Reinforcement Learning with Verifiable Rewards

Also known as RLVR

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59 papers in the last four weeks, up 119% on the four weeks before. 0.6% of all new papers.

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Oct 8, 2026cs.CL

Residual Advantage: Student-Relative Teacher Guidance for RL with Verifiable Rewards

Reinforcement learning with verifiable rewards (RLVR) and on-policy distillation (OPD) have become two main paradigms for post-training reasoning models. RLVR gives each response a single outcome label, leaving the steps inside it without separate credit. OPD provides token-level guidance at student-visited prefixes, but its pointwise signal does not directly reflect the pattern of teacher--student disagreement across the vocabulary. Dense, unbounded log-ratio supervision can amplify the teacher's influence, yet a strong solver is not necessarily a suitable guide when the student's solution paths depart from the teacher's. We propose Residual Advantage (\RA{}), which treats the teacher--student probability residual as a bounded one-step reward, subtracts the corresponding state value under the student policy to form a standard advantage, and centers the result within each response before adding it to the verifier advantage. The guidance term has zero mean within each response, so the verifier advantage remains the response's mean label and the teacher only redistributes credit among the steps within it. \CoRA{} further updates a teacher LoRA with verifier advantages on the same scored student batch and uses the updated teacher in the next iteration's residual, adapting guidance to the student's attempts. With Qwen3-1.7B-Base and Qwen3-4B-Base students and a Qwen3-8B teacher, \RA{} combined with GRPO or REINFORCE++ improves the underlying sequence-advantage algorithm in all 24 comparisons on three mathematical benchmarks, raising macro Avg@8 by 1.7--3.6 points and Pass@8 by 3.9--6.3 points. Both combinations surpass teacher-only OPD, and \CoRA{} adds a further 1.0--1.5 Avg@8 points.

Xiaobing Chen, Zhiqi Pang

Harbin Engineering University · Tencent · Harbin Institute of Technology

RL for Language Model Reasoning RL Post-Training Credit Assignment in RL Reinforcement Learning with Verifiable Rewards On-Policy Distillation

Oct 8, 2026cs.LG

Measuring and Mitigating Solution Mode Collapse in RLVR

A language model (LM) can usually answer the same question in more than one way, but reinforcement learning with verifiable rewards (RLVR) is indifferent to which correct answer a model produces. A solution will earn the same reward whether it is the thousandth copy of a familiar answer or one the model has never produced before. Yet, there is potential value in having the model retain multiple correct solutions as it is trained. For instance, multiple modes may give users a choice and provide problem-solving strategies that improve overall model performance. Here, we introduce ModeBench, a benchmark of multi-solution tasks in which the verifier returns both correctness and mode discovered. We then use ModeBench to measure how solution diversity changes under RLVR post-training. We find that RLVR post-training concentrates probability onto fewer correct modes even as accuracy holds or improves, and moreover, that frontier models are already highly concentrated. We then introduce our solution, Re:Max, which stores one verified example per discovered mode in a replay buffer and trains on those stored modes uniformly. A solution found once is, therefore, practiced as often as one found repeatedly. Across three model scales, two RL objectives, and harder task constructions, replay improves both how often a policy succeeds and how many different ways it can succeed.

Liv G. d'Aliberti, Marwa Abdulhai, Sofiia Druchyna +2

Princeton University

Mode Collapse RL Post-Training Reinforcement Learning with Verifiable Rewards Experience Replay

Oct 7, 2026cs.CV

VICO: Visual Environments Co-Evolving for Vision-Language Model Reasoning

Reinforcement learning with verifiable rewards (RLVR) has become a standard recipe for post-training vision-language models (VLMs), but it typically assumes a static training environment. As the actor improves, fixed tasks drift out of its learning frontier: many become trivial, others remain unsolvable; and the learning signal collapses. We argue that VLM post-training should evolve the visual environment alongside the actor, not just the actor itself. We propose VICO, a co-evolutionary framework in which an actor and an Environment-as-Rewriter (EnvRewriter) are trained jointly: the EnvRewriter edits verifiable image-side structures, such as scene graphs, chart tables, or protected region masks, and re-renders them to produce label-valid training samples whose difficulty is calibrated to the actor's current ability through a pass-rate-based reward. This loop continuously realigns task difficulty with actor capability without any additional human annotation. Across nine multimodal benchmarks spanning mathematical reasoning and visually grounded understanding, VICO-8B improves over its base model by up to +5.0% on out-of-domain tasks, surpasses the strongest self-evolution and text-editing co-evolution baselines by +4.3% and +8.4% respectively, and stays comparable to chart-specialized RLVR methods using 16-160 times fewer labeled samples. By shifting from human-labeled supervision to image-editing co-evolution, VICO offers a scalable path beyond static-corpus RLVR for visual reasoning.

Meng Lu, Ligeng Zhu, Olivia Xiao +8

Virginia Tech · Cisco · NVIDIA +6

VLM Reasoning RL for VLMs Reinforcement Learning with Verifiable Rewards

Oct 7, 2026cs.LG

Decoupling Exploration from Optimization in RLVR

Modern language models undergo reinforcement learning with verifiable rewards (RLVR) on top of already-trained checkpoints. A key promise of RLVR is the discovery of new reasoning strategies. In principle, a model can sample novel ideas absent from its prior training data. In practice, however, augmenting RLVR with strong novelty incentives has seen limited success and can degrade model quality. Because verifiable rewards supervise only a narrow slice of the model's knowledge and behavior, such degradations are difficult to recover from. Instead, we decouple exploration from optimization in a framework we call Exploration-Distillation (ExpDis). We train one or more explorer policies with a novelty bonus in the reward, filter their trajectories for correctness and quality, and distill them into a separate student policy. The student policy is then trained without a novelty bonus. We repeat the above procedure for several rounds, alternating between exploration and optimization. This decoupling allows us to aggressively scale exploration without degrading the student policy. Across seven mathematical reasoning benchmarks and two model families, ExpDis outperforms DAPO at the same wall-clock budget. Moreover, we observe improved pass@

k

scaling, indicating that ExpDis produces models that generate more diverse correct solutions.

Saif Punjwani, Micah Goldblum

Department of Computer Science, Columbia University · Department of Electrical Engineering, Columbia University

RL for Language Model Reasoning RL Exploration Reinforcement Learning with Verifiable Rewards

Oct 7, 2026cs.CL

Beyond Outcome Rewards: Constructing and Assigning Retrieval Credit for Search Agents

Search agents enable Large Language Models (LLMs) to iteratively retrieve and use information for complex multi-hop questions. Reinforcement Learning with Verifiable Rewards (RLVR) offers a promising approach for post-training such agents, but its reliance on sparse, outcome-based supervision can make credit assignment difficult and limit learning efficiency. In this paper, we systematically investigate how intermediate supervision can improve reinforcement learning for search agents. We study a range of reward-shaping and credit-assignment strategies that provide learning signals from intermediate retrieval steps. Building on these insights, we develop a training framework that combines intermediate signals with final outcome rewards to improve learning from multi-step search trajectories. Experiments across multiple benchmarks under matched training conditions demonstrate improvements in aggregate search-agent performance and show that both the choice of intermediate signal and where its credit is assigned affect training behaviour. These findings show that reward design and credit assignment are important design dimensions for training effective search agents.

Wenyu Huang, Xinyu Hou, Pavlos Vougiouklis +2

University of Edinburgh, UK · Huawei Technologies Research & Development (UK) Limited

Reward Design for RL RL for IR Step-Level Credit Assignment in RL Reinforcement Learning with Verifiable Rewards

Oct 7, 2026cs.AI

RewardWeaver: Long-Horizon Interactive Learning for Language Agents via Self-Evolving Reward Adaptation

Reinforcement learning with verifiable rewards (RLVR) has driven substantial progress in domains where task outcomes can be reliably evaluated, but long-horizon interaction remains challenging due to sparse terminal feedback and difficult credit assignment. Process rewards provide denser supervision, yet the capabilities most relevant for training can change as the policy evolves: a behavior that is easy to evaluate or frequently deficient need not be the bottleneck currently limiting task success. We introduce RewardWeaver, a self-evolving reward adaptation framework for language agents in long-horizon interaction. RewardWeaver maintains a validated capability space in which the semantics of admitted Rubrics remain fixed, and closes the loop between policy optimization, task evaluation, failure attribution, and reward adaptation. After each training stage, it performs outcome-grounded backward attribution on low-outcome trajectories, aggregates recurrent and policy-controlled capability bottlenecks, and dynamically selects the corresponding process rewards for the next stage. Recurrent failures not covered by the existing capability space trigger a separate, controlled expansion procedure. We evaluate REWARDWEAVER on SOTOPIA, Amazon?HistoryPrice, and a newly constructed Sales Benchmark. Across social interaction, bilateral bargaining, and domain-specific sales, REWARDWEAVER establishes new state-of-the-art (SOTA) results. Ablations further demonstrate the importance of dynamic reward allocation, failure-grounded attribution, and stable semantics for admitted capabilities.

Hengbo Xiao, Boyao Zhang, Purui Liu +4

RL for LLM Agents Rubric-Based RL Reward Shaping Process Reward Models Reinforcement Learning with Verifiable Rewards Credit Assignment in RL

Oct 6, 2026cs.AI

VeriFine: Scaling Verification for Self-Improvement in Embodied Reasoning

Self-improving policies continually expose new failure patterns, changing what their judges must be able to verify. However, current fixed judges constrain both optimization feedback and the discovery of useful training examples, limiting further self-improvement. This challenge is even more acute in embodied reasoning, where reliable evaluation must account for spatial grounding, causal reasoning, and safety-aware decision-making. We introduce VeriFine, an agent harness framework that scales verification through the co-evolution of the policy, training curriculum, and judge. The Policy Improvement Loop uses a rubric judge to diagnose recurring failures, construct an adaptive curriculum, and optimize the policy. When progress plateaus and verification becomes a bottleneck, the Judge Improvement Loop selectively queries human guidance on informative failure cases and refines the judge through coactive calibration, in which humans and agents resolve disagreements and converge toward the objective rubric of physical reasoning. The revised judge then guides the next stage of data selection and policy optimization. Experiments on driving and robot navigation tasks demonstrate continuous self-improvement in both policy and judge capability across reinforcement and supervised fine-tuning. These results show how scaling verification supports continuous self-improvement as policy failure patterns evolve.

Zewei Zhou, Rachel Luo, Yulong Cao +10

NVIDIA · UCLA · Work done during an internship at NVIDIA +2

Reinforcement Learning with Verifiable Rewards Self-Improving Agents Embodied Reasoning

Oct 6, 2026cs.AI

Self-Retrospection Distillation: Turning Post-hoc Experiences into Prior Foresight

Reinforcement learning with verifiable rewards (RLVR) turns agent experience into learning signals primarily through scalar outcome rewards after interaction. For group-relative objectives, however, this signal vanishes when all rollouts receive the same reward, even though their trajectories may reveal useful information about what the task requires and how the agent fails. We ask a complementary question: can hindsight teach an agent what it could have anticipated before acting? We introduce prospective learning, which uses post-hoc experience to supervise foresight predictions from the pre-interaction view, and instantiate it with Self-Retrospection Distillation (SRD). Intuitively, a completed trajectory reveals knowledge that would have been useful and pitfalls that should be avoided; SRD distills this privileged hindsight into trajectory-blind foresight of the same policy. Foresight serves only as a training target and need not be explicitly generated at inference time. Across 10 tool-integrated reasoning and long-horizon agentic tasks, SRD complements RLVR and self-distillation baselines with gains of up to 24.2 pp. Its advantage is especially pronounced when reward contrast is scarce: when 37--98% of rollout groups are reward-uniform across model scales, yet SRD can still exploit learning signal from sampled trajectories. In the 2B setting, where 98% of groups are all-failure, the RLVR training ends up at 0.0% success, while adding SRD reaches 60.6% under the same rollout budget. Our results suggest that post-hoc agent experience is useful not only for evaluating or improving behavior, but also for shaping predictive representations before available interaction.

Haoxiang Zhang, Qinglin Chen, Hiroaki Hayashi +9

Salesforce AI Research · UC San Diego · Texas A&M University +1

Agentic RL RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Self-Distillation Sparse-Reward RL

Oct 6, 2026cs.LG

FC-SWE: Failure-Conditioned RL for Long-Horizon Software Engineering Agents

Repository-level software engineering (SWE) is a challenging long-horizon setting: agents must reason over extended interactions, use tools, and adapt to stateful environments. Recent work trains SWE agents with reinforcement learning methods such as Group Relative Policy Optimization (GRPO), which independently sample multiple trajectories per issue, test the resulting patches, and compare terminal rewards within a fixed group. However, this training setup does not reuse verifier feedback from failed patches as context for subsequent attempts, even though this feedback contains valuable diagnostic information about what went wrong. Training on recovery trajectories is challenging because the preceding outcome determines whether the next trajectory is generated, while the failed execution determines its conditioning context. We introduce FC-SWE, a failure-conditioned RL framework that incorporates recovery attempts into policy training. After a patch fails verification, FC-SWE restores the repository to its original task state and uses the failed patch and verifier feedback as context for a recovery trajectory. FC-SWE adapts GRPO to these chains of complete, multi-turn tool-use trajectories through two mechanisms. Trajectory-local rewards preserve each attempt's verifier outcome, preventing recovery success from rewarding an earlier failed patch. Active-set advantage estimation forms a comparison group from all initial and recovery trajectories actually executed for the same issue, so failed attempts remain in the group while unexecuted attempts are excluded. On all 500 SWE-bench Verified tasks under a verifier-assisted protocol, FC-SWE with Qwen3.5-4B and SWE-agent achieves 41.7% Resolved@1 and 52.8% Resolved@2, compared with 38.9% and 48.5% for GRPO. Although trained with at most two attempts per chain, FC-SWE reaches 70.7% Resolved@11 under an eleven-attempt test-time budget.

Jia Liufu, Bin Hu, Linglin Jing +5

NVIDIA · Tsinghua University

RL for Code Generation Software Engineering Agents Reinforcement Learning Reinforcement Learning with Verifiable Rewards RL for Tool Use

Oct 5, 2026cs.LG

What pass@k Cannot Measure: Evaluating Diversity and Capability Retention after Post-Training

pass@

k

, the fraction of problems a model solves within

k

sampled attempts, is the field's default protocol for deciding whether reinforcement-learning (RL) post-training on verifiable rewards improved a model. At the population level, pass@

k

depends only on a problem's probability of a correct sample, with no term for how it is distributed across outputs. We show this gap is not academic. Training Qwen2.5-1.5B-Instruct on grade-school math with Group Relative Policy Optimization (GRPO) and with rejection-sampling fine-tuning (RFT, training on the model's own shortest verifier-passed rollout) moves three complementary diversity measures (token-level entropy, answer-level entropy, unique answers per prompt) in opposite directions, with zero overlap across three seeds per arm. The gap survives restricting to verifier-correct completions only (lexical diversity among correct solutions is 15% lower for GRPO, after controlling for length) and a count-controlled check isolating diversity among incorrect answers alone, ruling out that GRPO's higher accuracy alone explains it. Yet pass@8 and pass@32 show no consistent winner on GSM8K, and a hard MATH-500 subset shows the same pattern: separation only at low

k

. Compared against the starting checkpoint, no trained arm significantly improves hard-problem coverage: RFT is significantly worse, while GRPO is statistically indistinguishable from it - so GRPO's pass@1 edge over RFT reflects a smaller loss relative to Base, not a capability gain, a missing-control issue, not a failure of pass@

k

. On GSM8K, only pass@1, with no role in detecting diversity by construction, separates the arms cleanly, rewarding the arm whose correct solutions are least diverse. We argue this is a concrete instance of a standard evaluation protocol missing a property it is routinely used to certify.

Subham Rath, Raj Dandekar, Rajat Dandekar +1

Accenture Strategy & Consulting · Vizuara AI Labs (vizuara.com) · Vizuara AI Labs (vizuara.ai)

LLM Evaluation Language Model Generation Evaluation Reinforcement Learning with Verifiable Rewards RL Post-Training

Oct 5, 2026cs.LG

Minimal Witness Reinforcement Learning

``What are the irreducible conditions that are sufficient to produce an outcome?'' is one of the most common questions that recur across computation and science. Its answers, the minimal sufficient witnesses, are what we mean by explanations, mechanisms and reasons. These problems usually ask for multiple minimal witnesses, yet standard RL methods may reveal only one solution or redundant ones. We formalize this problem as minimal-witness identification and introduce Minimal-Witness Reinforcement Learning (MWRL). MWRL takes the union of the sets certified by successful proposals sampled from the policy and credits each proposal for the coverage the group union would lose without that proposal. This credit assignment, derived directly from the problem definition, unifies the demands for minimality and recovery of alternatives from a single black-box verifier bit. Under this principle, we derive a value iteration planner that recovers the entire family of witnesses and a policy gradient method that can scale to large language models. Across different experimental settings, MWRL recovers most minimal witnesses, while other methods return redundant supersets or a single witness. By making witness families learnable from verifier feedback, MWRL expands the scope of reinforcement learning beyond single-solution optimization. Our code is available at https://github.com/TSUITUENYUE/MWRL.

T. Y. Tsui, Zihao Ye, Pengxiang Cai +3

University of Pennsylvania · Shanghai AI Laboratory

RL for Language Model Reasoning Credit Assignment in RL Reinforcement Learning with Verifiable Rewards

Oct 5, 2026cs.LG

Reward-Driven Learning under Prompt-Level Differential Privacy

Reinforcement learning with verifiable rewards (RLVR) trains a language model on problems that may themselves be confidential, and the trained model can reveal which problems it saw. We study RLVR under prompt-level differential privacy: the released weights must be (ε,δ)-differentially private with respect to the presence of any one training problem. Taking the group of responses to one prompt as the privacy record, our method aggregates their gradients, clips the prompt's contribution once, adds Gaussian noise, and composes the privacy loss across updates, so the budget depends on neither the number of responses per prompt nor the clipping norm; to our knowledge this is the first differential privacy guarantee for RLVR training. We train Qwen2.5-1.5B-Instruct with LoRA at a per-run budget of ε=8 and compare, on the same prompts and at the same budget, a control that removes only the reward signal and two private supervised fine-tuning recipes. The reward signal improves accuracy over the control by 2.65 points on MATH and 3.24 on GSM8K, in every seed; the improvement survives a format-robust scorer, at 1.3 points on MATH, and is not explained by response length. At the same budget the private model outperforms both supervised recipes on MATH and GSM8K by 2.3 to 3.8 points, retains 85--90% of the gain of non-private GRPO on these tasks, and on MATH the noise of an eightfold tighter budget costs at most 1.2 points. The reward effect also carries to CommonsenseQA, an exploratory non-mathematical task. Verifier feedback thus remains a usable learning signal under prompt-level privacy.

Jiachen Zhao, Antonia Januszewicz, Taeho Jung

University of Notre Dame

Privacy-Preserving Language Models Differentially Private Stochastic Gradient Descent Reinforcement Learning with Verifiable Rewards Differential Privacy

Oct 5, 2026cs.AI

VERA: Scaling Verifiable Environments for Agentic co-Evolution

Competent agents need precise and verifiable environments, such as sandboxes that are resumable at any stage and evolve from observable evidence. However, most long-horizon work exposes how rare these are: for example, an agent in medical research must ground a finding, classify it, and write a report over dozens of dependent steps, yet recent environments score only the outcome. To address the challenges in stable training, we present VERA, which builds such environments at scale and lets agents evolve on them. VERA builds these environments from initial trajectories: an agent writes rubrics, executable checks, a judge verifies each sandbox, and only those that pass enter the training bank. On these environments, VERA alternates between two updates: train the model with rubric rewards, or edit the harness skills. We also create a verifier which gates model checkpoints and harness edits using explicit development-set acceptance criteria. This attribution distinguishes VERA's co-evolution from single-axis baselines: its updates target not only the cause but the outcome. With an open-source corpus of 9,000+ long-horizon verifiable environments, a 9B model paired with its co-evolved agent beats the strongest baseline by 10.3 and 13.0 points in the two domains. At 27B, it surpasses the baseline on AutoCoWorkBench (71.6) and AutoMedBench (80.7), transfers to unseen workflows, and retains general capabilities.

Junqi Liu, Yongyang Pan, Zhuosong Jiang +14

University of California, Santa Cruz · University of Illinois Urbana-Champaign · National University of Singapore +4

Reinforcement Learning with Verifiable Rewards Long-Horizon LLM Agents Self-Evolving Agents

Oct 4, 2026cs.AI

Beyond Instruction Following: Learning Grounded Skill-Following with Skill Contracts

Instruction following typically enforces discrete, response-level requirements, whereas an expert-authored skill prescribes procedural requirements spanning multiple phases and environment interactions. Given such a skill, we train the executor to execute all required phases instead of focusing solely on the final answer. We therefore introduce Grounded Skill-Following, which requires an agent to execute a fixed, expert-authored skill across its required phases by grounding decisions in environment observations. To achieve verifiable procedural execution, we formulate each skill as a skill contract combining visible skill instructions with an explicit contract runtime. The runtime specifies required phases, admissible actions, permitted transitions, and accepted termination. This structure provides a dense, verifiable training signal throughout execution. We leverage this by introducing Verified Progress Credit, which assigns rewards upon the initial completion of contract milestones and aggregates them into the trajectory return to guide policy optimization. During rollout, the contract runtime continuously tracks state transitions to provide Contract-State Feedback, which indicates whether the latest action is accepted and guides the agent toward valid next actions. To measure procedural compliance, we introduce the Protocol Completion Rate (PCR), defined as reaching accepted termination through all required phases, and decouple it from the final Task Outcome. Jointly trained with our framework, Qwen3.5-4B achieves Protocol Completion Rates of 99.27% on Math and 99.96% on Search, while slightly outperforming original baselines in Task Outcome (82.95% and 46.61%, respectively). Controlled studies examine how skill instructions, training signals, and contract-state feedback affect both metrics, while withholding interventions evaluate behavioral dependence on observation content.

Jianghan Shen, Zhenjie Liu, Yue Li +10

Nanjing University · Shanghai Artificial Intelligence Laboratory · Peking University +3

Reinforcement Learning LLM Agent Skill Learning Reinforcement Learning with Verifiable Rewards RL for LLM Agents

Oct 1, 2026cs.CL

KaliBench: A Fine-Grained Benchmark for Cybersecurity Tool Use on Kali Linux with Runtime-Free Verifiable Rewards

LLMs are increasingly applied to cybersecurity workflows, where they are expected to translate analysts' intent into tool invocations. However, existing evaluations focus on knowledge-based assessments or end-to-end agentic tasks, and do not directly measure LLMs' ability to generate executable commands for real-world cybersecurity tools. This gap is critical because cybersecurity operations rely on strict command-line interfaces (CLIs), where minor syntax errors, incorrect flag--value bindings, or argument misordering can invalidate execution. We introduce KaliBench, a fine-grained benchmark and dataset for natural-language--to--CLI translation on Kali Linux, comprising 8,504 query--command pairs spanning 1,642 tools across 23 capability dimensions and 5 security phases. KaliBench is constructed via a manuscript-grounded pipeline with deterministic canonicalization and alias-aware evaluation, enabling precise and reproducible assessment of tool selection and argument construction. To ensure both semantic correctness and practical executability, we develop a multi-stage verification pipeline that combines LLM-based validation, sandboxed terminal execution, and human-in-the-loop refinement. Building on these fine-grained, deterministic signals, KaliBench further enables runtime-free verifiable rewards for training. Across three evaluation modes and 24 configurations of general-purpose and security-focused open-weight models, no open-weight model exceeds 42% exact-command accuracy in the unrestricted setting, highlighting the difficulty of accurate CLI-based cybersecurity tool use without explicit tool hints. We further show that supervised fine-tuning and reinforcement learning with verifiable rewards derived from KaliBench significantly improve an 8B model and achieve performance comparable to a 685B MoE model.

Pengfei Li, Naufal Suryanto, Sicheng Zhang +1

Khalifa University · University of Western Australia

LLMs for Cybersecurity LLM Tool Use Reinforcement Learning with Verifiable Rewards

Oct 1, 2026cs.LG

On Language Drift during RLVR Post-Training

Recent advances in LLM reasoning models---driven primarily by the paradigm of post-training via reinforcement learning with verifiable reward (RLVR)---have enabled them to accomplish impressively complex tasks. However, in parallel with their rising capabilities, LLMs have increasingly displayed signs of language drift in their chains of thought (CoTs): unusual, non-standard, and seemingly nonsensical language use. Although it is well-documented---and can potentially impair CoT monitorability---the causes of language drift are thus far poorly understood. In this paper, we identify the conditions under which language drift occurs: we prove theoretically that RLVR optimization pressure permits unbounded language drift, while supervised fine-tuning does not. We then show empirically that language drift specifically arises during RLVR on novel reasoning tasks---i.e. when the target behavior cannot be drawn out of the base model. Finally, we prove that it is not possible to constrain language drift without constraining expected reward, suggesting that CoT monitorability cannot be improved without harming performance during RLVR post-training at the frontier.

Michael Sullivan, Alexander Koller

Department of Language Science and Technology, Saarland University, Saarbrücken, Germany.

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards CoT Monitoring

Oct 1, 2026cs.LG

Same Reward, Different Skills: When Multimodal RL Learns to Look

Reinforcement learning with verifiable rewards (RLVR) improves vision-language benchmark scores even without visual information during training. With images at test, blind-trained models recover roughly half of the real-image gain at 3B and nearly four fifths at 7B. Prolonged real-image training can erode grounding while benchmark gains persist. Both findings expose the same gap: an image in the prompt is not an image in the learning signal. Our design rule, visual resolvability, asks that visual evidence be necessary for a correct answer and that the task remain learnable. We test it on counterfactual coordinate scenes in which the question stays fixed and the target is never named, so a correct answer requires finding the target in the image. With standard GRPO and correctness-and-format rewards, a 7B model raises its accuracy at finding the target (discovery) from 0.425 to 0.875 on held-out scenes denser than any it trained on, and it improves on question types it never trained on. Two controls locate the source of the gain. Replacing test images with gray canvases drops discovery to zero; training on gray canvases instead, at matched step 30 and in each of four seeds, yields essentially none of the gain even when the model is then tested with real images. The learned skill carries over to grounding tasks built independently of the training corpus. A caption that answers the training question, added to the same images, reward and budget, cuts the gain by nearly two thirds. Changing what reward requires changes what RL learns.

Haocun Ye, Xinlong Jiang, Qile Chen +6

University of the Chinese Academy of Sciences · Institute of Computing Technology, Chinese Academy of Sciences · Independent Researcher

Reinforcement Learning with Verifiable Rewards Visual Grounding Visual RL RL for VLMs

Oct 1, 2026cs.AI

LineupRL: Verifiable Reinforcement Learning for Time Series Captioning via Caption-to-Series Identification

Time series captioning is a fundamental step in time series understanding and can also serve as the bridge between signal and natural language. Supervised fine-tuning (SFT) relies on a larger model's captions and cannot exceed their quality. Reinforcement learning (RL) can, but its rewards were designed for other modalities and other tasks, and they transfer poorly to open-ended generation in the time series domain. We address this by proposing LineupRL, a reinforcement learning with verifiable rewards (RLVR) pipeline whose reward is caption-to-series identification. The reward model is a frozen large language model (LLM) verifier that reads the generated caption and the candidate time series as raw values, never the chart, and must pick the described time series from multiple distractors. Matching is a far lighter demand on the verifier than writing questions or judging a caption, so an off-the-shelf LLM can supply the reward. Across two captioning benchmarks, and on forecasting and reconstruction where the predictor sees only the caption, LineupRL outperforms SFT and RL baselines on every metric. The 3B vision language model (VLM) trained by LineupRL also outperforms, at 1/24 of the parameters, the 72B VLM whose captions the SFT baseline is distilled from. Our case study shows that LineupRL resists reward hacking, and that the captioner it trains both traces the trend and names the values at key points.

Haochen Zhang, Laura Yao, Zachary Plotkin +2

University of North Carolina at Chapel Hill · Foci Labs

RL for Language Models Reinforcement Learning with Verifiable Rewards

Oct 1, 2026cs.LG

Function-Structured Reinforcement Learning with Executable Verifiers for Mathematical Reasoning

Algorithmic mathematical reasoning requires reliable decomposition, computation, and aggregation. Final-answer rewards provide limited guidance on intermediate errors, while successful execution does not guarantee mathematical correctness. This work proposes Function-Structured Graph Reinforcement Learning (FSG-RL), connecting subproblem graphs and Python implementations with multi-verifier feedback. The policy first learns to generate code from function graphs through supervised fine-tuning (SFT). Group Relative Policy Optimization (GRPO) then optimizes the policy using answer-gated rewards and span-level credit assignment. The framework also supports teacher supervision and structured memory. A benchmark curated from Grade School Math 8K (GSM8K), MathQA, MATH, and Omni-MATH pairs public function graphs with private verification specifications. Under a unified evaluation protocol, GRPO improves final-answer accuracy from 43.25% to 67.50% and full solution success from 32.25% to 52.25% over SFT. Continued reinforcement learning (RL) with teacher supervision yields additional gains. The gains extend beyond producing correctly formatted code, supporting verifier-guided reinforcement learning for mathematical reasoning. Code is available at https://github.com/ZihanLiummyycc/FSG-RL.

Zihan Liu, Xurong Xie

Department of Computer Science, University College London, UK · Institute of Software, Chinese Academy of Sciences, China

LLM Reasoning with Graphs RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

Oct 1, 2026cs.AI

Improving Math Reasoning through Value-guided Informative Search

Reinforcement learning with verifiable rewards (RLVR) has substantially improved the mathematical reasoning capabilities of large language models. Recent work introduces search into RLVR rollouts to increase trajectory diversity, but diversity alone does not ensure that the search-induced rollout policy improves upon the current policy. To address this gap, we propose APIVIS, a training-time framework that adapts finite-budget Gumbel search to chunk-level mathematical reasoning. APIVIS combines direct and searched responses within each rollout group, allowing improvements found by search to produce informative relative rewards. It further applies selective supervision to search-improved tokens, preserving a learning signal when uniform group rewards render GRPO ineffective. We show that exact value-guided selection improves the expected verifier reward at each searched state and that this guarantee extends to the complete rollout policy, with a corresponding approximate guarantee under bounded value-estimation error. Experiments on widely recognized mathematical reasoning benchmarks and different model scales demonstrate substantial improvements over competitive search-based methods, validating the effectiveness of APIVIS.

Shaohuai Liu, Yuning Wu, Haoran Liu +3

Amazon · Department of Electrical and Computer Engineering, Texas A&M University

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards LLM Mathematical Reasoning

Oct 1, 2026cs.LG

Adapter Thickets: Splitting an RLVR Budget Beats Concentrating It

Majority voting over sampled completions is the workhorse of test-time scaling, and reinforcement learning with verifiable rewards (RLVR) is the workhorse for making each completion better. The standard pipeline composes the two: train one policy with RLVR, then sample it many times and vote. We show that this composition is lossy. A vote can only overturn mistakes that its voters do not share, and RLVR sharpens a policy so that its samples increasingly make the same mistakes. With every method drawing exactly

160

completions per problem, training a single LoRA adapter on the full RLVR budget raises single-sample accuracy on every model we test (

1.5

B-

8

B). Yet on three of four models it leaves the majority vote below that of the untrained base model, by up to

4.8

points. The damage builds during training: voter errors grow steadily more correlated, and the majority vote accuracy peaks early before falling by up to

7.0

points. The cause is concentration, not RLVR itself. We split the same data and training budget across

K

LoRA adapters, each trained on its own random disjoint shard, and call the result an adapter thicket. Thickets out-vote the fully trained adapter in all

16

(model,

K

) settings, and for

K{\geq}4

they stay within

0.8

points of the base model or above it. A single adapter stopped early, at a thicket member's step count, is a strong control that matches thickets for small

K

. For

K{\geq}8

, thickets keep more of RLVR's single-sample gain and out-vote this control in six of eight settings. The cost of concentration also grows with the number of votes: from

16

to

160

votes, the thicket's lead over the fully trained adapter widens from

1.3

to

3.3

points. When the plan is to sample and vote, an RLVR budget is better spent broad than deep.

Jonathan Williams, Esin Tureci Karthik R. Narasimhan

Princeton University

RL for Language Model Reasoning Low-Rank Adaptation Reinforcement Learning with Verifiable Rewards Language Model Ensembles

Sep 30, 2026cs.LG

Semifactual Credit-Augmented Policy Optimization

Reinforcement learning with verifiable rewards (RLVR) has improved the reasoning capabilities of large language models (LLMs), yet their predictions remain sensitive to task-irrelevant prompt features. We investigate this sensitivity through semifactual prompt interventions that preserve the underlying problem and its answer. Our analysis reveals substantial variation in token-level sensitivity and shows that suppressing high-drift token candidates during decoding improves reasoning accuracy without updating model weights. These findings highlight a limitation of Group Relative Policy Optimization (GRPO), which assigns the same outcome-derived advantage to every response token and may reinforce potential spurious dependence alongside useful reasoning. Motivated by this observation, we introduce Semifactual Credit-Augmented Policy Optimization (SCAPO), a causally inspired variant of GRPO that incorporates semifactual stability into token-level credit assignment. SCAPO measures token probability drift for fixed responses under semifactual interventions and uses normalized stability scores to reduce advantages for relatively unstable tokens during early training, while granting no additional credit for stability alone. On Qwen3-4B-Base and Qwen3-1.7B-Base, SCAPO improves AIME 2024-2026 accuracy over GRPO by 5.63 and 4.17 percentage points, respectively. At both model scales, SCAPO achieves the best results on most evaluated mathematics benchmarks and all evaluated out-of-distribution benchmarks among the compared methods. These results suggest that semifactual stability provides an effective training signal for improving reasoning and generalization through finer-grained credit assignment in RLVR. The code is available at https://github.com/DtYXs/SCAPO.

Junshu Pan, Zhizhang Fu, Shulin Huang +5

Zhejiang University · Westlake University · Shanghai Innovation Institute +1

Prompt Sensitivity RL for Language Model Reasoning Credit Assignment in RL Step-Level Credit Assignment in RL Reinforcement Learning with Verifiable Rewards Causal Interventions in Language Models+1

Sep 30, 2026cs.AI

Unlearnable, or Unmeasured? On the Reliability of Difficulty Labels in RLVR

Reinforcement learning with verifiable rewards (RLVR) has become an important approach for improving reasoning during post-training. Recent work suggests that some difficult prompts remain resistant to learning even when they occasionally produce correct solutions. We revisit this unlearnability phenomenon and find that the affected prompts do improve, at roughly one third of the learnable rate, while the difficulty-defined set used to study them is much less reproducible than expected. These difficulty labels are estimated from a limited number of sampled responses. Combining them across seeds can further change which prompts are selected instead of simply reducing measurement noise. We develop a sampling-based framework for quantifying this instability and determining how much evaluation is required for difficulty assignments to reproduce reliably. We also revisit the gradient-similarity evidence proposed to explain unlearnability and show that part of the observed separation arises because difficult prompts provide fewer correct rollouts from which their gradients can be estimated. Matching this sample count weakens the gradient difference but does not remove it. Overall, the slow-learning phenomenon survives our reanalysis, while both the prompts used to define it and the evidence used to explain it require more careful measurement.

Chandak Chakma, Syed Nazmus Sakib, Nafiul Haque +1

Department of Robotics and Mechatronics Engineering, University of Dhaka

Reinforcement Learning RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards

Sep 30, 2026cs.CL

CATCH: A Controllable Analysis Testbed for Reward Hacking in Coding RL

During reinforcement learning with verifiable rewards (RLVR), large language models (LLMs) can exploit loopholes in their environments to obtain high rewards without improving the intended capabilities, i.e., reward hacking. Despite its risks to training efficiency and safety, monitoring and mitigating reward hacking during training remain challenging, which is limited by a lack of testbeds that reproduce hacking and reliably identify it. We introduce CATCH, a controllable testbed for studying reward hacking in coding RL. CATCH deliberately exposes environmental loopholes and provides execution-based gold labels by comparing success under a vulnerable evaluator with task correctness under an independent audit. It also can control the model's initial hacking tendency through supervised fine-tuning data mixtures and the difficulty of earning rewards through reward designing, enabling systematic comparisons of hacking dynamics and interventions. Experiments show that CATCH can produce diverse RL training trajectories with clear reward hacking, and analyses demonstrate that both initial models and reward difficulties shape the emergence of reward hacking. We further evaluate the effectiveness of different reward hacking detection and mitigation methods. A key finding is that a chain-of-thought monitor initially suppresses hacking, but this protection erodes as the policy model learn to mislead the monitor with code comments. This highlights the need to evaluate hacking mitigations throughout training with CATCH. The source code and resources are publicly released at https://github.com/THUAIS-Lab/CATCH.

Shouli Wang, Yanfeng Jia, Zhihao Ou +6

Tsinghua University · Beihang University · Southern University of Science and Technology +2

RL Benchmarks RL for Code Generation Reward Hacking Reinforcement Learning with Verifiable Rewards Reward Design for RL

Sep 30, 2026cs.LG

From Imitation to Reward Discovery: On-Policy Warmup for Agentic RL

Reinforcement learning with a verifiable reward (RLVR) offers a scalable approach to training language-model agents, yet sparse outcome rewards can leave early training with little signal for policy improvement. We identify an On-Policy Acceleration Phenomenon: in our main comparisons, RLVR initialized with on-policy distillation reaches high performance earlier in training and achieves both higher average performance during subsequent RLVR and higher final performance than the alternative baselines. Motivated by this observation, we study On-Policy Warmup (OPW), a teacher-guided stage in which the student trains with teacher supervision on its own interaction trajectories before transitioning to RLVR. Unlike imitation on fixed teacher-generated trajectories, OPW targets states induced by the student's own decisions, including imperfect actions and recovery situations. We provide a theoretical explanation by connecting on-policy reverse-KL distillation to trajectory-level distribution matching. Under a competent teacher and sufficiently small population distillation loss, this connection yields a lower bound on initial verifier success and a corresponding bound on reward-discovery complexity. For group-relative RLVR, we further characterize when increased success probability produces more reward-informative groups. Together, our findings support on-policy distillation as an effective warmup for agentic RLVR and identify initial reward discovery as a mechanism that can contribute to the observed acceleration.

Yitong Qiao, Tiantian He, Lei Liu +4

Zhejiang University · Ant Healthcare, Ant Group

Agentic RL Reinforcement Learning with Verifiable Rewards On-Policy Distillation Sparse-Reward RL Policy Learning RL for LLM Agents

Sep 30, 2026cs.AI

Advancing Entropy-Level Credit Assignment in RLVR via Proximal Entropy Policy Optimization

Value-model-free RLVR methods such as GRPO assign uniform advantages to all tokens in a rollout, ignoring that tokens contribute unequally. Recent methods use token entropy as an importance proxy but compute it globally across the batch, conflating importance with prompt difficulty and positional trends. We argue that importance should instead be measured relative to the local context of each token. We introduce proximal entropy, a local measure of token importance relative to neighboring tokens, and prove it is invariant to both confounders. Proximal Entropy Policy Optimization (PEPO) uses it to weight per-token advantages and outperforms GRPO and entropy-based baselines on mathematical reasoning across Qwen3-1.7B, Qwen3-4B, and Llama-3.2-3B-Instruct. We also show the formulation generalizes to other algorithms where substituting proximal entropy into existing methods improves, and applying it to single-stream RL succeeds where global entropy fails.

Yun Kim, Nojun Kwak

Seoul National University

RL for Language Model Reasoning Credit Assignment in RL Step-Level Credit Assignment in RL Reinforcement Learning with Verifiable Rewards

Sep 30, 2026cs.LG

From Search to Signal: Online Post-Training in Automatic Heuristic Design

Large language model (LLM)-based automatic heuristic design (AHD) iteratively proposes and refines heuristics, pairing design rationales with executable code. Task-specific evaluators assess programs; execution outcomes and performance scores guide search. Many AHD systems keep the generator frozen; EvoTune and Co-Evolution of Algorithms and Language Model (CALM) instead update it from evaluated candidates. When such outcomes drive reinforcement learning with verifiable rewards (RLVR), they create a search-coupled loop: the evaluated candidate stream supplies both search-state updates and training signals for the model that generates future candidates. Yet validity and performance do not uniquely determine useful model updates; converting them into learning signals must account for the prompt and evolving search state that produced each candidate. We formulate online post-training of small open-weight LLMs in AHD as context-dependent signal construction and develop alternative mappings from program validity, task performance, and generation context to update signals. Using shared evaluated rollouts and matched update budgets, controlled experiments across AHD tasks and model families compare these mappings with online post-training baselines, testing their effects on validity, performance among valid proposals, and the yield of valid proposals that improve under contextual comparisons. Complementary checkpoint, frozen-search, and live-system evaluations assess whether proposal-level gains appear in updated checkpoint behavior and subsequent search, rather than arising solely from accumulated search state. A resource-matched comparison under pre-specified cost accounting tests whether online updating adds value beyond additional search with a frozen generator. Together, this design avoids treating end-to-end search gains alone as evidence of stronger heuristic-design capabilities.

Yilun Yuan, Tianyu Zhou, Zhenzhou Tang

Wenzhou University

Automated Heuristic Design Reinforcement Learning with Verifiable Rewards LLM Post-Training Language Model Post-Training

Sep 30, 2026cs.AI

Reinforcing Multimodal Reasoning via Token-Level Perception-Grounded Advantage Estimation

Reinforcement Learning with Verifiable Rewards (RLVR) has improved the reasoning capabilities of Multimodal Large Language Models (MLLMs), yet existing frameworks rely on coarse, sequence-level reward signals that lack the fine-grained supervision over the visually-grounded steps within a multimodal reasoning chain. We investigate this gap through the lens of two token-level metrics: visual dependency (i.e. how much a token's prediction relies on the input image features) and predictive entropy. Our empirical analysis reveals two key findings: (1) correct reasoning chains exhibit a markedly sharper entropy reduction as visual grounding intensifies, compared to incorrect ones; (2) pivotal tokens, those whose misprediction triggers reasoning collapse, are statistical outliers in the joint distribution of visual dependency and predictive entropy derived from correct chains. Motivated by these findings, we propose token-level perception-grounded advantage estimation (TPAE), which estimates token-level advantages by measuring each token's statistical consistency with the vision-entropy patterns of correct rollouts. TPAE leverages this granular score to modulate the sequence-level advantage, producing a fine-grained supervision signal that can be integrated into various RLVR frameworks. Extensive experiments on seven benchmarks show that TPAE consistently outperforms leading strong baselines, yielding more stable and efficient optimization for multimodal reasoning. The code is publicly available at https://github.com/Zhihan72/TPAE.

Zhihan Zhang, Lizi Liao

Singapore Management University Singapore, Singapore

Token-Level Credit Assignment Multimodal Large Language Models Multimodal Reasoning Reinforcement Learning with Verifiable Rewards RL for VLMs

Sep 29, 2026cs.LG

Learning Beyond What You Sample: Off-Policy-Aware Cross-Model Trajectory Exchange for RLVR

Reinforcement Learning with Verifiable Rewards (RLVR) methods such as GRPO rely on successful self-generated trajectories, but finite rollout budgets can produce all-fail groups with no reward-based policy-gradient signal. While additional rollouts improve the chance of success at higher cost, successful trajectories missing from one model's rollouts may already have been discovered by another. Indeed, we observe that heterogeneous models often succeed on complementary prompts, creating opportunities for mutual learning without a designated stronger teacher. To exploit this complementarity, we propose GRAFT (Gated Replacement of Answer-Failed groups with peer Trajectories), an off-policy-aware framework that replaces all-fail groups with informative peer groups. GRAFT transfers both successful and unsuccessful peer responses with peer-computed advantages, while controlling cross-model mismatch through sequence-level compatibility weighting and token-level importance ratio clipping. Across three heterogeneous model pairs and five mathematical reasoning benchmarks, GRAFT consistently improves both models over GRPO with the same per-model rollout budget, gaining 2.1 points on average and up to 4.5 points in model-level average performance. Stored peer trajectories preserve most of the gains, improving over GRPO by 1.8 points on average without simultaneous co-training.

Doohyuk Jang, Yoonsik Park, Gyouk Chu +2

KAIST · AITRICS

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Off-Policy RL

Sep 29, 2026cs.LG

Privy to the Foil: Recasting Value Estimation with a Self-Privileged Critic for RLVR

Assigning credit to intermediate steps remains a central challenge in training Large Language Models (LLMs) on multi-step reasoning tasks with sparse terminal rewards, and actor-critic methods such as PPO address this by learning value functions to construct token-level advantages. Their effectiveness, however, hinges on reliable value estimation, a difficult task requiring the critic to both assess progress toward a correct solution and anticipate an evolving policy's future behavior; errors in either can compromise credit assignment and destabilize online training. In this paper, we revisit the standard state-only formulation of value estimation and propose

π

PPO, a self-privileged actor-critic framework. By reusing verified same-prompt rollouts as contrastive evidence,

π

PPO helps the critic assess intermediate reasoning against successful and failed attempts, while preserving standard policy optimization and the deployment interface. Experiments show that

π

PPO consistently improves value-estimation quality by a substantial margin and outperforms representative actor-critic and critic-free RLVR baselines on challenging mathematical reasoning benchmarks, while remaining effective even when paired with substantially smaller asymmetric critics.

Kun Liang, Chenming Tang, Clive Bai +5

School of Computer Science, Peking University · National Key Laboratory for Multimedia Information Processing, Peking University · Foundation Model Department, Tencent

Value Function Estimation RL for Language Model Reasoning Credit Assignment in RL Step-Level Credit Assignment in RL Reinforcement Learning with Verifiable Rewards Actor-Critic Methods+1