Reinforcement Learning with Verifiable Rewards

Also known as RLVR

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This field studies how to improve language and multimodal models by rewarding outputs whose correctness or task progress can be checked, for example with unit tests, mathematical answers, formal proofs, or simulated environments. Common approaches use policy-gradient methods such as PPO and GRPO, with outcome rewards supplemented by step-level feedback, rollout exploration, or replay to improve credit assignment and data efficiency. Work spans math and coding benchmarks as well as search, GUI, and other interactive agents. A central tension is that rewards must be scalable yet faithful: imperfect or narrow verifiers can be exploited, and a correct final answer does not by itself establish sound reasoning, visual grounding, or robust generalization.

Latest papers

692 papers137 papers in the last four weeks, up 111% on the four weeks before. 0.8% of all new papers.

May 9, 2026

Breaking the Impasse: Dual-Scale Evolutionary Policy Training for Social Language Agents

While Reinforcement Learning with Verifiable Rewards (RLVR) has proven effective for closed-ended tasks, extending it to open-ended social language games via self-play reveals a critical issue: evolution impasse. Due to the vast strategy space, language agents frequently converge to homogenized behaviors, leading to deterministic match outcomes that eliminate the gradient signals necessary for policy evolution. To tackle this issue, we propose Dual-scale Evolutionary Policy Training (DEPT) for social language games. DEPT introduces a time-scaled evolutionary perception mechanism that detects impasse by quantifying dual-scale value baseline divergence alongside match entropy. Upon perceiving the collapse, it then activates asymmetric advantage reshaping to dynamically modulate the optimization landscape for intervention. Thus, our method effectively restores gradient signals and enforces sustained strategic exploration. Extensive experiments on multiple social language games demonstrate that DEPT outperforms strong baselines, avoiding policy degeneration and driving the continuous evolution of social language agents.

Minzheng Wang, Run Luo, Yanbo Wang +6

MAIS, Institute of Automation, Chinese Academy of Sciences · School of Artificial Intelligence, University of Chinese Academy of Sciences · National University of Singapore +1

Reinforcement Learning Self-Play RL Reinforcement Learning with Verifiable Rewards Conversational Agents Evolutionary RL

May 9, 2026

The Cancellation Hypothesis in Critic-Free RL: From Outcome Rewards to Token Credits

A commonly accepted explanation of critic-free RL for LLMs, based on sequence-level rewards, is that it reinforces successful rollouts with a positive advantage while penalizing failed ones. In contrast, we study critic-free RL from a token-level perspective, revealing the token-flipping phenomenon: positive and negative rollouts exhibit remarkably similar proportions of tokens whose probabilities are boosted or suppressed during RL training. To explain this phenomenon, we further show that a token's change in probability is not fully determined by its own advantage; coupled gradient interactions with other tokens also play a non-negligible role. Specifically, these token coupling effects occur primarily between identical tokens that are both predicted with low confidence. Building upon this analysis, we propose the cancellation hypothesis: as a result of coupling, opposing signals cancel out for tokens shared by positive and negative rollouts, while tokens more specific to successful rollouts receive stronger reinforcement, thereby inducing hidden token-level credit assignment from rollout-level rewards. We support this hypothesis with complementary empirical evidence. (1) Compared with training on only positive rollouts, critic-free RL shifts updates from template and formatting tokens toward reasoning tokens; (2) Tokens boosted by critic-free RL consistently demonstrate higher value than suppressed tokens, regardless of whether they originate from positive or negative rollouts. Guided by this view, we implement two batching interventions to encourage or preserve cancellation in critic-free RL training: query-preserved mini-batching and reward-balanced batching. Despite their simplicity, these interventions improve RLVR training across multiple model scales, supporting cancellation as both an explanatory principle and a practical design criterion for critic-free RL training.

Tianhao Cheng, Zeyu Huang, Zihan Qiu +5

1Fudan University · 3The University of Edinburgh · 6Qwen Team, Alibaba Group +4

Reinforcement Learning Token-Level Credit Assignment RL for Language Model Reasoning Critic-Free RL Reinforcement Learning with Verifiable Rewards

May 8, 2026

PYTHALAB-MERA: Validation-Grounded Memory, Retrieval, and Acceptance Control for Frozen-LLM Coding Agents

Local LLM-based coding agents increasingly work in settings where correctness is earned through execution feedback, persistent state, and bounded repair, not through a single fluent answer. Static retrieval, long-context prompting, self-refinement, execution-feedback repair, and reinforcement learning over model weights each address part of this setting, but they do not jointly provide validation-grounded episodic memory, adaptive retrieval-action selection, delayed credit assignment, and structural skill reuse around a frozen local model. We introduce PYTHALAB-MERA, a lightweight external controller for local validation-conditioned code generation. The frozen language model proposes complete source files; the controller decides which memory records and AST-derived skills should enter the next prompt, validates each candidate through a fail-fast pipeline, converts validation outcomes into bounded shaped rewards, and propagates delayed credit through TD(lambda)-style eligibility traces. We evaluate the implementation as a local CLI artifact on reinforcement-learning coding tasks with strict validation gates. In the measured hard RL setting with three tasks, three repetitions, and a three-attempt budget, PYTHALAB-MERA passed 8/9 strict validations; the self-refinement baseline and the investigated GRACE extension each passed 0/9. These results support a deliberately bounded claim: in this recorded setting, the external memory-and-retrieval controller improved validation success. They do not establish general-purpose code synthesis, state-of-the-art performance, formal program correctness, or formal safety.

Mehmet Iscan

PythaLab, Yildiz Technical University

RL for Code Generation Execution-Guided Code Generation Reinforcement Learning with Verifiable Rewards Memory-Augmented Language Model Agents

May 8, 2026

DUET: Optimize Token-Budget Allocation for Reinforcement Learning with Verifiable Rewards

Reinforcement learning with verifiable rewards (RLVR) generates hundreds of thousands of tokens per training step, with rollout generation dominating the computational cost. The overall token budget can be controlled along two main dimensions: (i) deciding which prompts to allocate rollouts to, and (ii) deciding how long each rollout should be. Prior work has generally controlled only one of these dimensions at a time. We show that jointly tuning both decisions under a shared compute budget improves both reasoning quality and wall-clock training time. We instantiate this view as DUal-controlled tokEn allocaTion (DUET), a computationally efficient layer over GRPO that uses a lightweight pre-rollout surrogate of prompt informativeness to set how many rollouts each prompt receives, and a marker-gated abort rule with importance reweighting to set when to stop them. On Qwen3-1.7B trained on MATH, DUET outperforms full-budget GRPO and the other three budget-aware baseline methods. DUET's advantage further generalizes to other benchmarks across math and coding, and is on par with the best baseline on the scientific Q&A domain, while also achieving a

1.62\times

wall-clock speedup. More notably, using only 50% of the token budget, DUET still outperforms all baseline methods at their full budget, achieving an even higher

2.51\times

speedup over full-budget GRPO. We verify the high performance of DUET on other backbone LLMs, including Qwen3-4B to 14B and Llama-3.2-3B-Instruct. Notably, the gap between DUET and the strongest baseline widens as the budget tightens, contrary to the usual pattern in which efficient methods trade off quality as compute decreases. More broadly, these results suggest that DUET budget-aware control strategies are valuable not only for accelerating training, but also for improving the quality of the learning signal.

Haoyu Hu, Xuandong Zhao, Xuhai "Orson'' Xu +1

Cornell University · University of California, Berkeley · Columbia University

Reinforcement Learning Group Relative Policy Optimization Reinforcement Learning with Verifiable Rewards RL Fine-Tuning

May 8, 2026

AIPO: Learning to Reason from Active Interaction

Recent advances in LLMs have demonstrated strong reasoning capabilities, largely stimulated by RLVR. However, the exploration of existing RLVR algorithms remains largely constrained by the knowledge and reasoning strategies already accessible to the policy model. Although recent methods introduce external expert demonstrations to broaden exploration, they typically rely on complete trajectory-level guidance, which can be sample-inefficient, information-sparse, and insufficiently adaptive to intermediate reasoning bottlenecks. Inspired by collaborative multi-agent systems, we propose AIPO, an enhanced reinforcement learning framework that improves LLM reasoning through active multi-agent interaction during exploration. Specifically, when encountering reasoning bottlenecks, AIPO enables the policy model to proactively consult three functional collaborative agents, namely the verify agent, knowledge agent, and reasoning agent, thereby obtaining fine-grained and state-dependent guidance during rollout. The resulting mixed-policy trajectories expose the policy to reasoning directions that may be difficult to discover through isolated on-policy exploration. To learn effectively from collaborator-provided tokens, we further introduce a corrected importance sampling coefficient together with a lower-bound clipping strategy to mitigate off-policy discrepancy and vanishing gradients. After training, the policy model reasons independently without relying on collaborative agents. Extensive experiments across mathematical, scientific, coding, and puzzle reasoning benchmarks show that AIPO consistently improves reasoning performance and generalizes across different policy models, collaborator backbones, and RLVR algorithms.

Junnan Liu, Linhao Luo, Thuy-Trang Vu +1

Department of Data Science and AI, Faculty of Information Technology, Monash University, Australia

RL for Language Model Reasoning LLM-Guided RL Reinforcement Learning with Verifiable Rewards

May 8, 2026

Rubric-Grounded RL: Structured Judge Rewards for Generalizable Reasoning

We argue that decomposing reward into weighted, verifiable criteria and using an LLM judge to score them provides a partial-credit optimization signal: instead of a binary outcome or a single holistic score, each response is graded along multiple task-specific criteria. We formalize rubric-grounded reinforcement learning (RL): a framework in which the policy is optimized against a structured, multi-criterion reward produced by a frozen LLM judge that conditions on auxiliary grounding the policy never sees. We instantiate the framework by deriving rubrics from an Office of Scientific and Technical Information (OSTI)-derived corpus of roughly 100,000 scientific and technical documents and training Llama-3.1-8B-Instruct with Group Relative Policy Optimization (GRPO). With GRPO-based training, the model achieves

71.7\%

normalized reward on held-out rubric evaluation. The GRPO-tuned policy also improves over the base model on four reasoning benchmarks not derived from the training corpus -- GSM8K, MATH, GPQA Main, and GPQA Diamond. These results provide evidence that structured, document-grounded rewards can improve held-out rubric performance and induce transferable reasoning behaviors beyond the corpus used to construct the training environment.

Manish Bhattarai, Ismael Boureima, Nishath Rajiv Ranasinghe +2

Los Alamos National Laboratory, Los Alamos, NM, USA.

Reinforcement Learning RL for Language Model Reasoning Rubric-Based RL Reinforcement Learning with Verifiable Rewards Reward Design for RL

May 8, 2026

Interactive Critique-Revision Training for Reliable Structured LLM Generation

In structured decision-making workflows such as form filling, compliance checking, and maintenance reporting, LLM outputs must be locally correct, globally consistent, and auditable against task-specific rules. Existing refinement methods often rely on heuristic debate, self-play, or LLM-generated supervision, creating a second-order assurance problem. We propose DPA-GRPO (Dual Paired-Action Group-Relative Policy Optimization), a paired-action training method for a two-player generator--verifier game with structured verifier interventions. The generator proposes outputs and may revise them when challenged; the verifier either remains silent or raises a safety assurance case (SAC) containing a claim, argument, and evidence. These SAC/no-SAC and KEEP/REVISE decisions induce paired counterfactual action groups, which DPA-GRPO uses for role-specific KL-regularized GRPO updates. We analyze the unregularized game and show that positive probability on strictly lower-reward intervention or revision actions creates a profitable unilateral deviation. Under standard stochastic-approximation assumptions, DPA-GRPO tracks the corresponding game ODE, whose isolated asymptotically stable limit points are stationary and candidate local equilibria under role-wise local optimality. Experiments on TaxCalcBench TY24 show that DPA-GRPO improves structured decision accuracy over zero-shot generation and generator-only RL baselines across Qwen3-4B and Qwen3-8B. Training increases correct silent acceptance, reduces missed errors, and improves calibrated revision behavior, indicating gains for both generator and verifier.

Fei Xu Yu, Zuyuan Zhang, Mahdi Imani +2

The George Washington University · Northeastern University · United States Military Academy

LLM Self-Refinement Group Relative Policy Optimization Reinforcement Learning with Verifiable Rewards Structured Output Generation

May 8, 2026

Gradient Starvation in Binary-Reward GRPO: Why Group-Mean Centering Fails and Why the Simplest Fix Works

Group Relative Policy Optimization (GRPO) is a standard algorithm for reinforcement learning from verifiable rewards, but its group-mean-centered advantage can fail under binary rewards. The failure mode is gradient starvation: when every response in a group is correct or every response is wrong, the centered advantage is exactly zero and the policy receives no learning signal. We prove that the true degeneracy rate always exceeds the i.i.d. Bernoulli prediction by Jensen's inequality, and observe a 0.69 degeneracy rate at group size four in logged Qwen3.5-9B GSM8K training. We then show that the fixed-reference Sign advantage,

A=2r-1

, performs pass@

G

failure descent by increasing the probability that at least one sample in the group succeeds. On the full GSM8K test set across seven seeds, Sign reaches 73.8% accuracy versus 28.4% for standard normalized group-mean DrGRPO at group size four, a 45.4 point gain with

p<0.0001

. The effect is directionally consistent on Llama-3.1-8B and positive but underpowered on a MATH-500 transfer check. Pass@

k

analysis indicates that the main benefit is search compression rather than large capacity expansion, aligning the empirical gains with recent RLVR ceiling observations.

Wenhua Nie, Jianan Wu, Junlin Liu +6

National Taiwan University

Reinforcement Learning Group Relative Policy Optimization Reinforcement Learning with Verifiable Rewards

May 8, 2026

Your Language Model is Its Own Critic: Reinforcement Learning with Value Estimation from Actor's Internal States

Reinforcement learning with verifiable rewards (RLVR) for Large Reasoning Models rests on variance reduction, which requires both a reliable baseline and high prompt diversity within each training batch. This is especially difficult in multi-domain training for general reasoning models, where prompts from different tasks induce highly diverse gradient signals. Existing approaches fall short in different ways: GRPO estimates its baseline as the group mean over rollouts from the same prompt, so an accurate baseline leaves fewer distinct prompts in the batch, while PPO avoids this trade-off by training a policy scale critic, roughly doubling the cost of training. We introduce POISE (Policy Optimization with Internal State Value Estimation), a reinforcement learning algorithm that turns the model's internal states into a value model. A lightweight probe reads the signals already computed during the forward pass to predict the baseline, and is trained online alongside the policy. To preserve gradient unbiasedness, we introduce a cross-rollout construction that predicts each rollout's value from an independent rollout's internal states. On Qwen3-4B and OLMo3-7B-Instruct-DPO across a six-domain verifiable-reward corpus, POISE outperforms other RLVR baselines while achieving more stable training. Moreover, the probe matches a separate LLM-scale value model, generalizes to various tasks, and remains accurate as the policy scales. By leveraging the model's internal representations, POISE enables stable policy optimization.

Yunho Choi, Jongwon Lim, Woojin Ahn +3

Graduate School of Data Science Seoul National University · Computer Science and Engineering Seoul National University

Value Function Estimation RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Actor-Critic Methods Policy Learning

May 8, 2026

HTPO: Towards Exploration-Exploitation Balanced Policy Optimization via Hierarchical Token-level Objective Control

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a pivotal technique for enhancing the reasoning capabilities of Large Language Models (LLMs). However, the de facto practice of mainstream RL algorithms is to treat all tokens of one response equally and assign the same optimization objective to each token, failing to provide granular guidance for the reasoning process. While in Chain-of-Thought (CoT) reasoning, different tokens usually play distinct roles. Therefore, the current RL algorithms lack an effective mechanism to dynamically balance the exploration-exploitation trade-off during learning. To this end, we propose Hierarchical Token-level Objective Control Policy Optimization (HTPO), a novel RL algorithm that takes the divide-and-conquer idea to hierarchically partition the response tokens into specific functional groups from three aspects (i.e., prompt difficulty, answer correctness, and token entropy). Within each group, according to the contributions to exploration or exploitation, we design specialized optimization objectives to facilitate the effective execution of each token's expected functionality. In this way, HTPO can achieve a more balanced exploration-exploitation trade-off. Extensive experiments on challenging reasoning benchmarks validate the superiority of our HTPO algorithm, which significantly outperforms the strong DAPO baseline (e.g., +8.6% and +6.7% on AIME'24 and AIME'25, respectively). When scaling test-time compute, the HTPO-trained model maintains a consistent performance advantage over the DAPO baseline, and the gap widens as the sampling budget increases, validating that our adaptive token-level control method fosters effective exploration without sacrificing exploitation performance. Code will be at https://github.com/xcyao00/HTPO.

Xincheng Yao, Ruoqi Li, Cheng Chen +4

School of Information Science and Electronic Engineering, Shanghai Jiao Tong University · Xiaohongshu Inc. · MoE Key Lab of Artificial Intelligence, AI Institute, Shanghai Jiao Tong University.

RL for Language Models Exploration-Exploitation Tradeoff RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards

May 8, 2026

Implicit Compression Regularization: Concise Reasoning via Internal Shorter Distributions in RL Post-Training

Reinforcement learning with verifiable rewards improves LLM reasoning but often induces overthinking, where models generate unnecessarily long reasoning traces. Existing methods mainly rely on length penalties or early-exit strategies; however, the former may degrade accuracy and induce underthinking, whereas the latter assumes that substantial portions of reasoning traces can be safely truncated. To obtain a compression signal without these limitations, we revisit the training dynamics of existing compression methods. We observe that the length--accuracy correlation is initially negative but continually increases during compression, indicating that shorter responses are initially more likely to be correct but gradually lose this property as the policy moves toward underthinking. Based on this observation, we formalize overthinking: a negative correlation indicates an overthinking regime, while a positive one indicates underthinking. When overthinking, the shortest correct responses are shorter than the group-average response length in expectation, making them natural compression targets already present in on-policy rollouts. We therefore propose Implicit Compression Regularization (ICR), an on-policy regularization method whose compression signal comes from a virtual shorter distribution induced by the shortest correct responses in rollout groups, guiding the policy toward concise yet correct trajectories. Training dynamics show that ICR maintains a better length--accuracy correlation during compression, indicating that short responses remain better aligned with correctness instead of drifting toward underthinking. Experiments on three reasoning backbones and multiple mathematical and knowledge-intensive benchmarks show that ICR consistently shortens responses while preserving or improving accuracy, achieving a stronger accuracy--length Pareto frontier.

Chen Wang, Hexuan Deng, Yining Zhang +5

College of Software, Nankai University · 2Zhongguancun Academy · 3Harbin Institute of Technology +4

Overthinking in Language Models Reinforcement Learning with Verifiable Rewards RL Post-Training CoT Compression LLM Reasoning

May 8, 2026

Structured Role-Aware Policy Optimization for Multimodal Reasoning

Reinforcement learning from verifiable rewards (RLVR), especially with Group Relative Policy Optimization (GRPO), has shown strong potential for improving the reasoning capabilities of large vision-language models (LVLMs). However, in multimodal reasoning, final-answer rewards are typically assigned at the sequence level and do not distinguish the functional roles of different tokens, making it difficult to determine whether a correct answer is supported by task-relevant visual evidence. In this paper, we revisit multimodal RLVR from the perspective of role-aware token-level credit assignment, where structured responses are decomposed into perception tokens for extracting visual evidence and reasoning tokens for deriving answers from that evidence. Based on this perspective, we propose Structured Role-aware Policy Optimization (SRPO), which refines the sequence-level GRPO advantage into role-aware token-level advantages without changing the reward function. Specifically, SRPO assigns role-specific credit by using self-distilled on-policy contrasts: perception tokens are emphasized according to their visual dependency under original versus corrupted visual inputs, while reasoning tokens are emphasized according to their consistency with the generated perception. These role-specific signals are further unified through a shared trajectory-level baseline, yielding positive token weights that adjust relative update magnitudes while preserving the original GRPO reward and optimization direction, without requiring external reward models or separate teachers. Experiments across diverse multimodal reasoning benchmarks show that SRPO improves evidence-grounded reasoning, highlighting the importance of moving beyond uniform sequence-level credit toward role-aware optimization for reliable multimodal reasoning.

Bingqing Jiang, Difan Zou

School of Computing & Data Science, The University of Hong Kong · School of Computing & Data Science and Institute of Data Science, The University of Hong Kong

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

May 8, 2026

Can You Break RLVER? Probing Adversarial Robustness of RL-Trained Empathetic Agents

Reinforcement learning from verifiable emotion rewards RLVER has produced language models with strong empathetic performance, evaluated on benchmarks that assume cooperative, honest users. Yet real emotional interactions systematically violate this assumption: users gaslight, escalate, and pressure AI systems for unconditional validation, dynamics that cooperative benchmarks cannot surface. We construct the Adversarial Empathy Benchmark AEB and introduce the Emotional Consistency Score ECS to evaluate empathetic robustness under adversarial conditions. AEB comprises six psychologically grounded adversarial trajectory types with discriminative reward structures that penalize formulaic responses; ECS formally disentangles a model's capacity to track user emotional states from its capacity to improve them. In a controlled experiment across eight scenario-matched conditions (think and no-think conditions on 2 RLVER models, and 2 base models (Qwen 1.5B and 7B) with 480 adversarial dialogues), RLVER-PPO-Think substantially outperforms the same-scale untuned baseline (0.963 vs. 0.761,

p<0.001, r=0.688

), with zero dialogue collapses and 47% higher hidden-intention detection. However, ECS remains nearly flat and is not significantly different for RLVER-PPO-Think versus Base-7B-Think (

p=0.650

): RL training improves emotional responsiveness without measurable gains in observable state tracking. We interpret the ECS--FS (Final Score) gap as a behavioral/legibility dissociation inside this simulator family, not as evidence about internal understanding or clinical readiness.

Deeraj S K, Sadhana Devarajan, Krishna Mehra +1

Department of Artificial Intelligence, Sardar Vallabhbhai National Institute of Technology, Surat, India

Emotion Recognition in Conversations Adversarial Robustness of Language Models Reinforcement Learning with Verifiable Rewards

May 8, 2026

Adaptive Negative Reinforcement for LLM Reasoning:Dynamically Balancing Correction and Diversity in RLVR

Reinforcement learning with verifiable rewards (RLVR) has become a highly effective method for improving the reasoning abilities of Large Language Models (LLMs). Recent research shows that Negative Sample Reinforcement (NSR) -- which focuses on penalizing incorrect steps rather than simply rewarding correct ones -- can match or even exceed the performance of more complex frameworks like PPO and GRPO across the entire Pass@k spectrum. However, current NSR techniques usually apply a fixed penalty throughout the training process and treat every incorrect response with the same weight. To address these limitations, we propose two extensions to the NSR framework: Adaptive Negative Sample Reinforcement. Rather than using a fixed update rule, A-NSR uses time-dependent scheduling functions. In the initial training phases, the system focuses heavily on correcting errors to stabilize the model. As training continues, it shifts toward more subtle and controlled updates. We also introduce Confidence-Weighted Negative Reinforcement, which operates on the principle that different mistakes carry different levels of importance. CW-NSR assigns specific penalty weights based on the model's normalized sequence likelihood. If the model is highly confident in a wrong path, it receives a larger penalty and for uncertain errors -- where the model is effectively exploring -- are penalized less strictly. Our formal analysis shows how these mechanisms govern token-level updates, allowing the model to leverage prior-guided probability redistribution while providing a natural defense against overfitting. We evaluated these methods on difficult reasoning datasets, including MATH, AIME 2025, and AMC23, using the Qwen2.5-Math-1.5B architecture.

Yash Ingle, Jaival Chauhan, Ankit Yadav +1

Sardar Vallabhbhai National Institute of Technology (SVNIT), Surat, India

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

May 8, 2026

Where to Spend Rollouts: Hit-Utility Optimal Rollout Allocation for Group-Based RLVR

Reinforcement learning with verifiable rewards (RLVR) has emerged as a central paradigm for improving the reasoning capabilities of large language models. Group-based policy optimization methods, such as GRPO, typically allocate a fixed number of rollouts to every prompt. This uniform allocation can be inefficient: it over-allocates compute to prompts whose sampled groups are already saturated while under-exploring prompts for which additional samples may reveal useful correct trajectories. To address this limitation, we introduce hit utility, the posterior probability that at least one rollout in a proposed additional allocation for a prompt will be correct. Building on this notion, we propose Hit-Utility Optimal Rollout Allocation (HORA), a learning-free rollout allocation policy that maximizes total posterior hit utility within each allocation batch. HORA adaptively reallocates rollout budgets while leaving the downstream reward evaluation and group-based advantage estimator unchanged. Across two training sets, four mathematical reasoning benchmarks, and three model families at multiple scales, HORA preserves comparable Pass@1 and improves Pass@K over compute-matched GRPO, and it outperforms related allocation baselines at large K. It is also drop-in compatible with other group-based estimators such as RLOO. Ablation studies indicate that the uniform prior used by HORA is competitive with five prompt-conditioned learned-prior alternatives.

Tao Wang, Shuo Li, Yan Sun +2

University of Pennsylvania · Now at Google DeepMind · New Jersey Institute of Technology +1

RL for Language Model Reasoning Group Relative Policy Optimization Reinforcement Learning with Verifiable Rewards

May 7, 2026

Verifier-Backed Hard Problem Generation for Mathematical Reasoning

Large Language Models (LLMs) demonstrate strong capabilities for solving scientific and mathematical problems, yet they struggle to produce valid, challenging, and novel problems - an essential component for advancing LLM training and enabling autonomous scientific research. Existing problem generation approaches either depend on expensive human expert involvement or adopt naive self-play paradigms, which frequently yield invalid problems due to reward hacking. This work introduces VHG, a verifier-enhanced hard problem generation framework built upon three-party self-play. By integrating an independent verifier into the conventional setter-solver duality, our design constrains the setter's reward to be jointly determined by problem validity (evaluated by the verifier) and difficulty (assessed by the solver). We instantiate two verifier variants: a Hard symbolic verifier and a Soft LLM-based verifier, with evaluations conducted on indefinite integral tasks and general mathematical reasoning tasks. Experimental results show that VHG substantially outperforms all baseline methods by a clear margin.

Yuhang Lai, Jiazhan Feng, Yee Whye Teh +1

Department of Data Science, City University of Hong Kong · School of Intelligence Science and Technology, Peking University · Work done during Jiazhan’s visiting at the Department of Statistics, University of Oxford. +2

Mathematical Problem Generation Reinforcement Learning with Verifiable Rewards Multi-Agent LLMs LLM Mathematical Reasoning

May 7, 2026

Beyond Negative Rollouts: Positive-Only Policy Optimization with Implicit Negative Gradients

Reinforcement learning with verifiable rewards (RLVR), due to the deterministic verification, becomes a dominant paradigm for enhancing the reasoning ability of large language models (LLMs). The community witnesses the rapid change from the Proximal Policy Optimization (PPO) to Group Relative Policy Optimization (GRPO), in which GRPO reduces the complicated advantage estimation with simple estimation over grouped positive and negative rollouts. However, we note that negative rollouts may admit no gradation of failure severity, and the combinatorial vastness makes penalizing a few sampled negatives unlikely to cover a meaningful reward signal under sparse binary rewards. In this work, we propose Positive-Only Policy Optimization (POPO), a novel RLVR framework in which learning can occur exclusively via online positive rollouts. Specifically, POPO utilizes bounded importance sampling over the positive rollout set. Thus, no disjoint negative rollouts are used for the gradient guidance. We show that implicit negative gradients can emerge naturally through reinforcing the positive probability via rollouts redistribution. Next, POPO stabilizes the policy optimization through two mechanisms. First, it applies a siamese policy network with a momentum-based adaptation law for stabilized policy evolution. Second, we replace the KL-divergence with a bounded similarity penalty term in the siamese representation space. We conduct extensive experiments using publicly available, well-established text-LLM models, e.g., the Qwen family, across all-level mathematical benchmarks. Our experiment demonstrates that POPO achieves performance comparable to, or even superior to GRPO. Notably, we show that POPO can achieve 36.67% in AIME 2025 with Qwen-Math-7B, outperforming GRPO 30.00%. Our ablation and sweep studies further illustrate the necessity and robustness of POPO components.

Mingwei Xu, Hao Fang

University of Washington, Seattle, USA

Reinforcement Learning RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Policy Gradient Methods

May 7, 2026

On the Implicit Reward Overfitting and the Low-rank Dynamics in RLVR

Recent extensive research has demonstrated that the enhanced reasoning capabilities acquired by models through Reinforcement Learning with Verifiable Rewards (RLVR) are primarily concentrated within the rank-1 components. Predicated on this observation, we employed Periodic Rank-1 Substitution and identified a counterintuitive phenomenon: RLVR may exhibit implicit reward overfitting to the training dataset. Specifically, the model can achieve satisfactory performance on the test set even when its rewards remain relatively low during the training process. Furthermore, we characterize three distinct properties of RL training: (1) The effective rank-1 component in RLVR don't maintain other model knowledge except mathematical reasoning capability. (2) RLVR fundamentally functions by optimizing a specific singular spectrum. The distribution of singular values of almost all linear layers in RLVR-trained model behaves like heavy-tailed distribution. (3) the left singular vectors associated with rank-1 components demonstrate a stronger alignment tendency during training, which echoes the discovery that RLVR is optimizing sampling efficiency in essence. Taken together, our findings and analysis further reveal how RLVR shapes model parameters and offer potential insights for improving existing RL paradigms or other training paradigms to implement continual learning.

Hao Ye, Jisheng Dang, Junfeng Fang +7

Lanzhou University · National University of Singapore · Beijing Institute of Technology

Reward Over-Optimization Reinforcement Learning Reinforcement Learning with Verifiable Rewards Neural Network Training Dynamics

May 7, 2026

Skip What You Can Predict: Predictive Repositioning for Policy Optimization for Efficient LLM Training

Reinforcement learning with verifiable rewards (RLVR) can improve the reasoning ability of large language models, but repeatedly updating a policy on the same rollout batch is expensive: every additional update requires another backward pass, and multi-step methods pay for all intermediate optimization steps. We introduce Predictive Repositioning for Policy Optimization (PrePO), which uses two observed optimizer transitions to estimate a farther point along the same-batch optimization trajectory, moves partway toward that point, evaluates the original objective there, and applies a corrective update. This gives PrePO a fixed active cost that does not grow with the virtual optimization depth. Our analysis gives finite-horizon error bounds for AdamW and Muon and shows that sufficiently accurate endpoint estimates preserve the usual descent and convergence behavior of smooth gradient descent. In RLVR experiments, PrePO reaches matched performance targets in fewer optimization steps and lower wall-clock time than the corresponding baselines. We further evaluate the same update mechanism in supervised fine-tuning on a different dataset, showing that its use is not restricted to the original RLVR setting. Together, these results suggest that PrePO provides a practical mechanism for approximating repeated same-batch optimization while avoiding the cost of explicitly executing every intermediate update.

Ismam Nur Swapnil, Aranya Saha, Tasneea Zahra +3

Bangladesh University of Engineering and Technology · University of Maryland, College Park · Illinois Institute of Technology +1

RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Policy Gradient Methods RL Fine-Tuning

May 7, 2026

OPSD Compresses What RLVR Teaches: A Post-RL Compaction Stage for Reasoning Models

On-Policy Self-Distillation (OPSD) has recently emerged as an alternative to Reinforcement Learning with Verifiable Rewards (RLVR), promising higher accuracy and shorter responses through token-level credit assignment from a self-teacher conditioned on privileged context. However, this promise does not carry over to thinking-enabled mathematical reasoning, where reported accuracy gains shrink and sometimes turn negative. We hypothesize that hindsight supervision can specify better token-level alternatives in short thinking-disabled outputs, but in long thinking-enabled traces it more readily identifies redundancy than supplies better replacements. To test this, we applied OPSD separately to correct and incorrect rollout groups, so that compression and correction can be observed in isolation. Our results show that in thinking-enabled mathematical reasoning, OPSD behaves most reliably as a compression mechanism rather than a correction mechanism: training only on correct rollouts preserves accuracy while substantially shortening responses, whereas training only on incorrect rollouts damages accuracy. In light of these findings, we propose a revised post-training pipeline for thinking-enabled mathematical reasoning: SFT then RLVR then OPSD.

Jaehoon Kim, Dongha Lee

Yonsei University

Reinforcement Learning with Verifiable Rewards On-Policy Distillation RL Post-Training LLM Mathematical Reasoning CoT Compression

May 7, 2026

Listwise Policy Optimization: Group-based RLVR as Target-Projection on the LLM Response Simplex

Reinforcement learning with verifiable rewards (RLVR) has become a standard approach for large language models (LLMs) post-training to incentivize reasoning capacity. Among existing recipes, group-based policy gradient is prevalent, which samples a group of responses per prompt and updates the policy via group-relative advantage signals. This work reveals that these optimization strategies share a common geometric structure: each implicitly defines a target distribution on the response simplex and projects toward it via first-order approximation. Building on this insight, we propose Listwise Policy Optimization (LPO) to explicitly conduct the target-projection, which demystifies the implicit target by restricting the proximal RL objective to the response simplex, and then projects the policy via exact divergence minimization. This framework provides (i) monotonic improvement on the listwise objective with bounded, zero-sum, and self-correcting projection gradients, and (ii) flexibility in divergence selection with distinct structural properties through the decoupled projection step. On diverse reasoning tasks and LLM backbones, LPO consistently improves training performance over typical policy gradient baselines under matched targets, while intrinsically preserving optimization stability and response diversity.

Yun Qu, Qi Wang, Yixiu Mao +11

Department of Automation, Tsinghua University · 2LLM Department, Tencent

RL for Language Model Reasoning Group Relative Policy Optimization Policy Optimization Reinforcement Learning with Verifiable Rewards RL Post-Training

May 7, 2026

Selective Successor Eligibility Traces for RLVR

Reinforcement Learning with Verifiable Rewards (RLVR) has become a key approach for improving the reasoning abilities of large language models. However, widely used critic-free algorithms such as Group Relative Policy Optimization (GRPO) suffer from coarse-grained temporal credit assignment, as they assign the same trajectory-level advantage to all tokens and fail to distinguish the contributions of different reasoning steps. To address this challenge, we propose Selective Successor Eligibility Traces (S-trace), an critic-free eligibility trace method that enables more effective credit assignment. S-trace introduces successor importance reweighting to recover downstream credit signals under PPO-style clipping and propagates them to selected historical tokens through entropy-based token selection. Theoretically, we establish an upper bound on its discrepancy from the raw eligibility traces-based policy gradient estimator, jointly controlled by the KL divergence and the trace decay factor

λ

. Experiments show that S-trace achieves absolute gains in average pass@16 of 1.80% and 6.17% over GRPO, and 2.32% and 0.87% over trace-style GRPO(

λ

), on Qwen3-1.7B and Qwen3-4B, respectively, while maintaining robust gains on Qwen3-8B and producing shorter responses.

Chaoli Mou, Xinning Chen, Zhan Zhuang +3

Southern University of Science and Technology · City University of Hong Kong

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

May 7, 2026

AGPO: Asymmetric Group Policy Optimization for Verifiable Reasoning and Search Ads Relevance at JD

Reinforcement Learning with Verifiable Rewards (RLVR) has demonstrated notable success in enhancing the reasoning performance of large language models (LLMs). However, recent studies reveal that while current RLVR methods improve sampling efficiency towards correct paths, they do not elicit fundamentally new reasoning patterns. Instead, the reasoning capability boundary of trained models often narrows compared to their base models, with base models achieving higher coverage at large sample sizes. In this work, we propose Asymmetric Group Policy Optimization (AGPO) to counteract this boundary shrinkage. AGPO adopts a negative-dominant reinforcement strategy to suppress incorrect reasoning paths, maintaining the base model's exploration capacity. For positive reinforcement, AGPO adopts a group advantage mechanism, which scales positive updates based on intra-group variance, allowing the model to focus on rare correct paths while suppressing updates from trivial paths. Our experiments on five mathematical benchmarks demonstrate that AGPO achieves state-of-the-art accuracy while consistently improving pass@

k

performance at scale. In a large-scale industrial application for search ads relevance optimization, AGPO effectively enhances the quality of the data annotation, leading to substantial performance gains in downstream student models.

Yang Xu, Kun Yao, Yiming Deng +3

Nanjing University, Nanjing, China · Peking University, Beijing, China · JD.com, Beijing, China

RL for Language Model Reasoning Policy Optimization Mathematical Reasoning Reinforcement Learning with Verifiable Rewards

May 7, 2026

Nonsense Helps: Prompt Space Perturbation Broadens Reasoning Exploration

Reinforcement learning with verifiable rewards, particularly Group Relative Policy Optimization (GRPO), has significantly advanced the reasoning capabilities of Large Language Models (LLMs). However, in complex tasks, GRPO frequently suffers from the ``zero-advantage problem'': when all sampled rollouts for a query fail, the relative advantage collapses to zero. Consequently, the model loses effective training signals for these questions, wasting the training data and computational budget. While simply increasing the sampling budget for these questions is a common remedy, the static sampling policy inherently constrains reasoning exploration, limiting the success rate. In this paper, we propose Lorem Perturbation for Exploration (LoPE), a simple yet effective training framework to break this exploration bottleneck. We posit that task-irrelevant prompt-space perturbations can shift the model's output distribution enough to unlock orthogonal reasoning pathways for hard questions. Specifically, LoPE prepends sequences stochastically assembled from Lorem Ipsum vocabulary (a pseudo-Latin placeholder text) to the prompts before resampling. Experiments across 1.7B, 4B, and 7B models demonstrate that LoPE significantly outperforms resampling with the original prompts. Further analysis reveals that other Latin-based random sequences with low perplexity are also effective perturbations. Our results establish LoPE as a strong baseline for broadening exploration in LLM reinforcement learning.

Langlin Huang, Chengsong Huang, Jinyuan Li +3

Washington University in St. Louis

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

May 7, 2026

SPARK: Self-Play with Asymmetric Reward from Knowledge Graphs

Self-play reinforcement learning has shown strong performance in domains with formally verifiable structure, such as mathematics and coding, where both problem generation and reward computation can be grounded in explicit rules. Extending this paradigm to scientific literature is more challenging: the relationships among multi-modal elements within and across documents are rarely made explicit in text, which makes automatic generation of relational reasoning questions difficult and weakens the reliability of reward signals. We propose SPARK (Self-Play with Asymmetric Reward from Knowledge Graphs), a framework that automatically constructs a unified knowledge graph (KG) from multi-document scientific literature and uses it as the structural basis for self-play. KG paths over multimodal nodes serve as a source for generating relational reasoning questions, and structured facts stored in the KG provide a basis for verifiable reward computation. A single small vision-language model (sVLM) alternates between Proposer and Solver roles under information asymmetry against a fixed KG, a design that we believe can be naturally extended toward online adaptation in future work. We evaluate SPARK on public benchmarks and a self-constructed cross-document multi-hop QA dataset. Results show that SPARK consistently outperforms flat-corpus-based self-play baselines, and the performance gap widens as hop count increases, suggesting that KG-structure grounding contributes to relational multi-hop reasoning beyond what unstructured corpus grounding can provide.

Hyobin Park, Taeseop Kim, Dong-Geol Choi

ANTLAB, South Korea · Hanbat National University, South Korea

AI for Science Self-Play RL Multimodal Reasoning Reinforcement Learning with Verifiable Rewards Multi-Hop Reasoning

May 6, 2026

Rollout Pass-Rate Control: Steering Binary-Reward RL Toward Its Most Informative Regime

Agentic reinforcement learning (RL) for software engineering spends much of its compute on stateful trajectories whose grouped binary rewards are highly skewed and weakly contrastive. We frame this as pass-rate control and show that the binary reward-side signal is strongest near a 50% rollout pass rate under four criteria: reward entropy, group-filtering survival, leave-one-out (RLOO) advantage energy under Group Relative Policy Optimization (GRPO), and success-failure pair count. We propose Prefix Sampling (PS), which replays self-generated trajectory prefixes to steer skewed groups toward this regime: successful prefixes give mostly failing groups a head start, while failing prefixes handicap mostly passing groups. Replayed states are reconstructed through the existing rollout path, and replayed tokens are masked from the loss so optimization applies only to current-policy continuations. On SWE-bench Verified, PS reaches the baseline high-score regime within evaluation variability while delivering 2.01x and 1.55x end-to-end wall-clock speedups on Qwen3-14B and Qwen3-32B; the 14B peak improves from 0.274 to 0.295. AIME 2025 experiments on 4B and 8B show the same pass-rate-control pattern, and 4B ablations attribute gains to replay, bidirectional coverage, and adaptive control.

Tianshu Zhu, Wenyu Zhang, Xiaoying Zuo +8

Baidu

RL for Code Generation Reinforcement Learning Reinforcement Learning with Verifiable Rewards

May 6, 2026

EP-GRPO: Entropy-Progress Aligned Group Relative Policy Optimization with Implicit Process Guidance

Reinforcement learning with verifiable rewards (RLVR), particularly Group Relative Policy Optimization (GRPO), has advanced LLM reasoning. However, GRPO suffers from three credit assignment failures: uniform token-level granularity that ignores heterogeneous informational value, uniform polarity that penalizes correct steps and rewards incorrect ones, and zero-variance collapse that erases outcome-driven gradients. We systematically quantify these failures, revealing highly non-uniform token informativeness, widespread step-level polarity misalignment, and substantial training waste. To address these limitations, we propose Entropy-Progress Aligned GRPO (EP-GRPO), a framework that mines the model's intrinsic information flow for dense, self-supervised guidance. EP-GRPO integrates entropy-gated modulation to prioritize high entropy decision pivots, implicit process signals from policy divergence anchored to outcome advantages for directional token-level feedback without external reward models, and cumulative entropy mapping that enables progress-aligned advantage normalization, naturally maintaining gradient flow under zero reward variance. Extensive experiments on mathematical reasoning benchmarks demonstrate that EP-GRPO achieves superior accuracy and efficiency compared to GRPO and its variants. The code will be available.

Song Yu, Li Li, Wenwen Zhao +1

School of Computer and Information Science, Southwest University, Chongqing 400715, China

RL for Language Model Reasoning Group Relative Policy Optimization Credit Assignment in RL Reinforcement Learning with Verifiable Rewards

May 5, 2026

Correct Is Not Enough: Training Reasoning Planners with Executor-Grounded Rewards

Reinforcement learning with verifiable rewards has become a common way to improve explicit reasoning in large language models, but final-answer correctness alone does not reveal whether the reasoning trace is faithful, reliable, or useful to the model that consumes it. This outcome-only signal can reinforce traces that are right for the wrong reasons, overstate reasoning gains by rewarding shortcuts, and propagate flawed intermediate states in multi-step systems. To this end, we propose TraceLift, a planner-executor training framework that treats reasoning as a consumable intermediate artifact. During planner training, the planner emits tagged reasoning. A frozen executor turns this reasoning into the final artifact for verifier feedback, while an executor-grounded reward shapes the intermediate trace. This reward multiplies a rubric-based Reasoning Reward Model (RM) score by measured uplift on the same frozen executor, crediting traces that are both high-quality and useful. To make reasoning quality directly learnable, we introduce TRACELIFT-GROUPS, a rubric-annotated reason-only dataset built from math and code seed problems. Each example is a same-problem group containing a high-quality reference trace and multiple plausible flawed traces with localized perturbations that reduce reasoning quality or solution support while preserving task relevance. Extensive experiments on code and math benchmarks show that this executor-grounded reasoning reward improves the two-stage planner-executor system over execution-only training, suggesting that reasoning supervision should evaluate not only whether a trace looks good, but also whether it helps the model that consumes it. Our code is available at: https://github.com/MasaiahHan/TraceLift

Tianyang Han, Hengyu Shi, Junjie Hu +3

1D4 Lab · 2Independent Researcher

RL for Code Generation Process Reward Models RL for Language Model Reasoning Credit Assignment in RL Reinforcement Learning with Verifiable Rewards Reward Design for RL+1

May 4, 2026

Reference-Sampled Boltzmann Projection for KL-Regularized RLVR: Target-Matched Weighted SFT, Finite One-Shot Gaps, and Policy Mirror Descent

Online reinforcement learning with verifiable rewards (RLVR) turns checkable outcomes into a scalable training signal, but it keeps rollout generation, verifier scoring, and reference-policy evaluations on the optimization path. Static weighted supervised fine-tuning (SFT) on precomputed rollouts seems to remove this bottleneck, yet a weighted likelihood is not specified by rewards alone: its sampler and weights induce the policy being fit. This paper identifies the reference-sampled weighted-SFT objective whose induced policy equals the fixed-reference KL-regularized RLVR optimizer. The optimizer is the standard Boltzmann target policy, obtained by exponentially tilting the reference policy by verifier reward. Matching a weighted-SFT induced policy to this target forces density-ratio weights; in the reference-sampled subclass, this reduces uniquely, up to prompt scaling, to the prompt-normalized Boltzmann weight

\exp(r(x,y)/β)/Z(x)

. BOLT, a Boltzmann-Targeted SFT procedure, is the empirical estimator of this projection. The finite one-shot analysis separates the exact stored-support price

β\log(1/π^*(S_N\mid x))

from partition estimation, effective-sample-size variance, generalization, optimization, and approximation errors. This decomposition explains why extra SFT epochs cannot repair missing reference-policy coverage and exposes the temperature--coverage--variance frontier. When coverage needs adaptive sampling, refreshed Boltzmann projections become KL policy mirror descent; finite inner solves enter as additive drift from the exact mirror step. Single-run Qwen experiments provide projection evidence for the target-matched weight, one-shot saturation, refreshed-sampler gains, and optimization-time savings, within the stated single-run scope.

Yao Shu, Chenxing Wei, Hongbin Lin +2

Supervised Fine-Tuning Reinforcement Learning KL-Regularized RL Policy Optimization Reinforcement Learning with Verifiable Rewards

May 4, 2026

Binary Rewards and Reinforcement Learning: Fundamental Challenges

Reinforcement learning with verifiable rewards (RLVR) has become a standard approach for improving reasoning in language models, yet models trained with RLVR often suffer from diversity collapse: while single-sample accuracy improves, multi-sample coverage degrades, sometimes falling below the base model. We provide a structural account of this phenomenon grounded in the properties of binary rewards. Binary rewards create a fundamental degeneracy for policy gradient methods: the set of distributions maximizing expected reward is infinite, with no distinguished element. KL-control resolves this degeneracy by selecting, in the limit

β\to 0

, the filtered model

p_*:=a(\cdot\mid\mathcal{Y}_1)

-- the base model conditioned on validity -- which is the unique fully valid distribution closest to the base model in KL divergence. This selection operates through a nontrivial asymmetry: the tilted distribution

p_{[β]}\propto a(y)\,e^{v(y)/β}

converges to

p_*

in forward KL as

β\to 0

, yet

p_*

cannot serve as a direct optimization target because

\mathrm{KL}(q\,\|\,p_*)

is infinite for any full-support policy

q

. We develop explicit formulas relating the hyperparameter

β

to the more interpretable target validity rate

μ

. Under model misspecification -- the typical practical regime -- the pressure to decrease

β

drives the optimizer toward highly concentrated distributions over a small number of valid outputs, collapsing toward ever fewer as

β

decreases, rather than toward the filtered model. We illustrate this mechanism on a toy autoregressive experiment and discuss how alternative divergences that target

p_*

directly -- as pursued empirically by \citet{kruszewski_whatever_2026} -- avoid this failure mode by rewarding coverage of

p_*

's support rather than concentration on high-validity outputs.

Marc Dymetman

∗Scientific Consultant

Reinforcement Learning KL-Regularized RL RL for Language Model Reasoning Reinforcement Learning with Verifiable Rewards Reward Design for RL Behavioral Diversity in RL