≤R(T) = m(T) +\n√nw(T).\nWe now bound the regret from a truly unsafe bin before\nit is certified. Let ∆t := maxy∈{0,1} r(xt, y) −r(xt, yt)\nbe the instantaneous regret at time t.\nLemma 6.5 (Per-bin commit regret). On G, for any\nB ∈B with kB(T) ≥1 and µB < −(2L√n + 1)w(T),\nX\nt: xt∈B, yt=1\n∆t ≤2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n.\nProof idea. Lem. 6.3 shows that a negative bin is certi-\nfied after O(σ2\nw/(µB + L√nw(T))2) commits. Each\nsuch commit incurs at most O(|µB|) regret, but\nLem. 6.4 ensures that µB ≥−LR(T), so the loss per\ncommit is bounded. Multiplying the number of pre-\ncertification commits by the maximum per-step regret\nyields the stated bound.\n□\nNow that we have controlled the regret contribution\nof each individual bin, we sum across all bins that are\never visited and include the effect of near-margin bins\n(those with µB close to zero). Such bins may never\nbe certified, but their regret per commit is small, so\ntheir total contribution is still controlled.\nLemma 6.6 (Total commit regret inside the trusted\nregion). On G,\nX\nt: yt=1\n∆t ≤v1R(T)n\nw(T)n\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n+ (3L√n + 1)w(T)T.\nProof idea. We partition commits into bins with de-\ncisively negative mean and those near the decision\n\nRisk-Sensitive Abstention in Bandits\nboundary. For µB well below zero, Lem. 6.5 bounds\nthe regret before certification. Summing over all such\nbins gives at most |B| times the per-bin cost, and by\nthe packing bound |B|w(T)n ≤v1R(T)n, this gives the\nfirst term. For bins near the margin, the algorithm may\ncontinue committing longer, but Lipschitzness bounds\nthe per-round regret by (3L√n + 1)w(T), giving the\nsecond term after T rounds.\n□\nLem. 6.6 completes the analysis of commit regret\ninside the trusted region. Combining this with the\nabstention regret outside the ball of radius m(T), we\nyield the final rate in Thm. 5.2 as follows.\nProof idea of Thm. 5.2.\nRegret decomposes into (i)\nabstention outside the trusted ball; (ii) commits inside.\nFor (i), each input with ∥xt∥> m(T) contributes at\nmost 1, giving T ¯ν(m(T)), which is sublinear for any\nfixed ν since ¯ν(ln T) →0. For (ii), Lipschitz conti-\nnuity bounds the within-bin variation, and a uniform\nconcentration event (probability 1 −O(T −2)) ensures\nempirical means stay within confidence radii. Nega-\ntive bins are certified after O(σ2\nw/margin2) commits,\nso each contributes at most O(LR(T)+σ2\nw log T/w(T))\nregret. Summing across O((m(T)/w(T))n) bins gives\n˜O\n\u0010\nR(T)n\u0000LR(T)w(T)−n + σ2\nww(T)−(n+1)\u0001\u0011\n,\nand bins near the decision boundary contribute an\nadditional O(w(T)T).\n□\nIf we ignore log factors, the leading terms trade off\nR(T)nw(T)−(n+1)\n|\n{z\n}\nvariance-driven\nvs.\nw(T)T\n| {z }\nmargin-driven\n.\nBalancing these yields the optimal choice w(T) ≍\nT −1/(n+2). Independently, the radius m(T) trades off\nthe abstention term T ¯ν(m(T)) against the growth of\nthe volume factor R(T)n. Choosing m(T) = ln T makes\nT ¯ν(m(T)) sublinear for any fixed ν (since ¯ν(ln T) →0)\nwhile increasing R(T) only logarithmically.\n7\nCONCLUSION\nIn this work, we introduced a formal model for safe\nlearning under distribution shift in contextual bandits\nwith catastrophic tails, provided impossibility results\nthat clarify when sublinear regret is unattainable, and\ngave a cautious risk-sensitive algorithm with sublinear\nregret under suitable conditions. Our work has several\nlimitations, which also provide directions for future\nwork, including handling richer structure beyond\nLipschitz continuity, incorporating adaptive or learned\nmetrics, and extending the analysis to non-i.i.d. inputs\nor worst-case sequences.\nOur\nregret\nbound\ncan\nbe\nclose\nto\nlinear.\nIn Thm. 5.2, the abstention term T ¯ν(ln T) can dom-\ninate for heavy-tailed inputs (e.g., power laws). This\nis the price of caution: avoiding catastrophic far OOD\ncommits requires systematic abstention in the tails, and\nthe resulting regret can be unavoidable (see the impos-\nsibility in Thm. 4.2). Moreover, while the bound is sub-\nlinear for every fixed n, the exponent (n+1)/(n+2) →1\nas n →∞, which is a standard curse of dimensionality\nin Lipschitz contextual bandits (Slivkins et al., 2019,\nThms. 4.11–4.12); see also Plaut et al. (2025a, Thm. 10).\nWhile we do not expect to remove these dependencies\nentirely, future work could improve rates. The sim-\nplicity of Algorithm 1 is appealing but ignores useful\ninformation: commits inform not only their own bin but\nalso nearby bins via Lipschitz continuity, and certifying\na bin as positive could justify expanding the trusted re-\ngion around it. Additional structural assumptions, such\nas margin/low-noise conditions, intrinsic low dimension-\nality, or smoothness beyond Lipschitz, could also help.\nAssumptions may not always hold. Our guaran-\ntees here rely on i.i.d. inputs and Lipschitz continuity\nof the commit reward. In practice, inputs may drift\nor exhibit temporal dependence, and rewards may be\nonly piecewise smooth or even non-smooth. Extending\nthe analysis to weaker smoothness conditions or\ndrifting processes is an important direction. Moreover,\nAlgorithm 1 assumes knowledge of L, σ2, and T.\nWhile knowledge of T can be handled by the standard\ndoubling trick (see Slivkins et al. (2019, §1.5)), L and\nσ2 may be unknown. Thus, developing parameter-free\n(or adaptively tuned) algorithms that remain cautious\nwould increase robustness.\nNo unconditionally irreparable errors. Obtaining\nregret −T on a single time step is irreparable in the\nsense that it automatically implies linear regret on that\nrun. However, errors in our model are only irreparable\nfor a fixed T: for any error, there exists a large enough\nT that the error is no longer catastrophic. It may be\nworth considering alternative models of catastrophe\nsuch as inescapable trap states in MDPs which do\nallow for errors that are unconditionally catastrophic.\nBroader impact. This work is motivated by safety\nconcerns in the deployment of learning systems in high-\nstakes domains. We provide theoretical justification for\nabstention as a mechanism for averting catastrophic\nerrors under distribution shift, and abstention is also a\npractical choice for deployed systems. Agents that can\ndefer action when uncertain may be safer and more\ntrustworthy, but abstention mechanisms must be de-\nsigned carefully to avoid consequences such as excessive\nconservatism or over-reliance on human supervision.\n\nSarah Liaw, Benjamin Plaut\nAcknowledgments\nThis work was supported by a gift from Open Philan-\nthropy to the Center for Human-Compatible AI (CHAI)\nat UC Berkeley. This work was conducted while Sarah\nLiaw was a research intern at CHAI. We would also\nlike to thank Vamshi Bonagiri and Pavel Czempin for\nhelpful discussions and feedback.\nReferences\nAgrawal, S., Juneja, S. K., and Koolen, W. M. (2021).\nRegret minimization in heavy-tailed bandits.\nIn\nConference on Learning Theory, pages 26–62. PMLR.\nAltman, E. (1999). Constrained Markov Decision Pro-\ncesses, volume 7. CRC Press.\nAzar, M. G., Osband, I., and Munos, R. (2017). Min-\nimax regret bounds for reinforcement learning. 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Conservative bandits. In International\nConference on Machine Learning (ICML).\nYang, J., Jin, T., and Tan, V. Y. F. (2024). Multi-\narmed bandits with abstention.\nZhao, W., He, T., Chen, R., Wei, T., and Liu, C. (2023).\nState-wise Safe Reinforcement Learning: A Survey.\nvolume 6, pages 6814–6822. ISSN: 1045-0823.\n\nSarah Liaw, Benjamin Plaut\nA\nPROOF OF MAIN RESULT\nA.1\nProof notation\nThe proof will use the following notation:\n1. The true mean of bin B is µB = Ex∼ν[r(x, 1) | x ∈B].\n2. Let kB(t) denote the value of the variable kB in Algorithm 1 at the end of time step t.\n3. Let ˆµB(k) denote the value of the variable ˆµB in Algorithm 1 after k rewards from bin B have been observed.\n4. Let σw =\np\nnL2w(T)2 + σ2 for brevity.\n5. Define the confidence radius γ(k) =\nq\nc−1σ2w ln(2T 4)\nk\nwhere c is the absolute constant from Lemma A.2.\n6. Define the good event G = {∀t ∈[T], ∀B ∈B where kB(t) > 0 : |ˆµB(kB(t)) −µB| ≤γ(kB(t))}.\n7. A bin B is certified negative at time t if ˆµB(kB(t)) + γ(kB(t)) + L√nw(T) < 0.\n8. Let ¯ν be the radial survival function of ν. That is, for any y ∈R≥0, ¯ν(y) = Prx∼ν[∥x∥≥y].\n9. Let ∆t = maxy∗∈{0,1} r(xt, y∗) −r(xt, yt) be the single-step regret at time t.\n10. Let v1 be the volume of the unit ball {x ∈X : ∥x∥≤1}.\n11. Let R(T) = m(T) + √nw(T). This will be the maximum distance of any input in ∪B∈BB from the origin.\nLemma A.1 (Hoeffding’s Lemma, Lemma 2.2 in Boucheron et al., 2013). If Z is a random variable taking values\nin the bounded interval [a, b], then Z is ( b−a\n2 )-subgaussian.\nLemma A.2 (Hoeffding’s inequality, subgaussian version). Let X1, . . . , Xk be independent random variables with\nmean zero, where each Xi is σi-subgaussian for some σi > 0. Then there exists an absolute constant c > 0 such\nthat for any ε > 0,\nPr\n\"\f\f\f\f\f\nk\nX\ni=1\nXi\n\f\f\f\f\f > ε\n#\n≤2 exp\n \n−\ncε2\nPk\ni=1 σ2\ni\n!\nLemma A.3. If x ∈B ∈B, then |r(x, 1) −µB| ≤L√nw(T).\nProof. We must prove that r(x, 1) ≥µB −L√nw(T) and r(x, 1) ≤µB + L√nw(T). Let r−= infx′∈B r(x′, 1)\nand r+ = supx′∈B r(x′, 1). Then r−≤µB ≤r+ and r−≤r(x, 1) ≤r+. Next, for any ε > 0, there exists\nx−, x+ ∈B such that r(x−, 1) −ε < r−and r(x+, 1) + ε > r+ (if not, this contradicts r−and r+ being the\ninfimum and supremum). Then r(x, 1) and µB belong to the interval [r−, r+], which is a subset of the interval\n[r(x−, 1) −ε, r(x+, 1) + ε].\nSince x−and x+ belong to the same n-hypercube with side length w(T), ∥x−−x+∥≤√nw(T). Then by\nLipschitz continuity, |r(x+, 1) −r(x−, 1)| = r(x+, 1) −r(x−, 1) ≤L√nw(T). Therefore r(x, 1) and µB belong to\nthe same interval of length L√nw(T) + 2ε, so |r(x, 1) −µB| ≤L√nw(T) + 2ε. Since this holds for all ε > 0, we\nmust have |r(x, 1) −µB| ≤L√nw(T).\nLemma 6.1 (Per-bin concentration). For any B ∈B and t ∈[T], Pr [|ˆµB(kB(t)) −µB| > γ(kB(t))] ≤T −4.\nProof. Fix a bin B and t ∈[T]. Let k = kB(t) for brevity and let t1 < t2 < . . . tk be the set of time steps i ≤t\nwith xi ∈B and yi = 1. For each j ∈[k], define Zj = r(xtj, 1) −µB. The idea is to apply Lemma A.2 to\nZ1, . . . , Zk and ηt1, . . . , ηtk. To do so, we establish key three properties of Z1, . . . , Zk.\nProperty 1. Fix some j ∈[k]. Since xtj ∈B, Lemma A.3 implies that |r(xtj, 1) −µB| ≤L√nw(T). Thus\nthe random variable Zj = r(xtj, 1) −µB is always belongs to an interval of length 2L√nw(T): specifically,\n[−L√nw(T), L√nw(T)]. Then by Lemma A.1, Zj is (L√nw(T))-subgaussian.\nProperty 2. Observe that the algorithm’s behavior does not distinguish between inputs in the same bin. Thus for\nany i ∈[T], conditional on xi ∈B, xi is independent of y1, . . . , yi (though clearly not independent in general).\nBy assumption, xi is independent of x1, . . . , xi−1. Therefore\nE[r(xtj, 1)] = E\n\u0002\nr(xi, 1) | xi ∈B and yi = 1 and kB(i) = j −1\n\u0003\n= E[r(xi, 1) | xi ∈B]\n\nRisk-Sensitive Abstention in Bandits\n=\nE\nx∼ν[r(x, 1) | x ∈B]\n= µB\nTherefore E[Zj] = E[r(xtj, 1)] −µB = 0. Also, since xt1, . . . , xtk are iid, Z1, . . . , Zk are also iid.\nProperty 3. We claim that Z1, . . . , Zk are also independent of ηt1, . . . , ηtk. One way to see this is imagine that\nat t = 0, for each bin B, we take k samples ηt1, . . . , ηtk which are independent from each other and also from\nZ1, . . . , Zk. Then on each time step i ∈[t], if xi ∈B, we let ηi be equal to the next ηtj that has not already been\nused. This process is equivalent to randomly sampling ηi on each time step, and makes it clear that ηt1, . . . , ηtk\nare independent from Z1, . . . , Zk.2\nThus Z1, . . . , Zk, ηt1, . . . , ηtk are independent random variables with mean zero, where each Zj is (L√nw(T))-\nsubgaussian and each ηtj is σ-subgaussian. Then by Lemma A.2, for any ε > 0,\nPr\n\n\n\f\f\f\f\f\nk\nX\nj=1\nZj +\nk\nX\nj=1\nηtj\n\f\f\f\f\f > ε\n\n≤2 exp\n \n−\ncε2\nPk\nj=1(L√nw(T))2 + Pk\nj=1 σ2\n!\n= 2 exp\n\u0012\n−cε2\nkσ2w\n\u0013\nNote that the ˆµB(k) =\n1\nk\nPk\nj=1 rtj =\n1\nk\nPk\nj=1(r(xtj, 1) + ηtj). Then ˆµB(k) −µB =\n1\nk\nPk\nj=1(Zj + ηtj). Set\nε = kγ(k) =\np\nkc−1σ2w ln(2T 4) to get\nPr [|ˆµB(k) −µB| > γ(k)] = Pr\nh\nk|ˆµB(k) −µB| >\np\nkc−1σ2w ln(2T 4)\ni\n= Pr\n\n\n\f\f\f\f\f\nk\nX\nj=1\nZj +\nk\nX\nj=1\nηtj\n\f\f\f\f\f >\np\nkc−1σ2w ln(2T 4)\n\n\n≤2 exp(−ln(2T 4))\n= 2 exp\n\u0012\nln\n\u0012 1\n2T 4\n\u0013\u0013\n= T −4\nas required.\nLemma 6.2 (Uniform concentration bound). With probability at least 1−T −2, the good event G holds. Equivalently,\nPr(¬G) ≤T −2.\nProof. Let J be the the number of bins that receive at least one commit, i.e., J = |{B ∈B : ∃t ∈[T] s.t. xt ∈\nB, yt = 1}|. For each j ∈[J], let Bj be the jth bin to receive a commit. Then\nPr[¬G] = E[Pr[¬G | J, B1, . . . , BJ]]\n(Law of total expectation)\n= E\n\nPr\n\n\nJ[\nj=1\nT[\nt=1\n{|ˆµBj(kB(t)) −µBj| > γ(kB(t))}\n\n\n\f\f\f J, B1, . . . , BJ\n\n\n(Direct negation)\n≤E\n\n\nJ\nX\nj=1\nT\nX\nt=1\nPr[|ˆµBj(kB(t)) −µBj| > γ(kB(t))]\n\f\f\f J, B1, . . . , BJ\n\n\n(Union bound)\n≤E\n\n\nJ\nX\nj=1\nT\nX\nt=1\nT −4 \f\f\f J, B1, . . . , BJ\n\n\n(Lemma 6.1)\n≤E\nh\nT −2 \f\f\f J, B1, . . . , BJ\ni\n(J ∈[T])\n≤T −2\n(Expectation of a constant)\nas required.\n2This is similar to the “reward tape” argument used in Section 1.3.1 of Slivkins et al. (2019).\n\nSarah Liaw, Benjamin Plaut\nLemma 6.3 (Samples for negative certification). Consider any t ∈[T] and B ∈B. On G, if µB < −L√nw(T)\nand kB(t) >\n4c−1σ2\nw ln(2T 4)\n(µB+L√nw(T ))2 , then bin B is certified negative at time t.\nProof. Note that µB < −L√nw(T) implies that the denominator is well-defined. By assumption on kB(t),\nγ(kB(t)) =\ns\nc−1σ2w ln(2T 4)\nkB(t)\n<\nr\n(µB + L√nw(T))2\n4\n= |µB + L√nw(T)|\n2\n= −µB + L√nw(T)\n2\nBy definition of G, we have −γ(kB(t)) ≤ˆµB(kB(t)) −µB ≤γ(kB(t)), so\nˆµB(kB(t)) + γ(kB(t)) + L√nw(T) ≤µB + 2γ(kB(t)) + L√nw(T)\n< µB −(µB + L√nw(T) + L√nw(T)\n= 0\nso B is certified negative at time t.\nLemma 6.4 (Trusted cover is a slightly larger ball). Every x ∈S\nB∈B B satisfies ∥x∥≤R(T) = m(T) + √nw(T).\nProof. If x ∈B for some B ∈B, there must exist x′ ∈B such that ∥x′∥≤m(T). The maximum distance between\nany pair of points in an n-cube with side length w(T) is √nw(T). Thus by the triangle inequality, x satisfies\n∥x∥≤∥x′∥+ ∥x −x′∥≤m(T) + √nw(T) = R(T)\nas required.\nLemma 6.5 (Per-bin commit regret). On G, for any B ∈B with kB(T) ≥1 and µB < −(2L√n + 1)w(T),\nX\nt: xt∈B, yt=1\n∆t ≤2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n.\nProof. Since µB < −(2√nL+1)w(T) < −L√nw(T) and G holds, Lemma 6.3 implies that B is certified negative on\nthe first time step t such that kB(t) >\n4c−1σ2\nw ln(2T 4)\n(µB+L√nw(T ))2 . Therefore |{t ∈[T] : xt ∈B, yt = 1}| ≤⌈4c−1σ2\nw ln(2T 4)\n(µB+L√nw(T ))2 ⌉≤\n1 +\n4c−1σ2\nw ln(2T 4)\n(|µB|−L√nw(T ))2 . Since |µB| ≥(2L√n + 1)w(T) ≥2L√nw(T), we have\n|µB| = |µB|\n2\n+ |µB|\n2\n≥|µB|\n2\n+ L√nw(T)\nso |µB| −L√nw(T) ≥|µB/2|.\nTherefore (|µB| −L√nw(T))2 ≥µ2\nB/4, so |{t ∈[T] : xt ∈B, yt = 1}| ≤\n1 + 16c−1σ2\nw ln(2T 4)\nµ2\nB\n.\nFor any t ∈[T] such that yt = 1, either yt = 1 is optimal, in which case the single-step regret ∆t is 0, or yt = 0\nis optimal, if which case ∆t = −r(xt, 1). If xt ∈B, Lemma A.3 implies that r(xt, 1) ≥µB −L√nw(T). Since\nµB < −L√nw(T), we have r(xt, 1) ≥2µB. Hence\nX\nt:xt∈B,yt=1\n∆t ≤\nX\nt:xt∈B,yt=1\n(−2µB)\n= |{t ∈[T] : xt ∈B, yt = 1}| · 2|µB|\n≤\n\u0012\n1 + 16c−1σ2\nw ln(2T 4)\nµ2\nB\n\u0013\n· 2|µB|\n\nRisk-Sensitive Abstention in Bandits\n= 2|µB| + 32c−1σ2\nw ln(2T 4)\n|µB|\n≤2|µB| + 32c−1σ2\nw ln(2T 4)\n(2√nL + 1)w(T)\n≤2|µB| + 32c−1σ2\nw ln(2T 4)\nw(T)\nwith the last step due to 2√nL + 1 ≥1. By Lemma 6.4, any x ∈B satisfies ∥x∥≤R(T). Thus by Lipschitz con-\ntinuity, r(x, 1) ≥r(0, 1) −LR(T) > −LR(T) for all x ∈B. ThusµB = Ex∼ν[r(x, 1) | x ∈B] ≥Ex∼ν[−LR(T)] =\n−LR(T), so\nX\nt:xt∈B,yt=1\n∆t ≤2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\nas required.\nLemma 6.6 (Total commit regret inside the trusted region). On G,\nX\nt: yt=1\n∆t ≤v1R(T)n\nw(T)n\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n+ (3L√n + 1)w(T)T.\nProof. For each t ∈[T], let B(t) denote the bin to which xt belongs. Partition the time steps with commits into\nS1 = {t ∈[T] : yt = 1 and µB(t) < −(2L√n+1)w(T)} and S2 = {t ∈[T] : yt = 1 and µB(t) ≥−(2L√n+1)w(T)}.\nLet B1 = {B ∈B : ∃t ∈S1 s.t. B(t) = B} be the set of bins associated with time steps in S1. Then we can write\nX\nt∈S1\n∆t =\nX\nB∈B1\nX\nt∈S1:B(t)=B\n∆t\n=\nX\nB∈B1\nX\nt:xt∈B,yt=1\n∆t\nThen by Lemma 6.5,\nX\nt∈S1\n∆t ≤\nX\nB∈B1\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n≤|B1|\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n≤|B|\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\nBy Lemma 6.4, every x ∈∪B∈BB satisfies ∥x∥≤R(T). Thus ∪B∈BB is fully contained within an n-ball of radius\nr. The volume of such a ball is v1R(T)n. Each bin in B has side length w(T) so has volume w(T)n. Furthermore,\nthe bins in B have no volume overlap, so the total volume of bins in B is w(T)n|B|. Then w(T)n|B| ≤v1R(T)n.\nTherefore\nX\nt∈S1\n∆t ≤v1R(T)n\nw(T)n\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\nNow consider any t ∈S2. By definition, xt ∈B(t) ∈B, so Lemma A.3 implies that r(xt, 1) ≥µB(t) −L√nw(T).\nSince µB(t) ≥−(2L√n + 1)w(T) by construction of S2, we have r(xt, 1) ≥−(3L√n + 1)w(T). Therefore\nX\nt∈S2\n\u0012\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, 1)\n\u0013\n≤\nX\nt∈S2\n(3L√n + 1)w(T)\n= |S2|(3L√n + 1)w(T)\n≤(3L√n + 1)w(T)T\n\nSarah Liaw, Benjamin Plaut\nPutting it all together,\nX\nt:yt=1\n∆t =\nX\nt∈S1\n\u0012\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, 1)\n\u0013\n+\nX\nt∈S2\n\u0012\nmax\ny∗∈{0,1} r(xt, y∗) −r(xt, 1)\n\u0013\n≤v1R(T)n\nw(T)n\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n+ (3L√n + 1)w(T)T\nas required.\nTheorem 5.2. In the stochastic setting with xt ∼ν i.i.d., Algorithm 1 with w(T) = T −1/(n+2) and m(T) = ln T\nsatisfies\nE[Reg(T)] ∈O\n\u0010\n(L + σ2)T\nn+1\nn+2 (ln T)n+1 + T ¯ν(ln T)\n\u0011\n.\nProof. First assume G holds. Let S3 = {t ∈[T] : yt = 1 and r(xt, 1) < r(xt, 0)} be the time steps where we\ncommitted but we should have abstained, and let S4 = {t ∈[T] : yt = 0 and r(xt, 0) < r(xt, 1)} be the time steps\nwhere we should have committed but we abstained. Lemma 6.6 bounds the regret of time steps in S3. Since we\nalways commit whenever xt ∈B for some B ∈B, S4 can only occur when xt ̸∈B for all B ∈B. By construction,\nany such xt satisfies ∥xt∥> m(T) (otherwise the bin containing xt would be in B). Also, r(xt, 1) −r(xt, 0) ≤1\nby assumption. Hence\nReg(T) =\nT\nX\nt=1\n∆t\n=\nX\nt∈S3\n∆t +\nX\nt∈S4\n∆t\n≤v1R(T)n\nw(T)n\n\u0012\n2LR(T) + 32c−1σ2\nw ln(2T 4)\nw(T)\n\u0013\n+ (3L√n + 1)w(T)T +\nT\nX\nt=1\n1(∥xt∥> m(T))\n= 2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ (3L√n + 1)w(T)T +\nT\nX\nt=1\n1(∥xt∥> m(T))\nTherefore\nE[Reg(T) | G] ≤2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ (3L√n + 1)w(T)T +\nT\nX\nt=1\nPr[∥xt∥> m(T)]\n= 2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ (3L√n + 1)w(T)T +\nT\nX\nt=1\n¯ν(m(T))\n= 2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ (3L√n + 1)w(T)T + T ¯ν(m(T))\nNow suppose G does not hold.\nConsider an arbitrary t ∈[T].\nIf yt = 0, then the regret at time t is at\nmost 1.\nIf yt = 1, we still have ∥xt∥≤R(T), so by Lipschitz continuity, r(xt, 1) ≥−LR(T).\nTherefore\nE[Reg(T) | ¬G] ≤T + LR(T)T. Lemma 6.2 implies that Pr[¬G] ≤T −2, so by the law of expectation,\nE[Reg(T)] = Pr[¬G] E[Reg(T) | ¬G] + Pr[G] E[Reg(T) | G]\n≤1\nT 2 · (T + LR(T)T) + 2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ (3L√n + 1)w(T)T + T ¯ν(m(T))\n∈O\n\u00121 + LR(T)\nT\n+ 2Lv1R(T)n+1\nw(T)n\n+ 32v1c−1σ2\nwR(T)n ln(2T 4)\nw(T)n+1\n+ L√nw(T)T + T ¯ν(m(T))\n\u0013\nWe now plug in w(T) = T\n−1\nn+2 and m(T) = ln T. Since limT →∞w(T) = 0, we have σ2\nw = nL2w(T)2 + σ2 ∈O(σ2).\nSimilarly, for any k ≥0, R(T)k = (ln(T) + √nT\n−1\nn+2 )k ∈O((ln T)k). Thus\nE[Reg(T)] ∈O\n\u0012L ln T\nT\n+ L(ln T)n+1\nT\n−n\nn+2\n+ σ2(ln T)n+1\nT\n−n−1\nn+2\n+ LT\n−1\nn+2 T + T ¯ν(ln T)\n\u0013\n\n= O\n\u0010\n(L + σ2)T\nn+1\nn+2 (ln T)n+1 + T ¯ν(ln T)\n\u0011\nas required.\n"},"cleaned":{"kind":"string","value":"Learning When Not to Learn: Risk-Sensitive Abstention in Bandits with Unbounded Rewards Sarah Liaw∗ Benjamin Plaut∗ Harvard University -stakes AI applications, even a single action can cause irreparable damage. However, nearly all of sequential decision-making theory assumes that all errors are recoverable (e.g., by bounding rewards). Standard bandit algorithms that explore aggressively may cause irreparable damage when this assumption fails. Some prior work avoids irreparable errors by asking for help from a mentor, but a mentor may not always be available. In this work, we formalize a model of learning with unbounded rewards without a mentor as a two-action contextual bandit with an abstain option: at each round the agent observes an input and chooses either to abstain (always 0 reward) or to commit (execute a preexisting task policy). Committing yields rewards that are upper-bounded but can be arbitrarily negative, and the commit reward is assumed Lipschitz in the input. We propose a caution-based algorithm that learns when not to learn: it chooses a trusted region and commits only where the available evidence does not already certify harm. Under these conditions and i.i.d. inputs, we establish sublinear regret guarantees, theoretically demonstrating the effectiveness of cautious exploration for deploying learning agents safely in high-stakes environments. 1 INTRODUCTION With AI becoming ubiquitous, many learning systems are now deployed in unpredictable, safety-critical domains, such as process control and manufacturing robotics, autonomous driving, and surgical assistance. In these settings, a single ill-chosen action can cause irreparable and lasting damage with no opportunity ∗Equal contribution. for subsequent recovery. For instance, a self-driving car cannot compensate for a deadly crash by later driving more safely, nor can a medical robot undo a fatal mistake during surgery. Following Plaut et al. (2025a,b), we refer to such irreparable errors as catastrophes. Despite the risks such deployments pose, there is limited work (and limited theoretical work in particular) on how an agent can learn without ever incurring an irreparable error. The possibility of catastrophes challenges standard frameworks for sequential decision making, especially the familiar notion of optimism under uncertainty. Optimism effectively assumes that early mistakes can be offset (or be compensated for) by later gains, an assumption that is inappropriate when errors are irrecoverable. Instead, these settings call for pessimism under uncertainty: when evidence is insufficient, prefer inaction to risky action. One approach to mitigate these problems is to let the agent ask for help from a mentor in unfamiliar or risky situations. Such human-in-the-loop oversight can block unsafe actions and prevent irreparable errors (even if ordinary, recoverable errors still occur). However, this approach depends on the availability of a capable mentor, which can be costly or impractical at scale. This motivates a mentor-free alternative: can an agent avoid irreparable errors on its own by acting cautiously when inputs appear unfamiliar? We propose a model of learning in the presence of irreparable costs without a mentor but with an option to abstain from action. The key question is when to abstain, i.e., when not to learn. To focus on this question, we assume the agent has previously learned a baseline policy that works well in-distribution but behaves unpredictably elsewhere. This allows us to streamline the model to two actions: abstain (do nothing) and commit (follow the baseline policy). Abstaining yields a deterministic safe reward r(x, 0) = 0, while committing yields a reward r(x, 1) ∈(-∞, 1]1. 1The asymmetric bounds on the commit reward reflect that a single action can be catastrophic, whereas it is rare for a single action to yield arbitrarily large benefit. 16 Oct 2025 Risk-Sensitive Abstention in Bandits We treat the origin as fully \"in-distribution\" and assume the baseline policy is beneficial there: r(0, 1) > 0. We use the distance from the origin ∥x∥as a measure of how out-of-distribution (OOD) an input is. The commit reward is assumed L-Lipschitz, capturing the idea that similar inputs yield similar outcomes. For our main results we focus on a fixed distribution ν; for our impossibility results we also consider a T-dependent distribution νT . We formalize the tension between exploration and safety via two negative results. First, in the worst case, any algorithm that begins by always exploring (i.e., commits on the first round regardless of the input) can suffer infinite expected regret (Thm. 4.1). Second, when every input lies uniformly far OOD, there is no safe way to explore to identify a beneficial committing region, and sublinear regret is impossible (Thm. 4.2). Together, these results delineate both the necessity and the limits of caution. Motivated by this perspective, we develop a cautionbased algorithm that learns only when it can guarantee that an error is not catastrophic (which essentially corresponds to not-too-OOD inputs). This approach yields sublinear expected regret for i.i.d. inputs from any fixed distribution, with bounds that also reflect how often the agent encounters far OOD inputs, while prioritizing the avoidance of irreparable errors. Contributions. Our contributions can be summarized as follows: 1. We introduce a formal model of learning with irreparable costs and no external mentor. 2. We prove two impossibility results that delineate the necessity and limits of caution. 3. We develop a caution-based algorithm that achieves sublinear regret for any fixed input distribution. Organization. §3 introduces the formal model and notation. §4 presents the impossibility results (Thms. 4.1 and 4.2) and their implications for exploration. § 5 describes our caution-based learning algorithm and states the main regret bound. § 6 outlines the proof strategy and supporting lemmas. 2 RELATED WORK Most prior work on sequential decision-making and safe exploration focuses on settings where errors are ultimately recoverable; here we contrast this with our setting where individual actions can cause irreparable harm. 2.1 Sequential decision-making when all errors are recoverable The literature on sequential decision-making is vast, spanning bandit problems, reinforcement learning, and online learning. See Slivkins et al. (2019), Sutton et al. (1998), and Cesa-Bianchi and Lugosi (2006) for introductions to these (somewhat overlapping) topics, respectively. However, nearly all of this work assumes explicitly or implicitly that any error can be recovered from. This assumption enables the agent to ignore risk and simply try all possible behaviors, since no matter how badly it performs in the short term, it can always eventually make up for it. Indeed, most sequential decision-making algorithms with formal regret bounds have this general structure. This assumption can manifest in different ways. In bandit settings, it suffices to assume that rewards are bounded (or at least have bounded expectation). This assumption implies that the expected regret from any action on any time step is always bounded, which is sufficient for the risk-agnostic exploration mentioned above. In contrast, we allow unbounded negative rewards so that actions can be arbitrarily costly. Indeed, our first negative result (Thm. 4.1) relies on the expected regret for a single action potentially being infinite in our model. In Markov Decision Processes (MDPs), the agent's actions determine the next state via a transition function, so in addition to bounded rewards, one typically assumes that either the environment is reset at the start of each \"episode\" (e.g., Azar et al., 2017) or that any state is reachable from any other (e.g., Jaksch et al., 2010). The dependence of standard MDP algorithms on these assumptions was observed by Moldovan and Abbeel (2012a); Cohen et al. (2021), among others. Regardless of the specific form of this assumption, it clearly does not hold in safety-critical contexts where a single action can be catastrophic. 2.2 Safe exploration These issues have motivated a wide field of safe exploration. A full survey is beyond the scope of this paper (see Garc ́ıa and Fern ́andez, 2015; Gu et al., 2024; Krasowski et al., 2023; Tan et al., 2022 for surveys), so we cover only the most relevant prior work. Avoiding irreparable errors while learning has also been studied empirically across multiple domains (e.g., Saunders et al., 2017; Moldovan and Abbeel, 2012b; Wachi et al., 2023; Zhao et al., 2023; Perkins and Barto, 2003), but here we focus on theoretical work, which is most relevant to our setting. Safe exploration is modeled in two main ways. The first Sarah Liaw, Benjamin Plaut approach is to require the agent to satisfy some sort of constraint in addition to maximizing reward. The constraint can be entirely separate from reward, as in the case of constrained MDPs (Altman, 1999), or they can be related to the reward (e.g., the agent's reward must always exceed some baseline). When zero or near-zero constraint violation is required, these formalisms do capture the possibility of irreparable errors. The second approach treats reward as the sole objective, with safety as a necessary but not sufficient property for maximizing reward. Here, irreparable errors correspond to either unboundedly negative rewards (our work falls into this category) or inescapable \"trap\" states with poor reward. An agent that obtains very negative rewards or enters trap states clearly cannot obtain high reward. Both of these models must contend with a fundamental obstacle: how does one learn which actions are catastrophic without trying those actions directly? This can be formalized by the so called \"Heaven or Hell problem\". Suppose there are two available actions, where one has unbounded positive reward and the other has unbounded negative reward. In this case, the agent can do no better than simply guessing and can never guarantee good regret. This problem shows that some sort of additional assumption is necessary for any meaningful regret guarantees. Below, we categorize work within safe exploration based on which assumption(s) it uses for this purpose. Full prior knowledge. Perhaps the simplest approach is to assume that the agent knows the precise safety constraint upfront (see Zhao et al., 2023 for a survey). This immediately resolves the Heaven or Hell problem; indeed, it eliminates the need for the agent to \"learn when not to learn\" at all. However, full knowledge of the safety constraint may not hold in practice. In contrast, we only assume that the (1) baseline policy performs well in-distribution and (2) the agent can always safely abstain. Learning constraints using a safe fallback action. There is a growing body of work which shares our assumption of a safe fallback action. Liu et al. (2021); Stradi et al. (2024) use this approach in the constrained MDP model, while Wu et al. (2016); Kazerouni et al. (2017); Lin et al. (2022); Chen et al. (2022) require the reward to exceed a fixed baseline in a bandit model. These papers generally rely on a pair of subtle but crucial assumptions to obtain zero constraint violation: (1) the constraint violation on any given time step is bounded and (2) the baseline policy satisfies the constraints with a known amount of slack (this is called Slater's gap, although not all of the above papers use this term). This combination of assumptions enables the agent to still explore aggressively with some known probability. Furthermore, the resulting bounds typically depend inversely on Slater's gap. Our work is complementary to each of these two assumptions. First, rather than assuming global boundedness, we assume that rewards decrease at a bounded rate, i.e., rewards are Lipschitz continuous. Second, rather than dependence on the reward or cost function (in the form of Slater's gap), our bounds depend on the input distribution: specifically, our bounds degrade as the agent sees more OOD inputs. Our approach may be more or less realistic depending on the specific context, but it notably diverges from the typical way fallback actions are utilized. Asking for help. Perhaps the most common approach in this model is relying on external supervision. This is a growing body of work which uses limited queries to a mentor to prove formal regret guarantees in the presence of irreversible dynamics (Cohen et al., 2021; Cohen and Hutter, 2020; Kosoy, 2019; Maillard et al., 2019; Plaut et al., 2025b,a). However, as the number of deployed AI systems continues to grow, it may be impractical for each one to have a human supervisor. Even in cases where external help will eventually become available, the agent may need to behave safely on its own in the short-term. These considerations motivate our study of how to learn safely in the absence of external help. 2.3 Other related work We briefly discuss some topics that are less directly relevant but still worth mentioning. One is the heavytailed bandit model (Bubeck et al., 2013; Agrawal et al., 2021), which studies the case where reward distributions are not subgaussian and thus less predictable. While this model does incorporate elements of safety, as long as the expected reward from any action is bounded, risk-agnostic exploration remains valid (as discussed above). Another topic adjacent to our work is the standard Lipschitz bandit model with bounded rewards and bounded domain (see, e.g., Chapters 4 and 8 of Slivkins, 2011). This work shares some similarities with ours, like the algorithmic use of discretization. However, the core of our paper is removing the boundedness assumptions, which introduces a host of new challenges. Finally, there is complementary work on abstention with bounded rewards (Neu and Zhivotovskiy, 2020; Yang et al., 2024). While this line of work also demonstrates the benefits of abstention, it does not address the possibility of irreparable errors. 3 PRELIMINARIES We study a two-action contextual bandit model in which, on each round, the agent observes an input and Risk-Sensitive Abstention in Bandits chooses either to commit, thus executing a fixed task policy that may yield risky outcomes, or to abstain, receiving a safe default reward of zero. In this section, we introduce the formal notation and assumptions used throughout. For k ∈N, let [k] = {1, . . . , k}. Let X = Rn be the input space, T ∈N be the time horizon, and ∥· ∥be the Euclidean norm (though one could also consider a more general metric space). On each time step t ∈[T], the agent observes an input xt ∈X, chooses an action yt ∈{0, 1}, and receives a (noisy) scalar reward; the precise noise assumptions are stated below. Actions and Rewards. We interpret yt = 0 as \"abstaining\", a safe default which deterministically yields r(xt, 0) = 0 for any xt ∈X. We interpret yt = 1 as \"committing\", which executes a preexisting policy whose reward r(xt, 1) may be arbitrarily negative (catastrophic) but is assumed to have a constant upper bound (rescaled to 1 without loss of generality). This captures the asymmetry of high-stakes settings where catastrophic losses can be unbounded in magnitude, whereas gains typically saturate. Input models. We assume inputs are i.i.d. draws from an unknown distribution ν on X, i.e., x1, . . . , xT i.i.d. ∼ν. We typically take ν to be fixed, but in our impossibility results we also consider the case of T-dependent ν (denoted νT ). We assume bandit feedback: the agent observes only the realized reward of its chosen action. Abstaining provides no information about the counterfactual commit reward r(xt, 1), so the agent cannot \"learn by abstaining\". Formally, at round t the learner observes rt = r(xt, yt) + ηt, where (ηt)T t=1 are i.i.d. zero-mean σ-subgaussian noise variables, independent of (xt) and of the learner's internal randomness (specified formally in Def. 3.1). Definition 3.1 (σ-subgaussian). A random variable Z is σ-subgaussian if E[exp(λ(Z -E[Z]))] ≤exp σ2λ2 2 for all λ ∈R. Equivalently, Z -E[Z] has tails that are dominated by a centered Gaussian with variance proxy σ2. Regularity. We make two assumptions on the reward function: (i) the commit reward r(·, 1) is L-Lipschitz in the Euclidean norm, i.e., there exists L > 0 such that for all x, x′ ∈X, |r(x, 1) -r(x′, 1)| ≤L∥x -x′∥. This is a standard smoothness condition in Lipschitz bandit models (see, e.g., Slivkins et al., 2019) and captures the intuition that similar inputs yield similar commit rewards. Since r(x, 0) ≡0, the abstain reward is 0-Lipschitz. (ii) The in-distribution baseline input yields strictly positive reward when committing, i.e. r(0, 1) > 0. This guarantees that committing is beneficial somewhere (at the origin); without it, the optimal policy would be to always abstain and cautious learning would be impossible. Objective. The agent's goal is to minimize its (expected) regret, which is the difference between its cumulative reward and the optimal cumulative reward. Formally, define Reg(T) = T X t=1 max y∗∈{0,1} r(xt, y∗) -r(xt, yt) . We take the expectations over the input process (in the stochastic model), the observation noise, and the learner's internal randomness. The goal is to achieve sublinear expected regret, i.e., E[Reg(T)] = o(T), equivalently E[Reg(T)]/T →0 as T →∞. 4 THE VIRTUES AND LIMITS OF CAUTION In this section, we provide two impossibility results that demonstrate the importance and limitations of caution in high-stakes, unbounded reward bandits. First, caution is necessary: if an agent commits with non-negligible probability on inputs that are far OOD, catastrophic tail losses dominate-indeed, even a single risk-agnostic exploratory commit can incur infinite expected regret. This kind of \"incautious exploration\" is exactly how standard bandit algorithms behave when they begin by pulling every arm at least once. Second, caution has limits: when the input stream is uniformly far OOD, there is no way to explore cautiously to identify a beneficial committing region without risking catastrophe. In such settings, sublinear regret is not possible and the optimal strategy is to abstain on every time step. Theorem 4.1 (The need for caution). Let ν be any distribution over X such that Ex∼ν[∥x∥] = ∞and assume x1, . . . , xT ∼ν i.i.d. Then there exists a reward function r such that any algorithm which always commits on the first time step satisfies E[Reg(T)] = ∞. Proof. Define r(x, 1) = 1 -L∥x∥for all x ∈X. Then E[Reg(T)] = E \" T X t=1 max y∗∈{0,1} r(xt, y∗) -r(xt, yt) # ≥E max y∗∈{0,1} r(x1, y∗) -r(x1, y1) ≥E[0 -(1 -L∥x1∥)] Sarah Liaw, Benjamin Plaut = L E x∼ν[∥x∥] -1 = ∞ as required. The proof can easily be modified to handle the cases where the first commit is taken with constant probability (rather than probability 1) or where the algorithm abstains for a constant number of initial rounds. Essentially, this negative result applies to any algorithm that is not cautious, i.e., that explores without considering how OOD xt is. However, caution can only get us so far. While it prevents catastrophic first commits, some exploration is necessary to obtain sublinear regret. If all inputs are far OOD, then there is no safe way to explore, so the agent has no choice but to always abstain. Equivalently, this can be phrased by considering i.i.d. inputs from a Tdependent distribution νT supported on {x : ∥x∥= T}. Theorem 4.2 (The limits of caution). Let νT be any distribution supported on {x : ∥x∥= T}, and suppose x1, . . . , xT i.i.d. ∼νT . Then no algorithm can guarantee E[Reg(T)] ∈o(T). Proof. Define r-(x, 1) := 1 -L∥x∥and r+(x, 1) := 1, with r±(x, 0) := 0. Since we only care about asymptotics, we can restrict our attention to T > 1/L. Then for ∥xt∥= T, optimal behavior for r+ is to always commit, while optimal behavior for r-is to always abstain. We show maxr∈{r-,r+} E[Reg(T)] ∈Ω(T). To do so, we use a mild version of the probabilistic method. Let U(r-, r+) be the uniform distribution over {r-, r+}. It suffices to show Er∼U E[Reg(T)] ∈Ω(T), where the second expectation is over x1, . . . , xT and y1, . . . , yT . Let E be the event that the agent ever commits. If E holds, there exists i ∈[T] with yi = 1. Since yi is independent of r, E r E[Reg(T) | E] = E r E \" T X t=1 max y∗∈{0,1} r(xt, y∗) -r(xt, yt) | E # ≥Pr[r = r-] E max y∗∈{0,1} r(xi, y∗) -r(xi, yi) | r = r- = LT -1 2 On the other hand, if E does not occur, then E r E[Reg(T) | ¬E] ≥Pr[r = r+] E \" T X t=1 max y∗∈{0,1}r+(xt, y∗)-r+(xt, yt) |¬E # ≥T 2 . Then by the law of total expectation, E r E[Reg(T)] = Pr[E] E r E[Reg(T) | E] + Pr[¬E] E r E[Reg(T) | ¬E] ≥min Pr[E], Pr[¬E] min LT -1 2 , T 2 ∈Ω(T) as required. 5 ALGORITHM AND MAIN RESULT Following the negative results in § 4, we propose an algorithm (Algorithm 1) that operationalizes cautious learning: only learn in regions that are not too far OOD and where the available evidence does not already certify that committing is harmful. Informally, we define a trusted region around the origin whose radius grows with the time horizon, reflecting the maximum regret we are willing to tolerate-intuitively, this corresponds to allowing mistakes that are bad but not catastrophic. We then discretize the region into bins to exploit Lipschitz continuity. Within each bin, the commit reward cannot vary by more than a Lipschitz discretization error, so it suffices to estimate a single per-bin mean. The agent always abstains outside the trusted region, and inside it abstains in any bin whose pessimistic upper bound on reward is negative; otherwise it commits to gather information. More precisely, the algorithm defines a ball of radius m(T) around the origin, treating inputs outside this ball as too OOD to test. The ball is partitioned into n-dimensional hypercubes (bins) of side length w(T). By Lipschitz continuity, the variation of r(·, 1) within any bin B is at most L√nw(T). For each bin B, the algorithm maintains its empirical mean ˆμB and a confidence radius γ(k) after k commits in B. If ˆμB + γ(k) + L√nw(T) 0, w : N →R>0 H ←partition of X into n-cubes of side length w(T) B ←{B ∈H : ∃x ∈B with ∥x∥≤m(T)} σw ← p nL2w(T)2 + σ2 γ(k) := q c-1σ2w ln(2T 4) k where c is the absolute constant from Lemma A.2 (kB, ˆμB) = (0, 0) for all B ∈B for t = 1, . . . , T do if ∃B ∈B s.t. xt ∈B then ▷xt is not too OOD: it's safe to learn if ˆμB + γ(kB) + L√nw(T) 0, then T ̄ν(ln T) ≍T/(ln T)α = o(T). If one has prior knowledge of ν, the choice of m(T) can be tailored more precisely than our generic setting m(T) = ln T. In particular, for polynomial tails, setting m(T) = T c 0 x1 x2 w(T) grid (bins) bin ∈B (trusted) certified negative ∥x∥≤m(T) Figure 1: Trusted region of radius m(T) around the origin, partitioned into bins of side w(T). Any square intersecting the ball is shown fully green (bin ∈B). Certified negative bins are shown hatched red. The agent abstains outside the ball. for small c > 0 improves the bound to O(T 1-cα). Thus, the regret decomposes into a geometric/statistical term from discretization and concentration inside the trusted region, and a tail term from far OOD inputs; both are sublinear for any fixed ν. 6 PROOF SKETCH We now outline the logical structure of the proof of Thm. 5.2; we also provide the intuition behind each step. Full technical details and complete proofs of all lemmas are deferred to Appendix A. Let m(T), w(T) be as in Algorithm 1, and let B be the set of bins intersecting the ball of radius m(T). For any B ∈B, let μB = Ex∼ν[ r(x, 1) | x ∈B ] be its true mean commit reward, let kB(t) be the number of commits taken in B by the end of round t (so kB(0) = 0), and let ˆμB(k) be the empirical mean in B after k commits (i.e., the running mean from Algorithm 1 indexed by its commit count). To control estimation error we define the confidence radius γ(k) = r c-1σ2w ln(2T 4) k , σ2 w = nL2w(T)2 + σ2, where c > 0 is the absolute constant from Lemma A.2. Here σ2 w aggregates the combines observation noise σ2 with the Lipschitz-induced within-bin variation of (L√nw(T))2. Sarah Liaw, Benjamin Plaut Define the good event under which all per-bin estimates are uniformly accurate over the realized commit counts: G = n ∀B ∈B, ∀t ∈[T] : |ˆμB(kB(t))-μB| ≤γ(kB(t)) o . On G, each empirical mean is a reliable proxy for its bin's true mean at every commit count that occurs along the algorithm's trajectory. The analysis conditions on G (which holds with high probability by a union bound over realized bin-count pairs), and then decomposes regret into: (i) commits inside the trusted region (handled by certification of negative bins plus a margin term), and (ii) abstentions outside the ball of radius m(T) (quantified by the radial survival function). Lemma 6.1 (Per-bin concentration). For any B ∈B and t ∈[T], Pr [|ˆμB(kB(t)) -μB| > γ(kB(t))] ≤T -4. Proof idea. Fix B and t and condition on the kB(t) commit times t1 γ(kB(t))) ≤T -4. □ Lem. 6.1 gave concentration for each fixed bin and commit count. To extend this guarantee uniformly, we apply a union bound over the at most T bin-time pairs actually realized by the algorithm. Lemma 6.2 (Uniform concentration bound). With probability at least 1 -T -2, the good event G holds. Equivalently, Pr(¬G) ≤T -2. Proof idea. There are at most T 2 relevant pairs (B, t) along the trajectory of the algorithm. Each has failure probability T -4 by Lem. 6.1. A union bound yields failure probability at most T -2. □ Recall that the algorithm abstains permanently in any bin once ˆμB(kB(t)) + γ(kB(t)) + L√nw(T) 4c-1σ2 w ln(2T 4) (μB+L√nw(T ))2 , then bin B is certified negative at time t. Proof idea. On the good event G, certification in bin B occurs when ˆμB +γ(kB(t))+L√nw(T) m(T) contributes at most 1, giving T ̄ν(m(T)), which is sublinear for any fixed ν since ̄ν(ln T) →0. For (ii), Lipschitz continuity bounds the within-bin variation, and a uniform concentration event (probability 1 -O(T -2)) ensures empirical means stay within confidence radii. Negative bins are certified after O(σ2 w/margin2) commits, so each contributes at most O(LR(T)+σ2 w log T/w(T)) regret. Summing across O((m(T)/w(T))n) bins gives ̃O R(T)n LR(T)w(T)-n + σ2 ww(T)-(n+1) , and bins near the decision boundary contribute an additional O(w(T)T). □ If we ignore log factors, the leading terms trade off R(T)nw(T)-(n+1) | {z } variance-driven vs. w(T)T | {z } margin-driven . Balancing these yields the optimal choice w(T) ≍ T -1/(n+2). Independently, the radius m(T) trades off the abstention term T ̄ν(m(T)) against the growth of the volume factor R(T)n. Choosing m(T) = ln T makes T ̄ν(m(T)) sublinear for any fixed ν (since ̄ν(ln T) →0) while increasing R(T) only logarithmically. 7 CONCLUSION In this work, we introduced a formal model for safe learning under distribution shift in contextual bandits with catastrophic tails, provided impossibility results that clarify when sublinear regret is unattainable, and gave a cautious risk-sensitive algorithm with sublinear regret under suitable conditions. Our work has several limitations, which also provide directions for future work, including handling richer structure beyond Lipschitz continuity, incorporating adaptive or learned metrics, and extending the analysis to non-i.i.d. inputs or worst-case sequences. Our regret bound can be close to linear. In Thm. 5.2, the abstention term T ̄ν(ln T) can dominate for heavy-tailed inputs (e.g., power laws). This is the price of caution: avoiding catastrophic far OOD commits requires systematic abstention in the tails, and the resulting regret can be unavoidable (see the impossibility in Thm. 4.2). Moreover, while the bound is sublinear for every fixed n, the exponent (n+1)/(n+2) →1 as n →∞, which is a standard curse of dimensionality in Lipschitz contextual bandits (Slivkins et al., 2019, Thms. 4.11-4.12); see also Plaut et al. (2025a, Thm. 10). While we do not expect to remove these dependencies entirely, future work could improve rates. The simplicity of Algorithm 1 is appealing but ignores useful information: commits inform not only their own bin but also nearby bins via Lipschitz continuity, and certifying a bin as positive could justify expanding the trusted region around it. Additional structural assumptions, such as margin/low-noise conditions, intrinsic low dimensionality, or smoothness beyond Lipschitz, could also help. Assumptions may not always hold. Our guarantees here rely on i.i.d. inputs and Lipschitz continuity of the commit reward. In practice, inputs may drift or exhibit temporal dependence, and rewards may be only piecewise smooth or even non-smooth. Extending the analysis to weaker smoothness conditions or drifting processes is an important direction. Moreover, Algorithm 1 assumes knowledge of L, σ2, and T. While knowledge of T can be handled by the standard doubling trick (see Slivkins et al. (2019, §1.5)), L and σ2 may be unknown. Thus, developing parameter-free (or adaptively tuned) algorithms that remain cautious would increase robustness. No unconditionally irreparable errors. Obtaining regret -T on a single time step is irreparable in the sense that it automatically implies linear regret on that run. However, errors in our model are only irreparable for a fixed T: for any error, there exists a large enough T that the error is no longer catastrophic. It may be worth considering alternative models of catastrophe such as inescapable trap states in MDPs which do allow for errors that are unconditionally catastrophic. Broader impact. This work is motivated by safety concerns in the deployment of learning systems in highstakes domains. We provide theoretical justification for abstention as a mechanism for averting catastrophic errors under distribution shift, and abstention is also a practical choice for deployed systems. Agents that can defer action when uncertain may be safer and more trustworthy, but abstention mechanisms must be designed carefully to avoid consequences such as excessive conservatism or over-reliance on human supervision. Sarah Liaw, Benjamin Plaut Acknowledgments This work was supported by a gift from Open Philanthropy to the Center for Human-Compatible AI (CHAI) at UC Berkeley. This work was conducted while Sarah Liaw was a research intern at CHAI. 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Safe exploration in reinforcement learning: A generalized formulation and algorithms. Wu, Y., Shariff, R., Lattimore, T., and Szepesv ́ari, C. (2016). Conservative bandits. In International Conference on Machine Learning (ICML). Yang, J., Jin, T., and Tan, V. Y. F. (2024). Multiarmed bandits with abstention. Zhao, W., He, T., Chen, R., Wei, T., and Liu, C. (2023). State-wise Safe Reinforcement Learning: A Survey. volume 6, pages 6814-6822. ISSN: 1045-0823. Sarah Liaw, Benjamin Plaut A PROOF OF MAIN RESULT A.1 Proof notation The proof will use the following notation: 1. The true mean of bin B is μB = Ex∼ν[r(x, 1) | x ∈B]. 2. Let kB(t) denote the value of the variable kB in Algorithm 1 at the end of time step t. 3. Let ˆμB(k) denote the value of the variable ˆμB in Algorithm 1 after k rewards from bin B have been observed. 4. Let σw = p nL2w(T)2 + σ2 for brevity. 5. Define the confidence radius γ(k) = q c-1σ2w ln(2T 4) k where c is the absolute constant from Lemma A.2. 6. Define the good event G = {∀t ∈[T], ∀B ∈B where kB(t) > 0 : |ˆμB(kB(t)) -μB| ≤γ(kB(t))}. 7. A bin B is certified negative at time t if ˆμB(kB(t)) + γ(kB(t)) + L√nw(T) 0. Then there exists an absolute constant c > 0 such that for any ε > 0, Pr \" k X i=1 Xi > ε # ≤2 exp - cε2 Pk i=1 σ2 i ! Lemma A.3. If x ∈B ∈B, then |r(x, 1) -μB| ≤L√nw(T). Proof. We must prove that r(x, 1) ≥μB -L√nw(T) and r(x, 1) ≤μB + L√nw(T). Let r-= infx′∈B r(x′, 1) and r+ = supx′∈B r(x′, 1). Then r-≤μB ≤r+ and r-≤r(x, 1) ≤r+. Next, for any ε > 0, there exists x-, x+ ∈B such that r(x-, 1) -ε r+ (if not, this contradicts r-and r+ being the infimum and supremum). Then r(x, 1) and μB belong to the interval [r-, r+], which is a subset of the interval [r(x-, 1) -ε, r(x+, 1) + ε]. Since x-and x+ belong to the same n-hypercube with side length w(T), ∥x--x+∥≤√nw(T). Then by Lipschitz continuity, |r(x+, 1) -r(x-, 1)| = r(x+, 1) -r(x-, 1) ≤L√nw(T). Therefore r(x, 1) and μB belong to the same interval of length L√nw(T) + 2ε, so |r(x, 1) -μB| ≤L√nw(T) + 2ε. Since this holds for all ε > 0, we must have |r(x, 1) -μB| ≤L√nw(T). Lemma 6.1 (Per-bin concentration). For any B ∈B and t ∈[T], Pr [|ˆμB(kB(t)) -μB| > γ(kB(t))] ≤T -4. Proof. Fix a bin B and t ∈[T]. Let k = kB(t) for brevity and let t1 0, Pr k X j=1 Zj + k X j=1 ηtj > ε ≤2 exp - cε2 Pk j=1(L√nw(T))2 + Pk j=1 σ2 ! = 2 exp -cε2 kσ2w Note that the ˆμB(k) = 1 k Pk j=1 rtj = 1 k Pk j=1(r(xtj, 1) + ηtj). Then ˆμB(k) -μB = 1 k Pk j=1(Zj + ηtj). Set ε = kγ(k) = p kc-1σ2w ln(2T 4) to get Pr [|ˆμB(k) -μB| > γ(k)] = Pr h k|ˆμB(k) -μB| > p kc-1σ2w ln(2T 4) i = Pr k X j=1 Zj + k X j=1 ηtj > p kc-1σ2w ln(2T 4) ≤2 exp(-ln(2T 4)) = 2 exp ln 1 2T 4 = T -4 as required. Lemma 6.2 (Uniform concentration bound). With probability at least 1-T -2, the good event G holds. Equivalently, Pr(¬G) ≤T -2. Proof. Let J be the the number of bins that receive at least one commit, i.e., J = |{B ∈B : ∃t ∈[T] s.t. xt ∈ B, yt = 1}|. For each j ∈[J], let Bj be the jth bin to receive a commit. Then Pr[¬G] = E[Pr[¬G | J, B1, . . . , BJ]] (Law of total expectation) = E Pr J[ j=1 T[ t=1 {|ˆμBj(kB(t)) -μBj| > γ(kB(t))} J, B1, . . . , BJ (Direct negation) ≤E J X j=1 T X t=1 Pr[|ˆμBj(kB(t)) -μBj| > γ(kB(t))] J, B1, . . . , BJ (Union bound) ≤E J X j=1 T X t=1 T -4 J, B1, . . . , BJ (Lemma 6.1) ≤E h T -2 J, B1, . . . , BJ i (J ∈[T]) ≤T -2 (Expectation of a constant) as required. 2This is similar to the \"reward tape\" argument used in Section 1.3.1 of Slivkins et al. (2019). Sarah Liaw, Benjamin Plaut Lemma 6.3 (Samples for negative certification). Consider any t ∈[T] and B ∈B. On G, if μB 4c-1σ2 w ln(2T 4) (μB+L√nw(T ))2 , then bin B is certified negative at time t. Proof. Note that μB 4c-1σ2 w ln(2T 4) (μB+L√nw(T ))2 . Therefore |{t ∈[T] : xt ∈B, yt = 1}| ≤⌈4c-1σ2 w ln(2T 4) (μB+L√nw(T ))2 ⌉≤ 1 + 4c-1σ2 w ln(2T 4) (|μB|-L√nw(T ))2 . Since |μB| ≥(2L√n + 1)w(T) ≥2L√nw(T), we have |μB| = |μB| 2 + |μB| 2 ≥|μB| 2 + L√nw(T) so |μB| -L√nw(T) ≥|μB/2|. Therefore (|μB| -L√nw(T))2 ≥μ2 B/4, so |{t ∈[T] : xt ∈B, yt = 1}| ≤ 1 + 16c-1σ2 w ln(2T 4) μ2 B . For any t ∈[T] such that yt = 1, either yt = 1 is optimal, in which case the single-step regret ∆t is 0, or yt = 0 is optimal, if which case ∆t = -r(xt, 1). If xt ∈B, Lemma A.3 implies that r(xt, 1) ≥μB -L√nw(T). Since μB -LR(T) for all x ∈B. ThusμB = Ex∼ν[r(x, 1) | x ∈B] ≥Ex∼ν[-LR(T)] = -LR(T), so X t:xt∈B,yt=1 ∆t ≤2LR(T) + 32c-1σ2 w ln(2T 4) w(T) as required. Lemma 6.6 (Total commit regret inside the trusted region). On G, X t: yt=1 ∆t ≤v1R(T)n w(T)n 2LR(T) + 32c-1σ2 w ln(2T 4) w(T) + (3L√n + 1)w(T)T. Proof. For each t ∈[T], let B(t) denote the bin to which xt belongs. Partition the time steps with commits into S1 = {t ∈[T] : yt = 1 and μB(t) m(T) (otherwise the bin containing xt would be in B). Also, r(xt, 1) -r(xt, 0) ≤1 by assumption. Hence Reg(T) = T X t=1 ∆t = X t∈S3 ∆t + X t∈S4 ∆t ≤v1R(T)n w(T)n 2LR(T) + 32c-1σ2 w ln(2T 4) w(T) + (3L√n + 1)w(T)T + T X t=1 1(∥xt∥> m(T)) = 2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + (3L√n + 1)w(T)T + T X t=1 1(∥xt∥> m(T)) Therefore E[Reg(T) | G] ≤2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + (3L√n + 1)w(T)T + T X t=1 Pr[∥xt∥> m(T)] = 2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + (3L√n + 1)w(T)T + T X t=1 ̄ν(m(T)) = 2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + (3L√n + 1)w(T)T + T ̄ν(m(T)) Now suppose G does not hold. Consider an arbitrary t ∈[T]. If yt = 0, then the regret at time t is at most 1. If yt = 1, we still have ∥xt∥≤R(T), so by Lipschitz continuity, r(xt, 1) ≥-LR(T). Therefore E[Reg(T) | ¬G] ≤T + LR(T)T. Lemma 6.2 implies that Pr[¬G] ≤T -2, so by the law of expectation, E[Reg(T)] = Pr[¬G] E[Reg(T) | ¬G] + Pr[G] E[Reg(T) | G] ≤1 T 2 · (T + LR(T)T) + 2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + (3L√n + 1)w(T)T + T ̄ν(m(T)) ∈O 1 + LR(T) T + 2Lv1R(T)n+1 w(T)n + 32v1c-1σ2 wR(T)n ln(2T 4) w(T)n+1 + L√nw(T)T + T ̄ν(m(T)) We now plug in w(T) = T -1 n+2 and m(T) = ln T. Since limT →∞w(T) = 0, we have σ2 w = nL2w(T)2 + σ2 ∈O(σ2). Similarly, for any k ≥0, R(T)k = (ln(T) + √nT -1 n+2 )k ∈O((ln T)k). Thus E[Reg(T)] ∈O L ln T T + L(ln T)n+1 T -n n+2 + σ2(ln T)n+1 T -n-1 n+2 + LT -1 n+2 T + T ̄ν(ln T) = O (L + σ2)T n+1 n+2 (ln T)n+1 + T ̄ν(ln T) as required."}}},{"rowIdx":98,"cells":{"id":{"kind":"string","value":"2510.14888"},"raw":{"kind":"string","value":"Modeling nonlinear scales for dynamical dark energy cosmologies with COLA\nJoão Rebouças,1, 2 Victoria Lloyd,3 Jonathan Gordon,3, 4 Guilherme Brando,1 and Vivian Miranda4\n1CBPF - Brazilian Center for Research in Physics, Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil\n2Instituto de Física Teórica da Universidade Estadual Paulista,\nR. Dr. Bento Teobaldo Ferraz, 271, Bloco II, Barra-Funda - São Paulo/SP, Brazil\n3Department of Physics & Astronomy, Stony Brook University, Stony Brook, NY 11794, USA\n4C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA\n(Dated: October 17, 2025)\nUpcoming galaxy surveys will bring a wealth of information about the clustering of matter at small scales, but\nmodeling small-scale structure beyond ΛCDM remains computationally challenging. While accurate 𝑁-body\nemulators exist to model the matter power spectrum for ΛCDM and some limited extensions, it’s unfeasible\nto generate 𝑁-body simulation suites for all candidate models. Motivated by recent hints of an evolving dark\nenergy equation of state from galaxy surveys, we assess the viability of employing the COmoving Lagrangian\nAcceleration (COLA) method to generate simulation suites assuming the 𝑤0𝑤𝑎dynamical dark energy model.\nFollowing up on our previous work, we combine COLA simulations with an existing high-precision ΛCDM\nemulator to extend its predictions into new regions of parameter space. We assess the precision of our emulator\nat the level of the matter power spectrum, finding that our emulator can reproduce the nonlinear boosts from\nEuclidEmulator2 at less than 2% error. Moreover, we perform an analysis of a simulated cosmic shear survey\nakin to future data from the Legacy Survey of Space and Time (LSST) first year of observations, assessing the\ndifferences in parameter constraints between our COLA-based emulator and the benchmark emulator. We find\nour emulator to be in excellent agreement with the benchmark, achieving less than 0.3𝜎shifts in cosmological\nparameters across multiple fiducial cosmologies, and a 7D figure of bias of less than 0.35. We further compare\nour emulator’s performance to a commonly used approach: assuming the ΛCDM boost can be employed for\nextended parameter spaces without modification. We find that our emulator yields a significantly smaller Δ𝜒2\ndistribution, parameter constraint biases, and a more accurate figure of merit compared to this second approach.\nThese results demonstrate that COLA emulators provide a computationally efficient and physically motivated\npath forward for modeling nonlinear structure in extended cosmologies, offering a practical alternative to full\n𝑁-body suites in the era of precision cosmology.\nI.\nINTRODUCTION\nIn recent decades, galaxy surveys have been able to map the\nlarge-scale structure of the Universe with great precision, being\nas competitive as CMB surveys. Current photometric surveys,\nsuch as DES [1–10], KiDS [11–23], HSC [24–29], and spectro-\nscopic surveys, such as DESI [30–36], BOSS/eBOSS [37–50],\nand WiggleZ [51–55], have placed percent-level constraints\non ΛCDM parameters, demonstrating the success of the the-\noretical model in describing independent datasets. However,\nin recent years, tensions in the cosmological parameter con-\nstraints have begun to arise with the increase in precision.\nNotably, most galaxy surveys report a mildly lower value for\nthe structure growth parameter 𝑆8 than those from CMB mea-\nsurements (see e.g. [56]). Moreover, recent findings from\nthe DESI collaboration, as well as Type Ia supernovae datasets\nsuch as Pantheon+, Union3, and DESY5 [30, 34, 35, 57–59] fa-\nvor an evolution of the dark energy equation of state. Whether\nthese results truly indicate new physics is still under debate\n[60–64].\nForthcoming Stage-IV galaxy surveys, such as the Vera\nRubin Observatory’s LSST [65], Euclid [66], SphereX [67],\nand Roman [68], will enable higher-precision measurements,\nespecially at small scales where non-linearities in the mat-\nter density field become increasingly sizable [69], and will\nbe decisive for investigating dark energy dynamics.\nAnal-\nyses of galaxy surveys rely on a key theoretical prediction:\nthe matter power spectrum, as galaxies are biased tracers of\nthe underlying matter density field. At large scales and early\ntimes, the power spectrum can be quickly and accurately com-\nputed using Einstein-Boltzmann solvers such as camb [70] and\nclass [71, 72]. However, for small scales and low redshifts,\nlinear perturbation theory breaks down, and accurately mod-\neling non-linearities becomes a task of central importance.\nAlthough N-body simulations offer the most accurate pre-\ndictions in the nonlinear regime, they are computationally ex-\npensive — each one demanding tens of thousands of CPU\nhours [73–75].\nAs a result, integrating simulations into\nBayesian analyses is a prohibitive task, as theoretical predic-\ntions must be provided for O(105 −106) points in the param-\neter space. To address this issue, machine learning emulators\ntrained on N-body simulations have been developed for ΛCDM\nand simple, widely adopted extensions such as the phenomeno-\nlogical 𝑤0𝑤𝑎CDM [76, 77] parametrization for dark energy, as\nwell as massive neutrinos with their total mass as a free param-\neter. Examples include EuclidEmulator2 [78], bacco [79],\nCosmicEmu [80], the Dark Quest emulator [81, 82], the CSST\nEmulator [83–85], aemulus [86–89], among others. At the\nsame time, due to the sheer amount of candidate cosmologi-\ncal models and the high cost of running N-body simulations,\nemulators for broader extensions of ΛCDM are still scarce:\nexamples are emulators to specific modified gravity theories\n(e.g. [90–93]) where we have full N-Body simulations [94], as\nwell as some hydrodynamical emulators.\nA viable alternative to using full 𝑁-body simulations for\nconstructing matter power spectrum emulators is to use well-\nestablished approximate methods for extended cosmological\nmodels. These methods reduce the computational complexity\narXiv:2510.14888v1 [astro-ph.CO] 16 Oct 2025\n\n2\nof running hundreds of simulations to train emulators, at the\ncost of losing accuracy in deep non-linear scales. One com-\npelling approach is to use the COmoving Lagrangian Acceler-\nation (COLA) method, which combines Lagrangian Perturba-\ntion Theory with a particle-mesh (PM) [95] evolution scheme\nto approximate N-body results while being cheaper than the\nusual N-body methods by 1-2 orders of magnitude [96, 97].\nEmulators created using pure COLA simulations are prone to\nsmall-scale inaccuracies when compared directly to their N-\nbody counterparts. To mitigate this effect, the work of [98]\nintroduces an approach that combines COLA simulations with\npredictions from high-accuracy ΛCDM emulators or full N-\nbody results, leveraging COLA’s reduced computational cost\nand dramatically increasing small-scale accuracy simultane-\nously. Our previous work [99] has validated this approach,\ncreating an emulator for the matter power spectrum of COLA\nsimulations under the 𝑤CDM cosmological model, and testing\nit in a mock Stage-IV cosmic shear analysis. As such, we aim to\ndemonstrate here that the hybrid approach of COLA-based em-\nulators combined with high-resolution ΛCDM emulators can\nprovide unbiased cosmological parameter constraints when\ncompared to full N-body methods.\nIn this work, we now present a final validation of our COLA-\nbased emulators on the 𝑤0𝑤𝑎CDM cosmological model,\nwhere the dark energy equation of state evolves linearly with\nthe scale factor.\nThis parametrization represents the most\nwidely used and general extension of ΛCDM for which high-\naccuracy emulators currently exist, and remains central to on-\ngoing investigations into dynamical dark energy [30, 34]. We\nextend our machine learning pipeline, combining COLA simu-\nlations with ΛCDM emulators, to predict the nonlinear matter\nspectrum across the 𝑤0𝑤𝑎CDM parameter space. We train\na simple neural network to emulate the nonlinear correction\nfactor (i.e the boost) from COLA, correcting for small-scale\ninaccuracies by referencing boosts from a high-fidelity ΛCDM\nemulator. This hybrid approach enables fast predictions across\nan extended cosmological parameter space while maintaining\nconsistency with 𝑁-body precision. To validate our emulator\nin a cosmological inference setting, we perform a simulated\ncosmic shear analysis using survey specifications consistent\nwith LSST’s first year of observations (LSST-Y1) [65]. We\ncompare parameter constraints derived using both our pipeline\nand a benchmark 𝑁-body emulator, chosen as EuclidEmula-\ntor2, quantifying their disagreement with standard tension\nmetrics [100–102].\nAdditionally, we also benchmark our emulator against a\nwidely used approximation method in beyond-ΛCDM analy-\nses for models without dedicated nonlinear simulations [103–\n108]: projecting the nonlinear boost from the nearest ΛCDM\ncosmology. This projection method assumes that nonlinear\ncorrections calibrated in ΛCDM remain valid in nearby ex-\ntended cosmologies, providing a computationally inexpensive\nworkaround but at the cost of uncontrolled systematics. In con-\ntrast, we demonstrate that for dynamical dark energy models,\nwe find that the ΛCDM projection approach may introduce sig-\nnificant deviations in both goodness-of-fit and parameter con-\nstraints. Meanwhile, our COLA-based emulator reproduces\nthe predictions of high-precision 𝑁-body emulators without\nbias.\nThis paper is organized as follows: Section II describes\nthe COLA simulations, cosmological parameters, simulation\noutput processing, emulator construction, and validation; in\nSection III, we present our LSST-Y1 simulated cosmic shear\nanalysis and the tension metrics used to assess their differences;\nin Section IV, we present and discuss the results of the LSST-\nY1 simulated analysis; finally, we conclude in Section V.\nII.\nCOLA EMULATOR\nA.\nCOLA Simulation Suite\nThe COmoving Lagrangian Acceleration (COLA) algo-\nrithm [96] is a fast approximate method for 𝑁-body simu-\nlations, wherein particles evolve in a frame comoving with\ntrajectories calculated using Lagrangian Perturbation Theory\n(LPT), most commonly 2nd order Lagrangian perturbation the-\nory (2LPT). For small scales, the method computes the force\nby using a Particle-Mesh (PM) algorithm, where the residual\ndisplacements not captured by LPT are added to the trajecto-\nries.\nCOLA has been shown to agree with full 𝑁-body simula-\ntions, at the level of the power spectrum at up to 𝑘∼1ℎ/Mpc,\nas well as when predicting ratios of the modified gravity power\nspectrum and the ΛCDM one, the so-called boost function in\nmodified gravity [98, 109–111]. Despite being 1-2 orders of\nmagnitude faster than a full 𝑁-body run, the computational\ncost of these approximations is still too high for direct use in\nthe O(106) computations of the matter power spectrum re-\nquired for Monte Carlo searches. A practical alternative is\nto use a fixed set of COLA simulations to train emulators\nfor the matter power spectrum, enabling efficient interpola-\ntion across cosmological parameter space. Our previous work\ndemonstrated this approach for 𝑤CDM [99]; here, we extend\nit to 𝑤0𝑤𝑎CDM and evaluate its performance relative to the\nbenchmark EuclidEmulator2 [78], which achieves ≲1%\nprecision for 𝑤0𝑤𝑎CDM + Í 𝑚𝜈up to 𝑘= 10 ℎ/Mpc−1 and\n𝑧≤3.\n1.\nSimulation Settings\nWe use the COLA algorithm as implemented in the pub-\nlic fml1 code.\nEach simulation is performed in a box of\nsize 𝐿= 1024ℎ−1Mpc, populated with 𝑁part = 10243 parti-\ncles, initialized at 𝑧ini = 19, and evolved over 51 time steps\nchosen to maintain a uniform time resolution of Δ𝑎≈0.02.\nThe force grid uses 𝑁mesh = 20483 cells, and the power\nspectra are calculated on-the-fly using a 𝑁3\npk−mesh = 10243\ngrid.\nTherefore, the corresponding Nyquist frequency is\n𝑘Nyq = 𝜋𝑁pk−mesh/𝐿= 𝜋ℎ/Mpc.\nTo avoid aliasing, we\n1 https://github.com/HAWinther/FML\n\n3\nrestrict our analysis to 𝑘≤𝑘Nyq [112]. Our choices are based\non Reference [98] and are validated therein.\nInitial conditions are generated using 2LPT, and we employ\nthe forward approach [113] for our simulations. We provide\nthe linear transfer functions of matter density, velocity, and\nrelativistic species’ densities at each time step using class2 in\nsynchronous gauge, and convert to the 𝑁-body gauge [114–\n117] in COLA. Our simulations were run in the Seawulf3\ncluster. With these settings, and using 128 cores, one COLA\n𝑤0𝑤𝑎simulation takes approximately 40 minutes to finish,\nand requires a total RAM of approximately 950 GB.\nTo suppress sample variance from finite box effects at large\nscales (𝑘≈1/𝐿), we use the pairing-and-fixing method [118],\nin which we generate Gaussian random field modes with a\nfixed amplitude, 𝛿𝑖,lin, but with phase shifts of 𝜋with respect\nto one another. The initial overdensity fields are sampled as\n𝛿𝑖,lin =\n√︁\n𝑃𝑖𝑒i𝜃𝑖.\n(1)\nwhere 𝜃𝑖is a random phase and 𝑃𝑖the initial power spectrum.\nAveraging over each pair, we find that the result substantially\nsuppresses the effects of cosmic variance. This strategy was\nchosen for this work following [99] and [78].\n2.\nDefinition of the Parameter Space\nWe consider the cosmological 𝑤0𝑤𝑎CDM model, where the\ndark energy equation of state is parametrized as\n𝑤(𝑎) = 𝑤0 + 𝑤𝑎(1 −𝑎),\n(2)\nwith 𝑎being the scale factor, and 𝑤0 and 𝑤𝑎control the\npresent-day value and time derivative of the dark energy equa-\ntion of state, respectively.\nThe ΛCDM model is recovered\nin the limit 𝑤0 = −1 and 𝑤𝑎= 0. The free cosmological\nparameters are:\n• Ω𝑚, the total baryon density,\n• Ω𝑏, the total matter density,\n• ℎ= 𝐻0/(100 km s−1 Mpc−1), the dimensionless Hub-\nble parameter,\n• 𝐴𝑠, the amplitude of initial scalar fluctuations,\n• 𝑛𝑠, the scalar spectral index,\n• 𝑤0 and 𝑤𝑎, the dark energy equation of state parameters.\nWe fix the summed neutrino masses to the minimum value\nallowed by neutrino oscillation experiments, Í 𝑚𝜈= 0.058\neV, assuming three degenerate massive species [119, 120].\nThe parameter space boundaries are described in Table I, set\nto match those of EuclidEmulator2, which we have adopted\n2 https://github.com/lesgourg/class_public\n3 https://rci.stonybrook.edu/HPC\nΩ𝑚\nΩ𝑏\n𝑛𝑠\n𝐴𝑠× 109\nℎ\n𝑤0\n𝑤𝑎\nMin\n0.24\n0.04\n0.92\n1.7\n0.61\n−1.3\n−0.7\nMax\n0.40\n0.06\n1.00\n2.5\n0.73\n−0.7\n0.5\nCenter\n0.319\n0.05\n0.96\n2.1\n0.67\n−1\n0\nTABLE I. Parameter space validity bounds of our COLA-based em-\nulator. The training set is drawn from a slightly bigger hypercube,\nwhere each dimension is stretched by 10% in each direction (e.g.\n0.224 < Ω𝑚< 0.416). We also define a center cosmology chosen to\nagree with the EuclidEmulator2 reference cosmology [78].\nas our benchmark for comparison. We emphasize that this\nchoice is arbitrary, and our methodology is agnostic to the\nbenchmark we have chosen. To improve model performance\nnear parameter space edges, our COLA training simulations\nare sampled from an expanded box where each parameter inter-\nval has been stretched symmetrically by 10%. Cosmologies\nwithin this volume are selected using Latin hypercube sam-\npling to ensure uniform coverage for training and validation.\nB.\nEmulation of COLA Boosts\n1.\nEmulator Prototypes with halofit\nOur goal is for the emulation error (i.e., the error in recov-\nering COLA boosts from a predetermined test set excluded\nfrom training) to be significantly smaller than the intrinsic\nCOLA approximation error relative to full 𝑁-body simula-\ntions. To determine optimal hyperparameters, such as training\nset size and emulator architecture, we perform mock tests us-\ning halofit [121] boosts. We generated training datasets with\n𝑁train ∈[500, 600, 700, 800, 1000] halofit boosts and a test\ndataset with 𝑁test = 200. We found that 600 simulations were\nsufficient to achieve ∼0.1% error at 𝑘= 1 ℎ/Mpc; conserva-\ntively, we adopt 𝑁train = 700 and 𝑁test = 200 for our COLA\nemulator.\n2.\nPost-processing the Simulation Boosts\nWe define the nonlinear boost as\n𝐵X(𝑘, 𝑧|θ) ≡𝑃X(𝑘, 𝑧|θ)\n𝑃L(𝑘, 𝑧|θ) ,\n(3)\nwhere θ refers to a point in the 𝑤0𝑤𝑎CDM parameter space,\n𝑃𝑋(𝑘, 𝑧|θ) is the matter power spectrum for cosmology θ,\neither linear (denoted 𝑃L), or calculated using COLA or an-\nother N-body method (generically denoted 𝑃X).\nPrior to\ncomputing 𝐵COLA, we subtract the shot noise power spec-\ntrum, 𝑃SN = (𝐿/𝑁part)3 = 1(Mpc/ℎ)3, from 𝑃COLA.\nAt\nhigh redshift, 𝑧> 1.182, aliasing of the 𝑘modes near the\nNyquist frequency leads to a power spectrum less than the\nshot noise for some simulations, and the subtraction would\nlead to unphysical negative values [113]; for these redshifts,\n\n4\nwe choose to cut the scales at half of the Nyquist frequency,\n𝑘𝑧>1.182 ≤(𝜋/2) ℎ/Mpc, following our procedure in [99]\n(also see [112]).\nWe then perform several transformations to optimize the\ninputs and outputs of our emulator. For instance, machine\nlearning techniques are known to perform poorly if the features\nspan several orders of magnitude. To stabilize the following\nprocedures, we normalize the cosmological parameters θ to\n[−1, 1] according to\nθ𝑁= −1 + 2 θ −θmin\nθmax −θmin\n,\n(4)\nwhere minimum and maximum values correspond to the train-\ning set boundaries, i.e., stretching the intervals of Table I by\n10% in each direction. Furthermore, we standardize the boosts\nusing\n𝐵COLA\n𝑁\n(𝑘, 𝑧|θ) = 𝐵COLA(𝑘, 𝑧|θ) −¯𝐵COLA(𝑘, 𝑧)\n𝜎𝐵(𝑘, 𝑧)\n,\n(5)\nwhere 𝐵COLA\n𝑁\n(𝑘, 𝑧|θ) is the normalized COLA boost, ¯𝐵(𝑘, 𝑧)\nis the average of all boosts in the training set, and 𝜎𝐵(𝑘, 𝑧) is\ntheir standard deviation. We then perform a Principal Com-\nponent Analysis (PCA) decomposition of the COLA boosts\nusing scikit-learn [122] to reduce dimensionality. We retain\n𝑁PC = 15 components, which are sufficient to recover the test\nset boosts to within 0.2%.\n3.\nNeural Network Emulator\nAfter post-processing, we train our emulator with the nor-\nmalized cosmological parameters as input features and the\nprincipal components as targets. We use a fully connected\nneural network with three hidden layers, each with 1024 neu-\nrons, with a mean squared error loss function,\nL =\n𝑁train\n∑︁\n𝑖=1\n𝑁PC\n∑︁\n𝑗=1\n(𝛼𝑖,train\n𝑗\n−𝛼𝑖,pred\n𝑗\n)2,\n(6)\nwhere 𝛼𝑖,train\n𝑗\nis the 𝑗-th principal component coefficient of the\n𝑖-th cosmology in the training set, and 𝛼𝑖,pred\n𝑗\nthe corresponding\nprediction. We use the parametric activation function [123,\n124]\n𝑦𝑚+1\n𝑛\n=\n\u0014\n𝛾𝑚\n𝑛+ (1 −𝛾𝑚\n𝑛)\n1\n1 + 𝑒−𝛽𝑚\n𝑛𝑦𝑚\n𝑛\n\u0015\n˜𝑦𝑚\n𝑛,\n(7)\nwhere 𝑦𝑚+1\n𝑛\nis the value of the 𝑛-th neuron of the (𝑚+ 1)-th\nlayer, ˜𝑦𝑚\n𝑛the 𝑛-th neuron from the (𝑚+1)-th layer after the ap-\nplication of weights and biases, and 𝛾𝑚\n𝑛and 𝛽𝑚\n𝑛are parameters\nof the activation function that can be back-propagated during\ntraining. We use the Adam [125] optimizer to train the model\nparameters.\n4.\nBoost Errors\nWe perform a series of accuracy checks on the emulator out-\nputs. First, to assess the accuracy of our neural network, we\ncompare the emulator’s predictions for test set cosmologies,\nunseen in the training procedure, against the actual COLA\nsimulations. The relative errors are shown in the first panel\nof Figure 1. At 𝑘= 1 ℎ/Mpc, 90% of the test set cosmolo-\ngies have an emulation error within 0.1%. Comparing these\ndirect emulation errors to the errors on the corrected boosts\n˜𝐵COLA(𝑘, 𝑧) (second panel), we note that the errors between\nthe two differ by an order of magnitude. This indicates that,\nin the context of comparing COLA with high-precision simu-\nlations, the COLA emulator faithfully reproduces its simula-\ntions, and differences between the emulators can be attributed\nto the COLA approximation rather than the performance of\nthe machine learning model.\nAs per [98], COLA simulations increasingly lose power4 at\nprogressively smaller scales, leading to typical errors of 10% at\n𝑘= 1 ℎ/Mpc for raw COLA boosts 𝐵COLA, as defined in Equa-\ntion 3. However, this power loss is cosmology-independent.\nOur previous work [99] showed that the best technique to build\na COLA-based emulator is to leverage existing high-precision\nemulators in ΛCDM, using COLA only to extend the results\ninto new dimensions, i.e., extra model parameters. This idea\nis encoded in the following expression for the nonlinear boost,\n˜𝐵COLA(𝑘, 𝑧|θ) = 𝐵N-body(𝑘, 𝑧|θ𝑝) × 𝐵COLA(𝑘, 𝑧|θ)\n𝐵COLA(𝑘, 𝑧|θ𝑝) ,\n(8)\nwhere θ𝑝is the projection of θ in the ΛCDM subspace and\n𝐵N−body is the nonlinear boost obtained from our benchmark\n𝑁-body prescription. We choose EuclidEmulator2 as the\nbase N-body prescription.\nThe second panel of Figure 1 shows the relative difference\nbetween ˜𝐵(𝑘, 𝑧), calculated using Equation 8, and the bench-\nmark emulator predictions, 𝐵EE2(𝑘, 𝑧), for all test set cosmolo-\ngies. At 𝑘= 1 ℎ/Mpc, 90% of cosmologies have emulation\nerrors within 2%, with 50% of cosmologies contained well\nwithin 1%. This demonstrates that our method successfully\nmitigates the accumulation of errors typical of COLA simula-\ntions in the nonlinear regime, allowing us to generate accurate\npredictions across our target 𝑘range. Furthermore, the cos-\nmologies with larger errors are those with higher values of\n𝑤0 + 𝑤𝑎, a region of the parameter space excluded by current\ndata.\nFinally, we consider a third nonlinear prescription: using\nthe nonlinear boost from EuclidEmulator2 in the ΛCDM\nsubspace; this approach will be denoted as EE2 ΛCDM. For\nthis purpose, we compute the nonlinear boosts for 𝑤0𝑤𝑎CDM\ncosmologies using EuclidEmulator2, setting 𝑤0 = −1 and\n𝑤𝑎= 0. The third panel of Figure 1 shows relative errors\n4 This loss of power is well known to PM N-body codes, which fail to resolve\nthe internal dynamics of halos. This trend of losing power starts roughly at\na scale at which the pure 1-halo term of the halo model would dominate the\npower spectrum.\n\n5\n5.0\n2.5\n0.0\n2.5\n5.0\n1000×(B COLA\nNN\n/B COLA\n1)\n50%\n10 1\n100\nk (h/Mpc)\n5.0\n2.5\n0.0\n2.5\n5.0\n100×(BCOLA/B EE2\n1)\n90%\n10 1\n100\nk (h/Mpc)\n5.0\n2.5\n0.0\n2.5\n5.0\n100×(B EE2\nCDM/B EE2\n1)\n100%\nFIG. 1. From top to bottom: 1) Relative errors between the COLA\nboosts 𝐵COLA predicted by the emulator versus those obtained from\nthe test set simulations.\n2) Relative errors between ˜𝐵COLA (see\nEquation 8) and the boosts from EE2. 3) Relative errors between\n𝐵EE2 ΛCDM and EE2. Colors in all panels denote the percentile of\ncosmologies around the mean: blue contours enclose 50% of cos-\nmologies, red contours enclose 90% of cosmologies, and the outer\ngray lines enclose 100% of cosmologies. All panels show results for\n𝑧= 0, see Appendix A for the equivalent plots at higher redshifts.\nbetween 𝐵EE2 ΛCDM and the actual EuclidEmulator2 boosts.\nThe 90% percentile shows errors of the order of 7.5% at 𝑘=\n1 ℎ/Mpc, significantly worse than the COLA errors of the\npanel above.\nThese results support the viability of our emulator for pa-\nrameter inference in 𝑤0𝑤𝑎CDM, as the emulated boost ˜𝐵(𝑘, 𝑧)\nagrees with our 𝑁-body proxy at a level suitable for upcoming\nprecision cosmology experiments [126] while requiring sig-\nnificantly less computational expense compared to traditional\n𝑁-body methods. In the following, we investigate how the dif-\nferences shown in Figure 1 impact the parameter constraints\nfrom simulated cosmic shear analysis.\nParameter\nFiducial\nPrior\nSurvey specifications\nArea\n12300 deg2\n–\nShape noise per component\n0.26\n–\n𝑛sources\neff\n11.2 arcmin−2\n–\nPhotometric redshift offsets\nΔ𝑧𝑖\nsource\n0\nN[0, 0.002]\nIntrinsic alignment (NLA)\n𝑎1\n0.7\nU[-5, 5]\n𝜂1\n-1.7\nU[-5, 5]\nShear calibration\n𝑚𝑖\n0\nN[0, 0.005]\nTABLE II. Mock survey specifications for our simulated analysis, and\nnuisance parameter priors. U[𝑎, 𝑏] denotes an uniform distribution\nwith edges [𝑎, 𝑏], while N [𝑎, 𝑏] denotes a Gaussian distribution\nwith mean 𝑎and standard deviation 𝑏. Tomographic bin indices are\ndenoted by 𝑖, and all our priors are the same for all bins.\nIII.\nANALYSIS OF LSST-Y1 SIMULATED DATA\nA.\nSimulating Cosmic Shear Data\nWe simulate cosmic shear observations based on LSST-Y1,\nfollowing the methodology of [127, 128] and detailed in [99].\nSurvey specifications, source galaxy redshift distributions, and\nnuisance parameter priors are taken from the LSST DESC\nScience Requirements Document [65], and summarized in Ta-\nble II. The redshift distribution is modeled as a Smail distri-\nbution convolved with a Gaussian uncertainty 0.02(1 + 𝑧) and\ndivided into five tomographic bins with equal galaxy number\ndensities.\nThe cosmic shear two-point correlation functions 𝜉𝑖𝑗\n± (𝜃)\nare computed by first evaluating in Fourier space, 𝐶𝑖𝑗\n𝜅𝜅(ℓ),\nusing the nonlinear matter power spectrum via the Limber ap-\nproximation, then transforming to real space via the analytic\nfunctions in Appendix A of [6]. We compute 𝜉𝑖𝑗\n± in 26 loga-\nrithmically spaced angular bins between 2.5 and 900 arcmin,\naveraging over each bin. We include standard self-calibrating\nsystematics in our computation of 𝜉± — photometric redshift\nuncertainties, multiplicative shear calibration, and the non-\nlinear alignment (NLA) model of intrinsic galaxy alignments\n(see, e.g., [129, 130]).\nLikelihood analyses are performed using Cocoa, the\nCobaya-CosmoLike Joint Architecture5 [131–133]. Linear\npower spectra are computed with camb [134, 135], and nonlin-\near corrections are applied using either ˜𝐵(Eq. 8), 𝐵EE2 ΛCDM,\nor EuclidEmulator2. We use MCMC sampling to explore\nthe parameter space and assess convergence using the Gel-\nman–Rubin criterion (|𝑅−1| < 0.01) [136].\n5 https://github.com/CosmoLike/cocoa\n\n6\n𝜃\nΩ𝑚\n109𝐴𝑠\n𝑛𝑠\n𝑤0\n𝑤𝑎\n𝜃↑\n0.36\n2.3\n0.98\n-0.85\n0.25\n𝜃↓\n0.28\n1.9\n0.94\n-1.15\n-0.35\nTABLE III. \"High\" (↑) and \"low\" (↓) cosmological parameter values\nused to construct the fiducial cosmologies of our analyses. In this\nnotation, fiducial cosmologies are labeled in the text by the parameters\nshifted from the central values listed in Table I.\nCutoff\n⟨𝑧⟩\n0.33\n0.54\n0.74\n1.01\n1.62\nCutoff 1\n1.4\n1.1\n0.9\n0.9\n0.8\nCutoff 2\n2.9\n2.2\n1.9\n1.7\n1.6\nCutoff 3\n5.7\n4.3\n3.8\n3.4\n3.3\nTABLE IV. Approximate scales in wavenumber 𝑘, measured in\nℎMpc−1, for each galaxy source bin that correspond to the angu-\nlar cutoffs in 𝜉+ tested in our LSST-like cosmic shear analysis. The\nscales are approximated by computing the angular diameter distance\nto the mean redshift of each bin and converting to a wavenumber.\nTo evaluate the emulator’s accuracy in the parameter space,\nwe define 29 fiducial cosmologies within the emulator’s valid-\nity range. One data vector is generated at the center cosmology\nshown in Table I. We then vary cosmological parameters to\ntheir intermediate \"low\" (↓) and \"high\" (↑) values, shown in\nTable III. We define four cosmologies by varying both 𝑤0 and\n𝑤𝑎for each combination of their low and high values, keeping\nother parameters fixed at their central values. We further de-\nfine 24 fiducial cosmologies varying 𝑤0, 𝑤𝑎, and one of Ω𝑚,\n𝐴𝑠, or 𝑛𝑠, using all combinations of their low and high values.\nAll data vectors are generated using EuclidEmulator2 as the\nnonlinear prescription, and the covariance is computed analyt-\nically with CosmoCov [127]. We do not include a Gaussian\nnoise realization in the fiducial data vectors.\nTo mitigate biases from emulator inaccuracies on small\nscales, which may degrade the goodness-of-fit, we apply three\nangular scale cuts (C1-C3) following [99], removing 𝜉± mea-\nsurements below a minimum angular separation, 𝜃min. The\ncorresponding wavenumbers are shown in Table IV; cuts are\nmore aggressive for 𝜉−due to its sensitivity to smaller scales.\nFinally, to compute the cosmic shear integrals beyond the\nemulator’s 𝑘-range, we extrapolate log( ˜𝐵(𝑘)) versus log(𝑘)\nusing a linear fit for both COLA and EE2 emulators. A Sav-\nitsky–Golay filter (order 1, window length 5) is applied to\nthe last entries of the ˜𝐵COLA vector to suppress noise before\nextrapolation.\nB.\nQuantifying discrepancies\nTo quantify deviations between parameter constraints from\nthe LSST-Y1 simulated analyses using the nonlinear prescrip-\ntions 𝑋and 𝑌, we first evaluate the one-dimensional bias for\nkey parameters Ω𝑚, 𝑆8, 𝑤0, and 𝑤𝑎, defined as\nΔ𝜃𝑖\n𝜎𝜃𝑖\n=\n⟨𝜃𝑖⟩X −⟨𝜃𝑖⟩Y\n√︃\n𝜎2\n𝜃𝑖,X + 𝜎2\n𝜃𝑖,Y\n,\n(9)\nwhere 𝜃𝑖denotes one of Ω𝑚, 𝑆8, 𝑤, or 𝑤𝑎, and ⟨𝜃𝑖⟩and 𝜎2\n𝜃𝑖\nare, respectively, the sample mean and sample variance from\nMCMC posteriors.\nTo capture parameter correlations, we also compute the Fig-\nure of Bias (FoB), a multivariate generalization of the 1D bias\ndefined by\nFoB(θ) = [Δ⟨θ⟩T · (𝐶COLA + 𝐶EE2)−1 · Δ⟨θ⟩]1/2,\n(10)\nwhere θ denotes a vector of cosmological parameters, Δ⟨θ⟩=\n⟨θ⟩COLA −⟨θ⟩EE2 is the difference in sample means, and 𝐶X\ndenotes the parameter covariance matrices for prescription X.\nWe choose to calculate the FoB in selected 2D planes: Ω𝑚×𝑆8,\nΩ𝑚× 𝑤0 and Ω𝑚× 𝑤𝑎. Furthermore, we also calculate the\nFoB in the seven cosmological parameters. A bias of less than\n0.3 is considered negligible [137].\nChanging the nonlinear modeling of the cosmic shear data\nvector may lead to underestimating or overestimating cosmo-\nlogical parameters, compared to a fiducial model. The strength\nof the constraints is measured by the Figure of Merit (FoM)\nstatistic, defined as\nFoM = 𝛼det(𝐶)−1/2,\n(11)\nwhere 𝐶is the covariance matrix of cosmological parameters\nobtained from the MCMC, and 𝛼is a prefactor that depends on\nthe desired limits (i.e., 1𝜎or 2𝜎) and the number of parameters\nconsidered [138]. We report the FoM ratio between the COLA\nand benchmark analyses.\nTo assess whether a nonlinear prescription X may degrade\nthe goodness-of-fit when compared to the benchmark emula-\ntor, we compute the quantity\nΔ𝜒2 = (tX −tEE2)𝑇· 𝐶−1\ndata · (tX −tEE2),\n(12)\nwhere tX is the cosmic shear theory prediction calculated using\nthe nonlinear prescription X and 𝐶data is the data covariance\nmatrix. We compute this quantity for random points across\nthe parameter space.\nIV.\nRESULTS FOR LSST-Y1 SIMULATED ANALYSIS\nTo evaluate the accuracy of our emulator in practice, we be-\ngin by examining cosmological parameter constraints across\nthree nonlinear prescriptions: the EuclidEmulator2 bench-\nmark, our COLA-based emulator, and EE2 ΛCDM (i.e., using\nthe boost from the projected ΛCDM cosmology). Figure 2\nshows 1D and 2D posterior contours (68% and 95%) for the\nparameters Ω𝑚, 𝑆8, 𝑤0 and the sum 𝑤0 + 𝑤𝑎, assuming the\ncentral cosmology (see Table I) as the fiducial. Since the con-\nstraints on 𝑤𝑎are weak, we focus on 𝑤0 + 𝑤𝑎as the more\ninformative parameter combination.\n\n7\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.80\n0.84\n0.88\nS8\n0.80\n0.85\nS8\n1.2\n0.9\nw0\n2.0\n1.0\nw0 +wa\nCutoff 1\nFid. Cosmo.: Center\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.83\n0.86\nS8\n0.80\n0.85\nS8\n1.2\n0.9\nw0\n2.0\n1.0\nw0 +wa\nCutoff 2\nEE2 w0wa\nCOLA\nEE2 CDM\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.82\n0.84\n0.86\nS8\n0.80\n0.84\nS8\n1.2\n0.9\nw0\n2.0\n1.0\nw0 +wa\nCutoff 3\nFIG. 2. Cosmological parameter constraints (68% and 95%) from the LSST-Y1 simulated analyses assuming the center cosmology from Table I\nas the fiducial. Green-filled contours denote constraints obtained using EuclidEmulator2 as the nonlinear prescription, orange dashed and\ndotted contours use our COLA emulator, and blue dashed contours use EE2 ΛCDM prescription. The left, middle, and right panels show\nconstraints using the angular cutoffs C1, C2, and C3, respectively. We observe no shifts between the analyses; however, using cutoffs C2 and\nC3, the constraints obtained using COLA are slightly tighter than those found with EE2 for 𝑆8 and 𝑤0. For EE2 ΛCDM, this effect is amplified.\nBecause our fiducial cosmology is in the ΛCDM subspace,\nwhere all prescriptions are equivalent, we observe no signifi-\ncant biases between the different constraints, and all posteriors\npeak at the same point. However, an indication of failure out-\nside ΛCDM is observed in the size of the error bars. Employing\nthe most conservative cutoff (C1), the COLA and EuclidEm-\nulator2 contours are nearly indistinguishable. At the same\ntime, those of EE2 ΛCDM are significantly smaller than the\n\"true\" error bars of EE2, especially at the parameters 𝑆8 and\n𝑤0, positively correlated. These results suggest that, in the\ncontext of LSST-Y1 cosmic shear, COLA emulators are equiv-\nalent to high-precision N-body emulators up to 𝑘= 1 ℎ/Mpc.\nFigure 1 indicates that, as we advance towards smaller angu-\nlar scales, the disagreements in the boost predictions between\nCOLA and EuclidEmulator2 quickly increase. For Cutoffs\n2 and 3, the constraints are still in excellent agreement, but\nCOLA yields slightly tighter error bars on 𝑆8 and 𝑤0; this\neffect is much more pronounced in the case of EE2 ΛCDM.\nThe 1D marginalized constraints under Cutoff 2 are:\n• EuclidEmulator2:\n𝑆8\n=\n0.835 ± 0.013, 𝑤0\n=\n−1.02+0.15\n−0.18, 𝑤0 + 𝑤𝑎= −1.15+0.32\n−0.28;\n• COLA: 𝑆8 = 0.837 ± 0.012, 𝑤0 = −1.00 ± 0.14, 𝑤0 +\n𝑤𝑎= −1.13 ± 0.29;\n• EE2 ΛCDM: 𝑆8 = 0.838 ± 0.008, 𝑤0 = −1.00 ± 0.14,\n𝑤0 + 𝑤𝑎= −1.11 ± 0.26.\nUsing Cutoff 3, the constraints are:\n• EuclidEmulator2: 𝑆8 = 0.836+0.012\n−0.010, 𝑤0 = −1.01 ±\n0.14, 𝑤0 + 𝑤𝑎= −1.10 ± 0.26;\n• COLA: 𝑆8 = 0.838 ± 0.009, 𝑤0 = −1.00 ± 0.13, 𝑤0 +\n𝑤𝑎= −1.10 ± 0.26;\n• EE2 ΛCDM: 𝑆8 = 0.838 ± 0.007, 𝑤0 = −0.99 ± 0.13,\n𝑤0 + 𝑤𝑎= −1.09 ± 0.24.\nTo quantify the overestimation in 𝑆8 and 𝑤0, we compute\nthe figure of merit (FoM) in the 𝑆8 ×𝑤0 plane. Assuming Cut-\noff 2, relative to EuclidEmulator2, the COLA emulator in-\ncreases the FoM by 8%, indicating slightly tighter constraints,\nwhereas the projected EE2 ΛCDM boost inflates the FoM by\napproximately 47%. For the most \"aggressive\" Cutoff 3, the\nFoM obtained with COLA is 19% bigger than that of EE2,\nwhile EE2 ΛCDM increases the FoM by 58%. Remarkably, in\nterms of figure of merit, the COLA emulator performs better\nat Cutoff 3 than EE2 ΛCDM at Cutoff 1, where the FoM is\nincreased by 28%. From Figure 2, we observe that most of the\ndisagreement between COLA and EuclidEmulator2 lies in\nlow 𝑤0 and low 𝑆8 values, two parameters that are positively\ncorrelated in the analysis. This region of the parameter space\nis excluded by low-redshift geometric data from type-Ia super-\nnovae [58, 139] and BAO [30, 34]. Therefore, we expect that\nthis disagreement would not affect constraints obtained from\nthe combination of LSST cosmic shear data with supernovae\nand BAO distance measurements.\nFigure 3 further investigates the disagreements between\nCOLA and EuclidEmulator2, showing histograms of Δ𝜒2\n(see Equation 12) for COLA and EE2 ΛCDM compared to\nEuclidEmulator2, obtained from 10,000 cosmologies sam-\npled randomly from the emulation box (see Table I). Across all\ncutoffs, the EE2 ΛCDM prescription yields Δ𝜒2 distributions\nthat are substantially broader than those from our COLA emu-\nlator. As such, the use of COLA simulations can significantly\nimprove the consistency with the high-precision benchmark,\nwhile the projection approach fails to capture the nonlinear cor-\nrections required by dynamical dark energy models, leading to\ndegraded fits and potentially biased constraints. In Figure 4,\nwe investigate how these Δ𝜒2 values distribute around cosmo-\nlogical space; we find a clear trend of higher values of Δ𝜒2\ncorrelated with higher values of Ω𝑚and 𝜎8.\nTo assess the robustness of our results beyond the ΛCDM\n\n8\n0\n100\n200\n300\nNumber of Samples\n2\nCOLA <1: 89.5%\n2\nEE2\nCDM <1: 45.8%\nCutoff 1\nEE2 CDM\nCOLA\n0\n100\n200\n300\nNumber of Samples\n2\nCOLA <1: 77.5%\n2\nEE2\nCDM <1: 27.5%\nCutoff 2\n10 2\n10 1\n100\n101\n102\n2\n0\n100\n200\n300\nNumber of Samples\n2\nCOLA <1: 62.8%\n2\nEE2\nCDM <1: 18.4%\nCutoff 3\nFIG. 3. Histograms of Δ𝜒2\nX = (tX −tEE2)𝑇· 𝐶−1\ndata · (tX −tEE2) for\nprescription X ∈{COLA, EE2 ΛCDM} compared to EE2 at the\nfull 𝑤0𝑤𝑎cosmology, with random samples drawn from the prior.\nThe top, middle, and bottom panels show results for angular cut-\noffs C1, C2, and C3, respectively. The distribution of Δ𝜒2 values\ndemonstrates an order of magnitude difference between theory pre-\ndictions calculated using our COLA method compared to the tradi-\ntional EE2 ΛCDM approach, showing an improved fit from modeling\nthe extended parameters using COLA.\nsubspace, we repeat our analysis using the fiducials described\nin Section II, all of them with 𝑤0 ≠−1 and 𝑤𝑎≠0. We\nfind that the biases may increase as we shift 𝑤0 and 𝑤𝑎in\nthe same direction, either higher or lower, in the parameter\nspace of Table I. This is due to a known geometrical degener-\nacy along 𝑤0 + 𝑤𝑎, which can suppress modeling systematics\n0.25\n0.30\n0.35\n0.40\nm\n0.6\n0.8\n1.0\n8\n2\n4\n2\nCOLA\nFIG. 4. Spatial distribution of Δ𝜒2 in the Ω𝑚×𝜎8 plane. We observe\nhigher values of Δ𝜒2 at higher values of Ω𝑚and 𝜎8.\nwhen 𝑤0 +𝑤𝑎≈1. Figure 5 illustrates the results for a fiducial\nwith 𝑤↑\n0 and 𝑤↑\n𝑎(see Table III), keeping the other parameters\nfixed to their central values. In this case, we see a remarkable\nagreement between COLA and EE2 across all angular cut-\noffs. We focus on Cutoff 3, which yields the marginalized 1D\nconstraints:\n• EuclidEmulator2: 𝑆8 = 0.772+0.008\n−0.006, 𝑤0 > −0.882,\n𝑤0 + 𝑤𝑎= −0.76+0.26\n−0.15;\n• COLA: 𝑆8 = 0.770+0.008\n−0.007, 𝑤0 > −0.900, 𝑤0 + 𝑤𝑎=\n−0.79+0.26\n−0.17;\n• EE2 ΛCDM: 𝑆8 = 0.762 ± 0.006, 𝑤0 = −0.879+0.140\n−0.088,\n𝑤0 + 𝑤𝑎= −0.82+0.26\n−0.17.\nIn this case, the EE2 ΛCDM projection induces substantial\nbiases in 𝑆8, shifting up by nearly 1𝜎. By comparison, the\nCOLA emulator remains consistent to within 0.25 standard\ndeviations. This trend extends to the multidimensional figures\nof bias: COLA has a 7D figure of bias of FoB7D = 0.27\ncompared to the benchmark, while the projected EE2 ΛCDM\nreaches FoB7D = 1.04.\nUnlike the parameter constraints assuming the center cos-\nmology from Table I as the fiducial, we see in Figure 5 that\nthe posteriors calculated using our COLA-based emulator now\nclosely track those generated using our 𝑁-body proxy, rather\nthan overestimating 𝑆8 and 𝑤0 in any substantive way. We find\nthat in switching from the center cosmology to 𝑤↑\n0 and 𝑤↑\n𝑎, the\nrelative ratio between FoMs of COLA and EE2 in the 𝑆8 × 𝑤0\nplane is 0.97, while the same ratio is 1.21 for EE2 ΛCDM.\nTo investigate whether our COLA emulator can provide un-\nbiased constraints with FOMs similar to EE2 across the param-\neter space, Figure 6 shows the 1D biases of Equation 9 for Ω𝑚,\n𝑆8, 𝑤0 and 𝑤𝑎between the COLA emulator and our baseline\nEE2 for all scale cuts and all of the 29 cosmologies outlined\nin Section II. The fiducial cosmologies are listed in increasing\norder of their associated 𝜎8 values. All 1D biases are within\n\n9\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.72\n0.76\n0.80\nS8\n0.72\n0.78\nS8\n1.2\n0.9\nw0\n1.6\n0.8\nw0 +wa\nCutoff 1\nFid. Cosmo.: w0 ,wa\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.76\n0.79\nS8\n0.75\n0.80\nS8\n1.2\n0.9\nw0\n1.6\n0.8\nw0 +wa\nCutoff 2\nEE2 w0wa\nCOLA\nEE2 CDM\n0.30\n0.35\nm\n1.5\n1.0\n0.5\nw0 +wa\n1.2\n1.0\n0.8\nw0\n0.74\n0.76\n0.78\nS8\n0.75\n0.78\nS8\n1.2\n0.9\nw0\n1.6\n0.8\nw0 +wa\nCutoff 3\nFIG. 5. Cosmological parameter constraints (68% and 95%) from the LSST-Y1 simulated analyses assuming a fiducial cosmology with\n𝑤↑\n0 and 𝑤↑\n𝑎(see Table III), keeping the other parameters at their central values. Green-filled contours denote constraints obtained using\nEuclidEmulator2 as the nonlinear prescription, orange dashed and dotted contours use our COLA emulator, and blue dashed contours use\nEE2 ΛCDM prescription. The left, middle, and right panels show constraints using the angular cutoffs C1, C2, and C3, respectively. In this\ncase, the EE2 ΛCDM prescription can provide significant biases in 𝑆8, which are not present when using COLA.\n0.3𝜎, even for cosmologies with higher values of 𝜎8 and Ω𝑚,\nwhere Figure 4 shows our emulator performs worse. We com-\nputed the 7D figure of bias in cosmological parameters for all\nfiducial cosmologies using Cutoff 3, finding a maximum value\nof FoB7D = 0.35 at the cosmology Ω↑\n𝑚, 𝑤↑\n0, 𝑤↑\n𝑎. We conclude\nthat, for all scale cuts considered in this work, our emulator\nsucceeds in providing unbiased constraints on cosmological\nparameters when compared to a high-precision N-body em-\nulator in the context of dynamical dark energy models, even\nwith significant variations in cosmological parameters and ex-\ntreme values of 𝜎8. Further, Figure 6 indicates our measure of\nthe relative tightness of the parameter constraints in the 𝑆8 ×𝑤\nplane compared to EE2, FOMCOLA\n𝑆8×𝑤/FOMEE2\n𝑆8×𝑤, assuming Cut-\noff 3.\nThe highest ratio is 1.19 for the center cosmology,\nshown in Figure 2. The same remark from before applies to\nother fiducial cosmologies: the disagreement between COLA\nand EE2 is driven mainly by low values of 𝑤0 and 𝑆8, a region\nof the parameter space disallowed by low-redshift distance\nmeasurements.\nV.\nCONCLUSION\nIn light of recent hints of dynamical dark energy, constrain-\ning its equation of state has become a task of central impor-\ntance. While geometrical probes, such as BAO and supernovae\nmeasurements, are the primary probes of late dark energy be-\nhavior, galaxy surveys can also probe dark energy dynamics\nthrough their effects on the growth of large-scale structure.\nExtracting robust constraints from these surveys requires ac-\ncurate modeling of nonlinear gravitational effects in the mat-\nter power spectrum, which becomes increasingly challenging\nin extended cosmologies where 𝑤(𝑧) departs from a cosmo-\nlogical constant. While accurate modeling can be achieved\nwith 𝑁-body simulations, their high computational cost limits\ntheir applicability across the myriad candidate dynamical dark\nenergy models (e.g., quintessence), and, further, for models\nthat directly impact the growth of matter perturbations beyond\nmodifications in the Universe’s expansion.\nIn this context,\nCOLA is a fast approximate alternative to 𝑁-body simulations,\nwhich, when appropriately corrected, presents an avenue for\nconstructing accurate emulators for the nonlinear matter power\nspectrum at a fraction of the computational cost.\nIn this work, we have built an emulator for the nonlinear\nboost assuming the 𝑤0𝑤𝑎CDM cosmological model using a\nsuite of 1400 COLA simulations, two for each of the 700 cos-\nmologies in the training set to account for pairing-and-fixing.\nTo evaluate the accuracy of the neural network, we ran a pair\nof simulations for each of the 200 cosmologies in the test set.\nThe total computational cost of all simulations is estimated\nat 153.600 CPU-hours. A simple connected neural network\nwith a trainable activation function can reproduce the test set\nboosts at 0.1% error. The computational cost of the simula-\ntion suite could potentially be lowered by using an alternative\nsampling algorithm for the training set cosmologies: the Sobol\nsequence [140], which has been shown to improve emulation\nerrors compared to Latin hypercube sampling [83].\nWe have compared our COLA emulator to a benchmark\n𝑁-body emulator, chosen as EuclidEmulator2.\nWe test\nan additional nonlinear prescription common to analyses of\nextended cosmological models without 𝑁-body simulations:\nusing 𝑁-body boosts at the projected ΛCDM cosmology (i.e.,\nsetting 𝑤0 = −1 and 𝑤𝑎= 0), an approach we denote as\nEE2 ΛCDM. We compare nonlinear models in two manners:\nat the boost level, shown in Figure 1, and at the level of a\nsimulated cosmic shear analysis akin to LSST-Y1, assuming\nEuclidEmulator2 as the true nonlinear model. In the data\nanalysis, to account for possible variations in the cosmolog-\nical parameters when beyond-ΛCDM models are analyzed,\nwe define fiducial cosmologies scattered across the parameter\n\n10\n0.5\n0.0\n0.5\nm/\nm\nm,w0 ,wa\nAs ,w0 ,wa\nm,w0 ,wa\nns ,w0 ,wa\nm,w0 ,wa\nw0 ,wa\nns ,w0 ,wa\nAs ,w0 ,wa\nm,w0 ,wa\nAs ,w0 ,wa\nAs ,w0 ,wa\nns ,w0 ,wa\nw0 ,wa\nns ,w0 ,wa\nAs ,w0 ,wa\nCenter\nns ,w0 ,wa\nm,w0 ,wa\nw0 ,wa\nns ,w0 ,wa\nAs ,w0 ,wa\nns ,w0 ,wa\nw0 ,wa\nns ,w0 ,wa\nAs ,w0 ,wa\nm,w0 ,wa\nAs ,w0 ,wa\nm,w0 ,wa\nm,w0 ,wa\n0.5\n0.0\n0.5\nS8/ S8\n0.5\n0.0\n0.5\nw0/ w0\n0.5\n0.0\n0.5\n(w0 +wa)/ w0 +wa\n0.95 | 0.97 | 0.92\n1.04 | 1.05 | 0.96\n1.01 | 1.06 | 1.04\n0.99 | 1.05 | 1.01\n1.04 | 1.05 | 1.07\n1.00 | 1.05 | 0.94\n1.06 | 1.01 | 1.03\n1.02 | 1.04 | 1.14\n0.98 | 0.95 | 1.01\n1.00 | 0.98 | 1.06\n1.03 | 1.03 | 1.12\n0.99 | 1.04 | 1.10\n1.02 | 1.15 | 1.18\n1.06 | 1.10 | 1.17\n0.93 | 1.02 | 0.99\n1.06 | 1.08 | 1.19\n0.91 | 1.01 | 1.08\n1.07 | 1.10 | 1.02\n1.04 | 1.07 | 1.15\n1.07 | 1.05 | 1.14\n1.02 | 1.07 | 1.18\n1.00 | 0.96 | 0.97\n0.99 | 0.96 | 1.02\n1.07 | 0.97 | 1.04\n1.00 | 1.05 | 1.08\n1.04 | 1.11 | 1.13\n0.92 | 1.00 | 0.99\n0.95 | 0.98 | 0.99\n0.94 | 0.90 | 0.92\nFoMCOLA\nS8 ×w / FoMEE2\nS8 ×w\n C1 | C2 | C3 \nC1\nC2\nC3\nFIG. 6. One-dimensional biases from Equation 9, shown for the parameters Ω𝑚, 𝑆8, 𝑤0 and 𝑤𝑎, sorted in increasing order of their associated\n𝜎8 values. On the right-hand side, we show the ratios in figure of merit (see Equation 11) in the 𝑆8 × 𝑤0 plane between the analyses using\nCOLA and EE2, for the three angular cutoffs. The gray bands represent 0.3𝜎bias.\n\n11\nspace, including outside ΛCDM, with significant variations in\nΩ𝑚and 𝜎8. We have assessed the goodness-of-fit degradation,\nshown in Figure 3, and parameter constraint biases, shown in\nFigures 2, 5 and 6. We find that, at the boost level, our emu-\nlator can reproduce the benchmark emulator results with less\nthan 2% error at 𝑘= 1 ℎ/Mpc, while EE2 ΛCDM produces\n7.5% errors at the same scales. As for the simulated analysis,\nwe find that our COLA-based emulator can provide unbiased\nconstraints compared to EuclidEmulator2: all 1D biases\nare well within 0.3𝜎, even for the most aggressive angular\ncutoffs and exotic fiducial cosmologies with extreme values\nof Ω𝑚, 𝜎8, or outside ΛCDM. Furthermore, all 7D figures of\nbias are below 0.35. At the precision level expected for the\nfirst year of LSST observations, the COLA emulator yields\nconstraints equivalent to those obtained using EuclidEmula-\ntor2 for scales up to 𝑘≈3 ℎ/Mpc, comparable to our Cutoff\n2.\nOur results demonstrate that COLA, when combined with an\naccurate ΛCDM reference, offers a viable and flexible frame-\nwork for extending nonlinear modeling to dynamical dark en-\nergy and other beyond-ΛCDM scenarios. We emphasize that,\nwhile we use EuclidEmulator2 as the \"baseline\" ΛCDM\nemulator in Equation 8, any other emulator could be used to\nprovide ΛCDM boosts. Moreover, our methodology can be\napplied to more exotic models that also modify the growth of\nstructure directly, such as modified gravity or coupled dark en-\nergy [105]. We also remark that there are avenues to improve\nour methodology. One example is the choice of \"reference\"\ncosmology. Equation 8 uses the projected ΛCDM cosmology\nbecause, geometrically, it is the closest cosmology; however, a\npossible choice that improves accuracy is to use a 𝑤CDM cos-\nmology with the same value of 𝜎8, akin to what is done in the\nCasarini [141] prescription. Moreover, COLA can be com-\nbined with other analytical or semi-analytical prescriptions,\nsuch as the one proposed in [142], improving its accuracy at\nsmall scales.\nACKNOWLEDGEMENTS\nThe authors would like to thank Stony Brook Research Com-\nputing and Cyberinfrastructure and the Institute for Advanced\nComputational Science at Stony Brook University for access\nto the high-performance SeaWulf computing system.\nThis\nresearch was also supported by resources supplied by the Cen-\nter for Scientific Computing (NCC/GridUNESP) of the São\nPaulo State University (UNESP). This work made use of the\nCHE cluster, managed and funded by COSMO/CBPF/MCTI,\nwith financial support from FINEP and FAPERJ, and oper-\nating at the RSDC - Datacenter para Ciência Alfredo Mar-\nques de Oliveira/CBPF. JR acknowledges the financial support\nfrom FAPESP under grant 2020/03756-2, São Paulo Research\nFoundation (FAPESP) through ICTP-SAIFR. JR also acknowl-\nedges that this study was financed in part by the Coordenação\nde Aperfeiçoamento de Pessoal de Nível Superior – Brasil\n(CAPES) – Finance Code 001. VM is supported by the Roman\nProject Infrastructure Team “Maximizing Cosmological Sci-\nence with the Roman High Latitude Imaging Survey\" (NASA\ncontracts 80NM0018D0004-80NM0024F0012). V.M. is also\npartially supported by the Roman Project Infrastructure Team\n“A Roman Project Infrastructure Team to Support Cosmolog-\nical Measurements with Type Ia Supernovae\" (NASA contract\n80NSSC24M0023).\n[1] M. A. Troxel, N. MacCrann, J. Zuntz, T. F. Eifler, E. Krause,\net al., Dark Energy Survey Year 1 results: Cosmological con-\nstraints from cosmic shear, Phys. Rev. D 98, 043528 (2018),\narXiv:1708.01538 [astro-ph.CO].\n[2] T. M. C. Abbott, F. B. Abdalla, A. Alarcon, J. Aleksić, S. Al-\nlam, et al., Dark Energy Survey year 1 results: Cosmological\nconstraints from galaxy clustering and weak lensing, Phys. 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Pellejero-Ibañez, Web-Halo\nModel (WHM): Accurate non-linear matter power spec-\ntrum predictions without free parameters, arXiv e-prints ,\narXiv:2508.10902 (2025), arXiv:2508.10902 [astro-ph.CO].\nAppendix A: Simulations and Emulator Accuracy for Higher\nRedshifts\nIn this Appendix, we show relative errors from the simula-\ntions and the emulator for higher redshifts, ensuring that our\ndiscussion in Section II still holds. The results are presented\nin Figure 7. We find that for 𝑧≤3, the 𝑧= 0 results still hold\nin that at 𝑘= 1 ℎ/Mpc, 90% of cosmologies have emulation\nerrors within 2% on ˜𝐵(𝑘, 𝑧), and that 50% of cosmologies are\ncontained well within 1%. COLA emulators trained at differ-\nent redshifts demonstrate no notable change in the fidelity of\npredictions across the range 0 ≤𝑧≤3 (right panels), even\nthough the highest 10% of raw emulation errors (left panels)\nand COLA boost errors (middle panels) grow with increasing\n𝑧.\n\n17\nk (h/Mpc)\n5\n0\n5\nz=0.24\n50%\nk (h/Mpc)\n5\n0\n5\n90%\nk (h/Mpc)\n5\n0\n5\n100%\nk (h/Mpc)\n5\n0\n5\n1000×(B COLA\nNN\n/B COLA\n1)\nz=0.50\nk (h/Mpc)\n5\n0\n5\n100×(BCOLA/B EE2\n1)\nk (h/Mpc)\n5\n0\n5\n100×(B EE2\nCDM/B EE2\n1)\nk (h/Mpc)\n5\n0\n5\nz=0.74\nk (h/Mpc)\n5\n0\n5\nk (h/Mpc)\n5\n0\n5\n10 1\n100\nk (h/Mpc)\n5\n0\n5\nz=1.00\n10 1\n100\nk (h/Mpc)\n5\n0\n5\n10 1\n100\nk (h/Mpc)\n5\n0\n5\nFIG. 7. Left panel: Relative errors between the COLA boosts 𝐵COLA predicted by the emulator versus those obtained from the test set\nsimulations. Middle panel: Relative errors between ˜𝐵COLA (see Equation 8) versus the boosts from EuclidEmulator2. Right panel:\nRelative errors between 𝐵EE2 ΛCDM and EE2. Each row denotes a different redshift. Colors in all panels denote the percentile of cosmologies\naround the mean: blue contours enclose 50% of cosmologies, red contours enclose 90% of cosmologies, and the outer gray lines enclose 100%\nof cosmologies.\n"},"cleaned":{"kind":"string","value":"Modeling nonlinear scales for dynamical dark energy cosmologies with COLA João Rebouças,1, 2 Victoria Lloyd,3 Jonathan Gordon,3, 4 Guilherme Brando,1 and Vivian Miranda4 1CBPF - Brazilian Center for Research in Physics, Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil 2Instituto de Física Teórica da Universidade Estadual Paulista, R. Dr. Bento Teobaldo Ferraz, 271, Bloco II, Barra-Funda - São Paulo/SP, Brazil 3 11794, USA 4C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY, 11794, USA (Dated: October 17, 2025) Upcoming galaxy surveys will bring a wealth of information about the clustering of matter at small scales, but modeling small-scale structure beyond ΛCDM remains computationally challenging. While accurate N-body emulators exist to model the matter power spectrum for ΛCDM and some limited extensions, it's unfeasible to generate N-body simulation suites for all candidate models. Motivated by recent hints of an evolving dark energy equation of state from galaxy surveys, we assess the viability of employing the COmoving Lagrangian Acceleration (COLA) method to generate simulation suites assuming the w0wadynamical dark energy model. Following up on our previous work, we combine COLA simulations with an existing high-precision ΛCDM emulator to extend its predictions into new regions of parameter space. We assess the precision of our emulator at the level of the matter power spectrum, finding that our emulator can reproduce the nonlinear boosts from EuclidEmulator2 at less than 2% error. Moreover, we perform an analysis of a simulated cosmic shear survey akin to future data from the Legacy Survey of Space and Time (LSST) first year of observations, assessing the differences in parameter constraints between our COLA-based emulator and the benchmark emulator. We find our emulator to be in excellent agreement with the benchmark, achieving less than 0.3σshifts in cosmological parameters across multiple fiducial cosmologies, and a 7D figure of bias of less than 0.35. We further compare our emulator's performance to a commonly used approach: assuming the ΛCDM boost can be employed for extended parameter spaces without modification. We find that our emulator yields a significantly smaller Δχ2 distribution, parameter constraint biases, and a more accurate figure of merit compared to this second approach. These results demonstrate that COLA emulators provide a computationally efficient and physically motivated path forward for modeling nonlinear structure in extended cosmologies, offering a practical alternative to full N-body suites in the era of precision cosmology. I. INTRODUCTION In recent decades, galaxy surveys have been able to map the large-scale structure of the Universe with great precision, being as competitive as CMB surveys. Current photometric surveys, such as DES [1-10], KiDS [11-23], HSC [24-29], and spectroscopic surveys, such as DESI [30-36], BOSS/eBOSS [37-50], and WiggleZ [51-55], have placed percent-level constraints on ΛCDM parameters, demonstrating the success of the theoretical model in describing independent datasets. However, in recent years, tensions in the cosmological parameter constraints have begun to arise with the increase in precision. Notably, most galaxy surveys report a mildly lower value for the structure growth parameter S8 than those from CMB measurements (see e.g. [56]). Moreover, recent findings from the DESI collaboration, as well as Type Ia supernovae datasets such as Pantheon+, Union3, and DESY5 [30, 34, 35, 57-59] favor an evolution of the dark energy equation of state. Whether these results truly indicate new physics is still under debate [60-64]. Forthcoming Stage-IV galaxy surveys, such as the Vera Rubin Observatory's LSST [65], Euclid [66], SphereX [67], and Roman [68], will enable higher-precision measurements, especially at small scales where non-linearities in the matter density field become increasingly sizable [69], and will be decisive for investigating dark energy dynamics. Analyses of galaxy surveys rely on a key theoretical prediction: the matter power spectrum, as galaxies are biased tracers of the underlying matter density field. At large scales and early times, the power spectrum can be quickly and accurately computed using Einstein-Boltzmann solvers such as camb [70] and class [71, 72]. However, for small scales and low redshifts, linear perturbation theory breaks down, and accurately modeling non-linearities becomes a task of central importance. Although N-body simulations offer the most accurate predictions in the nonlinear regime, they are computationally expensive - each one demanding tens of thousands of CPU hours [73-75]. As a result, integrating simulations into Bayesian analyses is a prohibitive task, as theoretical predictions must be provided for O(105 -106) points in the parameter space. To address this issue, machine learning emulators trained on N-body simulations have been developed for ΛCDM and simple, widely adopted extensions such as the phenomenological w0waCDM [76, 77] parametrization for dark energy, as well as massive neutrinos with their total mass as a free parameter. Examples include EuclidEmulator2 [78], bacco [79], CosmicEmu [80], the Dark Quest emulator [81, 82], the CSST Emulator [83-85], aemulus [86-89], among others. At the same time, due to the sheer amount of candidate cosmological models and the high cost of running N-body simulations, emulators for broader extensions of ΛCDM are still scarce: examples are emulators to specific modified gravity theories (e.g. [90-93]) where we have full N-Body simulations [94], as well as some hydrodynamical emulators. A viable alternative to using full N-body simulations for constructing matter power spectrum emulators is to use wellestablished approximate methods for extended cosmological models. These methods reduce the computational complexity 16 Oct 2025 2 of running hundreds of simulations to train emulators, at the cost of losing accuracy in deep non-linear scales. One compelling approach is to use the COmoving Lagrangian Acceleration (COLA) method, which combines Lagrangian Perturbation Theory with a particle-mesh (PM) [95] evolution scheme to approximate N-body results while being cheaper than the usual N-body methods by 1-2 orders of magnitude [96, 97]. Emulators created using pure COLA simulations are prone to small-scale inaccuracies when compared directly to their Nbody counterparts. To mitigate this effect, the work of [98] introduces an approach that combines COLA simulations with predictions from high-accuracy ΛCDM emulators or full Nbody results, leveraging COLA's reduced computational cost and dramatically increasing small-scale accuracy simultaneously. Our previous work [99] has validated this approach, creating an emulator for the matter power spectrum of COLA simulations under the wCDM cosmological model, and testing it in a mock Stage-IV cosmic shear analysis. As such, we aim to demonstrate here that the hybrid approach of COLA-based emulators combined with high-resolution ΛCDM emulators can provide unbiased cosmological parameter constraints when compared to full N-body methods. In this work, we now present a final validation of our COLAbased emulators on the w0waCDM cosmological model, where the dark energy equation of state evolves linearly with the scale factor. This parametrization represents the most widely used and general extension of ΛCDM for which highaccuracy emulators currently exist, and remains central to ongoing investigations into dynamical dark energy [30, 34]. We extend our machine learning pipeline, combining COLA simulations with ΛCDM emulators, to predict the nonlinear matter spectrum across the w0waCDM parameter space. We train a simple neural network to emulate the nonlinear correction factor (i.e the boost) from COLA, correcting for small-scale inaccuracies by referencing boosts from a high-fidelity ΛCDM emulator. This hybrid approach enables fast predictions across an extended cosmological parameter space while maintaining consistency with N-body precision. To validate our emulator in a cosmological inference setting, we perform a simulated cosmic shear analysis using survey specifications consistent with LSST's first year of observations (LSST-Y1) [65]. We compare parameter constraints derived using both our pipeline and a benchmark N-body emulator, chosen as EuclidEmulator2, quantifying their disagreement with standard tension metrics [100-102]. Additionally, we also benchmark our emulator against a widely used approximation method in beyond-ΛCDM analyses for models without dedicated nonlinear simulations [103108]: projecting the nonlinear boost from the nearest ΛCDM cosmology. This projection method assumes that nonlinear corrections calibrated in ΛCDM remain valid in nearby extended cosmologies, providing a computationally inexpensive workaround but at the cost of uncontrolled systematics. In contrast, we demonstrate that for dynamical dark energy models, we find that the ΛCDM projection approach may introduce significant deviations in both goodness-of-fit and parameter constraints. Meanwhile, our COLA-based emulator reproduces the predictions of high-precision N-body emulators without bias. This paper is organized as follows: Section II describes the COLA simulations, cosmological parameters, simulation output processing, emulator construction, and validation; in Section III, we present our LSST-Y1 simulated cosmic shear analysis and the tension metrics used to assess their differences; in Section IV, we present and discuss the results of the LSSTY1 simulated analysis; finally, we conclude in Section V. II. COLA EMULATOR A. COLA Simulation Suite The COmoving Lagrangian Acceleration (COLA) algorithm [96] is a fast approximate method for N-body simulations, wherein particles evolve in a frame comoving with trajectories calculated using Lagrangian Perturbation Theory (LPT), most commonly 2nd order Lagrangian perturbation theory (2LPT). For small scales, the method computes the force by using a Particle-Mesh (PM) algorithm, where the residual displacements not captured by LPT are added to the trajectories. COLA has been shown to agree with full N-body simulations, at the level of the power spectrum at up to k∼1h/Mpc, as well as when predicting ratios of the modified gravity power spectrum and the ΛCDM one, the so-called boost function in modified gravity [98, 109-111]. Despite being 1-2 orders of magnitude faster than a full N-body run, the computational cost of these approximations is still too high for direct use in the O(106) computations of the matter power spectrum required for Monte Carlo searches. A practical alternative is to use a fixed set of COLA simulations to train emulators for the matter power spectrum, enabling efficient interpolation across cosmological parameter space. Our previous work demonstrated this approach for wCDM [99]; here, we extend it to w0waCDM and evaluate its performance relative to the benchmark EuclidEmulator2 [78], which achieves ≲1% precision for w0waCDM + Í mνup to k= 10 h/Mpc-1 and z≤3. 1. Simulation Settings We use the COLA algorithm as implemented in the public fml1 code. Each simulation is performed in a box of size L= 1024h-1Mpc, populated with Npart = 10243 particles, initialized at zini = 19, and evolved over 51 time steps chosen to maintain a uniform time resolution of Δa≈0.02. The force grid uses Nmesh = 20483 cells, and the power spectra are calculated on-the-fly using a N3 pk-mesh = 10243 grid. Therefore, the corresponding Nyquist frequency is kNyq = πNpk-mesh/L= πh/Mpc. To avoid aliasing, we 1 https://github.com/HAWinther/FML 3 restrict our analysis to k≤kNyq [112]. Our choices are based on Reference [98] and are validated therein. Initial conditions are generated using 2LPT, and we employ the forward approach [113] for our simulations. We provide the linear transfer functions of matter density, velocity, and relativistic species' densities at each time step using class2 in synchronous gauge, and convert to the N-body gauge [114117] in COLA. Our simulations were run in the Seawulf3 cluster. With these settings, and using 128 cores, one COLA w0wasimulation takes approximately 40 minutes to finish, and requires a total RAM of approximately 950 GB. To suppress sample variance from finite box effects at large scales (k≈1/L), we use the pairing-and-fixing method [118], in which we generate Gaussian random field modes with a fixed amplitude, δi,lin, but with phase shifts of πwith respect to one another. The initial overdensity fields are sampled as δi,lin = √︁ Pieiθi. (1) where θiis a random phase and Pithe initial power spectrum. Averaging over each pair, we find that the result substantially suppresses the effects of cosmic variance. This strategy was chosen for this work following [99] and [78]. 2. Definition of the Parameter Space We consider the cosmological w0waCDM model, where the dark energy equation of state is parametrized as w(a) = w0 + wa(1 -a), (2) with abeing the scale factor, and w0 and wacontrol the present-day value and time derivative of the dark energy equation of state, respectively. The ΛCDM model is recovered in the limit w0 = -1 and wa= 0. The free cosmological parameters are: • Ωm, the total baryon density, • Ωb, the total matter density, • h= H0/(100 km s-1 Mpc-1), the dimensionless Hubble parameter, • As, the amplitude of initial scalar fluctuations, • ns, the scalar spectral index, • w0 and wa, the dark energy equation of state parameters. We fix the summed neutrino masses to the minimum value allowed by neutrino oscillation experiments, Í mν= 0.058 eV, assuming three degenerate massive species [119, 120]. The parameter space boundaries are described in Table I, set to match those of EuclidEmulator2, which we have adopted 2 https://github.com/lesgourg/class_public 3 https://rci.stonybrook.edu/HPC Ωm Ωb ns As× 109 h w0 wa Min 0.24 0.04 0.92 1.7 0.61 -1.3 -0.7 Max 0.40 0.06 1.00 2.5 0.73 -0.7 0.5 Center 0.319 0.05 0.96 2.1 0.67 -1 0 TABLE I. Parameter space validity bounds of our COLA-based emulator. The training set is drawn from a slightly bigger hypercube, where each dimension is stretched by 10% in each direction (e.g. 0.224 1.182, aliasing of the kmodes near the Nyquist frequency leads to a power spectrum less than the shot noise for some simulations, and the subtraction would lead to unphysical negative values [113]; for these redshifts, 4 we choose to cut the scales at half of the Nyquist frequency, kz>1.182 ≤(π/2) h/Mpc, following our procedure in [99] (also see [112]). We then perform several transformations to optimize the inputs and outputs of our emulator. For instance, machine learning techniques are known to perform poorly if the features span several orders of magnitude. To stabilize the following procedures, we normalize the cosmological parameters θ to [-1, 1] according to θN= -1 + 2 θ -θmin θmax -θmin , (4) where minimum and maximum values correspond to the training set boundaries, i.e., stretching the intervals of Table I by 10% in each direction. Furthermore, we standardize the boosts using BCOLA N (k, z|θ) = BCOLA(k, z|θ) - ̄BCOLA(k, z) σB(k, z) , (5) where BCOLA N (k, z|θ) is the normalized COLA boost, ̄B(k, z) is the average of all boosts in the training set, and σB(k, z) is their standard deviation. We then perform a Principal Component Analysis (PCA) decomposition of the COLA boosts using scikit-learn [122] to reduce dimensionality. We retain NPC = 15 components, which are sufficient to recover the test set boosts to within 0.2%. 3. Neural Network Emulator After post-processing, we train our emulator with the normalized cosmological parameters as input features and the principal components as targets. We use a fully connected neural network with three hidden layers, each with 1024 neurons, with a mean squared error loss function, L = Ntrain ∑︁ i=1 NPC ∑︁ j=1 (αi,train j -αi,pred j )2, (6) where αi,train j is the j-th principal component coefficient of the i-th cosmology in the training set, and αi,pred j the corresponding prediction. We use the parametric activation function [123, 124] ym+1 n = γm n+ (1 -γm n) 1 1 + e-βm nym n ̃ym n, (7) where ym+1 n is the value of the n-th neuron of the (m+ 1)-th layer, ̃ym nthe n-th neuron from the (m+1)-th layer after the application of weights and biases, and γm nand βm nare parameters of the activation function that can be back-propagated during training. We use the Adam [125] optimizer to train the model parameters. 4. Boost Errors We perform a series of accuracy checks on the emulator outputs. First, to assess the accuracy of our neural network, we compare the emulator's predictions for test set cosmologies, unseen in the training procedure, against the actual COLA simulations. The relative errors are shown in the first panel of Figure 1. At k= 1 h/Mpc, 90% of the test set cosmologies have an emulation error within 0.1%. Comparing these direct emulation errors to the errors on the corrected boosts ̃BCOLA(k, z) (second panel), we note that the errors between the two differ by an order of magnitude. This indicates that, in the context of comparing COLA with high-precision simulations, the COLA emulator faithfully reproduces its simulations, and differences between the emulators can be attributed to the COLA approximation rather than the performance of the machine learning model. As per [98], COLA simulations increasingly lose power4 at progressively smaller scales, leading to typical errors of 10% at k= 1 h/Mpc for raw COLA boosts BCOLA, as defined in Equation 3. However, this power loss is cosmology-independent. Our previous work [99] showed that the best technique to build a COLA-based emulator is to leverage existing high-precision emulators in ΛCDM, using COLA only to extend the results into new dimensions, i.e., extra model parameters. This idea is encoded in the following expression for the nonlinear boost, ̃BCOLA(k, z|θ) = BN-body(k, z|θp) × BCOLA(k, z|θ) BCOLA(k, z|θp) , (8) where θpis the projection of θ in the ΛCDM subspace and BN-body is the nonlinear boost obtained from our benchmark N-body prescription. We choose EuclidEmulator2 as the base N-body prescription. The second panel of Figure 1 shows the relative difference between ̃B(k, z), calculated using Equation 8, and the benchmark emulator predictions, BEE2(k, z), for all test set cosmologies. At k= 1 h/Mpc, 90% of cosmologies have emulation errors within 2%, with 50% of cosmologies contained well within 1%. This demonstrates that our method successfully mitigates the accumulation of errors typical of COLA simulations in the nonlinear regime, allowing us to generate accurate predictions across our target krange. Furthermore, the cosmologies with larger errors are those with higher values of w0 + wa, a region of the parameter space excluded by current data. Finally, we consider a third nonlinear prescription: using the nonlinear boost from EuclidEmulator2 in the ΛCDM subspace; this approach will be denoted as EE2 ΛCDM. For this purpose, we compute the nonlinear boosts for w0waCDM cosmologies using EuclidEmulator2, setting w0 = -1 and wa= 0. The third panel of Figure 1 shows relative errors 4 This loss of power is well known to PM N-body codes, which fail to resolve the internal dynamics of halos. This trend of losing power starts roughly at a scale at which the pure 1-halo term of the halo model would dominate the power spectrum. 5 5.0 2.5 0.0 2.5 5.0 1000×(B COLA NN /B COLA 1) 50% 10 1 100 k (h/Mpc) 5.0 2.5 0.0 2.5 5.0 100×(BCOLA/B EE2 1) 90% 10 1 100 k (h/Mpc) 5.0 2.5 0.0 2.5 5.0 100×(B EE2 CDM/B EE2 1) 100% FIG. 1. From top to bottom: 1) Relative errors between the COLA boosts BCOLA predicted by the emulator versus those obtained from the test set simulations. 2) Relative errors between ̃BCOLA (see Equation 8) and the boosts from EE2. 3) Relative errors between BEE2 ΛCDM and EE2. Colors in all panels denote the percentile of cosmologies around the mean: blue contours enclose 50% of cosmologies, red contours enclose 90% of cosmologies, and the outer gray lines enclose 100% of cosmologies. All panels show results for z= 0, see Appendix A for the equivalent plots at higher redshifts. between BEE2 ΛCDM and the actual EuclidEmulator2 boosts. The 90% percentile shows errors of the order of 7.5% at k= 1 h/Mpc, significantly worse than the COLA errors of the panel above. These results support the viability of our emulator for parameter inference in w0waCDM, as the emulated boost ̃B(k, z) agrees with our N-body proxy at a level suitable for upcoming precision cosmology experiments [126] while requiring significantly less computational expense compared to traditional N-body methods. In the following, we investigate how the differences shown in Figure 1 impact the parameter constraints from simulated cosmic shear analysis. Parameter Fiducial Prior Survey specifications Area 12300 deg2 - Shape noise per component 0.26 - nsources eff 11.2 arcmin-2 - Photometric redshift offsets Δzi source 0 N[0, 0.002] Intrinsic alignment (NLA) a1 0.7 U[-5, 5] η1 -1.7 U[-5, 5] Shear calibration mi 0 N[0, 0.005] TABLE II. Mock survey specifications for our simulated analysis, and nuisance parameter priors. U[a, b] denotes an uniform distribution with edges [a, b], while N [a, b] denotes a Gaussian distribution with mean aand standard deviation b. Tomographic bin indices are denoted by i, and all our priors are the same for all bins. III. ANALYSIS OF LSST-Y1 SIMULATED DATA A. Simulating Cosmic Shear Data We simulate cosmic shear observations based on LSST-Y1, following the methodology of [127, 128] and detailed in [99]. Survey specifications, source galaxy redshift distributions, and nuisance parameter priors are taken from the LSST DESC Science Requirements Document [65], and summarized in Table II. The redshift distribution is modeled as a Smail distribution convolved with a Gaussian uncertainty 0.02(1 + z) and divided into five tomographic bins with equal galaxy number densities. The cosmic shear two-point correlation functions ξij ± (θ) are computed by first evaluating in Fourier space, Cij κκ(l), using the nonlinear matter power spectrum via the Limber approximation, then transforming to real space via the analytic functions in Appendix A of [6]. We compute ξij ± in 26 logarithmically spaced angular bins between 2.5 and 900 arcmin, averaging over each bin. We include standard self-calibrating systematics in our computation of ξ± - photometric redshift uncertainties, multiplicative shear calibration, and the nonlinear alignment (NLA) model of intrinsic galaxy alignments (see, e.g., [129, 130]). Likelihood analyses are performed using Cocoa, the Cobaya-CosmoLike Joint Architecture5 [131-133]. Linear power spectra are computed with camb [134, 135], and nonlinear corrections are applied using either ̃B(Eq. 8), BEE2 ΛCDM, or EuclidEmulator2. We use MCMC sampling to explore the parameter space and assess convergence using the Gelman-Rubin criterion (|R-1| -0.882, w0 + wa= -0.76+0.26 -0.15; • COLA: S8 = 0.770+0.008 -0.007, w0 > -0.900, w0 + wa= -0.79+0.26 -0.17; • EE2 ΛCDM: S8 = 0.762 ± 0.006, w0 = -0.879+0.140 -0.088, w0 + wa= -0.82+0.26 -0.17. In this case, the EE2 ΛCDM projection induces substantial biases in S8, shifting up by nearly 1σ. By comparison, the COLA emulator remains consistent to within 0.25 standard deviations. This trend extends to the multidimensional figures of bias: COLA has a 7D figure of bias of FoB7D = 0.27 compared to the benchmark, while the projected EE2 ΛCDM reaches FoB7D = 1.04. Unlike the parameter constraints assuming the center cosmology from Table I as the fiducial, we see in Figure 5 that the posteriors calculated using our COLA-based emulator now closely track those generated using our N-body proxy, rather than overestimating S8 and w0 in any substantive way. We find that in switching from the center cosmology to w↑ 0 and w↑ a, the relative ratio between FoMs of COLA and EE2 in the S8 × w0 plane is 0.97, while the same ratio is 1.21 for EE2 ΛCDM. To investigate whether our COLA emulator can provide unbiased constraints with FOMs similar to EE2 across the parameter space, Figure 6 shows the 1D biases of Equation 9 for Ωm, S8, w0 and wabetween the COLA emulator and our baseline EE2 for all scale cuts and all of the 29 cosmologies outlined in Section II. The fiducial cosmologies are listed in increasing order of their associated σ8 values. All 1D biases are within 9 0.30 0.35 m 1.5 1.0 0.5 w0 +wa 1.2 1.0 0.8 w0 0.72 0.76 0.80 S8 0.72 0.78 S8 1.2 0.9 w0 1.6 0.8 w0 +wa Cutoff 1 Fid. Cosmo.: w0 ,wa 0.30 0.35 m 1.5 1.0 0.5 w0 +wa 1.2 1.0 0.8 w0 0.76 0.79 S8 0.75 0.80 S8 1.2 0.9 w0 1.6 0.8 w0 +wa Cutoff 2 EE2 w0wa COLA EE2 CDM 0.30 0.35 m 1.5 1.0 0.5 w0 +wa 1.2 1.0 0.8 w0 0.74 0.76 0.78 S8 0.75 0.78 S8 1.2 0.9 w0 1.6 0.8 w0 +wa Cutoff 3 FIG. 5. Cosmological parameter constraints (68% and 95%) from the LSST-Y1 simulated analyses assuming a fiducial cosmology with w↑ 0 and w↑ a(see Table III), keeping the other parameters at their central values. Green-filled contours denote constraints obtained using EuclidEmulator2 as the nonlinear prescription, orange dashed and dotted contours use our COLA emulator, and blue dashed contours use EE2 ΛCDM prescription. The left, middle, and right panels show constraints using the angular cutoffs C1, C2, and C3, respectively. In this case, the EE2 ΛCDM prescription can provide significant biases in S8, which are not present when using COLA. 0.3σ, even for cosmologies with higher values of σ8 and Ωm, where Figure 4 shows our emulator performs worse. We computed the 7D figure of bias in cosmological parameters for all fiducial cosmologies using Cutoff 3, finding a maximum value of FoB7D = 0.35 at the cosmology Ω↑ m, w↑ 0, w↑ a. We conclude that, for all scale cuts considered in this work, our emulator succeeds in providing unbiased constraints on cosmological parameters when compared to a high-precision N-body emulator in the context of dynamical dark energy models, even with significant variations in cosmological parameters and extreme values of σ8. Further, Figure 6 indicates our measure of the relative tightness of the parameter constraints in the S8 ×w plane compared to EE2, FOMCOLA S8×w/FOMEE2 S8×w, assuming Cutoff 3. The highest ratio is 1.19 for the center cosmology, shown in Figure 2. The same remark from before applies to other fiducial cosmologies: the disagreement between COLA and EE2 is driven mainly by low values of w0 and S8, a region of the parameter space disallowed by low-redshift distance measurements. V. CONCLUSION In light of recent hints of dynamical dark energy, constraining its equation of state has become a task of central importance. While geometrical probes, such as BAO and supernovae measurements, are the primary probes of late dark energy behavior, galaxy surveys can also probe dark energy dynamics through their effects on the growth of large-scale structure. Extracting robust constraints from these surveys requires accurate modeling of nonlinear gravitational effects in the matter power spectrum, which becomes increasingly challenging in extended cosmologies where w(z) departs from a cosmological constant. While accurate modeling can be achieved with N-body simulations, their high computational cost limits their applicability across the myriad candidate dynamical dark energy models (e.g., quintessence), and, further, for models that directly impact the growth of matter perturbations beyond modifications in the Universe's expansion. In this context, COLA is a fast approximate alternative to N-body simulations, which, when appropriately corrected, presents an avenue for constructing accurate emulators for the nonlinear matter power spectrum at a fraction of the computational cost. In this work, we have built an emulator for the nonlinear boost assuming the w0waCDM cosmological model using a suite of 1400 COLA simulations, two for each of the 700 cosmologies in the training set to account for pairing-and-fixing. To evaluate the accuracy of the neural network, we ran a pair of simulations for each of the 200 cosmologies in the test set. The total computational cost of all simulations is estimated at 153.600 CPU-hours. A simple connected neural network with a trainable activation function can reproduce the test set boosts at 0.1% error. The computational cost of the simulation suite could potentially be lowered by using an alternative sampling algorithm for the training set cosmologies: the Sobol sequence [140], which has been shown to improve emulation errors compared to Latin hypercube sampling [83]. We have compared our COLA emulator to a benchmark N-body emulator, chosen as EuclidEmulator2. We test an additional nonlinear prescription common to analyses of extended cosmological models without N-body simulations: using N-body boosts at the projected ΛCDM cosmology (i.e., setting w0 = -1 and wa= 0), an approach we denote as EE2 ΛCDM. We compare nonlinear models in two manners: at the boost level, shown in Figure 1, and at the level of a simulated cosmic shear analysis akin to LSST-Y1, assuming EuclidEmulator2 as the true nonlinear model. In the data analysis, to account for possible variations in the cosmological parameters when beyond-ΛCDM models are analyzed, we define fiducial cosmologies scattered across the parameter 10 0.5 0.0 0.5 m/ m m,w0 ,wa As ,w0 ,wa m,w0 ,wa ns ,w0 ,wa m,w0 ,wa w0 ,wa ns ,w0 ,wa As ,w0 ,wa m,w0 ,wa As ,w0 ,wa As ,w0 ,wa ns ,w0 ,wa w0 ,wa ns ,w0 ,wa As ,w0 ,wa Center ns ,w0 ,wa m,w0 ,wa w0 ,wa ns ,w0 ,wa As ,w0 ,wa ns ,w0 ,wa w0 ,wa ns ,w0 ,wa As ,w0 ,wa m,w0 ,wa As ,w0 ,wa m,w0 ,wa m,w0 ,wa 0.5 0.0 0.5 S8/ S8 0.5 0.0 0.5 w0/ w0 0.5 0.0 0.5 (w0 +wa)/ w0 +wa 0.95 | 0.97 | 0.92 1.04 | 1.05 | 0.96 1.01 | 1.06 | 1.04 0.99 | 1.05 | 1.01 1.04 | 1.05 | 1.07 1.00 | 1.05 | 0.94 1.06 | 1.01 | 1.03 1.02 | 1.04 | 1.14 0.98 | 0.95 | 1.01 1.00 | 0.98 | 1.06 1.03 | 1.03 | 1.12 0.99 | 1.04 | 1.10 1.02 | 1.15 | 1.18 1.06 | 1.10 | 1.17 0.93 | 1.02 | 0.99 1.06 | 1.08 | 1.19 0.91 | 1.01 | 1.08 1.07 | 1.10 | 1.02 1.04 | 1.07 | 1.15 1.07 | 1.05 | 1.14 1.02 | 1.07 | 1.18 1.00 | 0.96 | 0.97 0.99 | 0.96 | 1.02 1.07 | 0.97 | 1.04 1.00 | 1.05 | 1.08 1.04 | 1.11 | 1.13 0.92 | 1.00 | 0.99 0.95 | 0.98 | 0.99 0.94 | 0.90 | 0.92 FoMCOLA S8 ×w / FoMEE2 S8 ×w C1 | C2 | C3 C1 C2 C3 FIG. 6. One-dimensional biases from Equation 9, shown for the parameters Ωm, S8, w0 and wa, sorted in increasing order of their associated σ8 values. On the right-hand side, we show the ratios in figure of merit (see Equation 11) in the S8 × w0 plane between the analyses using COLA and EE2, for the three angular cutoffs. The gray bands represent 0.3σbias. 11 space, including outside ΛCDM, with significant variations in Ωmand σ8. We have assessed the goodness-of-fit degradation, shown in Figure 3, and parameter constraint biases, shown in Figures 2, 5 and 6. We find that, at the boost level, our emulator can reproduce the benchmark emulator results with less than 2% error at k= 1 h/Mpc, while EE2 ΛCDM produces 7.5% errors at the same scales. As for the simulated analysis, we find that our COLA-based emulator can provide unbiased constraints compared to EuclidEmulator2: all 1D biases are well within 0.3σ, even for the most aggressive angular cutoffs and exotic fiducial cosmologies with extreme values of Ωm, σ8, or outside ΛCDM. Furthermore, all 7D figures of bias are below 0.35. At the precision level expected for the first year of LSST observations, the COLA emulator yields constraints equivalent to those obtained using EuclidEmulator2 for scales up to k≈3 h/Mpc, comparable to our Cutoff 2. Our results demonstrate that COLA, when combined with an accurate ΛCDM reference, offers a viable and flexible framework for extending nonlinear modeling to dynamical dark energy and other beyond-ΛCDM scenarios. We emphasize that, while we use EuclidEmulator2 as the \"baseline\" ΛCDM emulator in Equation 8, any other emulator could be used to provide ΛCDM boosts. Moreover, our methodology can be applied to more exotic models that also modify the growth of structure directly, such as modified gravity or coupled dark energy [105]. We also remark that there are avenues to improve our methodology. One example is the choice of \"reference\" cosmology. Equation 8 uses the projected ΛCDM cosmology because, geometrically, it is the closest cosmology; however, a possible choice that improves accuracy is to use a wCDM cosmology with the same value of σ8, akin to what is done in the Casarini [141] prescription. Moreover, COLA can be combined with other analytical or semi-analytical prescriptions, such as the one proposed in [142], improving its accuracy at small scales. ACKNOWLEDGEMENTS The authors would like to thank Stony Brook Research Computing and Cyberinfrastructure and the Institute for Advanced Computational Science at Stony Brook University for access to the high-performance SeaWulf computing system. This research was also supported by resources supplied by the Center for Scientific Computing (NCC/GridUNESP) of the São Paulo State University (UNESP). This work made use of the CHE cluster, managed and funded by COSMO/CBPF/MCTI, with financial support from FINEP and FAPERJ, and operating at the RSDC - Datacenter para Ciência Alfredo Marques de Oliveira/CBPF. JR acknowledges the financial support from FAPESP under grant 2020/03756-2, São Paulo Research Foundation (FAPESP) through ICTP-SAIFR. JR also acknowledges that this study was financed in part by the Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - Brasil (CAPES) - Finance Code 001. VM is supported by the Roman Project Infrastructure Team \"Maximizing Cosmological Science with the Roman High Latitude Imaging Survey\" (NASA contracts 80NM0018D0004-80NM0024F0012). 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COLA emulators trained at different redshifts demonstrate no notable change in the fidelity of predictions across the range 0 ≤z≤3 (right panels), even though the highest 10% of raw emulation errors (left panels) and COLA boost errors (middle panels) grow with increasing z. 17 k (h/Mpc) 5 0 5 z=0.24 50% k (h/Mpc) 5 0 5 90% k (h/Mpc) 5 0 5 100% k (h/Mpc) 5 0 5 1000×(B COLA NN /B COLA 1) z=0.50 k (h/Mpc) 5 0 5 100×(BCOLA/B EE2 1) k (h/Mpc) 5 0 5 100×(B EE2 CDM/B EE2 1) k (h/Mpc) 5 0 5 z=0.74 k (h/Mpc) 5 0 5 k (h/Mpc) 5 0 5 10 1 100 k (h/Mpc) 5 0 5 z=1.00 10 1 100 k (h/Mpc) 5 0 5 10 1 100 k (h/Mpc) 5 0 5 FIG. 7. Left panel: Relative errors between the COLA boosts BCOLA predicted by the emulator versus those obtained from the test set simulations. Middle panel: Relative errors between ̃BCOLA (see Equation 8) versus the boosts from EuclidEmulator2. Right panel: Relative errors between BEE2 ΛCDM and EE2. Each row denotes a different redshift. Colors in all panels denote the percentile of cosmologies around the mean: blue contours enclose 50% of cosmologies, red contours enclose 90% of cosmologies, and the outer gray lines enclose 100% of cosmologies."}}},{"rowIdx":99,"cells":{"id":{"kind":"string","value":"2510.14885"},"raw":{"kind":"string","value":"You May Speak Freely: Improving the Fine-Grained Visual Recognition\nCapabilities of Multimodal Large Language Models with Answer Extraction\nLogan Lawrence1 Oindrila Saha1 Megan Wei2 Chen Sun2 Subhransu Maji1 Grant Van Horn1\nUniversity of Massachusetts, Amherst1\nBrown University2\n{lclawrence, osaha, smaji, gvanhorn}@umass.edu1\n{meganwei, chen sun4}@brown.edu2\nAbstract\nDespite the renewed interest in zero-shot visual classifi-\ncation due to the rise of Multimodal Large Language Mod-\nels (MLLMs), the problem of evaluating free-form responses\nof auto-regressive models remains a persistent challenge.\nMost existing works focus on language-only tasks or don’t\nconsider Multiple Choice Questions (MCQs) beyond 5-way\noptions, both of which are critical capabilities to solve tasks\nin Fine-Grained Visual Classification (FGVC) where choice\ncounts are in the hundreds to thousands and the choices are\nhighly related. Furthermore, in this highly multi-way MCQ\nsetting it is not clear how to extend LLM choice extraction\nto retrieval-based problems, where computing probabilities\nover the choice set is computationally costly. In this work\nwe investigate nlg2choice, a simple two-stage method which\nfirst asks the MLLM an open-ended question for the task\nwith minimal constraints, then uses text-only constrained\ndecoding to predict the most likely choice. In retrieval set-\ntings, we compute the probability of the constrained re-\nsponse taking that choice with an early stopping method\nto significantly improve throughput. Our results show im-\nprovement over a suite of seven fine-grained visual datasets\nwhen evaluating in terms of classification and retrieval, and\nshow that this performance holds over the various ways that\nusers of LLMs can implement tasks in natural language.\n1. Introduction\nZero-shot visual classification has been revitalized by\nthe introduction of Multimodal Large Language Models\n(MLLMs) [1, 9, 20, 21], where one can formulate clas-\nsification datasets as Visual Question Answering (VQA)\ntasks with Multiple Choice Questions (MCQs) and a com-\nmon prompt. However, fine-grained visual classification\n(FGVC) poses a barrier to traditional MCQ problems due\nto two issues: (1) a vastly higher number of choices (of-\nten hundreds to thousands) and (2) the niche concepts that\neach choice represents. Recent evaluations have found that\nstate-of-the-art instruction-tuned MLLMs (e.g. LLaVA 1.5\nfree-form\nchoice\nThis bird is \nan Ivory Gull.\nCrested Auklet\nnlg2choice\nIvory Gull\n“What is the species of \nthis bird?”\n“What is the most likely species?”\n15x\n~\nFigure 1. Proposed evaluation setup of nlg2choice. Whereas\nprevious approaches for MLLM classification use methods like\nprobability sequences or constrained decoding, nlg2choice first\nprompts the MLLM for a natural language response, then performs\ntext-only constrained decoding on the text to extract classes. Accu-\nracy is computed as an average over multiple rewritings of prompts\nas a robustness measure. Correct MLLM responses are shown in\ngreen whereas incorrect responses are red .\n[20, 21], InstructBLIP [9], GPT-4V [1]) suffer dramatic\ndrops in accuracy when asked to name fine-grained cat-\negories.\nFor instance, LLaVA-1.5 (13B) achieves near-\nperfect accuracy on coarse categories (like “bird” vs “not\nbird”), but only 1–2% accuracy on fine-grained species la-\nbels in iNaturalist [15].\nUsing LLMs to extract choices from pre-existing text ap-\npears to be a potential solution to this problem. Constrain-\ning LLM’s output format to ease extraction can simplify\nscoring but might hinder the model’s reasoning [35, 41, 43],\narXiv:2510.14885v1 [cs.CV] 16 Oct 2025\n\n“What is the species of bird in this image? \nAnswer from: { full_species_list }”\n“What is the species of bird in this image?”\n“What is the species of bird in this image? \nAnswer with only the species name.”\nAccuracy\n𝜇 = 41.05\nσ = 0.74\n“It’s an Ivory Gull.”\n“It’s a Herring Gull.”\n“What species of \nbird is this?”\n𝜇 = 47.19\nσ = 2.37\n𝜇 = 63.36\nσ = 1.26\n“What species of \ngull is this?”\n“bird” → “gull”\nFigure 2.\nSeemingly innocuous instruction changes greatly affect MLLM performance.\nWhen testing on the gull subset of\nCUB200[40], replacing one word can change the predictions of a model (left). When rewriting an instruction in a semantically equiv-\nalent way, performance varies substantially for Qwen-2.5VL (right). Each distribution comprises 15 semantically equivalent ways of\nwriting the same instruction. These cases highlight the need to test LLMs over various equivalent inputs to ensure robustness.\nwhereas allowing free-form explanations improves reason-\ning at the cost of harder extraction [27, 35].\nMany ap-\nproaches prompt a strong LLM [8, 18] such as GPT4 [2]\nto perform second-stage answer extraction. However, the\nworks that propose LLMs as choice extractor are either fo-\ncused only on language-only tasks [24, 44] or do not extend\nbeyond simple lettering or numbering schemes [23]. Fur-\nthermore, no significant treatment of retrieval nor the ro-\nbustness of these results under input prompt variation has\nbeen considered.\nIn this work, we conduct a systematic analysis into con-\nstrained decoding with LLMs as answer extractors on free-\nform responses to fine-grained visual recognition tasks. We\ncome to the following conclusions:\n1. Answer extraction on free-form responses improves\nvisual recognition capabilities.\nWe find that perfor-\nmance improves substantially across 7 FGVC benchmarks\nwhen viewed from both classification (Tab. 2) and retrieval\n(Tab. 6) perspectives. It additionally improves misclassifi-\ncations, with most responses being within the same genus\n(Tab. 3). To overcome the computational costs associated\nwith generating probabilities over hundreds to thousands of\nchoices, we propose an early stopping method (Sec. 3.3)\nwhich increases throughput substantially (Tab. 1).\n2. Answer extraction is robust to user variation. We test\nrobustness by generating additional semantically equivalent\nways of writing an instruction (Sec. 3.1) and see that per-\nformance improvements are statistically significant when\nviewing user writing as a source of randomness (Fig. 3).\nWe also show that instructions which constrict the response\ndecrease performance (Tab. 4) and that methods which rely\non forcing the LLM to generate some initial tokens to stimu-\nlate Chain-of-Thought (CoT) reasoning do not consistently\nincrease performance (Tab. 5).\n3. Constrained decoding is a reliable answer extractor\nwithout additional training. We gauge the answer extrac-\ntion abilities of LLMs out-of-the-box by labeling spans and\nassigning answers to free-form NLG responses (Sec. A).\nWe find that the main bottleneck is not answer extrac-\ntion (Tab. 7) as studied in previous works, but a lack of\nextractable content from free-form responses themselves\n(Fig. 4). We make this data publicly available1.\n2. Related Work\nForcing MLLMs to Generate Valid Choices\nThe earli-\nest works first prompted an LLM directly for a response\nthen did regex parsing to extract choices [42], but it was\nshown that direct open-ended querying often underperforms\non fine-grained tasks [15]. While generating the full prob-\nability the text of each choice sequence is as an option, it\nremains difficult to implement in practice due to its com-\nputational cost [31].\nThe most dominant choice proba-\nbility generation rank the log probabilities assigned to the\noption IDs (e.g. ”A/B/C/D”) from the first token predic-\ntion of the model.\nThis approach is widely adopted in\nmany different benchmarks and LLM evaluation studies\n[14, 19, 33, 34, 45].\nParallel to this, constrained decoding techniques seek to\nguarantee consistent outputs by guiding the generation pro-\ncess during sampling by constricting the output vocabulary\n[5, 22]. There are many open-source tools that implement\nthis approach [10, 11, 13, 46] and while these methods en-\nsure model responses lie within the choice set, recent evalu-\nations have shown that they decrease performance [35, 41].\nOur work sits in opposition to these recent results, where we\nposit that constrained decoding is advantageous in the fine-\ngrained setting where previous approaches like first token\nprediction fail.\nMLLMs as Answer Extractors\nRecent works have\nshown a mismatch difference between text-based outputs\n1https://github.com/Racerx250/nlg2choice\n\nand token-probability metrics of LLMs on language-only\ntasks, revealing that a model’s first-token likelihood often\nfails to correspond with its generated answer [24, 43, 44].\nEvaluating a model by reading its full text answer instead\nof only the first-token or forced letter has been shown to\nimprove robustness, reinforcing the value of answer extrac-\ntion methods [43]. Many works use large proprietary mod-\nels like GPT-4 [2] to extract answers from free-form text\n[8, 18, 23].\nSome recent methods have addressed the problem of an-\nswer extraction directly. xFinder [47] propose a lightweight\nmodel fine-tuned specifically to map an LLM’s free-form\nanswer to the correct choice, reporting that their method\noutperforms a GPT-4 based extractor struggled in terms\nof extraction accuracy.\nSimilarly, SLOT [41] proposes\na model-agnostic, lightweight “Structured LLM Output\nTransformer” that is fine-tuned to post-process any LLM’s\nfree-form text into JSON that exactly matches an arbi-\ntrary schema, eliminating the need to modify or constrain\nthe generating model itself.\nxVerify [7] is an efficient\nanswer verifier for reasoning model evaluations that ex-\ntracts final answers from complex, multi-step LLM outputs\nand performs robust equivalence checking—trained on the\nVAR dataset of multi-model, multi-round–annotated ques-\ntion–answer pairs and achieving over 95% accuracy and\nF1, demonstrating strong generalizability. Contrary to these\napproaches the answer extraction method proposed in this\nwork requires no additional training.\nWe argue that an-\nswer extraction is not a critical bottleneck when using con-\nstrained decoding for fine-grained visual classification and\nshow that LLMs perform decent answer extraction out-of-\nthe-box on popular benchmarks.\n3. Methodology\n3.1. Simulating User Variation\nGiven the brittleness of LLMs to prompt variations (Fig. 2),\nwe wish to test the robustness of LLMs as choice extractors\nby generating a range of semantically equivalent writings\nof prompts for a given task. We do this by starting with a\nsmall set of hand-written templates for a classification task.\nLetting [choice list] be the list of classes for a given\ndataset separated by newlines, we define [templatei] to\nbe equal to the following:\n(Template 1) ”What is the species of bird in this image?”\n(Template 2) ”What is the species of bird in this image? An-\nswer only with species name and no other text.”\n(Template 3) ”What is the species of bird in this image?\nAnswer only from the following list with no other text:\n[choice list]”\nWhere the words ”species” and ”bird” are updated to\nsuite the benchmark dataset. Given [templatei], we use\no3-high [29] to generate 14 additional prompt variations:\n(Generate Variations) What are 14 semantically equivalent\nways of writing the following question?\nPrompt: [templatei]\nNext, we perform a manual check on this set of prompts\nfor semantic equivalence to the input prompt, resulting in\na total of 45 variations for each dataset. For the remainder\nof the paper, accuracies reported are across these variations.\nFor the full list of these prompts, see Sec. C.\n3.2. nlg2choice for Classification\nOur approach to classification involves two stages:\n(1)\nprompting the LLM for a free-form response to a question\ndefined by Sec. 3.1 and (2) prompting the same model again\nto select the most likely species given the previous response.\nOn the second step, we enforce a valid output by using con-\nstrained decoding across the choice set. Additionally, the\nsecond stage is text-only, namely it does not see the image\nwhich produced the original response. It receives the fol-\nlowing prompt:\n(Text-Only Choice) What is the most likely species of bird in-\ndicated in this response?\nResponse: [nlg]\nAnswer from the following: [choice list]\nWhere [nlg] is the free-form response to the dataset in-\nstruction and [choice list] is newline-separated list of\nclass names. Similarly, ”species” and ”bird” are updated ac-\ncordingly for the given dataset.\n3.3. nlg2choice for Retrieval\nIn thresholding-based scenarios one desires a continuous\nscore for every query. Most commonly this is done by tak-\ning the probability of \"Yes\" and \"No\" for each image,\nclass pair xi ∈X, yi ∈Y, ie.:\nˆyi = softmax(\n\u0014p(\"Yes\"|xi, [prompt])\np(\"No\"|xi, [prompt])\n\u0015\nWhere p(·) is the next-token probability distribution of\na MLLM and [prompt] describes a two-way predic-\ntion task on one class of the dataset, eg. \"Is this a\n[cname]?\" and [cname] is the name of some class.\nHowever, this requires |X|·|Y| forward passes of the model.\nTo improve throughput, we propose to update the\nsecond step of nlg2choice to stop generating proba-\nbilities when the remaining tokens for a given choice\ndo not occur in any other choices.\nFor example,\ntake the case of \"Baltimore Oriole\" which can\n\nMethod\nFull Prob\nYes/No\nTruncated Prob\nCUB200\n687\n200\n47\n(+0%)\n(+244%)\n(+1362%)\nFlowers\n257\n102\n12\n(+0%)\n(+152%)\n(+2042%)\nAircrafts\n479\n100\n79\n(+0%)\n(+379%)\n(+506%)\nCars\n1668\n196\n125\n(+0%)\n(+751%)\n(+1234%)\nFood\n269\n101\n24\n(+0%)\n(+166%)\n(+1021%)\nNABirds\n3294\n555\n701\n(+0%)\n(+494%)\n(+370%)\niNat-Birds\n5450\n1486\n480\n(+0%)\n(+267%)\n(+1035%)\nTable 1. Number of forward passes needed for different meth-\nods in the second-stage answer extraction of nlg2choice. ”Full\nProb” refers to the probability of a full class name p([cname])\nunder the LLM, where [cname] is a common name of a class.\n“Yes/No” refers to the common |X| · |Y| method of calculating\nprobability over all the classes yi ∈Y. “Truncated Prob” refers to\nthe process outlined in Sec. 3.3. The percentages in the “Yes/No”\nand ”Truncated Prob” columns refer to the throughput increase\nover caculating the full sequence probabilities ”Full Prob.” Every\nsetting for truncation is faster than “Yes/No” except for one bench-\nmark - NABirds [38]. The fastest method for each benchmark is\nbolded.\nbe decomposed into [\"B\", \"altimore\", \" Ori\",\n\"ole\"].\nWe can see that in the CUB200 species list\nthere are many names which share the first token \"B\", ie.\n\"Bewick Wren\" or \"Belted Kingfisher\". How-\never, \"Baltimore Oriole\" is the only choice name\nwhich has \"altimore\" as its second token. Thus, the\nprobabilities for [\" Ori\", \"ole\"] are not calculated.\nIn terms of probabilities, this is equivalent to:\npfull(\"Baltimore Oriole\") = p(\"B\")\n(1)\n× p(\"altimore\"|\"B\")\n× p(\" Ori\"|\"Baltimore\")\n× p(\"ole\"|\"Baltimore Ori\")\nptrunc(\"Baltimore Oriole\") = p(\"B\")\n(2)\n× p(\"altimore\"|\"B\")\n×((((((((((((\np(\" Ori\"|\"Baltimore\")\n×\n((((((((((((((\np(\"ole\"|\"Baltimore Ori\")\nOn many datasets,\nthis change produces substantial\nthroughput increase as many choice names can easily be de-\ntermined by their first or second tokens. The special prop-\nerty of truncated probability sequences is that they have the\nsame probabilities as sampling with constrained genera-\ntion. This throughput increase is depicted in Tab. 1.\n4. Experiments\nDatasets\nWe conduct experiments over a benchmark of 7\nFGVC datasets: CUB [40] (200 classes), Flowers 102 [28]\n(102 classes), Stanford Cars [17] (196 classes), FGVC\nAircrafts [25] (100 classes) and Food101 [6] (101 classes),\nNABirds [38] (555 classes), and iNaturalist-Birds [39]\n(1486 classes). We choose these benchmarks because they\nare well-known for fine-grained visual classification and are\ntraditionally difficult for zero-shot visual models to solve.\niNaturalist-Birds is the validation split of the iNat2021\ndataset where the the ”class” field of each row is equal to\n”Aves,” resulting in a test size of 14960. For each dataset\nwe use the full test set and the training split remains unused.\nEvaluation Metrics\nFor the classification setting we use\naccuracy averaged over 15 semantically equivalent prompts,\nwhereas for the retrieval setting we use Mean Average Pre-\ncision (mAP). Choice lists are the names of classes them-\nselves rather than letters or numbers. As compared to previ-\nous works [36], we perform the full-way class prediction\nrather than subsetting the choices to a 4-way prediction.\nHowever, we provide an experiment evaluating this setting\nin Sec. B.\nImplementation Details\nConstrained decoding is imple-\nmented using the Outlines2 library [46] and classes for each\nsetting are detailed in Sec. C. We start by generating 15\nprompts described in Sec. 3.1, then replacing the key fea-\ntures like domain (”birds,” ”flowers,” etc.) and class types\n(”species,” ”variant,” etc.) for each dataset. This allows us\nto compare directly between questions for a given model\nor dataset, and we also test the performance of appending\nclass lists and common CoT methods. For our experiments,\nwe test only open-source models. Specifically, we test the\nQwen-2.5VL [4], Llama-3.2 [12], and InternVL3 [49] ar-\nchitectures. For hyperparameters, we use the recommended\ngeneration parameters as defined by the HuggingFace card\nand take max new tokens to be 512. Experiments were ran\nin a heterogeneous environment of single-GPU nodes. All\nnodes required a GPU with at least 23GB VRAM, 16 CPU\ncores, and 64GB RAM.\n5. Results\n5.1. Zero-Shot Classification Performance\nAnswer extraction is better than directly answering.\nIn\nTab. 2 we report the performance of nlg2choice across\nvarious architectures and datasets.\nAll architectures en-\njoyed a bump in performance across individual datasets\nand on average (+17.46, +8.49, +6.87 across Qwen-2.5VL,\nLlama-3.2V, and Intern3), but model-specific differences\ncan be clearly seen. Likewise, we find that removing ex-\n2https://github.com/dottxt-ai/outlines\n\nMethod\nBirds\nFlowers\nAircrafts\nCars\nFoods\nNABirds\niNat-Birds\nAverage\nQwen-2.5VL-7B\nchoice\n40.08\n51.29\n44.97\n26.80\n57.93\n37.77\n17.29\n39.45\nnlg2choice\n47.72\n63.22\n59.96\n37.52\n68.99\n44.88\n21.47\n49.11\n(+7.64)\n(+11.93)\n(+14.99)\n(+10.72)\n(+11.06)\n(+7.11)\n(+4.18)\n(+9.66)\nnlg2choiceopen\n63.36\n78.03\n61.54\n47.13\n72.26\n50.19\n25.86\n56.91\n(+23.28)\n(+26.74)\n(+16.57)\n(+20.33)\n(+14.33)\n(+12.42)\n(+8.57)\n(+17.46)\nLlama-3.2-Vision-11B\nchoice\n42.27\n45.68\n31.38\n24.68\n64.70\n36.69\n13.82\n37.03\nnlg2choice\n46.96\n51.16\n30.75\n32.72\n64.52\n41.34\n15.81\n40.47\n(+4.69)\n(+5.48)\n(-0.63)\n(+8.04)\n(-0.18)\n(+4.65)\n(+1.99)\n(+3.43)\nnlg2choiceopen\n49.59\n62.24\n37.12\n36.78\n72.72\n46.37\n13.85\n45.52\n(+7.32)\n(+16.56)\n(+5.74)\n(+12.10)\n(+8.02)\n(+9.68)\n(+0.03)\n(+8.49)\nIntern3VL-8B\nchoice\n11.27\n24.68\n15.84\n14.78\n44.12\n6.90\n2.28\n17.12\nnlg2choice\n13.81\n32.52\n18.63\n18.34\n43.36\n7.06\n1.95\n19.38\n(+2.54)\n(+7.84)\n(+2.79)\n(+3.56)\n(-0.76)\n(+0.16)\n(-0.33)\n(+2.26)\nnlg2choiceopen\n18.47\n37.50\n21.69\n22.15\n51.48\n12.39\n4.29\n24.00\n(+7.20)\n(+12.82)\n(+5.85)\n(+7.37)\n(+7.36)\n(+5.49)\n(+2.01)\n(+6.87)\nTable 2. Performance across fine-grained visual classification (FGVC) benchmarks and architectures. The “Birds” dataset refers to\nCUB200. The first row of each model refers to using constrained decoding directly on the model across the class list of each dataset using\nthe prompt in the header. “choice” is performing constrained decoding on the prompt \"What is the {type} of {domain} in\nthis image?\nAnswer with {type} only.\" “nlg2choice” refers to letting the model freely respond to the prompt, then using\nconstrained decoding in a text-only approach. “nlg2choiceopen” refers to removing the appended response-style instruction, eg. \"Answer\nwith {type} only.\" All models are their instructed-tuned version. Best performance is bolded for each model.\nMethod\nGenus Matches\n% of Data\nQwen-2.5VL-7B\nchoice\n45.99\n59.92\nnlg2choice\n56.57 (+10.56)\n52.28\nnlg2choiceopen\n62.74 (+16.75)\n36.64\nLlama-3.2-Vision-11B\nchoice\n42.52\n57.73\nnlg2choice\n61.71 (+19.19)\n53.04\nnlg2choiceopen\n66.37 (+23.85)\n50.94\nIntern3VL-8B\nchoice\n36.77\n88.73\nnlg2choice\n48.79 (+12.02)\n86.19\nnlg2choiceopen\n60.36 (+23.59)\n81.53\nTable 3. Accuracy of genus-level predictions on misclassified\nexamples. ”Genus Matches” refers to the accuracy of model with\nrespect to the translating each species to its corresponding genus.\n”% of Data” refers to the amount of data misclassified by the\nmodel and is equivalent to 1 −accuracy in Tab. 2. The species to\ngenus mapping manually created.\nplicit answer formatting from the model (changing from\n“nlg2choice” and “nlg2choiceopen”). We find the Aves sub-\nset of iNaturalist [39] to be the hardest among all the bench-\nmarks, resulting in the smallest increase in performance\nacross all models.\nFor model-specific differences, we find that Qwen-\n2.5VL performs the best on average using nlg2choice, hav-\ning an average accuracy of 56.91 across the 7 datasets.\nWe also find that Qwen-2.5VL experiences the greatest\ngain, having an average accuracy difference of +17.46 from\nthe base constrained decoding method to nlg2choiceopen.\nComparatively, we see that Intern3-VL performs the worst\nacross architectures, having the overall smallest increase\nwhen using nlg2choiceopen (+6.87) and the lowest perfor-\nmance (24.00). However, we do still see improvements over\nall benchmarks.\nMisclassifications are more aligned with ground truth.\nA natural question is to ask how the classifications qualita-\ntively change under nlg2choice. For an example, take Fig. 1\nwhere the ground truth label is \"Ivory Gull\": under\nconstrained decoding the model responds with \"Crested\nAuklet\" whereas a higher quality misclassificaiton would\nbe \"Herring Gull\", \"California Gull\", ..., ie. a\nclass within the same genus as Ivory Gull. In Tab. 3 we\nqualitatively gauge how misclassifications change when us-\ning nlg2choice by measuring its genus-level performance.\nWe construct a species to genus mapping on the CUB200\ndataset manually, then map the species prediction in previ-\nous results to genus. Simply put, we find that nlg2choice\nmakes more reasonable errors, switching to nlg2choice in-\ncurs a genus-level performance improvement of +16.75,\n+23.85, +23.59 for Qwen-2.5VL, Llama-3.2V-11B, and\nIntern3VL-8B.\n\nΔ Accuracy\n-0.05\n0.00\n0.05\nIntern3VL-8B\nLlama-3.2V-11B\nQwen-2.5VL-8B\n0.10\n0.15\n0.20\n0.25\n0.30\n𝜇 = 7.38\nσ = 1.69\n𝜇 = 7.72\nσ = 2.95\n𝜇 = 22.06\nσ = 1.92\nFigure 3.\nDifference in accuracies on the question-level be-\ntween nlg2choice and constrained decoding.\nEach distribu-\ntion is made up of the performance difference when switching\nto nlg2choice over 15 questions. nlg2choice uses an open-ended\nprompt whereas constrained decoding uses an instruction to in-\ncrease brevity (\"...\nAnswer with species only.\").\nImprovements are statistically significant. Rather than\nask about the average performance of different methods\nunder user variation, we wish to quantify how much per-\nformance is gained for a specific question by switching\nto nlg2choice.\nWe calculate this by taking the differ-\nence between the performance of constrained decoding and\nnlg2choiceopen for each question, then compute summary\nstatistics over the set of questions. This forms a distribution\nof changes in accuracy, with significant results having a pos-\nitive mean and 95% confidence interval. The performance\ndifference for the various settings in outlined in Fig. 3.\nWe\nfind\nthat\nfor\nthe\nproposed\nsettings,\nswitch-\ning\nfrom\nconstrained\ndecoding\nwith\nan\ninstruction\nto increase brevity (\"...\nAnswer with species\nonly.\") to nlg2choiceopen comes with an average µ =\n22.06, 7.38, 7.72 percentage increase in accuracy for\nQwen-2.5VL, Llama-3.2V-11B, and Intern3VL-8B, re-\nspectively.\nAdditionally, we find that these increases\nare within the bounds of statistical significance (σ\n=\n1.92, 1.69, 2.95).\nConstrictive instructions reduce performance. A com-\nmon means of steering models to produce valid choices\nis through appending explicit instructions onto the instruc-\ntion which enumerate the available choices, typically by\nmeans of lettering or enumeration. We wish to quantify\nthe effect of adding this explicit instruction to the model.\nIn Tab. 4 we show the performance of constrained decod-\nMethod\nBirds\nFlowers\n\"What is the {type} of {domain} in this image?\"\nQwen-2.5VL\n0.92\n2.53\n+ nlg2choice\n63.36 (+62.44)\n78.03 (+75.5)\nLlama-3.2-Vision\n7.67\n27.50\n+ nlg2choice\n49.59 (+41.92)\n62.24 (+34.74)\nIntern3-VL\n2.28\n5.01\n+ nlg2choice\n18.47 (+16.19)\n37.5 (+32.49)\n\"...\nAnswer with {type} only.\"\nQwen-2.5VL\n40.08\n51.29\n+ nlg2choice\n47.72 (+7.64)\n63.22 (+11.93)\nLlama-3.2-Vision\n42.27\n45.68\n+ nlg2choice\n46.96 (+4.69)\n51.16 (+5.48)\nIntern3-VL\n11.27\n24.68\n+ nlg2choice\n13.81 (+2.54)\n32.52 (+7.84)\n\"...\nAnswer from {choice list}.\"\nQwen-2.5VL\n38.71\n69.19\n+ nlg2choice\n41.05 (+2.34)\n72.48 (+3.29)\nLlama-3.2-Vision\n13.50\n44.83\n+ nlg2choice\n46.68 (+33.18)\n66.03 (+21.2)\nIntern3-VL\n22.34\n35.24\n+ nlg2choice\n23.48 (+1.14)\n35.73 (+0.49)\nTable 4.\nClassification performance across instruction tem-\nplates with varying levels of constrictiveness. The model name\nrefers to default constrained decoding. The difference in perfor-\nmance is shown in parentheses. The best performing model and\nmethod for each input instruction is bolded. “nlg2choiceopen”\ncorresponds to the top section of the table.\ning and nlg2choice under three variations of instruction\nsteering: (1) no explicit choice steering (\"What is the\n{type} of {domain} in this image?\"), (2) en-\ncouraging succinct prediction (\"...\nAnswer with\n{type} only\"), and (3) explicitly enumerating choices\n(\"...\nAnswer from {choice list}\").\nWe see that for every template, switching from con-\nstrained decoding to nlg2choice improves performance.\nThe most stark setting is the most open-ended prompt,\nwhere we see an average increase of +68.97, +38.33, and\n+24.34 for Qwen-2.5VL, Llama-3.2V, and Intern3-VL, re-\nspectively. Surprisingly, explicit choice instruction shows\nthe weakest improvement for Qwen-2.5VL and Intern3-VL\n(+2.82 and +0.81) but strong improvement for Llama-3.2V\n(+27.19). Otherwise, keeping in line with previous results,\nwe find that Qwen-2.5VL is the greatest benefactor of per-\nformance.\nCoT doesn’t consistently improve performance. Next,\nwe ask how much performance can be improved by us-\ning CoT-style instructions. We investigate methods which\ninduce reasoning by forcing the model to generate some\ninitial tokens that are common to thought-out reasoning\n(eg.\n\"Let’s think step by step\"), then letting\nthe model freely respond. In Tab. 5 we choose three of\n\nMethod\nBirds\nFlowers\n\"What is the {type} of {domain} in this image?\"\nQwen-2.5VL\n63.36\n78.03\nLlama-3.2-Vision\n49.59\n62.24\nIntern3-VL\n18.47\n37.50\n...\n\"Let’s think step by step.\" [16]\nQwen-2.5VL\n54.93 (-8.43)\n70.16 (-7.87)\nLlama-3.2-Vision\n39.87 (-9.72)\n39.22 (-23.02)\nIntern3-VL\n19.11 (+0.64)\n30.82 (-6.68)\n...\n\"First,\" [3]\nQwen-2.5VL\n53.51 (-9.85)\n72.80 (-5.23)\nLlama-3.2-Vision\n41.84 (-7.75)\n45.31 (-16.93)\nIntern3-VL\n19.65 (+1.18)\n33.46 (-4.04)\n...\n\"Let’s solve this problem by splitting\nit into steps.\" [30]\nQwen-2.5VL\n54.69 (-8.67)\n59.56 (-18.47)\nLlama-3.2-Vision\n40.11 (-9.48)\n38.08 (-24.16)\nIntern3-VL\n19.68 (+1.21)\n31.06 (-6.44)\nTable 5. Classification performance when using assistant out-\nput prepending CoT methods. All results are reported using\nnlg2choice as an answer extraction method. The proposed set-\nting is the first section in gray. Forced assistant prepending text is\nshown in blue .\nthese methods and report their performance and difference\nbetween the default nlg2choiceopen setting.\nIn general, we see that these methods do not consistently\nimprove performance, resulting in an average decrease of\n-9.75, -15.18, and -2.36 across Qwen-2.5VL, Llama-3.2V,\nand Intern3-VL, respectively. The one exception to this rule\nis the CUB200 dataset with Intern3-VL, where we see an\naverage increase of +1.01.\n5.2. Zero-Shot Retrieval Performance\nAnswer extraction is better than directly answering.\nIn\nTab. 6, we report the performance of various models when\nviewing FGVC datasets as a one-vs-rest retrieval task. Like\nthe classification setting, we find that when compared to\nexhaustive “Yes/No” questioning, nlg2choice generally im-\nproves with the exception of the Stanford Cars [17], in\nwhich performance across all models decreases. We again\nfind Qwen-2.5VL to receive the greatest improvement, with\n+8.16 average mAP increase across the 5 datasets. Like-\nwise, we find Intern3-VL receives the least improvement,\nwith an average increase of +4.43.\nUnlike classification, we do not find that changing the\nprompt to be open-ended (nlg2choiceopen, ie.\nremoving\n\"Answer from {species list}\" from the end of\nthe input prompt) does not uniformly increase performance.\nHowever, we do still find that open-ended prompts gener-\nally perform better on average with nlg2choice (+8.16 vs.\n+6.96) and are less variable across all models.\nMethod\nBirds\nFlowers\nAircrafts\nCars\nFoods\nBaseline\nRandom\n0.64\n1.24\n1.23\n0.62\n1.02\nQwen-2.5VL-7B\nYes/No\n51.01\n61.19\n51.07\n77.58\n82.67\nnlg2choice\n57.79\n82.56\n65.14\n68.61\n84.30\n(+6.78)\n(+21.37)\n(+14.07)\n(-8.97)\n(+1.63)\nnlg2choiceopen\n59.01\n86.42\n64.03\n69.82\n85.05\n(+8.00)\n(+25.23)\n(+12.96)\n(-7.76)\n(+2.38)\nLlama-3.2-Vision-11B\nYes/No\n41.24\n63.59\n34.24\n66.03\n78.81\nnlg2choice\n51.14\n68.64\n39.39\n56.46\n86.73\n(+9.90)\n(+5.05)\n(+5.15)\n(-9.57)\n(+7.92)\nnlg2choiceopen\n52.61\n74.55\n40.00\n59.35\n88.84\n(+11.37)\n(+10.96)\n(+5.76)\n(-6.68)\n(+10.03)\nIntern3VL-8B\nYes/No\n15.22\n38.32\n17.93\n31.83\n61.46\nnlg2choice\n26.97\n45.13\n20.47\n25.81\n66.55\n(+11.75)\n(+6.81)\n(+2.54)\n(-6.02)\n(+5.09)\nnlg2choiceopen\n30.51\n41.12\n20.69\n26.84\n67.76\n(+15.29)\n(+2.80)\n(+2.76)\n(-4.99)\n(+6.30)\nTable 6.\nPerformance across fine-grained visual classifica-\ntion (FGVC) benchmarks and architectures when interpreted\nas retrieval tasks. Numbers represent Mean Average Precision\n(mAP). Every query is a one-vs-rest task, eg. for CUB200 (Birds)\nthe prior for a single PR curve is 1/200.\n5.3. Answer Extraction Performance\nIn Sec. A we detail the process of labeling data for species\nextraction from free-form responses.\nFirst, labelers are\nshown responses from various models to a dataset instruc-\ntion and image, then are instructed to highlight the first in-\ncidence of the predicted species within the generated text.\nThen, they are instructed to confirm whether the species\nhighlighted matches the choice selection by the model,\nand in the failure case to indicate the species predicted or\nwhether there was a schema failure. This process results in a\nset of common outcomes, ie. “schema failure,” “no species\npredicted,” “answer={species}”. In Fig. 4 we show the\nmakeup of the labeled data in terms of these various out-\ncomes.\nModels often respond out of schema. Depicted in Fig. 4,\nwe see that 34.64% of free-form responses by models have\nan answer within their text that does not occur within the\ndataset schema. In other words, the worst-case scenario\nis that only 65.35% of examples are correctly classifiable.\nHowever, this turns out to not be the case due to the fact\nthat a large percentage of these responses have real species\nwithin them (70.75% of failures) which can be resolved to\nthe correct answer, but that the predicted species itself can-\nnot be easily put within the given schema. We hypothe-\n\nFigure 4. Breakdown of the small labeled set of natural lan-\nguage responses. ”Other” contains two main categories, ”Refused\nto answer” and ”No discernable information.” Examples of these\nare given in Sec. A.\nsize that many of these responses are grouped into the cor-\nrect answer due to either genus-level answers in the dataset,\neg. CUB200’s labels include the genus \"Mockingbird\"\nwhich matches to \"Northern Mockingbird\" within\nthe free-form text, or being a closely-related but technically\ndifferent species, eg. \"Glaucous-winged Gull\" in\nschema versus \"Glaucous Gull\" in free-form text.\nMotivation of different methods.\nIn Tab. 7 we show\nthe performance of main architectures in terms of extrac-\ntion performance over various methods when sub-setting\nthe data to examples where an answer within the schema\nis clearly apparent. We include two additional variations:\n(1) nlg2nlg, the free-form response given the instruction\nto rephrase the MLLM answer and (2) nlg2nlg2choice, the\nconstrained decoding off one extra round of rephrasing. We\ninclude nlg2nlg due to the fact that we wish to test the direct\ndistance to the highlighted text created during labeling and\ninclude nlg2nlg2choice to gauge the distance between the\ninterpretation of that text into an answer.\nLLMs are decent answer extractors out-of-the-box.\nIn the default nlg2choiceopen setting, we find high an-\nswer extraction success rates for Qwen-2.5VL (97.93)\nand Intern3-VL (93.26), but lower results for Llama-3.2V\n(79.02). We find that the raw text extracted by the models\n(”nlg2nlgexact”) seldom fits neatly into the dataset schema,\nhaving large performance drops (-16.47, 50.77, -25.91).\nHowever, the output of nlg2nlgexact retains the relevant\ninformation from the original free-form text, as indicated\nby the nlg2nlg2choice scores which has much smaller per-\nMethod\nQwen2.5VL\nLlama3.2V\nIntern3VL\nOverall\nxFinder [47]\nQwen-1.5-0.5B [37]\n92.05\n88.67\n71.76\n86.27\nLlama-3-8B-Instruct [12]\n80.13\n80.67\n75.29\n79.27\nQwen-2.5VL 8B\nnlg2choice\n98.01\n98.67\n96.47\n97.93\nnlg2nlgexact\n80.79\n84.00\n78.82\n81.61\n(-17.22)\n(-14.67)\n(-17.65)\n(-16.32)\nnlg2nlg2choice\n98.01\n97.33\n100.00\n98.19\n(+0.00)\n(-1.34)\n(+3.53)\n(+0.26)\nLlama-3.2-Vision 11B\nnlg2choice\n77.48\n82.00\n76.47\n79.02\nnlg2nlgexact\n31.79\n26.00\n25.88\n28.24\n(-45.69)\n(-56.00)\n(-50.59)\n(-50.78)\nnlg2nlg2choice\n76.82\n80.00\n70.59\n76.68\n(-0.66)\n(-2.00)\n(-5.88)\n(-2.34)\nIntern3-VL 8B\nnlg2choice\n94.04\n94.00\n90.59\n93.26\nnlg2nlgexact\n59.60\n78.67\n62.35\n67.62\n(-34.44)\n(-15.33)\n(-28.24)\n(-25.64)\nnlg2nlg2choice\n90.06\n85.33\n90.59\n88.34\n(-3.98)\n(-8.67)\n(+0.00)\n(-4.92)\nTable 7. Performance of various MLLMs across a small labeled\nset. Differences are measured from the “nlg2choice“ row of each\nmodel.\nformance differences (+0.61, -2.72, and -4.39).\nWe hy-\npothesize that the nlg2nlg text still contains artifacts such\nas \"The species is ...\" or scientific names, which\nlower exact match performance.\n6. Limitations\nThis work has three main limitations. First, our experiments\nwere conducted primarily with medium-sized models (8B-\n11B) and we only used open-source LLMs from a few dif-\nferent families: Qwen, Llama, and Intern. Secondly, while\nnlg2choice avoids the need for specialized training data or\narchitectural modifications and is more aligned with the true\nresponses of the model, the method still requires significant\ncomputational resources to generate and process the tex-\ntual descriptions. Lastly, while nlg2choice shows promise\nin classification, it still remains to be seen whether the ap-\nproach scales to multi-label problems.\n7. Conclusion\nIn this work, we presented nlg2choice, a simple approach\nfor zero-shot fine-grained visual classification and retrieval\nthat extracts fine-grained answers from MLLM free-form\nresponses. Through experiments across multiple architec-\ntures and datasets, we demonstrated that nlg2choice con-\nsistently outperforms constrained decoding and exhaustive\nquestioning methods in both classification accuracy and re-\ntrieval performance. 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Labeling Species Answer Extraction Data\nMotivation.\nFor labeling answer extraction data, we pre-\npare a user interface built with Label Studio which shown\nin Fig. 5. On the right contains information on the inputs\nto the model, specifically the question, image, and ground\ntruth label. On the left is main interface, which consists a\nfree-form response from a model to the example displayed\non the right, as well as the choice predicted by the model\nvia nlg2choice. Labeling consists of two pieces: (1) iden-\ntification and highlighting of text spans containing answers\nto the input question and (2) matching the highlighted text\nspan to a class within the given schema.\nText span labeling.\nFor the text span labeling task, label-\ners read through the text to identify the first incidence of\na predicted species. Before highlighting the species found\nwithin the text, the prediction is confirmed to be a real\nspecies through a brief internet search. If there are any is-\nsues with highlighting the predicted species, a text descrip-\ntion of the issue is written in the input text box at the bottom\nand the example is skipped.\n\nFigure 5. Labeling interface for efficacy of the answer extraction stage of nlg2choice. Labelers are asked to highlight the name\npredicted in free-form responses from various models and whether the name occurs easily within the dataset choice list. Above shows\nan interesting example: for the Chipping Sparrow on the right, the model freely responded with ”Painted Bunting” and answer extraction\nmade an additional mistake by predicting ”Scarlet Tanager.”\nChoice assignment.\nOnce the span is labeled, we next\nwish to either assign the span to a class within the dataset\nor tag the span as a schema failure. For expediency, the la-\nbelers are also equipped with the nlg2choice prediction and\na tool for sorting most like class names with fuzzy match-\ning score. When the nlg2choice predict matches the text\nspan, the prediction is ”Consistent with nlg” and the label-\ners check the button. Otherwise, ”Inconsistent with nlg”\nis checked and a natural language explanation is put in the\nbottom text box. When the nlg2choice is inconsistent, “an-\nswer={species}” is put within that text box.\nB. Zero-Shot 4-Way VQA Performance\nAnswer extraction does not consistently increase per-\nformance.\nFollowing recent work [36],\nwe evalute\nnlg2choice on popular fine-graiend classification datasets\nas a 4-way VQA task. iNat-Animal and iNat-Plant are the\nrows of iNat21 [39] when subset to rows which whose king-\ndom matches ”Animalia” and ”Plantae.” ImgNet-Animal\nand ImgNet-Artifact are the rows of ImageNet-1K [32] sub-\nset to rows which have an anceter of ”Animal” and ”Arti-\nfact” in the WordNet hierarchy [26]. The choices for each\nexample are from the top 3 SigLIP [48] predictions and the\ncorrect class. For the nlg2choice prompt we use the same\nprompt as the previous work. We report the open-ended\nquestion performance in Tab. 8 where we find that it slightly\nMethod\nCUB200\niNat-Animal\niNat-Plant\nImgNet-Animal\nImgNet-Artifact\nQwen-2.5VL-7B\nA/B/C/D\n65.50\n41.33\n41.61\n85.20\n80.01\nnlg2choiceopen\n67.43\n40.19\n40.03\n84.03\n81.81\n(+1.93)\n(-1.14)\n(-1.58)\n(-1.17)\n(+1.80)\nLlama-3.2-Vision-11B\nA/B/C/D\n65.52\n32.44\n31.88\n79.93\n75.89\nnlg2choiceopen\n68.79\n31.52\n31.57\n81.39\n75.29\n(+3.27)\n(-0.92)\n(-0.31) (+1.46)\n(-0.60)\nIntern3VL-8B\nA/B/C/D\n50.52\n35.40\n36.39\n77.50\n69.41\nnlg2choiceopen\n47.30\n35.86\n35.64\n81.91\n70.02\n(-3.22)\n(+0.46) (-0.75) (+4.41) (+0.61)\nTable 8. 4-way VQA accuracy across fine-grained classifcation\ntasks.\nunderperforms or has not effect on the lettering approach.\nSpecifically, we find that accuracy changes by -0.03, +0.58,\nand +0.30 on average for Qwen-2.5VL, Llama-3.2V, and\nIntern3VL, respectively.\nDecreasing the choice set improves performance. We are\n\nalso able to compare the 4-way VQA setting of CUB200\ndirectly to the many-way prediction displayed in Tab. 2. We\nsee that subsetting the full choice set to four choices incurs\na performance improvement of +4.07, +19.20, and +28.83\nfor Qwen-2.5VL, Llama-3.2V, and Intern3VL, respectively.\nC. Prompt Variations\n\"What { type } is this { domain }?\"\n\"What is the { type } of this { domain }?\"\n\"What is the { type } of the { domain }?\"\n\"What is the { type } of the { domain } in this image?\"\n\"What is the { type } of the { domain } in the image?\"\n\"Identify this { domain }’s { type }.\"\n\"Name the { type } shown in the image.\"\n\"Which { domain } { type } is pictured here?\"\n\"Classify the { type } of this { domain }.\"\n\"What { domain } { type } does the photo depict?\"\n\"Determine the { type } of the { domain } in view.\"\n\"Provide the common name of this { domain }.\"\n\"To which { type } does this { domain } belong?\"\n\"Label the { type } of the { domain } shown.\"\n\"Recognize and state this { domain }’s { type }.\"\nTable 9. Base template variations generated by Sec. 3.1\nDataset\nType\nDomain\nCUB200\nspecies\nbird\nFlowers\nspecies\nflower\nAircrafts\nvariant\naircraft\nCars\nyear, make, and model\ncar\nFoods\nname\nfood\nNABirds\nspecies\nbird\niNat-Birds\nspecies\nbird\nTable 10. Dataset variables for filling out prompt templates.\n"},"cleaned":{"kind":"string","value":"You May Speak Freely: Improving the Fine-Grained Visual Recognition Capabilities of Multimodal Large Language Models with Answer Extraction Logan Lawrence1 Oindrila Saha1 Megan Wei2 Chen Sun2 Subhransu Maji1 Grant Van Horn1 1 Brown University2 {lclawrence, osaha, smaji, {meganwei, chen Abstract Despite the renewed interest in zero-shot visual classification due to the rise of Multimodal Large Language Models (MLLMs), the problem of evaluating free-form responses of auto-regressive models remains a persistent challenge. Most existing works focus on language-only tasks or don't consider Multiple Choice Questions (MCQs) beyond 5-way options, both of which are critical capabilities to solve tasks in Fine-Grained Visual Classification (FGVC) where choice counts are in the hundreds to thousands and the choices are highly related. Furthermore, in this highly multi-way MCQ setting it is not clear how to extend LLM choice extraction to retrieval-based problems, where computing probabilities over the choice set is computationally costly. In this work we investigate nlg2choice, a simple two-stage method which first asks the MLLM an open-ended question for the task with minimal constraints, then uses text-only constrained decoding to predict the most likely choice. In retrieval settings, we compute the probability of the constrained response taking that choice with an early stopping method to significantly improve throughput. Our results show improvement over a suite of seven fine-grained visual datasets when evaluating in terms of classification and retrieval, and show that this performance holds over the various ways that users of LLMs can implement tasks in natural language. 1. Introduction Zero-shot visual classification has been revitalized by the introduction of Multimodal Large Language Models (MLLMs) [1, 9, 20, 21], where one can formulate classification datasets as Visual Question Answering (VQA) tasks with Multiple Choice Questions (MCQs) and a common prompt. However, fine-grained visual classification (FGVC) poses a barrier to traditional MCQ problems due to two issues: (1) a vastly higher number of choices (often hundreds to thousands) and (2) the niche concepts that each choice represents. Recent evaluations have found that state-of-the-art instruction-tuned MLLMs (e.g. LLaVA 1.5 free-form choice This bird is an Ivory Gull. Crested Auklet nlg2choice Ivory Gull \"What is the species of this bird?\" \"What is the most likely species?\" 15x ~ Figure 1. Proposed evaluation setup of nlg2choice. Whereas previous approaches for MLLM classification use methods like probability sequences or constrained decoding, nlg2choice first prompts the MLLM for a natural language response, then performs text-only constrained decoding on the text to extract classes. Accuracy is computed as an average over multiple rewritings of prompts as a robustness measure. Correct MLLM responses are shown in green whereas incorrect responses are red . [20, 21], InstructBLIP [9], GPT-4V [1]) suffer dramatic drops in accuracy when asked to name fine-grained categories. For instance, LLaVA-1.5 (13B) achieves nearperfect accuracy on coarse categories (like \"bird\" vs \"not bird\"), but only 1-2% accuracy on fine-grained species labels in iNaturalist [15]. Using LLMs to extract choices from pre-existing text appears to be a potential solution to this problem. Constraining LLM's output format to ease extraction can simplify scoring but might hinder the model's reasoning [35, 41, 43], 16 Oct 2025 \"What is the species of bird in this image? Answer from: { full_species_list }\" \"What is the species of bird in this image?\" \"What is the species of bird in this image? Answer with only the species name.\" Accuracy μ = 41.05 σ = 0.74 \"It's an Ivory Gull.\" \"It's a Herring Gull.\" \"What species of bird is this?\" μ = 47.19 σ = 2.37 μ = 63.36 σ = 1.26 \"What species of gull is this?\" \"bird\" → \"gull\" Figure 2. Seemingly innocuous instruction changes greatly affect MLLM performance. When testing on the gull subset of CUB200[40], replacing one word can change the predictions of a model (left). When rewriting an instruction in a semantically equivalent way, performance varies substantially for Qwen-2.5VL (right). Each distribution comprises 15 semantically equivalent ways of writing the same instruction. These cases highlight the need to test LLMs over various equivalent inputs to ensure robustness. whereas allowing free-form explanations improves reasoning at the cost of harder extraction [27, 35]. Many approaches prompt a strong LLM [8, 18] such as GPT4 [2] to perform second-stage answer extraction. However, the works that propose LLMs as choice extractor are either focused only on language-only tasks [24, 44] or do not extend beyond simple lettering or numbering schemes [23]. Furthermore, no significant treatment of retrieval nor the robustness of these results under input prompt variation has been considered. In this work, we conduct a systematic analysis into constrained decoding with LLMs as answer extractors on freeform responses to fine-grained visual recognition tasks. We come to the following conclusions: 1. Answer extraction on free-form responses improves visual recognition capabilities. We find that performance improves substantially across 7 FGVC benchmarks when viewed from both classification (Tab. 2) and retrieval (Tab. 6) perspectives. It additionally improves misclassifications, with most responses being within the same genus (Tab. 3). To overcome the computational costs associated with generating probabilities over hundreds to thousands of choices, we propose an early stopping method (Sec. 3.3) which increases throughput substantially (Tab. 1). 2. Answer extraction is robust to user variation. We test robustness by generating additional semantically equivalent ways of writing an instruction (Sec. 3.1) and see that performance improvements are statistically significant when viewing user writing as a source of randomness (Fig. 3). We also show that instructions which constrict the response decrease performance (Tab. 4) and that methods which rely on forcing the LLM to generate some initial tokens to stimulate Chain-of-Thought (CoT) reasoning do not consistently increase performance (Tab. 5). 3. Constrained decoding is a reliable answer extractor without additional training. We gauge the answer extraction abilities of LLMs out-of-the-box by labeling spans and assigning answers to free-form NLG responses (Sec. A). We find that the main bottleneck is not answer extraction (Tab. 7) as studied in previous works, but a lack of extractable content from free-form responses themselves (Fig. 4). We make this data publicly available1. 2. Related Work Forcing MLLMs to Generate Valid Choices The earliest works first prompted an LLM directly for a response then did regex parsing to extract choices [42], but it was shown that direct open-ended querying often underperforms on fine-grained tasks [15]. While generating the full probability the text of each choice sequence is as an option, it remains difficult to implement in practice due to its computational cost [31]. The most dominant choice probability generation rank the log probabilities assigned to the option IDs (e.g. \"A/B/C/D\") from the first token prediction of the model. This approach is widely adopted in many different benchmarks and LLM evaluation studies [14, 19, 33, 34, 45]. Parallel to this, constrained decoding techniques seek to guarantee consistent outputs by guiding the generation process during sampling by constricting the output vocabulary [5, 22]. There are many open-source tools that implement this approach [10, 11, 13, 46] and while these methods ensure model responses lie within the choice set, recent evaluations have shown that they decrease performance [35, 41]. Our work sits in opposition to these recent results, where we posit that constrained decoding is advantageous in the finegrained setting where previous approaches like first token prediction fail. MLLMs as Answer Extractors Recent works have shown a mismatch difference between text-based outputs 1https://github.com/Racerx250/nlg2choice and token-probability metrics of LLMs on language-only tasks, revealing that a model's first-token likelihood often fails to correspond with its generated answer [24, 43, 44]. Evaluating a model by reading its full text answer instead of only the first-token or forced letter has been shown to improve robustness, reinforcing the value of answer extraction methods [43]. Many works use large proprietary models like GPT-4 [2] to extract answers from free-form text [8, 18, 23]. Some recent methods have addressed the problem of answer extraction directly. xFinder [47] propose a lightweight model fine-tuned specifically to map an LLM's free-form answer to the correct choice, reporting that their method outperforms a GPT-4 based extractor struggled in terms of extraction accuracy. Similarly, SLOT [41] proposes a model-agnostic, lightweight \"Structured LLM Output Transformer\" that is fine-tuned to post-process any LLM's free-form text into JSON that exactly matches an arbitrary schema, eliminating the need to modify or constrain the generating model itself. xVerify [7] is an efficient answer verifier for reasoning model evaluations that extracts final answers from complex, multi-step LLM outputs and performs robust equivalence checking-trained on the VAR dataset of multi-model, multi-round-annotated question-answer pairs and achieving over 95% accuracy and F1, demonstrating strong generalizability. Contrary to these approaches the answer extraction method proposed in this work requires no additional training. We argue that answer extraction is not a critical bottleneck when using constrained decoding for fine-grained visual classification and show that LLMs perform decent answer extraction out-ofthe-box on popular benchmarks. 3. Methodology 3.1. Simulating User Variation Given the brittleness of LLMs to prompt variations (Fig. 2), we wish to test the robustness of LLMs as choice extractors by generating a range of semantically equivalent writings of prompts for a given task. We do this by starting with a small set of hand-written templates for a classification task. Letting [choice list] be the list of classes for a given dataset separated by newlines, we define [templatei] to be equal to the following: (Template 1) \"What is the species of bird in this image?\" (Template 2) \"What is the species of bird in this image? Answer only with species name and no other text.\" (Template 3) \"What is the species of bird in this image? Answer only from the following list with no other text: [choice list]\" Where the words \"species\" and \"bird\" are updated to suite the benchmark dataset. Given [templatei], we use o3-high [29] to generate 14 additional prompt variations: (Generate Variations) What are 14 semantically equivalent ways of writing the following question? Prompt: [templatei] Next, we perform a manual check on this set of prompts for semantic equivalence to the input prompt, resulting in a total of 45 variations for each dataset. For the remainder of the paper, accuracies reported are across these variations. For the full list of these prompts, see Sec. C. 3.2. nlg2choice for Classification Our approach to classification involves two stages: (1) prompting the LLM for a free-form response to a question defined by Sec. 3.1 and (2) prompting the same model again to select the most likely species given the previous response. On the second step, we enforce a valid output by using constrained decoding across the choice set. Additionally, the second stage is text-only, namely it does not see the image which produced the original response. It receives the following prompt: (Text-Only Choice) What is the most likely species of bird indicated in this response? Response: [nlg] Answer from the following: [choice list] Where [nlg] is the free-form response to the dataset instruction and [choice list] is newline-separated list of class names. Similarly, \"species\" and \"bird\" are updated accordingly for the given dataset. 3.3. nlg2choice for Retrieval In thresholding-based scenarios one desires a continuous score for every query. Most commonly this is done by taking the probability of \"Yes\" and \"No\" for each image, class pair xi ∈X, yi ∈Y, ie.: ˆyi = softmax( p(\"Yes\"|xi, [prompt]) p(\"No\"|xi, [prompt]) Where p(·) is the next-token probability distribution of a MLLM and [prompt] describes a two-way prediction task on one class of the dataset, eg. \"Is this a [cname]?\" and [cname] is the name of some class. However, this requires |X|·|Y| forward passes of the model. To improve throughput, we propose to update the second step of nlg2choice to stop generating probabilities when the remaining tokens for a given choice do not occur in any other choices. For example, take the case of \"Baltimore Oriole\" which can Method Full Prob Yes/No Truncated Prob CUB200 687 200 47 (+0%) (+244%) (+1362%) Flowers 257 102 12 (+0%) (+152%) (+2042%) Aircrafts 479 100 79 (+0%) (+379%) (+506%) Cars 1668 196 125 (+0%) (+751%) (+1234%) Food 269 101 24 (+0%) (+166%) (+1021%) NABirds 3294 555 701 (+0%) (+494%) (+370%) iNat-Birds 5450 1486 480 (+0%) (+267%) (+1035%) Table 1. Number of forward passes needed for different methods in the second-stage answer extraction of nlg2choice. \"Full Prob\" refers to the probability of a full class name p([cname]) under the LLM, where [cname] is a common name of a class. \"Yes/No\" refers to the common |X| · |Y| method of calculating probability over all the classes yi ∈Y. \"Truncated Prob\" refers to the process outlined in Sec. 3.3. The percentages in the \"Yes/No\" and \"Truncated Prob\" columns refer to the throughput increase over caculating the full sequence probabilities \"Full Prob.\" Every setting for truncation is faster than \"Yes/No\" except for one benchmark - NABirds [38]. The fastest method for each benchmark is bolded. be decomposed into [\"B\", \"altimore\", \" Ori\", \"ole\"]. We can see that in the CUB200 species list there are many names which share the first token \"B\", ie. \"Bewick Wren\" or \"Belted Kingfisher\". However, \"Baltimore Oriole\" is the only choice name which has \"altimore\" as its second token. Thus, the probabilities for [\" Ori\", \"ole\"] are not calculated. In terms of probabilities, this is equivalent to: pfull(\"Baltimore Oriole\") = p(\"B\") (1) × p(\"altimore\"|\"B\") × p(\" Ori\"|\"Baltimore\") × p(\"ole\"|\"Baltimore Ori\") ptrunc(\"Baltimore Oriole\") = p(\"B\") (2) × p(\"altimore\"|\"B\") ×(((((((((((( p(\" Ori\"|\"Baltimore\") × (((((((((((((( p(\"ole\"|\"Baltimore Ori\") On many datasets, this change produces substantial throughput increase as many choice names can easily be determined by their first or second tokens. The special property of truncated probability sequences is that they have the same probabilities as sampling with constrained generation. This throughput increase is depicted in Tab. 1. 4. Experiments Datasets We conduct experiments over a benchmark of 7 FGVC datasets: CUB [40] (200 classes), Flowers 102 [28] (102 classes), Stanford Cars [17] (196 classes), FGVC Aircrafts [25] (100 classes) and Food101 [6] (101 classes), NABirds [38] (555 classes), and iNaturalist-Birds [39] (1486 classes). We choose these benchmarks because they are well-known for fine-grained visual classification and are traditionally difficult for zero-shot visual models to solve. iNaturalist-Birds is the validation split of the iNat2021 dataset where the the \"class\" field of each row is equal to \"Aves,\" resulting in a test size of 14960. For each dataset we use the full test set and the training split remains unused. Evaluation Metrics For the classification setting we use accuracy averaged over 15 semantically equivalent prompts, whereas for the retrieval setting we use Mean Average Precision (mAP). Choice lists are the names of classes themselves rather than letters or numbers. As compared to previous works [36], we perform the full-way class prediction rather than subsetting the choices to a 4-way prediction. However, we provide an experiment evaluating this setting in Sec. B. Implementation Details Constrained decoding is implemented using the Outlines2 library [46] and classes for each setting are detailed in Sec. C. We start by generating 15 prompts described in Sec. 3.1, then replacing the key features like domain (\"birds,\" \"flowers,\" etc.) and class types (\"species,\" \"variant,\" etc.) for each dataset. This allows us to compare directly between questions for a given model or dataset, and we also test the performance of appending class lists and common CoT methods. For our experiments, we test only open-source models. Specifically, we test the Qwen-2.5VL [4], Llama-3.2 [12], and InternVL3 [49] architectures. For hyperparameters, we use the recommended generation parameters as defined by the HuggingFace card and take max new tokens to be 512. Experiments were ran in a heterogeneous environment of single-GPU nodes. All nodes required a GPU with at least 23GB VRAM, 16 CPU cores, and 64GB RAM. 5. Results 5.1. Zero-Shot Classification Performance Answer extraction is better than directly answering. In Tab. 2 we report the performance of nlg2choice across various architectures and datasets. All architectures enjoyed a bump in performance across individual datasets and on average (+17.46, +8.49, +6.87 across Qwen-2.5VL, Llama-3.2V, and Intern3), but model-specific differences can be clearly seen. Likewise, we find that removing ex2https://github.com/dottxt-ai/outlines Method Birds Flowers Aircrafts Cars Foods NABirds iNat-Birds Average Qwen-2.5VL-7B choice 40.08 51.29 44.97 26.80 57.93 37.77 17.29 39.45 nlg2choice 47.72 63.22 59.96 37.52 68.99 44.88 21.47 49.11 (+7.64) (+11.93) (+14.99) (+10.72) (+11.06) (+7.11) (+4.18) (+9.66) nlg2choiceopen 63.36 78.03 61.54 47.13 72.26 50.19 25.86 56.91 (+23.28) (+26.74) (+16.57) (+20.33) (+14.33) (+12.42) (+8.57) (+17.46) Llama-3.2-Vision-11B choice 42.27 45.68 31.38 24.68 64.70 36.69 13.82 37.03 nlg2choice 46.96 51.16 30.75 32.72 64.52 41.34 15.81 40.47 (+4.69) (+5.48) (-0.63) (+8.04) (-0.18) (+4.65) (+1.99) (+3.43) nlg2choiceopen 49.59 62.24 37.12 36.78 72.72 46.37 13.85 45.52 (+7.32) (+16.56) (+5.74) (+12.10) (+8.02) (+9.68) (+0.03) (+8.49) Intern3VL-8B choice 11.27 24.68 15.84 14.78 44.12 6.90 2.28 17.12 nlg2choice 13.81 32.52 18.63 18.34 43.36 7.06 1.95 19.38 (+2.54) (+7.84) (+2.79) (+3.56) (-0.76) (+0.16) (-0.33) (+2.26) nlg2choiceopen 18.47 37.50 21.69 22.15 51.48 12.39 4.29 24.00 (+7.20) (+12.82) (+5.85) (+7.37) (+7.36) (+5.49) (+2.01) (+6.87) Table 2. Performance across fine-grained visual classification (FGVC) benchmarks and architectures. The \"Birds\" dataset refers to CUB200. The first row of each model refers to using constrained decoding directly on the model across the class list of each dataset using the prompt in the header. \"choice\" is performing constrained decoding on the prompt \"What is the {type} of {domain} in this image? Answer with {type} only.\" \"nlg2choice\" refers to letting the model freely respond to the prompt, then using constrained decoding in a text-only approach. \"nlg2choiceopen\" refers to removing the appended response-style instruction, eg. \"Answer with {type} only.\" All models are their instructed-tuned version. Best performance is bolded for each model. Method Genus Matches % of Data Qwen-2.5VL-7B choice 45.99 59.92 nlg2choice 56.57 (+10.56) 52.28 nlg2choiceopen 62.74 (+16.75) 36.64 Llama-3.2-Vision-11B choice 42.52 57.73 nlg2choice 61.71 (+19.19) 53.04 nlg2choiceopen 66.37 (+23.85) 50.94 Intern3VL-8B choice 36.77 88.73 nlg2choice 48.79 (+12.02) 86.19 nlg2choiceopen 60.36 (+23.59) 81.53 Table 3. Accuracy of genus-level predictions on misclassified examples. \"Genus Matches\" refers to the accuracy of model with respect to the translating each species to its corresponding genus. \"% of Data\" refers to the amount of data misclassified by the model and is equivalent to 1 -accuracy in Tab. 2. The species to genus mapping manually created. plicit answer formatting from the model (changing from \"nlg2choice\" and \"nlg2choiceopen\"). We find the Aves subset of iNaturalist [39] to be the hardest among all the benchmarks, resulting in the smallest increase in performance across all models. For model-specific differences, we find that Qwen2.5VL performs the best on average using nlg2choice, having an average accuracy of 56.91 across the 7 datasets. We also find that Qwen-2.5VL experiences the greatest gain, having an average accuracy difference of +17.46 from the base constrained decoding method to nlg2choiceopen. Comparatively, we see that Intern3-VL performs the worst across architectures, having the overall smallest increase when using nlg2choiceopen (+6.87) and the lowest performance (24.00). However, we do still see improvements over all benchmarks. Misclassifications are more aligned with ground truth. A natural question is to ask how the classifications qualitatively change under nlg2choice. For an example, take Fig. 1 where the ground truth label is \"Ivory Gull\": under constrained decoding the model responds with \"Crested Auklet\" whereas a higher quality misclassificaiton would be \"Herring Gull\", \"California Gull\", ..., ie. a class within the same genus as Ivory Gull. In Tab. 3 we qualitatively gauge how misclassifications change when using nlg2choice by measuring its genus-level performance. We construct a species to genus mapping on the CUB200 dataset manually, then map the species prediction in previous results to genus. Simply put, we find that nlg2choice makes more reasonable errors, switching to nlg2choice incurs a genus-level performance improvement of +16.75, +23.85, +23.59 for Qwen-2.5VL, Llama-3.2V-11B, and Intern3VL-8B. Δ Accuracy -0.05 0.00 0.05 Intern3VL-8B Llama-3.2V-11B Qwen-2.5VL-8B 0.10 0.15 0.20 0.25 0.30 μ = 7.38 σ = 1.69 μ = 7.72 σ = 2.95 μ = 22.06 σ = 1.92 Figure 3. Difference in accuracies on the question-level between nlg2choice and constrained decoding. Each distribution is made up of the performance difference when switching to nlg2choice over 15 questions. nlg2choice uses an open-ended prompt whereas constrained decoding uses an instruction to increase brevity (\"... Answer with species only.\"). Improvements are statistically significant. Rather than ask about the average performance of different methods under user variation, we wish to quantify how much performance is gained for a specific question by switching to nlg2choice. We calculate this by taking the difference between the performance of constrained decoding and nlg2choiceopen for each question, then compute summary statistics over the set of questions. This forms a distribution of changes in accuracy, with significant results having a positive mean and 95% confidence interval. The performance difference for the various settings in outlined in Fig. 3. We find that for the proposed settings, switching from constrained decoding with an instruction to increase brevity (\"... Answer with species only.\") to nlg2choiceopen comes with an average μ = 22.06, 7.38, 7.72 percentage increase in accuracy for Qwen-2.5VL, Llama-3.2V-11B, and Intern3VL-8B, respectively. Additionally, we find that these increases are within the bounds of statistical significance (σ = 1.92, 1.69, 2.95). Constrictive instructions reduce performance. A common means of steering models to produce valid choices is through appending explicit instructions onto the instruction which enumerate the available choices, typically by means of lettering or enumeration. We wish to quantify the effect of adding this explicit instruction to the model. In Tab. 4 we show the performance of constrained decodMethod Birds Flowers \"What is the {type} of {domain} in this image?\" Qwen-2.5VL 0.92 2.53 + nlg2choice 63.36 (+62.44) 78.03 (+75.5) Llama-3.2-Vision 7.67 27.50 + nlg2choice 49.59 (+41.92) 62.24 (+34.74) Intern3-VL 2.28 5.01 + nlg2choice 18.47 (+16.19) 37.5 (+32.49) \"... Answer with {type} only.\" Qwen-2.5VL 40.08 51.29 + nlg2choice 47.72 (+7.64) 63.22 (+11.93) Llama-3.2-Vision 42.27 45.68 + nlg2choice 46.96 (+4.69) 51.16 (+5.48) Intern3-VL 11.27 24.68 + nlg2choice 13.81 (+2.54) 32.52 (+7.84) \"... Answer from {choice list}.\" Qwen-2.5VL 38.71 69.19 + nlg2choice 41.05 (+2.34) 72.48 (+3.29) Llama-3.2-Vision 13.50 44.83 + nlg2choice 46.68 (+33.18) 66.03 (+21.2) Intern3-VL 22.34 35.24 + nlg2choice 23.48 (+1.14) 35.73 (+0.49) Table 4. Classification performance across instruction templates with varying levels of constrictiveness. The model name refers to default constrained decoding. The difference in performance is shown in parentheses. The best performing model and method for each input instruction is bolded. \"nlg2choiceopen\" corresponds to the top section of the table. ing and nlg2choice under three variations of instruction steering: (1) no explicit choice steering (\"What is the {type} of {domain} in this image?\"), (2) encouraging succinct prediction (\"... Answer with {type} only\"), and (3) explicitly enumerating choices (\"... Answer from {choice list}\"). We see that for every template, switching from constrained decoding to nlg2choice improves performance. The most stark setting is the most open-ended prompt, where we see an average increase of +68.97, +38.33, and +24.34 for Qwen-2.5VL, Llama-3.2V, and Intern3-VL, respectively. Surprisingly, explicit choice instruction shows the weakest improvement for Qwen-2.5VL and Intern3-VL (+2.82 and +0.81) but strong improvement for Llama-3.2V (+27.19). Otherwise, keeping in line with previous results, we find that Qwen-2.5VL is the greatest benefactor of performance. CoT doesn't consistently improve performance. Next, we ask how much performance can be improved by using CoT-style instructions. We investigate methods which induce reasoning by forcing the model to generate some initial tokens that are common to thought-out reasoning (eg. \"Let's think step by step\"), then letting the model freely respond. In Tab. 5 we choose three of Method Birds Flowers \"What is the {type} of {domain} in this image?\" Qwen-2.5VL 63.36 78.03 Llama-3.2-Vision 49.59 62.24 Intern3-VL 18.47 37.50 ... \"Let's think step by step.\" [16] Qwen-2.5VL 54.93 (-8.43) 70.16 (-7.87) Llama-3.2-Vision 39.87 (-9.72) 39.22 (-23.02) Intern3-VL 19.11 (+0.64) 30.82 (-6.68) ... \"First,\" [3] Qwen-2.5VL 53.51 (-9.85) 72.80 (-5.23) Llama-3.2-Vision 41.84 (-7.75) 45.31 (-16.93) Intern3-VL 19.65 (+1.18) 33.46 (-4.04) ... \"Let's solve this problem by splitting it into steps.\" [30] Qwen-2.5VL 54.69 (-8.67) 59.56 (-18.47) Llama-3.2-Vision 40.11 (-9.48) 38.08 (-24.16) Intern3-VL 19.68 (+1.21) 31.06 (-6.44) Table 5. Classification performance when using assistant output prepending CoT methods. All results are reported using nlg2choice as an answer extraction method. The proposed setting is the first section in gray. Forced assistant prepending text is shown in blue . these methods and report their performance and difference between the default nlg2choiceopen setting. In general, we see that these methods do not consistently improve performance, resulting in an average decrease of -9.75, -15.18, and -2.36 across Qwen-2.5VL, Llama-3.2V, and Intern3-VL, respectively. The one exception to this rule is the CUB200 dataset with Intern3-VL, where we see an average increase of +1.01. 5.2. Zero-Shot Retrieval Performance Answer extraction is better than directly answering. In Tab. 6, we report the performance of various models when viewing FGVC datasets as a one-vs-rest retrieval task. Like the classification setting, we find that when compared to exhaustive \"Yes/No\" questioning, nlg2choice generally improves with the exception of the Stanford Cars [17], in which performance across all models decreases. We again find Qwen-2.5VL to receive the greatest improvement, with +8.16 average mAP increase across the 5 datasets. Likewise, we find Intern3-VL receives the least improvement, with an average increase of +4.43. Unlike classification, we do not find that changing the prompt to be open-ended (nlg2choiceopen, ie. removing \"Answer from {species list}\" from the end of the input prompt) does not uniformly increase performance. However, we do still find that open-ended prompts generally perform better on average with nlg2choice (+8.16 vs. +6.96) and are less variable across all models. Method Birds Flowers Aircrafts Cars Foods Baseline Random 0.64 1.24 1.23 0.62 1.02 Qwen-2.5VL-7B Yes/No 51.01 61.19 51.07 77.58 82.67 nlg2choice 57.79 82.56 65.14 68.61 84.30 (+6.78) (+21.37) (+14.07) (-8.97) (+1.63) nlg2choiceopen 59.01 86.42 64.03 69.82 85.05 (+8.00) (+25.23) (+12.96) (-7.76) (+2.38) Llama-3.2-Vision-11B Yes/No 41.24 63.59 34.24 66.03 78.81 nlg2choice 51.14 68.64 39.39 56.46 86.73 (+9.90) (+5.05) (+5.15) (-9.57) (+7.92) nlg2choiceopen 52.61 74.55 40.00 59.35 88.84 (+11.37) (+10.96) (+5.76) (-6.68) (+10.03) Intern3VL-8B Yes/No 15.22 38.32 17.93 31.83 61.46 nlg2choice 26.97 45.13 20.47 25.81 66.55 (+11.75) (+6.81) (+2.54) (-6.02) (+5.09) nlg2choiceopen 30.51 41.12 20.69 26.84 67.76 (+15.29) (+2.80) (+2.76) (-4.99) (+6.30) Table 6. Performance across fine-grained visual classification (FGVC) benchmarks and architectures when interpreted as retrieval tasks. Numbers represent Mean Average Precision (mAP). Every query is a one-vs-rest task, eg. for CUB200 (Birds) the prior for a single PR curve is 1/200. 5.3. Answer Extraction Performance In Sec. A we detail the process of labeling data for species extraction from free-form responses. First, labelers are shown responses from various models to a dataset instruction and image, then are instructed to highlight the first incidence of the predicted species within the generated text. Then, they are instructed to confirm whether the species highlighted matches the choice selection by the model, and in the failure case to indicate the species predicted or whether there was a schema failure. This process results in a set of common outcomes, ie. \"schema failure,\" \"no species predicted,\" \"answer={species}\". In Fig. 4 we show the makeup of the labeled data in terms of these various outcomes. Models often respond out of schema. Depicted in Fig. 4, we see that 34.64% of free-form responses by models have an answer within their text that does not occur within the dataset schema. In other words, the worst-case scenario is that only 65.35% of examples are correctly classifiable. However, this turns out to not be the case due to the fact that a large percentage of these responses have real species within them (70.75% of failures) which can be resolved to the correct answer, but that the predicted species itself cannot be easily put within the given schema. We hypotheFigure 4. Breakdown of the small labeled set of natural language responses. \"Other\" contains two main categories, \"Refused to answer\" and \"No discernable information.\" Examples of these are given in Sec. A. size that many of these responses are grouped into the correct answer due to either genus-level answers in the dataset, eg. CUB200's labels include the genus \"Mockingbird\" which matches to \"Northern Mockingbird\" within the free-form text, or being a closely-related but technically different species, eg. \"Glaucous-winged Gull\" in schema versus \"Glaucous Gull\" in free-form text. Motivation of different methods. In Tab. 7 we show the performance of main architectures in terms of extraction performance over various methods when sub-setting the data to examples where an answer within the schema is clearly apparent. We include two additional variations: (1) nlg2nlg, the free-form response given the instruction to rephrase the MLLM answer and (2) nlg2nlg2choice, the constrained decoding off one extra round of rephrasing. We include nlg2nlg due to the fact that we wish to test the direct distance to the highlighted text created during labeling and include nlg2nlg2choice to gauge the distance between the interpretation of that text into an answer. LLMs are decent answer extractors out-of-the-box. In the default nlg2choiceopen setting, we find high answer extraction success rates for Qwen-2.5VL (97.93) and Intern3-VL (93.26), but lower results for Llama-3.2V (79.02). We find that the raw text extracted by the models (\"nlg2nlgexact\") seldom fits neatly into the dataset schema, having large performance drops (-16.47, 50.77, -25.91). However, the output of nlg2nlgexact retains the relevant information from the original free-form text, as indicated by the nlg2nlg2choice scores which has much smaller perMethod Qwen2.5VL Llama3.2V Intern3VL Overall xFinder [47] Qwen-1.5-0.5B [37] 92.05 88.67 71.76 86.27 Llama-3-8B-Instruct [12] 80.13 80.67 75.29 79.27 Qwen-2.5VL 8B nlg2choice 98.01 98.67 96.47 97.93 nlg2nlgexact 80.79 84.00 78.82 81.61 (-17.22) (-14.67) (-17.65) (-16.32) nlg2nlg2choice 98.01 97.33 100.00 98.19 (+0.00) (-1.34) (+3.53) (+0.26) Llama-3.2-Vision 11B nlg2choice 77.48 82.00 76.47 79.02 nlg2nlgexact 31.79 26.00 25.88 28.24 (-45.69) (-56.00) (-50.59) (-50.78) nlg2nlg2choice 76.82 80.00 70.59 76.68 (-0.66) (-2.00) (-5.88) (-2.34) Intern3-VL 8B nlg2choice 94.04 94.00 90.59 93.26 nlg2nlgexact 59.60 78.67 62.35 67.62 (-34.44) (-15.33) (-28.24) (-25.64) nlg2nlg2choice 90.06 85.33 90.59 88.34 (-3.98) (-8.67) (+0.00) (-4.92) Table 7. Performance of various MLLMs across a small labeled set. Differences are measured from the \"nlg2choice\" row of each model. formance differences (+0.61, -2.72, and -4.39). We hypothesize that the nlg2nlg text still contains artifacts such as \"The species is ...\" or scientific names, which lower exact match performance. 6. Limitations This work has three main limitations. First, our experiments were conducted primarily with medium-sized models (8B11B) and we only used open-source LLMs from a few different families: Qwen, Llama, and Intern. Secondly, while nlg2choice avoids the need for specialized training data or architectural modifications and is more aligned with the true responses of the model, the method still requires significant computational resources to generate and process the textual descriptions. Lastly, while nlg2choice shows promise in classification, it still remains to be seen whether the approach scales to multi-label problems. 7. Conclusion In this work, we presented nlg2choice, a simple approach for zero-shot fine-grained visual classification and retrieval that extracts fine-grained answers from MLLM free-form responses. Through experiments across multiple architectures and datasets, we demonstrated that nlg2choice consistently outperforms constrained decoding and exhaustive questioning methods in both classification accuracy and retrieval performance. We also validated the answer extraction directly through labeling predicted species in unstructured text, where we showed that models are capable of extraction their own answers without any labeled data. References [1] Gpt-4v(ision) system card. 2023. 1 [2] Josh Achiam, Steven Adler, Sandhini Agarwal, Lama Ahmad, Ilge Akkaya, Florencia Leoni Aleman, Diogo Almeida, Janko Altenschmidt, Sam Altman, Shyamal Anadkat, et al. 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Self-consistency improves chain of thought reasoning in language models. In The Eleventh International Conference on Learning Representations, ICLR 2023, Kigali, Rwanda, May 1-5, 2023. OpenReview.net, 2023. 2 [43] Xinpeng Wang, Chengzhi Hu, Bolei Ma, Paul R ̈ottger, and Barbara Plank. Look at the text: Instruction-tuned language models are more robust multiple choice selectors than you think. arXiv preprint , 2024. 1, 3 [44] Xinpeng Wang, Bolei Ma, Chengzhi Hu, Leon WeberGenzel, Paul R ̈ottger, Frauke Kreuter, Dirk Hovy, and Barbara Plank. \"my answer is c\": First-token probabilities do not match text answers in instruction-tuned language models. arXiv preprint , 2024. 2, 3 [45] Yubo Wang, Xueguang Ma, Ge Zhang, Yuansheng Ni, Abhranil Chandra, Shiguang Guo, Weiming Ren, Aaran Arulraj, Xuan He, Ziyan Jiang, et al. Mmlu-pro: A more robust and challenging multi-task language understanding benchmark. 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For labeling answer extraction data, we prepare a user interface built with Label Studio which shown in Fig. 5. On the right contains information on the inputs to the model, specifically the question, image, and ground truth label. On the left is main interface, which consists a free-form response from a model to the example displayed on the right, as well as the choice predicted by the model via nlg2choice. Labeling consists of two pieces: (1) identification and highlighting of text spans containing answers to the input question and (2) matching the highlighted text span to a class within the given schema. Text span labeling. For the text span labeling task, labelers read through the text to identify the first incidence of a predicted species. Before highlighting the species found within the text, the prediction is confirmed to be a real species through a brief internet search. If there are any issues with highlighting the predicted species, a text description of the issue is written in the input text box at the bottom and the example is skipped. Figure 5. Labeling interface for efficacy of the answer extraction stage of nlg2choice. Labelers are asked to highlight the name predicted in free-form responses from various models and whether the name occurs easily within the dataset choice list. Above shows an interesting example: for the Chipping Sparrow on the right, the model freely responded with \"Painted Bunting\" and answer extraction made an additional mistake by predicting \"Scarlet Tanager.\" Choice assignment. Once the span is labeled, we next wish to either assign the span to a class within the dataset or tag the span as a schema failure. For expediency, the labelers are also equipped with the nlg2choice prediction and a tool for sorting most like class names with fuzzy matching score. When the nlg2choice predict matches the text span, the prediction is \"Consistent with nlg\" and the labelers check the button. Otherwise, \"Inconsistent with nlg\" is checked and a natural language explanation is put in the bottom text box. When the nlg2choice is inconsistent, \"answer={species}\" is put within that text box. B. Zero-Shot 4-Way VQA Performance Answer extraction does not consistently increase performance. Following recent work [36], we evalute nlg2choice on popular fine-graiend classification datasets as a 4-way VQA task. iNat-Animal and iNat-Plant are the rows of iNat21 [39] when subset to rows which whose kingdom matches \"Animalia\" and \"Plantae.\" ImgNet-Animal and ImgNet-Artifact are the rows of ImageNet-1K [32] subset to rows which have an anceter of \"Animal\" and \"Artifact\" in the WordNet hierarchy [26]. The choices for each example are from the top 3 SigLIP [48] predictions and the correct class. For the nlg2choice prompt we use the same prompt as the previous work. We report the open-ended question performance in Tab. 8 where we find that it slightly Method CUB200 iNat-Animal iNat-Plant ImgNet-Animal ImgNet-Artifact Qwen-2.5VL-7B A/B/C/D 65.50 41.33 41.61 85.20 80.01 nlg2choiceopen 67.43 40.19 40.03 84.03 81.81 (+1.93) (-1.14) (-1.58) (-1.17) (+1.80) Llama-3.2-Vision-11B A/B/C/D 65.52 32.44 31.88 79.93 75.89 nlg2choiceopen 68.79 31.52 31.57 81.39 75.29 (+3.27) (-0.92) (-0.31) (+1.46) (-0.60) Intern3VL-8B A/B/C/D 50.52 35.40 36.39 77.50 69.41 nlg2choiceopen 47.30 35.86 35.64 81.91 70.02 (-3.22) (+0.46) (-0.75) (+4.41) (+0.61) Table 8. 4-way VQA accuracy across fine-grained classifcation tasks. underperforms or has not effect on the lettering approach. Specifically, we find that accuracy changes by -0.03, +0.58, and +0.30 on average for Qwen-2.5VL, Llama-3.2V, and Intern3VL, respectively. Decreasing the choice set improves performance. We are also able to compare the 4-way VQA setting of CUB200 directly to the many-way prediction displayed in Tab. 2. We see that subsetting the full choice set to four choices incurs a performance improvement of +4.07, +19.20, and +28.83 for Qwen-2.5VL, Llama-3.2V, and Intern3VL, respectively. C. Prompt Variations \"What { type } is this { domain }?\" \"What is the { type } of this { domain }?\" \"What is the { type } of the { domain }?\" \"What is the { type } of the { domain } in this image?\" \"What is the { type } of the { domain } in the image?\" \"Identify this { domain }'s { type }.\" \"Name the { type } shown in the image.\" \"Which { domain } { type } is pictured here?\" \"Classify the { type } of this { domain }.\" \"What { domain } { type } does the photo depict?\" \"Determine the { type } of the { domain } in view.\" \"Provide the common name of this { domain }.\" \"To which { type } does this { domain } belong?\" \"Label the { type } of the { domain } shown.\" \"Recognize and state this { domain }'s { type }.\" Table 9. Base template variations generated by Sec. 3.1 Dataset Type Domain CUB200 species bird Flowers species flower Aircrafts variant aircraft Cars year, make, and model car Foods name food NABirds species bird iNat-Birds species bird Table 10. 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gr-qc/0606075 | Tetrads in low-energy weak interactions
Alcides Garat11
1. Instituto de F´ısica, Facultad de Ciencias, Igu´a 4225, esq. Mataojo, Montevideo,
Uruguay.a)
(Dated: June 15th, 2006)
Tetrads are introduced in order to study the relationship between tetrad gauge states
of spacetime and particle interactions, specially in weak... | Tetrads in low-energy weak interactions Alcides Garat11 1. Instituto de F ́ısica, Facultad de Ciencias, Igu ́a 4225, esq. Mataojo, Montevideo, Uruguay.a) (Dated: June 15th, 2006) Tetrads are introduced in order to study the relationship between tetrad gauge states of spacetime and particle interactions, specially in we... |
math/0104241 | arXiv:math/0104241v1 [math.CO] 25 Apr 2001
THE LAURENT PHENOMENON
SERGEY FOMIN AND ANDREI ZELEVINSKY
Abstract. A composition of birational maps given by Laurent polynomi-
als need not be given by Laurent polynomials; however, sometimes—quite
unexpectedly—it does. We suggest a unified treatment of this phenomenon,
whic... | arXiv:math/0104241v1 [math.CO] 25 Apr 2001 THE LAURENT PHENOMENON SERGEY FOMIN AND ANDREI ZELEVINSKY Abstract. A composition of birational maps given by Laurent polynomials need not be given by Laurent polynomials; however, sometimes-quite unexpectedly-it does. We suggest a unified treatment of this phenomenon, which c... |
2510.14979 | FROM PIXELS TO WORDS – TOWARDS NATIVE VISION-
LANGUAGE PRIMITIVES AT SCALE
Haiwen Diao1
Mingxuan Li2
Silei Wu3
Linjun Dai3
Xiaohua Wang2
Hanming Deng3
Lewei Lu3
Dahua Lin3
Ziwei Liu1
1S-Lab, Nanyang Technological University
2Xi’an Jiaotong University
3SenseTime Research
Website:
https://github.com/EvolvingLMMs-Lab/NEO
... | FROM PIXELS TO WORDS - TOWARDS NATIVE VISIONLANGUAGE PRIMITIVES AT SCALE Haiwen Diao1 Mingxuan Li2 Silei Wu3 Linjun Dai3 Xiaohua Wang2 Hanming Deng3 Lewei Lu3 Dahua Lin3 Ziwei Liu1 1S-Lab, Nanyang Technological University 2Xi'an Jiaotong University 3SenseTime Research Website: https://github.com/EvolvingLMMs-Lab/NEO AB... |
2510.14981 | Preprint - Under Review
COUPLED DIFFUSION SAMPLING FOR TRAINING-FREE
MULTI-VIEW IMAGE EDITING
Hadi Alzayer1,2
Yunzhi Zhang1
Chen Geng1
Jia-Bin Huang2
Jiajun Wu1
1Stanford University
2University of Maryland, College Park
ABSTRACT
We present an inference-time diffusion sampling method to perform multi-view
consistent ima... | Preprint - Under Review COUPLED DIFFUSION SAMPLING FOR TRAINING-FREE MULTI-VIEW IMAGE EDITING Hadi Alzayer1,2 Yunzhi Zhang1 Chen Geng1 Jia-Bin Huang2 Jiajun Wu1 1Stanford University 2 -time diffusion sampling method to perform multi-view consistent image editing using pre-trained 2D image editing models. These models c... |
2510.14978 | Preprint
LEARNING AN IMAGE EDITING MODEL
WITHOUT IMAGE EDITING PAIRS
Nupur Kumari1
Sheng-Yu Wang1
Nanxuan Zhao2
Yotam Nitzan2
Yuheng Li2
Krishna Kumar Singh2
Richard Zhang2
Eli Shechtman2
Jun-Yan Zhu1
Xun Huang2
1Carnegie Mellon University
2Adobe
ABSTRACT
Recent image editing models have achieved impressive results whi... | Preprint LEARNING AN IMAGE EDITING MODEL WITHOUT IMAGE EDITING PAIRS Nupur Kumari1 Sheng-Yu Wang1 Nanxuan Zhao2 Yotam Nitzan2 Yuheng Li2 Krishna Kumar Singh2 Richard Zhang2 Eli Shechtman2 Jun-Yan Zhu1 Xun Huang2 1Carnegie Mellon University 2Adobe ABSTRACT Recent image editing models have achieved impressive results whi... |
2510.14976 | Ponimator: Unfolding Interactive Pose for Versatile Human-human
Interaction Animation
Shaowei Liu∗
Chuan Guo2†
Bing Zhou2†
Jian Wang2†
1University of Illinois Urbana-Champaign
2Snap Inc.
https://stevenlsw.github.io/ponimator/
Input
Generated Interaction Animation (left→right: time steps)
Two-person image
Estimated inte... | Ponimator: Unfolding Interactive Pose for Versatile Human-human Interaction Animation Shaowei Liu∗ Chuan Guo2† Bing Zhou2† Jian Wang2† 1 -Champaign 2Snap Inc. https://stevenlsw.github.io/ponimator/ Input Generated Interaction Animation (left→right: time steps) Two-person image Estimated interactive pose Single-person i... |
2510.14980 | Technical Report
AGENTIC DESIGN OF COMPOSITIONAL MACHINES
Wenqian Zhang1
Weiyang Liu2,*
Zhen Liu1,*,†
1The Chinese University of Hong Kong (Shenzhen)
2The Chinese University of Hong Kong
*Equal Advising
†Corresponding Author
besiegefield.github.io
Eureka！
Throw a Boulder
Machine Code
<boulder parent=bucket_1 attach_to=... | Technical Report AGENTIC DESIGN OF COMPOSITIONAL MACHINES Wenqian Zhang1 Weiyang Liu2,* Zhen Liu1,*,† 1The Chinese (Shenzhen) 2The Chinese *Equal Advising †Corresponding Author besiegefield.github.io Eureka! Throw a Boulder Machine Code ... Code Edit ... Figure 1: The task of compositional machine design is illustrated... |
2510.14971 | ON THE INVARIANTS OF FINITE GROUPS ARISING IN A
TOPOLOGICAL QUANTUM FIELD THEORY
CHRISTOPHER A. SCHROEDER AND HUNG P. TONG-VIET
Abstract. In this paper, we consider the properties of finite groups that are witnessed by
group invariants arising in the context of Dijkgraaf–Witten theory, a topological quantum field
theor... | ON THE INVARIANTS OF FINITE GROUPS ARISING IN A TOPOLOGICAL QUANTUM FIELD THEORY CHRISTOPHER A. SCHROEDER AND HUNG P. TONG-VIET Abstract. In this paper, we consider the properties of finite groups that are witnessed by group invariants arising in the context of Dijkgraaf-Witten theory, a topological quantum field theor... |
2510.14973 | ATTENTION IS ALL YOU NEED
FOR KV CACHE IN DIFFUSION LLMS
Quan Nguyen-Tri∗
FPT AI Residency
Hanoi, Vietnam
quannt40@fpt.com
Mukul Ranjan∗& Zhiqiang Shen
VILA Lab, MBZUAI
Abu Dhabi, UAE
{mukul.ranjan,zhiqiang.shen}@mbzuai.ac.ae
Project page: https://vila-lab.github.io/elastic-cache-webpage/
ABSTRACT
This work studies how... | ATTENTION IS ALL YOU NEED FOR KV CACHE IN DIFFUSION LLMS Quan Nguyen-Tri∗ FPT AI Residency Hanoi, Vietnam Mukul Ranjan∗& Zhiqiang Shen VILA Lab, MBZUAI Abu Dhabi, UAE { Project page: https://vila-lab.github.io/elastic-cache-webpage/ ABSTRACT This work studies how to adaptively recompute key-value (KV) caches for diffus... |
2510.14975 | WithAnyone: Towards Controllable and ID Consistent Image Generation
Hengyuan Xu1,2
Wei Cheng2,†
Peng Xing2
Yixiao Fang2
Shuhan Wu2
Rui Wang2
Xianfang Zeng2
Daxin Jiang2
Gang Yu2,‡
Xingjun Ma1,‡
Yu-Gang Jiang1
1 Fudan University
2 StepFun
Project Page
MultiID-2M
MultiID-Bench
Models
Code
Figure 1. Showcases of WithAnyon... | WithAnyone: Towards Controllable and ID Consistent Image Generation Hengyuan Xu1,2 Wei Cheng2,† Peng Xing2 Yixiao Fang2 Shuhan Wu2 Rui Wang2 Xianfang Zeng2 Daxin Jiang2 Gang Yu2,‡ Xingjun Ma1,‡ Yu-Gang Jiang1 1 Fudan University 2 StepFun Project Page MultiID-2M MultiID-Bench Models Code Figure 1. Showcases of WithAnyon... |
2510.14969 | LLMs as Scalable, General-Purpose Simulators
For Evolving Digital Agent Training
Yiming Wang2,⋆,♠, Da Yin1,⋆,♠,♡, Yuedong Cui1,⋆, Ruichen Zheng1,⋆
Zhiqian Li1, Zongyu Lin1, Di Wu1, Xueqing Wu1, Chenchen Ye1, Yu Zhou1, Kai-Wei Chang1,♡
1UCLA
2Harvard University
⋆Co-First Authors
♠Co-LeadAlphabetical Order
♡Equal Advise
... | LLMs as Scalable, General-Purpose Simulators For Evolving Digital Agent Training Yiming Wang2,⋆,♠, Da Yin1,⋆,♠,♡, Yuedong Cui1,⋆, Ruichen Zheng1,⋆ Zhiqian Li1, Zongyu Lin1, Di Wu1, Xueqing Wu1, Chenchen Ye1, Yu Zhou1, Kai-Wei Chang1,♡ 1UCLA 2Harvard University ⋆Co-First Authors ♠Co-LeadAlphabetical Order ♡Equal Advise ... |
2510.14977 | Preprint
TERRA: EXPLORABLE NATIVE 3D WORLD MODEL
WITH POINT LATENTS
Yuanhui Huang1
Weiliang Chen1
Wenzhao Zheng1
Xin Tao2
Pengfei Wan2
Jie Zhou1
Jiwen Lu1
1Tsinghua University
2Kuaishou Technology
1
2
4
3
5
6
5
RGB Point Cloud
Gaussian Primitives
P2G Encoder
Generative
World Modeling
P2G Decoder
Point Latents
4
3
2
1
... | Preprint TERRA: EXPLORABLE NATIVE 3D WORLD MODEL WITH POINT LATENTS Yuanhui Huang1 Weiliang Chen1 Wenzhao Zheng1 Xin Tao2 Pengfei Wan2 Jie Zhou1 Jiwen Lu1 1Tsinghua University 2Kuaishou Technology 1 2 4 3 5 6 5 RGB Point Cloud Gaussian Primitives P2G Encoder Generative World Modeling P2G Decoder Point Latents 4 3 2 1 6... |
2510.14966 | Identity-Link IRT for Label-Free LLM Evaluation:
Preserving Additivity in TVD-MI Scores
Zachary Robertson
Computer Science
Stanford University
zroberts@stanford.edu
Abstract
Pairwise comparisons of large language models using total variation distance
mutual information (TVD-MI) produces binary critic decisions per pair... | Identity-Link IRT for Label-Free LLM Evaluation: Preserving Additivity in TVD-MI Scores Zachary Robertson Computer Science Stanford University Abstract Pairwise comparisons of large language models using total variation distance mutual information (TVD-MI) produces binary critic decisions per pair. We show that averagi... |
2510.14972 | TOKDRIFT: When LLM Speaks in Subwords
but Code Speaks in Grammar
Yinxi Li, Yuntian Deng, Pengyu Nie
University of Waterloo
{yinxi.li, yuntian, pynie}@uwaterloo.ca
Abstract
Large language models (LLMs) for code rely
on subword tokenizers, such as byte-pair en-
coding (BPE), learned from mixed natural lan-
guage text and... | TOKDRIFT: When LLM Speaks in Subwords but Code Speaks in Grammar Yinxi Li, Yuntian Deng, Pengyu Nie {yinxi.li, yuntian, Abstract Large language models (LLMs) for code rely on subword tokenizers, such as byte-pair encoding (BPE), learned from mixed natural language text and programming language code but driven by statis... |
2510.14967 | INFORMATION GAIN-BASED POLICY OPTIMIZATION:
A SIMPLE AND EFFECTIVE APPROACH FOR MULTI-
TURN LLM AGENTS
Guoqing Wang1*, Sunhao Dai2*, Guangze Ye3*, Zeyu Gan2, Wei Yao2, Yong Deng1, Xiaofeng
Wu1, and Zhenzhe Ying1
1Ant Group
2Renmin University of China
3Individual Author
GitHub:
https://github.com/GuoqingWang1/IGPO
ABSTR... | INFORMATION GAIN-BASED POLICY OPTIMIZATION: A SIMPLE AND EFFECTIVE APPROACH FOR MULTITURN LLM AGENTS Guoqing Wang1*, Sunhao Dai2*, Guangze Ye3*, Zeyu Gan2, Wei Yao2, Yong Deng1, Xiaofeng Wu1, and Zhenzhe Ying1 1Ant Group 2Renmin 3Individual Author GitHub: https://github.com/GuoqingWang1/IGPO ABSTRACT Large language mod... |
2510.14970 | Biology-informed neural networks learn nonlinear
representations from omics data to improve genomic
prediction and interpretability
Katiana Kontolati∗
Bayer Crop Science
katiana.kontolati@bayer.com
Rini Jasmine Gladstone
Bayer Crop Science
rinijasmine.gladstone@bayer.com
Ian W. Davis
Bayer Crop Science
ian.davis@bayer.... | Biology-informed neural networks learn nonlinear representations from omics data to improve genomic prediction and interpretability Katiana Kontolati∗ Bayer Crop Science Rini Jasmine Gladstone Bayer Crop Science Ian W. Davis Bayer Crop Science Ethan Pickering∗ Bayer Crop Science Abstract We extend biologically-informed... |
2510.14968 | RDD: Retrieval-Based Demonstration Decomposer for
Planner Alignment in Long-Horizon Tasks
Mingxuan Yan1
Yuping Wang1,2
Zechun Liu3
Jiachen Li1∗
1University of California, Riverside
2University of Michigan
3Meta AI
{myan035, yuping.wang, jiachen.li}@ucr.edu
zechunliu@meta.com
Abstract
To tackle long-horizon tasks, recen... | RDD: Retrieval-Based Demonstration Decomposer for Planner Alignment in Long-Horizon Tasks Mingxuan Yan1 Yuping Wang1,2 Zechun Liu3 Jiachen Li1∗ 1 2 3Meta AI {myan035, yuping.wang, Abstract To tackle long-horizon tasks, recent hierarchical vision-language-action (VLAs) frameworks employ vision-language model (VLM)-based... |
2510.14963 | Orders matter: tight bounds on the precision of sequential quantum estimation
for multiparameter models
Gabriele Fazio,1, ∗Jiayu He,2, † and Matteo G. A. Paris1, ‡
1Dipartimento di Fisica Aldo Pontremoli,
Universit`a degli Studi di Milano, I-20133 Milano, Italy
2QTF Centre of Excellence, Department of Physics,
Universi... | Orders matter: tight bounds on the precision of sequential quantum estimation for multiparameter models Gabriele Fazio,1, ∗Jiayu He,2, † and Matteo G. A. Paris1, ‡ 1Dipartimento di Fisica Aldo Pontremoli, Universit`a degli Studi di Milano, I-20133 Milano, Italy 2QTF Centre of Excellence, -00014 Helsinki, Finland (Dated... |
2510.14974 | Preprint
PI-FLOW:
POLICY-BASED FEW-STEP GENERATION
VIA IMITATION DISTILLATION
Hansheng Chen1 Kai Zhang2 Hao Tan2 Leonidas Guibas1 Gordon Wetzstein1 Sai Bi2
1Stanford University
2Adobe Research
https://github.com/Lakonik/piFlow
Figure 1: High quality 4-NFE text-to-image generations by π-Flow, distilled from FLUX.1-12B
(... | Preprint PI-FLOW: POLICY-BASED FEW-STEP GENERATION VIA IMITATION DISTILLATION Hansheng Chen1 Kai Zhang2 Hao Tan2 Leonidas Guibas1 Gordon Wetzstein1 Sai Bi2 1Stanford University 2Adobe Research https://github.com/Lakonik/piFlow Figure 1: High quality 4-NFE text-to-image generations by π-Flow, distilled from FLUX.1-12B (... |
2510.14965 | CHANGINGGROUNDING: 3D VISUAL GROUNDING IN
CHANGING SCENES
Miao Hu1, Zhiwei Huang2, Tai Wang4, Jiangmiao Pang4, Dahua Lin3,4,
Nanning Zheng1B, Runsen Xu3,4B
1Xi’an Jiaotong University, 2Zhejiang University, 3The Chinese University of Hong Kong,
4Shanghai AI Laboratory
ABSTRACT
Real-world robots localize objects from nat... | CHANGINGGROUNDING: 3D VISUAL GROUNDING IN CHANGING SCENES Miao Hu1, Zhiwei Huang2, Tai Wang4, Jiangmiao Pang4, Dahua Lin3,4, Nanning Zheng1B, Runsen Xu3,4B 1Xi'an Jiaotong University, 2Zhejiang University, 3The Chinese 4Shanghai AI Laboratory ABSTRACT Real-world robots localize objects from natural-language instruction... |
2510.14964 | Efficient and Flexible Multirate Temporal Adaptivity
Daniel R. Reynoldsa,1,∗, Sylvia Amiherea,1, Dashon Mitchella,1, Vu Thai
Luanb
aDepartment of Mathematics and Statistics, University of Maryland Baltimore
County, Baltimore, Maryland, USA
bDepartment of Mathematics and Statistics, Texas Tech University, Lubbock, Texas... | Efficient and Flexible Multirate Temporal Adaptivity Daniel R. Reynoldsa,1,∗, Sylvia Amiherea,1, Dashon Mitchella,1, Vu Thai Luanb a - mal (MRI) time integration methods for adapting time steps when solving problems with multiple time scales. We compare these controllers against competing approaches on two benchmark pr... |
2510.14961 | EFFICIENT PARALLEL SAMPLERS FOR RECURRENT-
DEPTH MODELS AND THEIR CONNECTION TO DIFFU-
SION LANGUAGE MODELS
Jonas Geiping
ELLIS Institute Tübingen &
Max-Planck Institute for Intelligent Systems,
Tübingen AI Center
jonas@tue.ellis.eu
Xinyu Yang
Electrical and Computer Engineering
Carnegie Mellon University
Guinan Su
ELL... | EFFICIENT PARALLEL SAMPLERS FOR RECURRENTDEPTH MODELS AND THEIR CONNECTION TO DIFFUSION LANGUAGE MODELS Jonas Geiping ELLIS Institute Tübingen & Max-Planck Institute for Intelligent Systems, Tübingen AI Center Xinyu Yang Electrical and Computer Engineering Carnegie Mellon University Guinan Su ELLIS Institute Tübingen &... |
2510.14962 | Preprint
RAINDIFF: END-TO-END PRECIPITATION NOWCAST-
ING VIA TOKEN-WISE ATTENTION DIFFUSION
Thao Nguyen, Jiaqi Ma, Fahad Khan, Souhaib Ben Taieb, Salman Khan
Mohamed Bin Zayed University of AI
{thao.nguyen,salman.khan}@mbzuai.ac.ae
ABSTRACT
Precipitation nowcasting, predicting future radar echo sequences from current o... | Preprint RAINDIFF: END-TO-END PRECIPITATION NOWCASTING VIA TOKEN-WISE ATTENTION DIFFUSION Thao Nguyen, Jiaqi Ma, Fahad Khan, Souhaib Ben Taieb, Salman Khan Mohamed Bin Zayed { ABSTRACT Precipitation nowcasting, predicting future radar echo sequences from current observations, is a critical yet challenging task due to t... |
2510.14958 | MathCanvas: Intrinsic Visual Chain-of-Thought for Multimodal
Mathematical Reasoning
Weikang Shi1* Aldrich Yu1* Rongyao Fang1*† Houxing Ren1 Ke Wang1 Aojun Zhou1 Changyao Tian1
Xinyu Fu2 Yuxuan Hu1 Zimu Lu1 Linjiang Huang3 Si Liu3 Rui Liu2‡ Hongsheng Li1‡
1Multimedia Laboratory (MMLab), The Chinese University of Hong Ko... | MathCanvas: Intrinsic Visual Chain-of-Thought for Multimodal Mathematical Reasoning Weikang Shi1* Aldrich Yu1* Rongyao Fang1*† Houxing Ren1 Ke Wang1 Aojun Zhou1 Changyao Tian1 Xinyu Fu2 Yuxuan Hu1 Zimu Lu1 Linjiang Huang3 Si Liu3 Rui Liu2‡ Hongsheng Li1‡ 1Multimedia Laboratory (MMLab), The Chinese 2Huawei Research, 3BU... |
2510.14956 | ON WEIGHTED AND BOUNDED MULTIDIMENSIONAL CATALAN
NUMBERS
RYOTA INAGAKI AND DIMANA PRAMATAROVA
Abstract. We define a weighted analog for the multidimensional Catalan numbers, obtain
matrix-based recurrences for some of them, and give conditions under which they are peri-
odic. Building on this framework, we introduce tw... | ON WEIGHTED AND BOUNDED MULTIDIMENSIONAL CATALAN NUMBERS RYOTA INAGAKI AND DIMANA PRAMATAROVA Abstract. We define a weighted analog for the multidimensional Catalan numbers, obtain matrix-based recurrences for some of them, and give conditions under which they are periodic. Building on this framework, we introduce two ... |
2510.14960 | C4D: 4D Made from 3D through Dual Correspondences
Shizun Wang1
Zhenxiang Jiang1
Xingyi Yang2
Xinchao Wang1*
1National University of Singapore
2The Hong Kong Polytechnic University
{shizun.wang, zhenxiang.jiang}@nus.u.edu, xingyi.yang@polyu.edu.hk, xinchao@nus.edu.sg
Monocular Video
Per-frame dense Point Cloud & Camera ... | C4D: 4D Made from 3D through Dual Correspondences Shizun Wang1 Zhenxiang Jiang1 Xingyi Yang2 Xinchao Wang1* 1National 2The Hong Kong Polytechnic University {shizun.wang, , , Monocular Video Per-frame dense Point Cloud & Camera Poses Camera Intrinsics C4D Motion Masks 2D Point Tracking 3D Point Tracking Figure 1. Given ... |
2510.14957 | Phantom Mirage from Axion Dark Energy
Rayne Liu,1 Yijie Zhu,2 Wayne Hu,1 and Vivian Miranda3
1Kavli Institute for Cosmological Physics, University of Chicago, Chicago, IL 60637, USA
2Department of Physics and Astronomy, Stony Brook University, Stony Brook, NY 11794, USA
3C. N. Yang Institute for Theoretical Physics, St... | Phantom Mirage from Axion Dark Energy Rayne Liu,1 Yijie Zhu,2 Wayne Hu,1 and Vivian Miranda3 1Kavli Institute for Cosmological Physics, 60637, USA 2 11794, USA 3C. N. Yang Institute for Theoretical Physics, Stony Brook University, Stony Brook, NY 11794, USA (Dated: October 17, 2025) Supernova (SN) and baryon acoustic o... |
2510.14959 | CBF-RL: Safety Filtering Reinforcement Learning in Training with
Control Barrier Functions
*Lizhi Yang, *Blake Werner, Massimiliano de Sa, Aaron D. Ames
Abstract— Reinforcement learning (RL), while powerful and
expressive, can often prioritize performance at the expense
of safety. Yet safety violations can lead to cata... | CBF-RL: Safety Filtering Reinforcement Learning in Training with Control Barrier Functions *Lizhi Yang, *Blake Werner, Massimiliano de Sa, Aaron D. Ames Abstract- Reinforcement learning (RL), while powerful and expressive, can often prioritize performance at the expense of safety. Yet safety violations can lead to cata... |
2510.14954 | OMNIMOTION: MULTIMODAL MOTION GENERATION
WITH CONTINUOUS MASKED AUTOREGRESSION
Zhe Li1,2∗, Weihao Yuan2†, Weichao Shen2, Siyu Zhu3, Zilong Dong2, Chang Xu1†
1 University of Sydney
2 Alibaba Group
3 Fudan University
ABSTRACT
Whole-body multi-modal human motion generation poses two primary challenges:
creating an effecti... | OMNIMOTION: MULTIMODAL MOTION GENERATION WITH CONTINUOUS MASKED AUTOREGRESSION Zhe Li1,2∗, Weihao Yuan2†, Weichao Shen2, Siyu Zhu3, Zilong Dong2, Chang Xu1† 1 2 Alibaba Group 3 Fudan University ABSTRACT Whole-body multi-modal human motion generation poses two primary challenges: creating an effective motion generation ... |
2510.14951 | A universal description of Mott insulators:
Characterizing quantum phases beyond broken symmetries
Matheus de Sousa,1, ∗Zhiyu Fan,1, ∗and Wei Ku (顧威)1, †
1School of Physics and Astronomy, Shanghai Jiao Tong University, Shanghai 200240, China
(Dated: October 17, 2025)
Using Mott insulators as a prototypical example, we ... | A universal description of Mott insulators: Characterizing quantum phases beyond broken symmetries Matheus de Sousa,1, ∗Zhiyu Fan,1, ∗and Wei Ku (顧威)1, † 1 200240, China (Dated: October 17, 2025) Using Mott insulators as a prototypical example, we demonstrate a dynamics-based characterization of quantum phases of matte... |
2510.14950 | PREPRINT VERSION
1
A formative measurement validation
methodology for survey questionnaires
Mark Dominique Dalipe Muñoz
Mathematics Department
(Iloilo Science and Technology University – Main Campus)
Iloilo City, Philippines
markdominique.munoz@students.isatu.edu.ph
__________________________________________... | PREPRINT VERSION 1 A formative measurement validation methodology for survey questionnaires Mark Dominique Dalipe Muñoz Mathematics Department (Iloilo Science and Technology University - Main Campus) Iloilo City, Philippines ___________________________________________________________________________ Abstract: Model mis... |
2510.14955 | REALDPO: REAL OR NOT REAL, THAT IS THE PREF-
ERENCE
Guo Cheng3∗
Danni Yang1∗
Ziqi Huang2†
Jianlou Si5
Chenyang Si4
Ziwei Liu2B
1Shanghai Artificial Intelligence Laboratory
2S-Lab, Nanyang Technological University
3University of Electronic Science and Technology of China
4Nanjing University
5SenseTime Research
Project P... | REALDPO: REAL OR NOT REAL, THAT IS THE PREFERENCE Guo Cheng3∗ Danni Yang1∗ Ziqi Huang2† Jianlou Si5 Chenyang Si4 Ziwei Liu2B 1Shanghai Artificial Intelligence Laboratory 2S-Lab, Nanyang Technological University 3 4Nanjing University 5SenseTime Research Project Page: Vchitect.github.io/RealDPO-Project (b) Real Video vs ... |
2510.14953 | Dark Matter Subhalos and Higher Order Catastrophes in Gravitational Wave Lensing
Luka Vujeva,1, ∗Jose Mar´ıa Ezquiaga,1 Daniel Gilman,2 Srashti Goyal,3 and Miguel Zumalac´arregui3
1Center of Gravity, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark
2Department of Astronomy & Astrophysics, University of Ch... | Dark Matter Subhalos and Higher Order Catastrophes in Gravitational Wave Lensing Luka Vujeva,1, ∗Jose Mar ́ıa Ezquiaga,1 Daniel Gilman,2 Srashti Goyal,3 and Miguel Zumalac ́arregui3 1Center of Gravity, Niels Bohr Institute, Blegdamsvej 17, 2100 Copenhagen, Denmark 2 60637, USA 3Max Planck Institute for Gravitational Ph... |
2510.14952 | FROM LANGUAGE TO LOCOMOTION:
RETARGETING-FREE HUMANOID CONTROL VIA
MOTION LATENT GUIDANCE
Zhe Li1∗, Cheng Chi2∗†, Yangyang Wei3, Boan Zhu4, Yibo Peng2, Tao Huang5
Pengwei Wang2, Zhongyuan Wang2, Shanghang Zhang2,6B, Chang Xu1B
1 University of Sydney, 2 BAAI, 3 Harbin Institute of Technology
4 Hong Kong University of Sc... | FROM LANGUAGE TO LOCOMOTION: RETARGETING-FREE HUMANOID CONTROL VIA MOTION LATENT GUIDANCE Zhe Li1∗, Cheng Chi2∗†, Yangyang Wei3, Boan Zhu4, Yibo Peng2, Tao Huang5 Pengwei Wang2, Zhongyuan Wang2, Shanghang Zhang2,6B, Chang Xu1B 1 2 BAAI, 3 Harbin 4 Hong Kong 5 Shanghai Jiao Tong University 6 Peking University Prompt Mot... |
2510.14948 | FAIR COMMISSIONING - TOWARDS FIRST SCIENCE
S. Reimann∗, H. Albers, R. W. Assmann1, P. Gasik, O. Geithner, F. Hagenbuck, A. Herlert,
P. Hofmann, V. Kamerdzhiev, M. Kauschke, H. Kollmus, S. Pietri, N. Pyka, T. Radon, C. Schroeder,
S. Schwarz, H. Simon, P. Spiller, K. Vogt
GSI Helmholtz Centre, Darmstadt, Germany
1also at... | FAIR COMMISSIONING - TOWARDS FIRST SCIENCE S. Reimann∗, H. Albers, R. W. Assmann1, P. Gasik, O. Geithner, F. Hagenbuck, A. Herlert, P. Hofmann, V. Kamerdzhiev, M. Kauschke, H. Kollmus, S. Pietri, N. Pyka, T. Radon, C. Schroeder, S. Schwarz, H. Simon, P. Spiller, K. Vogt GSI Helmholtz Centre, Darmstadt, Germany 1also at... |
2510.14949 | DIALECTGEN: BENCHMARKING AND IMPROVING DI-
ALECT ROBUSTNESS IN MULTIMODAL GENERATION
Yu Zhou
∗†, Sohyun An
∗, Haikang Deng
∗, Da Yin, Clark Peng, Cho-Jui Hsieh,
Kai-Wei Chang, Nanyun Peng
University of California, Los Angeles
{yuzhou, kwchang, violetpeng}@cs.ucla.edu
Stable Diffusion 3.5
Two red packets on a table
(Sta... | DIALECTGEN: BENCHMARKING AND IMPROVING DIALECT ROBUSTNESS IN MULTIMODAL GENERATION Yu Zhou ∗†, Sohyun An ∗, Haikang Deng ∗, Da Yin, Clark Peng, Cho-Jui Hsieh, Kai-Wei Chang, Nanyun Peng {yuzhou, kwchang, Stable Diffusion 3.5 Two red packets on a table (Standard American English) Two ang pows on a table (Singaporean Eng... |
2510.14945 | 3D Scene Prompting for Scene-Consistent Camera-Controllable Video Generation
3D SCENE PROMPTING FOR SCENE-CONSISTENT
CAMERA-CONTROLLABLE VIDEO GENERATION
JoungBin Lee1∗
Jaewoo Jung1∗
Jisang Han1∗
Takuya Narihira2
Kazumi Fukuda2
Junyoung Seo1
Sunghwan Hong3
Yuki Mitsufuji2,4†
Seungryong Kim1†
1KAIST AI
2Sony AI
3ETH Z¨u... | 3D Scene Prompting for Scene-Consistent Camera-Controllable Video Generation 3D SCENE PROMPTING FOR SCENE-CONSISTENT CAMERA-CONTROLLABLE VIDEO GENERATION JoungBin Lee1∗ Jaewoo Jung1∗ Jisang Han1∗ Takuya Narihira2 Kazumi Fukuda2 Junyoung Seo1 Sunghwan Hong3 Yuki Mitsufuji2,4† Seungryong Kim1† 1KAIST AI 2Sony AI 3ETH Z ̈... |
2510.14943 | 2025-10-17
LaSeR: Reinforcement Learning with Last-Token Self-Rewarding
Wenkai Yang1,∗, Weijie Liu2, Ruobing Xie2, Yiju Guo1,
Lulu Wu2, Saiyong Yang2, Yankai Lin1,†
1Gaoling School of Artificial Intelligence, Renmin University of China
2LLM Department, Tencent
B {wenkaiyang,yankailin}@ruc.edu.cn
Abstract
Reinforcement ... | 2025-10-17 LaSeR: Reinforcement Learning with Last-Token Self-Rewarding Wenkai Yang1,∗, Weijie Liu2, Ruobing Xie2, Yiju Guo1, Lulu Wu2, Saiyong Yang2, Yankai Lin1,† 1Gaoling 2LLM Department, Tencent B { Abstract Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a core paradigm for enhancing ... |
2510.14947 | Architecture Is All You Need: Diversity-Enabled Sweet Spots for
Robust Humanoid Locomotion
*Blake Werner, *Lizhi Yang, Aaron D. Ames
Abstract— Robust humanoid locomotion in unstructured en-
vironments requires architectures that balance fast low-level
stabilization with slower perceptual decision-making. We show
that a... | Architecture Is All You Need: Diversity-Enabled Sweet Spots for Robust Humanoid Locomotion *Blake Werner, *Lizhi Yang, Aaron D. Ames Abstract- Robust humanoid locomotion in unstructured environments requires architectures that balance fast low-level stabilization with slower perceptual decision-making. We show that a s... |
2510.14946 | EdgeNavMamba: Mamba-Optimized Object
Detection for Energy-Efficient Edge Devices
Romina Aalishah, Mozhgan Navardi, and Tinoosh Mohsenin
Department of Electrical and Computer Engineering
Johns Hopkins University, Baltimore, MD, USA
Abstract—Deployment of efficient and accurate Deep Learning
models has long been a challe... | EdgeNavMamba: Mamba-Optimized Object Detection for Energy-Efficient Edge Devices Romina Aalishah, Mozhgan Navardi, and Tinoosh Mohsenin -Deployment of efficient and accurate Deep Learning models has long been a challenge in autonomous navigation, particularly for real-time applications on resource-constrained edge devi... |
2510.14942 | GroundedPRM: Tree-Guided and Fidelity-Aware
Process Reward Modeling for Step-Level Reasoning
Yao Zhang 1,6 ∗†
Yu Wu 2 †
Haowei Zhang 3 ‡
Weiguo Li 4
Haokun Chen1
Jingpei Wu1
Guohao Li5
Zhen Han 1 ∗
Volker Tresp1,6
1 LMU Munich
2 Technical University of Munich
3 Fudan University
4 University Heidelberg
5 University of O... | GroundedPRM: Tree-Guided and Fidelity-Aware Process Reward Modeling for Step-Level Reasoning Yao Zhang 1,6 ∗† Yu Wu 2 † Haowei Zhang 3 ‡ Weiguo Li 4 Haokun Chen1 Jingpei Wu1 Guohao Li5 Zhen Han 1 ∗ Volker Tresp1,6 1 LMU Munich 2 Technical 3 Fudan University 4 University Heidelberg 5 6 Munich Center for Machine Learning... |
2510.14940 | Invisible neutron decay and light BSM particles
J.C. Helo∗and T. Ota†
Departamento de F´ısica, Facultad de Ciencias, Universidad
de La Serena, Avenida Cisternas 1200, La Serena, Chile,
Millennium Institute for Subatomic Physics at the High Energy
Frontier (SAPHIR), Fern´andez Concha 700, Santiago, Chile
M. Hirsch‡
Inst... | Invisible neutron decay and light BSM particles J.C. Helo∗and T. Ota† Departamento de F ́ısica, Facultad de Ciencias, Universidad de La Serena, Avenida Cisternas 1200, La Serena, Chile, Millennium Institute for Subatomic Physics at the High Energy Frontier (SAPHIR), Fern ́andez Concha 700, Santiago, Chile M. Hirsch‡ In... |
2510.14941 | Non-Minimally Coupled Quintessence
in Light of DESI
Samuel S´anchez L´opez,a,b Alexandros Karam,c Dhiraj Kumar
Hazraa,b,d
aThe Institute of Mathematical Sciences, HBNI, CIT Campus, Chennai 600113, India
bHomi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai
400094, India
cNational Institute o... | Non-Minimally Coupled Quintessence in Light of DESI Samuel S ́anchez L ́opez,a,b Alexandros Karam,c Dhiraj Kumar Hazraa,b,d aThe 600113, India bHomi Bhabha National Institute, Training School Complex, Anushakti Nagar, Mumbai 400094, India cNational ̈avala 10, 10143 Tallinn, Estonia dINAF/OAS Bologna, Osservatorio di As... |
2510.14939 | Decoding in the presence of ISI without
interleaving – ORBGRAND-AI
Ken R. Duffy
Dept. of ECE & Dept. Mathematics
Northeastern University
Boston, USA
k.duffy@northeastern.edu
Moritz Grundei
Electrical and Computer Engineering
Technical University of Munich
Munich, Germany
moritz.grundei@tum.de
Jane A. Millward
Research ... | Decoding in the presence of ISI without interleaving - ORBGRAND-AI Ken R. Duffy Dept. of ECE & Dept. Mathematics Northeastern University Boston, USA Moritz Grundei Electrical and Computer Engineering Technical . Millward Research Laboratory of Electronics Massachusetts ́edard Research Laboratory of Electronics Massachu... |
2510.14944 | MetaBench: A Multi-task Benchmark for Assessing LLMs in Metabolomics
Yuxing Lu1,2, Xukai Zhao3, J. Ben Tamo4, Micky C. Nnamdi4,
Rui Peng2, Shuang Zeng1,2, Xingyu Hu5, Jinzhuo Wang2,*, May D. Wang 1,4,*,
1Wallace H. Coulter Department of Biomedical Engineering, Georgia Institute of Technology and Emory University
2Colle... | MetaBench: A Multi-task Benchmark for Assessing LLMs in Metabolomics Yuxing Lu1,2, Xukai Zhao3, J. Ben Tamo4, Micky C. Nnamdi4, Rui Peng2, Shuang Zeng1,2, Xingyu Hu5, Jinzhuo Wang2,*, May D. Wang 1,4,*, 1Wallace H. Coulter 2 3 4 5 : , Abstract Large Language Models (LLMs) have demonstrated remarkable capabilities on ge... |
2510.14937 | AI-Powered Early Diagnosis of Mental Health Disorders from Real-World Clinical
Conversations
Jianfeng Zhu1, Julina Maharjan1, Xinyu Li1, Karin G. Coifman2, Ruoming Jin1
1 Department of Computer Science, Kent State University
2 Department of Psychological Sciences, Kent State University
{jzhu6, jmaharja, xli5, rjin}@ken... | AI-Powered Early Diagnosis of Mental Health Disorders from Real-World Clinical Conversations Jianfeng Zhu1, Julina Maharjan1, Xinyu Li1, Karin G. Coifman2, Ruoming Jin1 1 2 {jzhu6, jmaharja, xli5, , Abstract Mental health disorders remain among the leading cause of disability worldwide, yet conditions such as depressio... |
2510.14935 | On Complexity of Model-Based Derivative-Free Methods.
A. Chaudhry‡
K. Scheinberg‡
Abstract. In many applications of mathematical optimization, one may wish to optimize an objective
function without access to its derivatives. These situations call for derivative-free optimization (DFO) methods.
Among the most successful... | On Complexity of Model-Based Derivative-Free Methods. A. Chaudhry‡ K. Scheinberg‡ Abstract. In many applications of mathematical optimization, one may wish to optimize an objective function without access to its derivatives. These situations call for derivative-free optimization (DFO) methods. Among the most successful... |
2510.14934 | TASLA: Text-Aligned Speech Tokens with Multiple Layer-Aggregation
Ming-Hao Hsu1
Liang-Hsuan Tseng2
Hung-yi Lee2
Zhizheng Wu1
1The Chinese University of Hong Kong, Shenzhen
2National Taiwan University
hsuminghao1006@gamil.com, wuzhizheng@cuhk.edu.cn
f11921067@ntu.edu.tw, hungyilee@ntu.edu.tw
Abstract
We propose Text-Ali... | TASLA: Text-Aligned Speech Tokens with Multiple Layer-Aggregation Ming-Hao Hsu1 Liang-Hsuan Tseng2 Hung-yi Lee2 Zhizheng Wu1 1The Chinese 2National Taiwan University , , Abstract We propose Text-Aligned Speech Tokens with Multiple Layer-Aggregation (TASLA), which is a text-aligned speech tokenization framework that aim... |
2510.14936 | CIRCUIT INSIGHTS:
TOWARDS INTERPRETABILITY
BEYOND ACTIVATIONS
Elena Golimblevskaia1, Aakriti Jain1, Bruno Puri1,2, Ammar Ibrahim1,
Wojciech Samek1,2,3, Sebastian Lapuschkin1,4
1Department of Artificial Intelligence, Fraunhofer Heinrich Hertz Institute
2Department of Electrical Engineering and Computer Science, Technisc... | CIRCUIT INSIGHTS: TOWARDS INTERPRETABILITY BEYOND ACTIVATIONS Elena Golimblevskaia1, Aakriti Jain1, Bruno Puri1,2, Ammar Ibrahim1, Wojciech Samek1,2,3, Sebastian Lapuschkin1,4 1 2 ̈at Berlin 3BIFOLD - Berlin Institute for the Foundations of Learning and Data 4Centre of eXplainable Artificial Intelligence, Technological... |
2510.14938 | Astronomy & Astrophysics manuscript no. current
©ESO 2025
October 17, 2025
X-ray panorama of the SS 433/W50 complex by SRG/eROSITA
Rashid Sunyaev1, 2, Ildar Khabibullin3, 1, 2, Eugene Churazov1, 2, Marat Gilfanov2, 1, Pavel Medvedev2, and
Sergey Sazonov2
1 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. ... | Astronomy & Astrophysics manuscript no. current October 17, 2025 X-ray panorama of the SS 433/W50 complex by SRG/eROSITA Rashid Sunyaev1, 2, Ildar Khabibullin3, 1, 2, Eugene Churazov1, 2, Marat Gilfanov2, 1, Pavel Medvedev2, and Sergey Sazonov2 1 Max Planck Institute for Astrophysics, Karl-Schwarzschild-Str. 1, D-85741... |
2510.14933 | Nucleon Electric Dipole Moments in Paramagnetic Molecules through Effective Field
Theory
Wouter Dekens,1 Jordy de Vries,2, 3 Lemonia Gialidi,2, 3 Javier Menéndez,4, 5 Heleen Mulder,3, 6 and Beatriz Romeo7
1Institute for Nuclear Theory, University of Washington, Seattle WA 91195-1550, USA
2Institute for Theoretical Phys... | Nucleon Electric Dipole Moments in Paramagnetic Molecules through Effective Field Theory Wouter Dekens,1 Jordy de Vries,2, 3 Lemonia Gialidi,2, 3 Javier Menéndez,4, 5 Heleen Mulder,3, 6 and Beatriz Romeo7 1Institute for Nuclear Theory, 91195-1550, USA 2Institute for Theoretical Physics Amsterdam and Delta Institute for... |
2510.14931 | Bo Wang1
Department of Mechanical Engineering,
The City College of New York,
The City University of New York,
New York, NY 10031, USA
email: bwang1@ccny.cuny.edu
Tianyu Han
Department of Mechanical Engineering,
The City College of New York,
The City University of New York,
New York, NY 10031, USA
email: than000@citymai... | Bo Wang1 10031, USA email: Tianyu Han 10031, USA email: Guangwei Wang 550025, China email: Further Results on Safety-Critical Stabilization of Force-Controlled Nonholonomic Mobile Robots In this paper, we address the stabilization problem for force-controlled nonholonomic mobile robots under safety-critical constraints... |
2510.14930 | VT-Refine: Learning Bimanual Assembly with
Visuo-Tactile Feedback via Simulation Fine-Tuning
Binghao Huang1
Jie Xu2
Iretiayo Akinola2
Wei Yang2
Balakumar Sundaralingam2
Rowland O’Flaherty2
Dieter Fox2
Xiaolong Wang2,3
Arsalan Mousavian2
Yu-Wei Chao2†
Yunzhu Li1†
1Columbia University
2NVIDIA
3University of California, S... | VT-Refine: Learning Bimanual Assembly with Visuo-Tactile Feedback via Simulation Fine-Tuning Binghao Huang1 Jie Xu2 Iretiayo Akinola2 Wei Yang2 Balakumar Sundaralingam2 Rowland O'Flaherty2 Dieter Fox2 Xiaolong Wang2,3 Arsalan Mousavian2 Yu-Wei Chao2† Yunzhu Li1† 1Columbia University 2NVIDIA 3 †Equal advising Abstract: ... |
2510.14932 | Sarker et al.
PAPER
Crossmark
RECEIVED
XX XX XXXX
REVISED
XX XX XXXX
Resonance Engineering via Harnessing Anti-Parallel Dipole
Image Coupling
Dip Sarker
and Abdoulaye Ndao∗
Department of Electrical and Computer Engineering, University of California, San Diego, La Jolla, California 92093,
USA
E-mail: ∗a1ndao@ucsd.edu
Ke... | Sarker et al. PAPER Crossmark RECEIVED XX XX XXXX REVISED XX XX XXXX Resonance Engineering via Harnessing Anti-Parallel Dipole Image Coupling Dip Sarker and Abdoulaye Ndao∗ 92093, USA E-mail: ∗ Keywords: Plasmonic, Sub-Wavelength Confinement, Anti-Parallel Dipole Image, EM Coupling, Tunable device, Bi-directional Reson... |
2510.14928 | Instruction Set Migration at Warehouse Scale
ERIC CHRISTOPHER, KEVIN CROSSAN, WOLFF DOBSON, CHRIS KENNELLY, DREW
LEWIS, KUN LIN, MARTIN MAAS, PARTHASARATHY RANGANATHAN, EMMA RAPATI,
BRIAN YANG, Google, USA
Migrating codebases from one instruction set architecture (ISA) to another is a major engineering challenge.
A rec... | Instruction Set Migration at Warehouse Scale ERIC CHRISTOPHER, KEVIN CROSSAN, WOLFF DOBSON, CHRIS KENNELLY, DREW LEWIS, KUN LIN, MARTIN MAAS, PARTHASARATHY RANGANATHAN, EMMA RAPATI, BRIAN YANG, Google, USA Migrating codebases from one instruction set architecture (ISA) to another is a major engineering challenge. A rec... |
2510.14926 | Current fluctuations in nonequilibrium open quantum systems beyond weak coupling:
a reaction coordinate approach
Khalak Mahadeviya,1, ∗Saulo V. Moreira,1 Sheikh Parvez Mandal,2
Mahasweta Pandit,2 Javier Prior,2 and Mark T. Mitchison1, 3, †
1School of Physics, Trinity College Dublin, College Green, Dublin 2, D02 K8N4, I... | Current fluctuations in nonequilibrium open quantum systems beyond weak coupling: a reaction coordinate approach Khalak Mahadeviya,1, ∗Saulo V. Moreira,1 Sheikh Parvez Mandal,2 Mahasweta Pandit,2 Javier Prior,2 and Mark T. Mitchison1, 3, † 1 2, D02 K8N4, Ireland 2Departamento de F ́ısica - CIOyN, Universidad de Murcia,... |
2510.14929 | Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX7.0.1
StarStream: Automatic detection algorithm for stellar streams
Yingtian Chen
,1 Oleg Y. Gnedin
,1 Adrian M. Price-Whelan
,2 and Colin Holm-Hansen
1
1Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
2Center for Comp... | Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX7.0.1 StarStream: Automatic detection algorithm for stellar streams Yingtian Chen ,1 Oleg Y. Gnedin ,1 Adrian M. Price-Whelan ,2 and Colin Holm-Hansen 1 1 48109, USA 2Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010... |
2510.14925 | Stable but Miscalibrated: A Kantian View on Overconfidence
from Filters to Large Language Models
Akira Okutomi
ToppyMicroServices OÜ, Tallinn, Estonia
Abstract
We reinterpret Kant’s Critique of Pure Reason as a theory of feedback stability, viewing
reason as a regulator that keeps inference within the bounds of possibl... | Stable but Miscalibrated: A Kantian View on Overconfidence from Filters to Large Language Models Akira Okutomi ToppyMicroServices OÜ, Tallinn, Estonia Abstract We reinterpret Kant's Critique of Pure Reason as a theory of feedback stability, viewing reason as a regulator that keeps inference within the bounds of possibl... |
2510.14924 | Draft version October 17, 2025
Typeset using LATEX twocolumn style in AASTeX7.0.1
StarStream on Gaia: Stream discovery and mass loss rate of globular clusters
Yingtian Chen
,1 Oleg Y. Gnedin
,1 and Adrian M. Price-Whelan
2
1Department of Astronomy, University of Michigan, Ann Arbor, MI 48109, USA
2Center for Computatio... | Draft version October 17, 2025 Typeset using LATEX twocolumn style in AASTeX7.0.1 StarStream on Gaia: Stream discovery and mass loss rate of globular clusters Yingtian Chen ,1 Oleg Y. Gnedin ,1 and Adrian M. Price-Whelan 2 1 48109, USA 2Center for Computational Astrophysics, Flatiron Institute, New York, NY 10010, USA ... |
2510.14922 | TRI-DEP: A TRIMODAL COMPARATIVE STUDY FOR DEPRESSION DETECTION USING
SPEECH, TEXT, AND EEG
Annisaa Fitri Nurfidausi∗, Eleonora Mancini*, Paolo Torroni
DISI, University of Bologna, Italy
ABSTRACT
Depression is a widespread mental health disorder, yet its au-
tomatic detection remains challenging.
Prior work has explored... | TRI-DEP: A TRIMODAL COMPARATIVE STUDY FOR DEPRESSION DETECTION USING SPEECH, TEXT, AND EEG Annisaa Fitri Nurfidausi∗, Eleonora Mancini*, Paolo Torroni DISI, - tomatic detection remains challenging. Prior work has explored unimodal and multimodal approaches, with multimodal systems showing promise by leveraging compleme... |
2510.14923 | FINITE ELEMENT METHODS FOR ELECTRONEUTRAL
MULTICOMPONENT ELECTROLYTE FLOWS∗
AARON BAIER-REINIO†, PATRICK E. FARRELL‡, AND CHARLES W. MONROE§
Abstract. We present a broad family of high-order finite element algorithms for simulating the
flow of electroneutral electrolytes. The governing partial differential equations th... | FINITE ELEMENT METHODS FOR ELECTRONEUTRAL MULTICOMPONENT ELECTROLYTE FLOWS∗ AARON BAIER-REINIO†, PATRICK E. FARRELL‡, AND CHARLES W. MONROE§ Abstract. We present a broad family of high-order finite element algorithms for simulating the flow of electroneutral electrolytes. The governing partial differential equations th... |
2510.14921 | Sound Masking Strategies for Interference with Mosquito Hearing
Justin Faber∗
Department of Physics & Astronomy, University of California, Los Angeles, California, USA
Alexandros C Alampounti and Marcos Georgiades
University College London Ear Institute: London, UK
Joerg T Albert
University College London Ear Institute... | Sound Masking Strategies for Interference with Mosquito Hearing Justin Faber∗ : London, UK Joerg T Albert University College London Ear Institute: London, UK and Cluster of Excellence Hearing4all, Sensory Physiology & Behaviour Group, Department for Neuroscience, ̈at Oldenburg, Oldenburg, Germany Dolores Bozovic (Dated... |
2510.14920 | Rank of Matrices Arising out of Singular Kernel Functions∗
Sumit Singh
†4 and Sivaram Ambikasaran
‡1,2,3,4
1Wadhwani School of Data Science and Artificial Intelligence
2Robert Bosch Centre for Data Science and Artificial Intelligence
3Department of Data Science and Artificial Intelligence, IIT Madras, Chennai, India
4D... | Rank of Matrices Arising out of Singular Kernel Functions∗ Sumit Singh †4 and Sivaram Ambikasaran ‡1,2,3,4 1Wadhwani 2Robert Bosch Centre for Data Science and Artificial Intelligence 3 4 . In this work, we study the rank of matrices arising out of the kernel function K : X × Y 7→R, where the sets X, Y ∈Rd are hypercube... |
2510.14917 | Cumulants, Moments and Selection: The
Connection Between Evolution and
Statistics
Hasan Ahmed, Deena Goodgold, Khushali Kothari, Rustom Antia*
Department of Biology, Emory University, Atlanta, GA 30322, USA.
*Corresponding author. Email: rantia@emory.edu
Abstract
Cumulants and moments are closely related to the ... | Cumulants, Moments and Selection: The Connection Between Evolution and Statistics Hasan Ahmed, Deena Goodgold, Khushali Kothari, Rustom Antia* 30322, USA. *Corresponding author. Email: Abstract Cumulants and moments are closely related to the basic mathematics of continuous and discrete selection (respectively). These ... |
2510.14918 | Tree-Like Shortcuttings of Trees
Hung Le∗
Lazar Milenković†
Shay Solomon‡
Cuong Than§
Abstract
Sparse shortcuttings of trees—equivalently, sparse 1-spanners for tree metrics with bounded
hop-diameter—have been studied extensively (under different names and settings), since the
pioneering works of [Yao82, Cha87, AS87, B... | Tree-Like Shortcuttings of Trees Hung Le∗ Lazar Milenković† Shay Solomon‡ Cuong Than§ Abstract Sparse shortcuttings of trees-equivalently, sparse 1-spanners for tree metrics with bounded hop-diameter-have been studied extensively (under different names and settings), since the pioneering works of [Yao82, Cha87, AS87, B... |
2510.14919 | Predicting Task Performance with Context-aware Scaling Laws
Kyle Montgomery1*, David Park2*, Jianhong Tu1,
Michael Bendersky3, Beliz Gunel4, Dawn Song5, Chenguang Wang1†
1 UC Santa Cruz, 2 Washington University in St. Louis, 3Databricks,
4Google DeepMind, 5UC Berkeley
{kylemontgomery, chenguangwang}@ucsc.edu
Abstract
S... | Predicting Task Performance with Context-aware Scaling Laws Kyle Montgomery1*, David Park2*, Jianhong Tu1, Michael Bendersky3, Beliz Gunel4, Dawn Song5, Chenguang Wang1† 1 UC Santa Cruz, 2 Washington University in St. Louis, 3Databricks, 4Google DeepMind, 5UC Berkeley {kylemontgomery, Abstract Scaling laws have transfo... |
2510.14915 | Harmonizing Diverse Models: A Layer-wise Merging Strategy for
Consistent Generation
Xujun Peng, Anoop Kumar, Jingyu Wu, Parker Glenn, Daben Liu
AI Foundations, Capital One
McLean, VA, USA
{xujun.peng, anoop.kumar, jingyu.wu, parker.glenn, daben.liu}@capitalone.com
Abstract
Retrieval-Augmented Generation (RAG) sys-
tems... | Harmonizing Diverse Models: A Layer-wise Merging Strategy for Consistent Generation Xujun Peng, Anoop Kumar, Jingyu Wu, Parker Glenn, Daben Liu AI Foundations, Capital One McLean, VA, USA {xujun.peng, anoop.kumar, jingyu.wu, parker.glenn, Abstract Retrieval-Augmented Generation (RAG) systems leverage Large Language Mod... |
2510.14927 | Astronomy & Astrophysics manuscript no. aa55684-25corr
©ESO 2025
October 17, 2025
NIRPS and TESS reveal a peculiar system around the M dwarf
TOI-756: A transiting sub-Neptune and a cold eccentric giant
Léna Parc1, ∗, François Bouchy1, Neil J. Cook2, Nolan Grieves1, Étienne Artigau2, 3, Alexandrine L’Heureux2, René
Doyo... | Astronomy & Astrophysics manuscript no. aa55684-25corr October 17, 2025 NIRPS and TESS reveal a peculiar system around the M dwarf TOI-756: A transiting sub-Neptune and a cold eccentric giant Léna Parc1, ∗, François Bouchy1, Neil J. Cook2, Nolan Grieves1, Étienne Artigau2, 3, Alexandrine L'Heureux2, René Doyon2, 3, Yur... |
2510.14911 |
Dude, Where’s My (Autonomous) Car? Defining an Accessible Description Logic for
Blind and Low Vision Travelers Using Autonomous Vehicles
Paul D. S. Fink1,2; 0000-0003-2915-1331
Justin R. Brown1; 0000-0002-3359-8411
Rachel Coombs1; 0009-0004-8593-559X
Emily A. Hamby1; 0009-0005-0318-1446
Kyle J. Ja... | Dude, Where's My (Autonomous) Car? Defining an Accessible Description Logic for Blind and Low Vision Travelers Using Autonomous Vehicles Paul D. S. Fink1,2; 0000-0003-2915-1331 Justin R. Brown1; 0000-0002-3359-8411 Rachel Coombs1; 0009-0004-8593-559X Emily A. Hamby1; 0009-0005-0318-1446 Kyle J. James1; 0009-0003-3205-8... |
2510.14912 | 1
Decoherence-Aware Entangling and Swapping
Strategy Optimization for Entanglement Routing in
Quantum Networks
Shao-Min Huang∥, Cheng-Yang Cheng∥, Ming-Huang Chien∥, Jian-Jhih Kuo, and Chih-Yu Wang
Abstract—Quantum teleportation enables high-security com-
munications through end-to-end quantum entangled pairs. End-
to-... | 1 Decoherence-Aware Entangling and Swapping Strategy Optimization for Entanglement Routing in Quantum Networks Shao-Min Huang∥, Cheng-Yang Cheng∥, Ming-Huang Chien∥, Jian-Jhih Kuo, and Chih-Yu Wang Abstract-Quantum teleportation enables high-security communications through end-to-end quantum entangled pairs. Endto-end ... |
2510.14914 |
1
Design of Paper Robot Building Kits
Ruhan Yang
ATLAS Institute, University of Colorado Boulder
ruhan.yang@colorado.edu
Ellen Yi-Luen Do
ATLAS Institute, University of Colorado Boulder
ellen.do@colorado.edu
Figure 1: Design space of paper robots
Building robots is an engaging activity that provides op... | 1 Design of Paper Robot Building Kits Ruhan Yang ATLAS Institute, -Luen Do ATLAS Institute, 1: Design space of paper robots Building robots is an engaging activity that provides opportunities for hands-on learning. However, traditional robot-building kits are usually costly with limited functionality due to material an... |
2510.14916 | EFFICIENT AND ROBUST CARATH´EODORY-STEINITZ
PRUNING OF POSITIVE DISCRETE MEASURES
FILIP BˇEL´IK, JESSE CHAN, AND AKIL NARAYAN
Abstract. In many applications, one seeks to approximate integration against
a positive measure of interest by a positive discrete measure: a numerical quad-
rature rule with positive weights. O... | EFFICIENT AND ROBUST CARATH ́EODORY-STEINITZ PRUNING OF POSITIVE DISCRETE MEASURES FILIP BˇEL ́IK, JESSE CHAN, AND AKIL NARAYAN Abstract. In many applications, one seeks to approximate integration against a positive measure of interest by a positive discrete measure: a numerical quadrature rule with positive weights. O... |
2510.14909 | *Corresponding Author’s Email:*annieliy@mit.edu
Proceedings of the Global Public Health Conference, Vol. 8, Issue 1, 2025, pp.17-29
Copyright © 2025 Li Y
ISSN 2613-8417 online
DOI: https://doi.org/10.17501/26138417.2025.8102
THE IMPACT OF MEDICAID COVERAGE ON MENTAL
HEALTH, WHY INSURANCE MAKES PEOPLE HAPPIER
IN O... | *Corresponding Author's Proceedings of the Global Public Health Conference, Vol. 8, Issue 1, 2025, pp.17-29 Li Y ISSN 2613-8417 online THE IMPACT OF MEDICAID COVERAGE ON MENTAL HEALTH, WHY INSURANCE MAKES PEOPLE HAPPIER IN OHIE: BY SPENDING LESS OR BY SPENDING MORE? Yangyang Li* : The Oregon Health Insurance Experiment... |
2510.14910 | VORTEX LINES INTERACTION IN THE THREE-DIMENSIONAL
MAGNETIC GINZBURG–LANDAU MODEL
CARLOS ROM´AN, ETIENNE SANDIER, AND SYLVIA SERFATY
Abstract. We complete our study of the three dimensional Ginzburg–Landau functional with
magnetic field, in the asymptotic regime of a small inverse Ginzburg–Landau parameter ε, and
near t... | VORTEX LINES INTERACTION IN THE THREE-DIMENSIONAL MAGNETIC GINZBURG-LANDAU MODEL CARLOS ROM ́AN, ETIENNE SANDIER, AND SYLVIA SERFATY Abstract. We complete our study of the three dimensional Ginzburg-Landau functional with magnetic field, in the asymptotic regime of a small inverse Ginzburg-Landau parameter ε, and near ... |
2510.14913 | Preprint
BUDGET-AWARE TEST-TIME SCALING VIA DISCRIMI-
NATIVE VERIFICATION
Kyle Montgomery1∗, Sijun Tan2∗, Yuqi Chen1, Siyuan Zhuang2, Tianjun Zhang2,
Raluca Ada Popa2, Chenguang Wang1†
1UC Santa Cruz, 2UC Berkeley
{kylemontgomery, chenguangwang}@ucsc.edu, sijuntan@berkeley.edu
ABSTRACT
Test-time scaling is a powerful s... | Preprint BUDGET-AWARE TEST-TIME SCALING VIA DISCRIMINATIVE VERIFICATION Kyle Montgomery1∗, Sijun Tan2∗, Yuqi Chen1, Siyuan Zhuang2, Tianjun Zhang2, Raluca Ada Popa2, Chenguang Wang1† 1UC Santa Cruz, 2UC Berkeley {kylemontgomery, , ABSTRACT Test-time scaling is a powerful strategy for boosting the performance of large l... |
2510.14908 | Fermi Bubbles Without AGN: γ-Ray Bubbles in MHD Galaxy Formation Simulations
with Full Cosmic Ray Spectra
Isabel S. Sands, Philip F. Hopkins, and Sam B. Ponnada
TAPIR, California Institute of Technology, Mailcode 350-17, Pasadena, CA 91125, USA
(Dated: October 17, 2025)
For the first time, we show in MHD simulations wi... | Fermi Bubbles Without AGN: γ-Ray Bubbles in MHD Galaxy Formation Simulations with Full Cosmic Ray Spectra Isabel S. Sands, Philip F. Hopkins, and Sam B. Ponnada TAPIR, California 350-17, Pasadena, CA 91125, USA (Dated: October 17, 2025) For the first time, we show in MHD simulations with cosmological initial conditions... |
2510.14907 | Learnable Mixed Nash Equilibria are Collectively Rational
Geelon So
Yi-An Ma
University of California, San Diego
Abstract
We extend the study of learning in games to dynamics that exhibit non-asymptotic stability. We do so
through the notion of uniform stability, which is concerned with equilibria of individually utili... | Learnable Mixed Nash Equilibria are Collectively Rational Geelon So Yi-An Ma -asymptotic stability. We do so through the notion of uniform stability, which is concerned with equilibria of individually utility-seeking dynamics. Perhaps surprisingly, it turns out to be closely connected to economic properties of collecti... |
2510.14906 | A Hard-Label Black-Box Evasion Attack against
ML-based Malicious Traffic Detection Systems
Zixuan Liu∗, Yi Zhao†B, Zhuotao Liu∗‡, Qi Li∗‡, Chuanpu Fu∗, Guangmeng Zhou∗, Ke Xu∗‡B
∗Tsinghua University, †Beijing Institute of Technology, ‡Zhongguancun Lab
Abstract—Machine Learning (ML)-based malicious traffic de-
tection i... | A Hard-Label Black-Box Evasion Attack against ML-based Malicious Traffic Detection Systems Zixuan Liu∗, Yi Zhao†B, Zhuotao Liu∗‡, Qi Li∗‡, Chuanpu Fu∗, Guangmeng Zhou∗, Ke Xu∗‡B ∗Tsinghua University, †Beijing ‡Zhongguancun Lab Abstract-Machine Learning (ML)-based malicious traffic detection is a promising security para... |
2510.14903 | Skyrmion behavior in attractive-repulsive square array of pinning
centers
L. Basseto1, N. P. Vizarim2, J. C. Bellizotti Souza2 and P. A. Venegas1
1 Department of Physics, Schools of Sciences, S˜ao Paulo State University - UNESP, 17033-360 Bauru, SP,
Brazil
2 ”Gleb Wataghin” Institute of Physics, Department of Condensed... | Skyrmion behavior in attractive-repulsive square array of pinning centers L. Basseto1, N. P. Vizarim2, J. C. Bellizotti Souza2 and P. A. Venegas1 1 ̃ao Paulo State University - UNESP, 17033-360 Bauru, SP, Brazil 2 "Gleb Wataghin" - UNICAMP, 13083-859, Campinas, SP, Brazil E-mail: Abstract. We investigate the driven dyn... |
2510.14905 | Continuous-time quantum walk on a random graph
using quantum circuits
Sabyasachi Chakraborty ,1, ∗Rohit Sarma Sarkar ,2, †
Sonjoy Majumder ,1, ‡ and Rohit Kishan Ray
3, 4, §
1Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur, West Bengal 721302, India
2International Centre for Theoretical Scien... | Continuous-time quantum walk on a random graph using quantum circuits Sabyasachi Chakraborty ,1, ∗Rohit Sarma Sarkar ,2, † Sonjoy Majumder ,1, ‡ and Rohit Kishan Ray 3, 4, § 1 721302, India 2International Centre for Theoretical Sciences (ICTS-TIFR), Bengaluru 560089, India 3 24061, USA 4Center for Theoretical Physics o... |
2510.14902 | VLA2: Empowering Vision-Language-Action Models with an Agentic
Framework for Unseen Concept Manipulation
Han Zhao‡,1,2, Jiaxuan Zhang‡,2,3, Wenxuan Song4, Pengxiang Ding1,2, Donglin Wang*,2,
1Zhejiang University, China 2MiLAB, Westlake University, China
3Southern University of Science and Technology, China
4Hong Kong U... | VLA2: Empowering Vision-Language-Action Models with an Agentic Framework for Unseen Concept Manipulation Han Zhao‡,1,2, Jiaxuan Zhang‡,2,3, Wenxuan Song4, Pengxiang Ding1,2, Donglin Wang*,2, 1Zhejiang University, China 2MiLAB, Westlake University, China 3Southern 4Hong Kong (Guangzhou), China Abstract- Current vision-l... |
2510.14904 | MASKCAPTIONER: LEARNING TO JOINTLY SEGMENT
AND CAPTION OBJECT TRAJECTORIES IN VIDEOS
Gabriel Fiastre1∗
Antoine Yang2
Cordelia Schmid1
1Inria, École Normale Supérieure, CNRS, PSL Research University
2Google Deepmind
ABSTRACT
Dense Video Object Captioning (DVOC) is the task of jointly detecting, track-
ing, and captionin... | MASKCAPTIONER: LEARNING TO JOINTLY SEGMENT AND CAPTION OBJECT TRAJECTORIES IN VIDEOS Gabriel Fiastre1∗ Antoine Yang2 Cordelia Schmid1 1Inria, École Normale Supérieure, CNRS, PSL Research University 2Google Deepmind ABSTRACT Dense Video Object Captioning (DVOC) is the task of jointly detecting, tracking, and captioning ... |
2510.14900 | Mapping Smarter, Not Harder: A Test-Time Reinforcement Learning
Agent That Improves Without Labels or Model Updates
Wen-Kwang Tsao*, Yao-Ching Yu*, Chien-Ming Huang
AI Lab, TrendMicro
{spark_tsao,yaoching_yu}@trendmicro.com
Abstract
The Enterprise Intelligence Platform must in-
tegrate logs from numerous third-party ve... | Mapping Smarter, Not Harder: A Test-Time Reinforcement Learning Agent That Improves Without Labels or Model Updates Wen-Kwang Tsao*, Yao-Ching Yu*, Chien-Ming Huang AI Lab, TrendMicro { Abstract The Enterprise Intelligence Platform must integrate logs from numerous third-party vendors in order to perform various downst... |
2510.14897 | Forecasting Quantum Observables:
A Compressed Sensing Approach with Performance Guarantees
Víctor Valls,1 Albert Akhriev,1 Olatz Sanz Larrarte,2 Javier Oliva del Moral,2, 3
Štěpán Šmíd,4, 5 Josu Etxezarreta Martinez,2 Sergiy Zhuk,1 and Dmytro Mishagli1, 5
1IBM Quantum, IBM Research Europe – Dublin
2Department of Basic ... | Forecasting Quantum Observables: A Compressed Sensing Approach with Performance Guarantees Víctor Valls,1 Albert Akhriev,1 Olatz Sanz Larrarte,2 Javier Oliva del Moral,2, 3 Štěpán Šmíd,4, 5 Josu Etxezarreta Martinez,2 Sergiy Zhuk,1 and Dmytro Mishagli1, 5 1IBM Quantum, IBM Research Europe - Dublin 2 - 20018 San Sebasti... |
2510.14898 | CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN
GENERAL ACTION SPACES
DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH
Abstract. We prove the stability and global convergence of a coupled actor-critic gradient flow for
infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous sta... | CONVERGENCE OF ACTOR-CRITIC FOR ENTROPY REGULARISED MDPS IN GENERAL ACTION SPACES DENIS ZORBA, DAVID ˇSIˇSKA, AND LUKASZ SZPRUCH Abstract. We prove the stability and global convergence of a coupled actor-critic gradient flow for infinite-horizon and entropy-regularised Markov decision processes (MDPs) in continuous sta... |
2510.14901 | Reasoning with Sampling:
Your Base Model is Smarter Than You Think
Aayush Karan1, Yilun Du1
1Harvard University
Website
Code
Abstract
Frontier reasoning models have exhibited incredible capabilities across a wide
array of disciplines, driven by posttraining large language models (LLMs) with
reinforcement learning (... | Reasoning with Sampling: Your Base Model is Smarter Than You Think Aayush Karan1, Yilun Du1 1Harvard University Website Code Abstract Frontier reasoning models have exhibited incredible capabilities across a wide array of disciplines, driven by posttraining large language models (LLMs) with reinforcement learning (... |
2510.14896 | Leveraging Multimodal LLM Descriptions of Activity for Explainable
Semi-Supervised Video Anomaly Detection
Furkan Mumcu1,†
Michael J. Jones2,∗
Anoop Cherian3,∗
Yasin Yilmaz4,†
†University of South Florida
∗Mitsubishi Electric Research Laboratories (MERL)
1furkan@usf.edu
2mjones@merl.com
3cherian@merl.com
4yasiny@usf.ed... | Leveraging Multimodal LLM Descriptions of Activity for Explainable Semi-Supervised Video Anomaly Detection Furkan Mumcu1,† Michael J. Jones2,∗ Anoop Cherian3,∗ Yasin Yilmaz4,† † ∗Mitsubishi Electric Research Laboratories (MERL) Abstract Existing semi-supervised video anomaly detection (VAD) methods often struggle with ... |
2510.14892 | A Comprehensive Framework for Efficient Court Case
Management and Prioritization
Shubham Varma
shubham.varma@somaiya.edu
Ananya Warior
ananya.warior@somaiya.edu
Dipti Pawade
diptipawade@somaiya.edu
Avani Sakhapara
avanisakhapara@somaiya.edu
Abstract
The Indian judicial system faces a critical challenge with approximate... | A Comprehensive Framework for Efficient Court Case Management and Prioritization Shubham Varma Ananya Warior Dipti Pawade Avani Sakhapara Abstract The Indian judicial system faces a critical challenge with approximately 52 [1] million pending cases, causing significant delays that impact socio-economic stability. This ... |
2510.14894 | Secure Sparse Matrix Multiplications and their Applications
to Privacy-Preserving Machine Learning
Marc Damie
m.f.d.damie@utwente.nl
University of Twente, The Netherlands
Inria, France
Florian Hahn
University of Twente, The Netherlands
Andreas Peter
University of Oldenburg, Germany
Jan Ramon
Inria, France
Abstract
To p... | Secure Sparse Matrix Multiplications and their Applications to Privacy-Preserving Machine Learning Marc Damie -party computation (MPC) enables executing Machine Learning (ML) algorithms on secret-shared or encrypted data. However, existing MPC frameworks are not optimized for sparse data. This makes them unsuitable for... |
2510.14895 | Fast
and
fault-tolerant
logical
measurements:
Auxiliary hypergraphs and transversal surgery
Alexander Cowtan1, Zhiyang He2, Dominic J. Williamson3,4, and Theodore J. Yoder5
1Department of Computer Science, University of Oxford, Oxford OX1 3QD, UK
2Department of Mathematics, Massachusetts Institute of Technology, Cambri... | Fast and fault-tolerant logical measurements: Auxiliary hypergraphs and transversal surgery Alexander Cowtan1, Zhiyang He2, Dominic J. Williamson3,4, and Theodore J. Yoder5 1 1 3QD, UK 2 02139, USA 3 2006, Australia 4IBM Quantum, IBM Almaden Research Center, San Jose, CA 95120, USA 5IBM Quantum, IBM T.J. Watson Researc... |
2510.14891 | A Performance Portable Matrix Free Dense
MTTKRP in GenTen*
Gabriel Kosmacher1, Eric T. Phipps2, Sivasankaran Rajamanickam2
Abstract—We extend the GenTen tensor decomposi-
tion package by introducing an accelerated dense ma-
tricized tensor times Khatri-Rao product (MTTKRP),
the workhorse kernel for canonical polyadic (... | A Performance Portable Matrix Free Dense MTTKRP in GenTen* Gabriel Kosmacher1, Eric T. Phipps2, Sivasankaran Rajamanickam2 Abstract-We extend the GenTen tensor decomposition package by introducing an accelerated dense matricized tensor times Khatri-Rao product (MTTKRP), the workhorse kernel for canonical polyadic (CP) ... |
2510.14899 | Electric field controlled second-order anomalous Hall effect in altermagnets
Arnob Mukherjee, Biplab Sanyal, Annica M. Black-Schaffer, and Ankita Bhattacharya∗
Department of Physics and Astronomy, Uppsala University, Box-516, S-75120 Uppsala, Sweden
(Dated: October 17, 2025)
Altermagnets are a recently discovered class... | Electric field controlled second-order anomalous Hall effect in altermagnets Arnob Mukherjee, Biplab Sanyal, Annica M. Black-Schaffer, and Ankita Bhattacharya∗ -516, S-75120 Uppsala, Sweden (Dated: October 17, 2025) Altermagnets are a recently discovered class of compensated magnets with momentum-dependent spin splitti... |
2510.14889 | Detecting Early and Implicit Suicidal Ideation via Longitudinal and
Information Environment Signals on Social Media
Soorya Ram Shimgekar1, Ruining Zhao1, Agam Goyal1, Violeta J. Rodriguez1,
Paul A. Bloom2, Hari Sundaram1, Koustuv Saha1
1University of Illinois Urbana-Champaign, {sooryas2, ruining, agamg2, vjrodrig, hs1,... | Detecting Early and Implicit Suicidal Ideation via Longitudinal and Information Environment Signals on Social Media Soorya Ram Shimgekar1, Ruining Zhao1, Agam Goyal1, Violeta J. Rodriguez1, Paul A. Bloom2, Hari Sundaram1, Koustuv Saha1 1 -Champaign, {sooryas2, ruining, agamg2, vjrodrig, hs1, 2Columbia University, New Y... |
2510.14886 | Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems
Jer´onimo Duarte,1 Ignacio Garc´ıa-Mata,1 and Diego A. Wisniacki2
1Instituto de Investigaciones F´ısicas de Mar del Plata (IFIMAR),
Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata
& CONICET, Funes 3350 (7600) Ma... | Ruelle-Pollicott Decay of Out-of-Time-Order Correlators in Many-Body Systems Jer ́onimo Duarte,1 Ignacio Garc ́ıa-Mata,1 and Diego A. Wisniacki2 1Instituto de Investigaciones F ́ısicas de Mar del Plata (IFIMAR), Facultad de Ciencias Exactas y Naturales, Universidad Nacional de Mar del Plata & CONICET, Funes 3350 (7600)... |
2510.14893 | STITCHER: Constrained Trajectory Planning in Complex
Environments with Real-Time Motion Primitive Search
Helene J. Levy and Brett T. Lopez
Abstract—Autonomous high-speed navigation through large,
complex environments requires real-time generation of agile
trajectories that are dynamically feasible, collision-free, and
... | STITCHER: Constrained Trajectory Planning in Complex Environments with Real-Time Motion Primitive Search Helene J. Levy and Brett T. Lopez Abstract-Autonomous high-speed navigation through large, complex environments requires real-time generation of agile trajectories that are dynamically feasible, collision-free, and ... |
2510.14890 | EM Approaches to Nonparametric
Estimation for Mixture of Linear Regressions
Andrew Welbaum
Department of Statistics, George Mason University
and
Wanli Qiao
Department of Statistics, George Mason University
October 17, 2025
Abstract
In a mixture of linear regression model, the regression coefficients are treated as
rand... | EM Approaches to Nonparametric Estimation for Mixture of Linear Regressions Andrew Welbaum 17, 2025 Abstract In a mixture of linear regression model, the regression coefficients are treated as random vectors that may follow either a continuous or discrete distribution. We propose two Expectation-Maximization (EM) algor... |
2510.14887 | Prediction-Specific Design of Learning-Augmented Algorithms
Sizhe Li*1, Nicolas Christianson†2, and Tongxin Li‡1
1School of Data Science, The Chinese University of Hong Kong, Shenzhen
2Department of Management Science and Engineering, Stanford University
Abstract
Algorithms with predictions has emerged as a powerful fr... | Prediction-Specific Design of Learning-Augmented Algorithms Sizhe Li*1, Nicolas Christianson†2, and Tongxin Li‡1 1 2 -driven performance benefits of machine-learned (ML) predictions. However, most existing approaches in this paradigm are overly conservative, as they do not leverage problem structure to optimize perform... |
2510.14884 | Learning When Not to Learn:
Risk-Sensitive Abstention in Bandits with Unbounded Rewards
Sarah Liaw∗
Benjamin Plaut∗
Harvard University
University of California, Berkeley
Abstract
In high-stakes AI applications, even a single
action can cause irreparable damage. How-
ever, nearly all of sequential decision-making
theory... | Learning When Not to Learn: Risk-Sensitive Abstention in Bandits with Unbounded Rewards Sarah Liaw∗ Benjamin Plaut∗ Harvard University -stakes AI applications, even a single action can cause irreparable damage. However, nearly all of sequential decision-making theory assumes that all errors are recoverable (e.g., by bo... |
2510.14888 | Modeling nonlinear scales for dynamical dark energy cosmologies with COLA
João Rebouças,1, 2 Victoria Lloyd,3 Jonathan Gordon,3, 4 Guilherme Brando,1 and Vivian Miranda4
1CBPF - Brazilian Center for Research in Physics, Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil
2Instituto de Física Teórica da Uni... | Modeling nonlinear scales for dynamical dark energy cosmologies with COLA João Rebouças,1, 2 Victoria Lloyd,3 Jonathan Gordon,3, 4 Guilherme Brando,1 and Vivian Miranda4 1CBPF - Brazilian Center for Research in Physics, Xavier Sigaud st. 150, zip 22290-180, Rio de Janeiro, RJ, Brazil 2Instituto de Física Teórica da Uni... |
2510.14885 | You May Speak Freely: Improving the Fine-Grained Visual Recognition
Capabilities of Multimodal Large Language Models with Answer Extraction
Logan Lawrence1 Oindrila Saha1 Megan Wei2 Chen Sun2 Subhransu Maji1 Grant Van Horn1
University of Massachusetts, Amherst1
Brown University2
{lclawrence, osaha, smaji, gvanhorn}@uma... | You May Speak Freely: Improving the Fine-Grained Visual Recognition Capabilities of Multimodal Large Language Models with Answer Extraction Logan Lawrence1 Oindrila Saha1 Megan Wei2 Chen Sun2 Subhransu Maji1 Grant Van Horn1 1 Brown University2 {lclawrence, osaha, smaji, {meganwei, chen Abstract Despite the renewed inte... |
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