Daily Papers

Reward models (RMs) used in reinforcement learning from human feedback (RLHF) are vulnerable to reward hacking: as the policy maximizes a learned proxy reward, true quality plateaus or degrades. We make the assumption that reward hacking is often caused by flipped advantage signs: instead of reducing the likelihood of a bad response, a flipped sign causes the update to increase it. By considering an adversarial perturbation in the RM parameter space, we can derive a certified sign-preservation radius, which is the smallest perturbation that can flip the advantage sign during policy optimization. Based on this formulation, we propose Sign-Certified Policy Optimization (SignCert-PO), down-weighting non-robust completions in the policy gradient update. Unlike prior approaches that require multiple RMs or access to the RM training data, SignCert-PO is lightweight and operates purely at the policy optimization stage using only the RM parameters and on-policy completions. On TL;DR summarization and AlpacaFarm benchmarks, SignCert-PO consistently achieves a better win rate than baselines and reduces reward hacking.

We describe an iterative procedure for optimizing policies, with guaranteed monotonic improvement. By making several approximations to the theoretically-justified procedure, we develop a practical algorithm, called Trust Region Policy Optimization (TRPO). This algorithm is similar to natural policy gradient methods and is effective for optimizing large nonlinear policies such as neural networks. Our experiments demonstrate its robust performance on a wide variety of tasks: learning simulated robotic swimming, hopping, and walking gaits; and playing Atari games using images of the screen as input. Despite its approximations that deviate from the theory, TRPO tends to give monotonic improvement, with little tuning of hyperparameters.

Reinforcement learning from verifiable rewards has emerged as a powerful technique for enhancing the complex reasoning abilities of Large Language Models (LLMs). However, these methods are fundamentally constrained by the ''learning cliff'' phenomenon: when faced with problems far beyond their current capabilities, models consistently fail, yielding a persistent zero-reward signal. In policy optimization algorithms like GRPO, this collapses the advantage calculation to zero, rendering these difficult problems invisible to the learning gradient and stalling progress. To overcome this, we introduce Scaf-GRPO (Scaffolded Group Relative Policy Optimization), a progressive training framework that strategically provides minimal guidance only when a model's independent learning has plateaued. The framework first diagnoses learning stagnation and then intervenes by injecting tiered in-prompt hints, ranging from abstract concepts to concrete steps, enabling the model to construct a valid solution by itself. Extensive experiments on challenging mathematics benchmarks demonstrate Scaf-GRPO's effectiveness, boosting the pass@1 score of the Qwen2.5-Math-7B model on the AIME24 benchmark by a relative 44.3% over a vanilla GRPO baseline. This result demonstrates our framework provides a robust and effective methodology for unlocking a model's ability to solve problems previously beyond its reach, a critical step towards extending the frontier of autonomous reasoning in LLM.

Model-free reinforcement learning algorithms have seen remarkable progress, but key challenges remain. Trust Region Policy Optimization (TRPO) is known for ensuring monotonic policy improvement through conservative updates within a trust region, backed by strong theoretical guarantees. However, its reliance on complex second-order optimization limits its practical efficiency. Proximal Policy Optimization (PPO) addresses this by simplifying TRPO's approach using ratio clipping, improving efficiency but sacrificing some theoretical robustness. This raises a natural question: Can we combine the strengths of both methods? In this paper, we introduce Simple Policy Optimization (SPO), a novel unconstrained first-order algorithm. By slightly modifying the policy loss used in PPO, SPO can achieve the best of both worlds. Our new objective improves upon ratio clipping, offering stronger theoretical properties and better constraining the probability ratio within the trust region. Empirical results demonstrate that SPO outperforms PPO with a simple implementation, particularly for training large, complex network architectures end-to-end.

Reinforcement learning with verifiable reward has recently emerged as a central paradigm for post-training large language models (LLMs); however, prevailing mean-based methods, such as Group Relative Policy Optimization (GRPO), suffer from entropy collapse and limited reasoning gains. We argue that these issues stem from overemphasizing high-probability output sequences while neglecting rare but informative reasoning paths. To address these challenges, we propose Risk-based Policy Optimization (RiskPO), which substitutes classical mean-based objectives with principled risk measures. Specifically, we introduce a Mixed Value-at-Risk objective that integrates weighted attention over multiple regions of the reward distribution, thereby amplifying gradient signals on challenging instances and preventing overconfident convergence. We further design a bundling scheme that aggregates multiple questions into bundles, thus enriching the feedback signal and yielding more stable and informative training dynamics. Theoretically, we prove that the risk-averse update alleviates entropy collapse and promotes exploration. Numerically, RiskPO achieves consistent and significant improvements in mathematical reasoning, multi-modal reasoning, and code generation benchmarks, surpassing GRPO and its variants on both Pass@1 and Pass@k metrics. Our results demonstrate that risk-based optimization provides a rigorous and effective paradigm for enhancing LLM reasoning capabilities.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as an important paradigm for unlocking reasoning capabilities in large language models, exemplified by the success of OpenAI o1 and DeepSeek-R1. Currently, Group Relative Policy Optimization (GRPO) stands as the dominant algorithm in this domain due to its stable training and critic-free efficiency. However, we argue that GRPO suffers from a structural limitation: it imposes a uniform, static trust region constraint across all samples. This design implicitly assumes signal homogeneity, a premise misaligned with the heterogeneous nature of outcome-driven learning, where advantage magnitudes and variances fluctuate significantly. Consequently, static constraints fail to fully exploit high-quality signals while insufficiently suppressing noise, often precipitating rapid entropy collapse. To address this, we propose Elastic Trust Regions (ETR), a dynamic mechanism that aligns optimization constraints with signal quality. ETR constructs a signal-aware landscape through dual-level elasticity: at the micro level, it scales clipping boundaries based on advantage magnitude to accelerate learning from high-confidence paths; at the macro level, it leverages group variance to implicitly allocate larger update budgets to tasks in the optimal learning zone. Extensive experiments on AIME and MATH benchmarks demonstrate that ETR consistently outperforms GRPO, achieving superior accuracy while effectively mitigating policy entropy degradation to ensure sustained exploration.

Current research on the Decompose-Then-Verify paradigm for evaluating the factuality of long-form text typically treats decomposition and verification in isolation, overlooking their interactions and potential misalignment. We find that existing decomposition policies, typically hand-crafted demonstrations, do not align well with downstream verifiers in terms of atomicity -- a novel metric quantifying information density -- leading to suboptimal verification results. We formulate finding the optimal decomposition policy for optimal verification as a bilevel optimization problem. To approximate a solution for this strongly NP-hard problem, we propose dynamic decomposition, a reinforcement learning framework that leverages verifier feedback to learn a policy for dynamically decomposing claims to verifier-preferred atomicity. Experimental results show that dynamic decomposition outperforms existing decomposition policies, improving verification confidence by 0.07 and accuracy by 0.12 (on a 0-1 scale) on average across varying verifiers, datasets, and atomcities of input claims.

On-policy reinforcement learning (RL) has become a popular framework for solving sequential decision problems due to its computational efficiency and theoretical simplicity. Some on-policy methods guarantee every policy update is constrained to a trust region relative to the prior policy to ensure training stability. These methods often require computationally intensive non-linear optimization or require a particular form of action distribution. In this work, we show that applying KL penalization alone is nearly sufficient to enforce such trust regions. Then, we show that introducing a "fixup" phase is sufficient to guarantee a trust region is enforced on every policy update while adding fewer than 5% additional gradient steps in practice. The resulting algorithm, which we call FixPO, is able to train a variety of policy architectures and action spaces, is easy to implement, and produces results competitive with other trust region methods.

While large language models (LLMs) have demonstrated exceptional capabilities in challenging tasks such as mathematical reasoning, existing methods to enhance reasoning ability predominantly rely on supervised fine-tuning (SFT) followed by reinforcement learning (RL) on reasoning-specific data after pre-training. However, these approaches critically depend on external supervisions--such as human labelled reasoning traces, verified golden answers, or pre-trained reward models--which limits scalability and practical applicability. In this work, we propose Entropy Minimized Policy Optimization (EMPO), which makes an early attempt at fully unsupervised LLM reasoning incentivization. EMPO does not require any supervised information for incentivizing reasoning capabilities (i.e., neither verifiable reasoning traces, problems with golden answers, nor additional pre-trained reward models). By continuously minimizing the predictive entropy of LLMs on unlabeled user queries in a latent semantic space, EMPO enables purely self-supervised evolution of reasoning capabilities with strong flexibility and practicality. Our experiments demonstrate competitive performance of EMPO on both mathematical reasoning and free-form commonsense reasoning tasks. Specifically, without any supervised signals, EMPO boosts the accuracy of Qwen2.5-Math-7B Base from 30.7\% to 48.1\% on mathematical benchmarks and improves truthfulness accuracy of Qwen2.5-7B Instruct from 87.16\% to 97.25\% on TruthfulQA.

Reinforcement Learning, particularly through policy gradient methods, has played a central role in enabling reasoning capabilities of Large Language Models. However, the optimization stability of policy gradients in this setting remains understudied. As a result, existing implementations often resort to conservative hyperparameter choices to ensure stability, which requires more training samples and increases computational costs. Hence, developing models for reliably tracking the underlying optimization dynamics and leveraging them into training enables more sample-efficient regimes and further unleashes scalable post-training. We address this gap by formalizing the stochastic optimization problem of policy gradients with explicit consideration of second-order geometry. We propose a tractable computational framework that tracks and leverages curvature information during policy updates. We further employ this framework to design interventions in the optimization process through data selection. The resultant algorithm, Curvature-Aware Policy Optimization (CAPO), identifies samples that contribute to unstable updates and masks them out. Theoretically, we establish monotonic improvement guarantees under realistic assumptions. On standard math reasoning benchmarks, we empirically show that CAPO ensures stable updates under aggressive learning regimes where baselines catastrophically fail. With minimal intervention (rejecting fewer than 8% of tokens), CAPO achieves up to 30x improvement in sample efficiency over standard GRPO for LLM reasoning.

Primal-dual safe RL methods commonly perform iterations between the primal update of the policy and the dual update of the Lagrange Multiplier. Such a training paradigm is highly susceptible to the error in cumulative cost estimation since this estimation serves as the key bond connecting the primal and dual update processes. We show that this problem causes significant underestimation of cost when using off-policy methods, leading to the failure to satisfy the safety constraint. To address this issue, we propose conservative policy optimization, which learns a policy in a constraint-satisfying area by considering the uncertainty in cost estimation. This improves constraint satisfaction but also potentially hinders reward maximization. We then introduce local policy convexification to help eliminate such suboptimality by gradually reducing the estimation uncertainty. We provide theoretical interpretations of the joint coupling effect of these two ingredients and further verify them by extensive experiments. Results on benchmark tasks show that our method not only achieves an asymptotic performance comparable to state-of-the-art on-policy methods while using much fewer samples, but also significantly reduces constraint violation during training. Our code is available at https://github.com/ZifanWu/CAL.

Reinforcement Learning with Verifiable Rewards has recently advanced the capabilities of Large Language Models in complex reasoning tasks by providing explicit rule-based supervision. Among RLVR methods, GRPO and its variants have achieved strong empirical performance. Despite their success, we identify that they suffer from Gradient Misassignment in Positives and Gradient Domination in Negatives, which lead to inefficient and suboptimal policy updates. To address these issues, we propose Rewards as Labels (REAL), a novel framework that revisits verifiable rewards as categorical labels rather than scalar weights, thereby reformulating policy optimization as a classification problem. Building on this, we further introduce anchor logits to enhance policy learning. Our analysis reveals that REAL induces a monotonic and bounded gradient weighting, enabling balanced gradient allocation across rollouts and effectively mitigating the identified mismatches. Extensive experiments on mathematical reasoning benchmarks show that REAL improves training stability and consistently outperforms GRPO and strong variants such as DAPO. On the 1.5B model, REAL improves average Pass@1 over DAPO by 6.7%. These gains further scale to 7B model, REAL continues to outperform DAPO and GSPO by 6.2% and 1.7%, respectively. Notably, even with a vanilla binary cross-entropy, REAL remains stable and exceeds DAPO by 4.5% on average.

Behavior constrained policy optimization has been demonstrated to be a successful paradigm for tackling Offline Reinforcement Learning. By exploiting historical transitions, a policy is trained to maximize a learned value function while constrained by the behavior policy to avoid a significant distributional shift. In this paper, we propose our closed-form policy improvement operators. We make a novel observation that the behavior constraint naturally motivates the use of first-order Taylor approximation, leading to a linear approximation of the policy objective. Additionally, as practical datasets are usually collected by heterogeneous policies, we model the behavior policies as a Gaussian Mixture and overcome the induced optimization difficulties by leveraging the LogSumExp's lower bound and Jensen's Inequality, giving rise to a closed-form policy improvement operator. We instantiate offline RL algorithms with our novel policy improvement operators and empirically demonstrate their effectiveness over state-of-the-art algorithms on the standard D4RL benchmark. Our code is available at https://cfpi-icml23.github.io/.

Reinforcement learning (RL) has emerged as a powerful framework for improving the reasoning capabilities of large language models (LLMs). However, most existing RL approaches rely on sparse outcome rewards, which fail to credit correct intermediate steps in partially successful solutions. Process reward models (PRMs) offer fine-grained step-level supervision, but their scores are often noisy and difficult to evaluate. As a result, recent PRM benchmarks focus on a more objective capability: detecting the first incorrect step in a reasoning path. However, this evaluation target is misaligned with how PRMs are typically used in RL, where their step-wise scores are treated as raw rewards to maximize. To bridge this gap, we propose Verifiable Prefix Policy Optimization (VPPO), which uses PRMs only to localize the first error during RL. Given an incorrect rollout, VPPO partitions the trajectory into a verified correct prefix and an erroneous suffix based on the first error, rewarding the former while applying targeted penalties only after the detected mistake. This design yields stable, interpretable learning signals and improves credit assignment. Across multiple reasoning benchmarks, VPPO consistently outperforms sparse-reward RL and prior PRM-guided baselines on both Pass@1 and Pass@K.

Policy constraint methods to offline reinforcement learning (RL) typically utilize parameterization or regularization that constrains the policy to perform actions within the support set of the behavior policy. The elaborative designs of parameterization methods usually intrude into the policy networks, which may bring extra inference cost and cannot take full advantage of well-established online methods. Regularization methods reduce the divergence between the learned policy and the behavior policy, which may mismatch the inherent density-based definition of support set thereby failing to avoid the out-of-distribution actions effectively. This paper presents Supported Policy OpTimization (SPOT), which is directly derived from the theoretical formalization of the density-based support constraint. SPOT adopts a VAE-based density estimator to explicitly model the support set of behavior policy and presents a simple but effective density-based regularization term, which can be plugged non-intrusively into off-the-shelf off-policy RL algorithms. SPOT achieves the state-of-the-art performance on standard benchmarks for offline RL. Benefiting from the pluggable design, offline pretrained models from SPOT can also be applied to perform online fine-tuning seamlessly.

The Adam optimizer is widely used for transformer optimization in practice, which makes understanding the underlying optimization mechanisms an important problem. However, due to the Adam's complexity, theoretical analysis of how it optimizes transformers remains a challenging task. Fortunately, Sign Gradient Descent (SignGD) serves as an effective surrogate for Adam. Despite its simplicity, theoretical understanding of how SignGD optimizes transformers still lags behind. In this work, we study how SignGD optimizes a two-layer transformer -- consisting of a softmax attention layer with trainable query-key parameterization followed by a linear layer -- on a linearly separable noisy dataset. We identify four stages in the training dynamics, each exhibiting intriguing behaviors. Based on the training dynamics, we prove the fast convergence but poor generalization of the learned transformer on the noisy dataset. We also show that Adam behaves similarly to SignGD in terms of both optimization and generalization in this setting. Additionally, we find that the poor generalization of SignGD is not solely due to data noise, suggesting that both SignGD and Adam requires high-quality data for real-world tasks. Finally, experiments on synthetic and real-world datasets empirically support our theoretical results.

We propose Process-Aware Policy Optimization (PAPO), a method that integrates process-level evaluation into Group Relative Policy Optimization (GRPO) through decoupled advantage normalization, to address two limitations of existing reward designs. Outcome reward models (ORM) evaluate only final-answer correctness, treating all correct responses identically regardless of reasoning quality, and gradually lose the advantage signal as groups become uniformly correct. Process reward models (PRM) offer richer supervision, but directly using PRM scores causes reward hacking, where models exploit verbosity to inflate scores while accuracy collapses. PAPO resolves both by composing the advantage from an outcome component Aout, derived from ORM and normalized over all responses, and a process component Aproc, derived from a rubric-based PRM and normalized exclusively among correct responses. This decoupled design ensures that Aout anchors training on correctness while Aproc differentiates reasoning quality without distorting the outcome signal. Experiments across multiple model scales and six benchmarks demonstrate that PAPO consistently outperforms ORM, reaching 51.3% vs.\ 46.3% on OlympiadBench while continuing to improve as ORM plateaus and declines.

Adversarial training is well-known to produce high-quality neural network models that are empirically robust against adversarial perturbations. Nevertheless, once a model has been adversarially trained, one often desires a certification that the model is truly robust against all future attacks. Unfortunately, when faced with adversarially trained models, all existing approaches have significant trouble making certifications that are strong enough to be practically useful. Linear programming (LP) techniques in particular face a "convex relaxation barrier" that prevent them from making high-quality certifications, even after refinement with mixed-integer linear programming (MILP) and branch-and-bound (BnB) techniques. In this paper, we propose a nonconvex certification technique, based on a low-rank restriction of a semidefinite programming (SDP) relaxation. The nonconvex relaxation makes strong certifications comparable to much more expensive SDP methods, while optimizing over dramatically fewer variables comparable to much weaker LP methods. Despite nonconvexity, we show how off-the-shelf local optimization algorithms can be used to achieve and to certify global optimality in polynomial time. Our experiments find that the nonconvex relaxation almost completely closes the gap towards exact certification of adversarially trained models.

Lion (Evolved Sign Momentum), a new optimizer discovered through program search, has shown promising results in training large AI models. It performs comparably or favorably to AdamW but with greater memory efficiency. As we can expect from the results of a random search program, Lion incorporates elements from several existing algorithms, including signed momentum, decoupled weight decay, Polak, and Nesterov momentum, but does not fit into any existing category of theoretically grounded optimizers. Thus, even though Lion appears to perform well as a general-purpose optimizer for a wide range of tasks, its theoretical basis remains uncertain. This lack of theoretical clarity limits opportunities to further enhance and expand Lion's efficacy. This work aims to demystify Lion. Based on both continuous-time and discrete-time analysis, we demonstrate that Lion is a theoretically novel and principled approach for minimizing a general loss function f(x) while enforcing a bound constraint |x|_infty leq 1/lambda. Lion achieves this through the incorporation of decoupled weight decay, where lambda represents the weight decay coefficient. Our analysis is made possible by the development of a new Lyapunov function for the Lion updates. It applies to a broader family of Lion-kappa algorithms, where the sign(cdot) operator in Lion is replaced by the subgradient of a convex function kappa, leading to the solution of a general composite optimization problem of min_x f(x) + kappa^*(x). Our findings provide valuable insights into the dynamics of Lion and pave the way for further improvements and extensions of Lion-related algorithms.

Recent advancements in Large Reasoning Models (LRMs), exemplified by DeepSeek-R1, have underscored the potential of scaling inference-time compute through Group Relative Policy Optimization (GRPO). However, GRPO frequently suffers from gradient signal attenuation when encountering problems that are either too trivial or overly complex. In these scenarios, the disappearance of inter-group advantages makes the gradient signal susceptible to noise, thereby jeopardizing convergence stability. While variants like DAPO attempt to rectify gradient vanishing, they do not alleviate the substantial computational overhead incurred by exhaustive rollouts on low-utility samples. In this paper, we propose Difficulty-Estimated Policy Optimization (DEPO), a novel framework designed to optimize the efficiency and robustness of reasoning alignment. DEPO integrates an online Difficulty Estimator that dynamically assesses and filters training data before the rollout phase. This mechanism ensures that computational resources are prioritized for samples with high learning potential. Empirical results demonstrate that DEPO achieves up to a 2x reduction in rollout costs without compromising model performance. Our approach significantly lowers the computational barrier for training high-performance reasoning models, offering a more sustainable path for reasoning scaling. Code and data will be released upon acceptance.

Reinforcement Learning with Verifiable Rewards (RLVR) offers a promising framework for optimizing large language models in reasoning tasks. However, existing RLVR algorithms focus on different granularities, and each has complementary strengths and limitations. Group Relative Policy Optimization (GRPO) updates the policy with token-level importance ratios, which preserves fine-grained credit assignment but often suffers from high variance and instability. In contrast, Group Sequence Policy Optimization (GSPO) applies single sequence-level importance ratios across all tokens in a response that better matches sequence-level rewards, but sacrifices token-wise credit assignment. In this paper, we propose Dynamic Hybrid Policy Optimization (DHPO) to bridge GRPO and GSPO within a single clipped surrogate objective. DHPO combines token-level and sequence-level importance ratios using weighting mechanisms. We explore two variants of the mixing mechanism, including an averaged mixing and an entropy-guided mixing. To further stabilize training, we employ a branch-specific clipping strategy that constrains token-level and sequence-level ratios within separate trust regions before mixing, preventing outliers in either branch from dominating the update. Across seven challenging mathematical reasoning benchmarks, experiments on both dense and MoE models from the Qwen3 series show that DHPO consistently outperforms GRPO and GSPO. We will release our code upon acceptance of this paper.

Proximal constraints are fundamental to the stability of the Large Language Model reinforcement learning. While the canonical clipping mechanism in PPO serves as an efficient surrogate for trust regions, we identify a critical bottleneck: fixed bounds strictly constrain the upward update margin of low-probability actions, disproportionately suppressing high-advantage tail strategies and inducing rapid entropy collapse. To address this, we introduce Band-constrained Policy Optimization (BandPO). BandPO replaces canonical clipping with Band, a unified theoretical operator that projects trust regions defined by f-divergences into dynamic, probability-aware clipping intervals. Theoretical analysis confirms that Band effectively resolves this exploration bottleneck. We formulate this mapping as a convex optimization problem, guaranteeing a globally optimal numerical solution while deriving closed-form solutions for specific divergences. Extensive experiments across diverse models and datasets demonstrate that BandPO consistently outperforms canonical clipping and Clip-Higher, while robustly mitigating entropy collapse.

Reinforcement Learning has emerged as the fundamental technique for enhancing reasoning in LLMs. However, existing algorithms apply uniform optimization to all tokens, ignoring their different roles in reasoning process. To address this limitation, we introduce Heterogeneous Adaptive Policy Optimization (HAPO), a comprehensive token-aware algorithm that dynamically adapts optimization based on token entropy. For rollout sampling, we propose Adaptive Temperature Sampling, which adjusts sampling temperature in real time, promoting exploration at high-entropy tokens while preserving coherence at low-entropy ones. For advantage calculation, we introduce Token Level Group Average that normalizes advantages at token level, jointly accounting for sequence-length as in token-mean loss while preserving non-biased treatment. We then develop Differential Advantage Redistribution that leverages entropy and importance ratios to modulate rewards-adjusting updates for tokens with clear signals. For clipping loss, we design Asymmetric Adaptive Clipping, allowing aggressive probability reduction for noisy low-entropy tokens while enabling exploration for high-entropy tokens. Through systematic investigation between entropy and training dynamics, we embedded token-level treatment into every stages to achieve fine-grained control. Extensive experiments demonstrate that HAPO consistently outperforms DAPO across multiple model scales. Our code can be found in https://github.com/starriver030515/HAPO.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as an indispensable paradigm for enhancing reasoning in Large Language Models (LLMs). However, standard policy optimization methods, such as Group Relative Policy Optimization (GRPO), often converge to low-entropy policies, leading to severe mode collapse and limited output diversity. We analyze this issue from the perspective of sampling probability dynamics, identifying that the standard objective disproportionately reinforces the highest-likelihood paths, thereby suppressing valid alternative reasoning chains. To address this, we propose a novel Advantage Re-weighting Mechanism (ARM) designed to equilibrate the confidence levels across all correct responses. By incorporating Prompt Perplexity and Answer Confidence into the advantage estimation, our method dynamically reshapes the reward signal to attenuate the gradient updates of over-confident reasoning paths, while redistributing probability mass toward under-explored correct solutions. Empirical results demonstrate that our approach significantly enhances generative diversity and response entropy while maintaining competitive accuracy, effectively achieving a superior trade-off between exploration and exploitation in reasoning tasks. Empirical results on Qwen2.5 and DeepSeek models across mathematical and coding benchmarks show that ProGRPO significantly mitigates entropy collapse. Specifically, on Qwen2.5-7B, our method outperforms GRPO by 5.7% in Pass@1 and, notably, by 13.9% in Pass@32, highlighting its superior capability in generating diverse correct reasoning paths.

Reinforcement Learning with Verifiable Rewards (RLVR) has improved the reasoning abilities of Large Language Models (LLMs) by using rule-based binary feedback, helping to mitigate reward hacking. However, current RLVR methods typically treat whole responses as single actions, assigning the same reward to every token. This coarse-grained feedback hampers precise credit assignment, making it hard for models to identify which reasoning steps lead to success or failure, and often results in suboptimal policies and inefficient learning. Methods like PPO provide credit assignment through value estimation, but often yield inaccurate and unverifiable signals due to limited sampling. On the other hand, methods using Process Reward Models can provide step-by-step judgments for each reasoning step, but they require high-quality process supervision labels and are time-consuming when applied in online reinforcement learning (RL). To overcome these limitations, we introduce a simple but efficient method Credit Assignment Policy Optimization (CAPO). Given a reasoning response rollout from the policy model, CAPO directly leverages an off-the-shelf, general-purpose LLM as a Generative Process Reward Model (LLM-as-GenPRM) to generate all step-wise critique by one pass, thereby providing verifiable token-level rewards to refine the tokens that were originally assigned identical rule-based rewards. This enables more fine-grained credit assignment in an effective way. Furthermore, to enhance the accuracy and robustness of CAPO, we employ voting mechanisms that scale with the number of generated critiques. Extensive experiments using different backbones like Llama and Qwen models and in different sizes show that CAPO consistently outperforms supervised learning-based and RL-based fine-tuning methods across six challenging mathematical benchmarks and three out-of-domain benchmarks.

Critic-free methods like GRPO reduce memory demands by estimating advantages from multiple rollouts but tend to converge slowly, as critical learning signals are diluted by an abundance of uninformative samples and tokens. To tackle this challenge, we propose the Dynamic Dual-Level Down-Sampling (D^3S) framework that prioritizes the most informative samples and tokens across groups to improve the efficient of policy optimization. D^3S operates along two levels: (1) the sample-level, which selects a subset of rollouts to maximize advantage variance (Var(A)). We theoretically proven that this selection is positively correlated with the upper bound of the policy gradient norms, yielding higher policy gradients. (2) the token-level, which prioritizes tokens with a high product of advantage magnitude and policy entropy (|A_{i,t}|times H_{i,t}), focusing updates on tokens where the policy is both uncertain and impactful. Moreover, to prevent overfitting to high-signal data, D^3S employs a dynamic down-sampling schedule inspired by curriculum learning. This schedule starts with aggressive down-sampling to accelerate early learning and gradually relaxes to promote robust generalization. Extensive experiments on Qwen2.5 and Llama3.1 demonstrate that integrating D^3S into advanced RL algorithms achieves state-of-the-art performance and generalization while requiring fewer samples and tokens across diverse reasoning benchmarks. Our code is added in the supplementary materials and will be made publicly available.

Reinforcement Learning with Verifiable Rewards (RLVR) has proven effective for enhancing the reasoning capabilities of Large Language Models (LLMs). However, dominant approaches like Group Relative Policy Optimization (GRPO) face critical stability challenges: they suffer from high estimator variance under computational constraints (small group sizes) and vanishing gradient signals in saturated failure regimes where all responses yield identical zero rewards. To address this, we propose Empirical Bayes Policy Optimization (EBPO), a novel framework that regularizes local group-based baselines by borrowing strength from the policy's accumulated global statistics. Instead of estimating baselines in isolation, EBPO employs a shrinkage estimator that dynamically balances local group statistics with a global prior updated via Welford's online algorithm. Theoretically, we demonstrate that EBPO guarantees strictly lower Mean Squared Error (MSE), bounded entropy decay, and non-vanishing penalty signals in failure scenarios compared to GRPO. Empirically, EBPO consistently outperforms GRPO and other established baselines across diverse benchmarks, including AIME and OlympiadBench. Notably, EBPO exhibits superior training stability, achieving high-performance gains even with small group sizes, and benefits significantly from difficulty-stratified curriculum learning.

Reinforcement learning has been central to recent advances in large language model reasoning, but most algorithms rely on on-policy training that demands fresh rollouts at every update, limiting efficiency and scalability. Asynchronous RL systems alleviate this by decoupling rollout generation from training, yet their effectiveness hinges on tolerating large staleness in rollout data, a setting where existing methods either degrade in performance or collapse. We revisit this challenge and uncover a prosperity-before-collapse phenomenon: stale data can be as informative as on-policy data if exploited properly. Building on this insight, we introduce M2PO (Second-Moment Trust Policy Optimization), which constrains the second moment of importance weights to suppress only extreme outliers while preserving informative updates. Notably, M2PO sharply reduces the fraction of clipped tokens under high staleness (from 1.22% to 0.06% over training), precisely masking high-variance tokens while maintaining stable optimization. Extensive evaluation across six models (from 1.7B to 32B) and eight benchmarks shows that M2PO delivers stable off-policy training even with data stale by at least 256 model updates and matches on-policy performance.

Group-relative policy optimization methods train language models by generating multiple rollouts per prompt and normalizing rewards with a shared mean reward baseline. In resource-constrained settings where the rollout budget is small, accuracy often degrades. We find that noise in the shared baseline induces advantage sign flips, where some rollouts receive an incorrect advantage sign, and the update direction is reversed. To address this, we propose Median-Centered Group Relative Policy Optimization (MC-GRPO), a simple and effective solution for small-rollout training. Our main idea is to replace the mean baseline with a median baseline: the median is far less sensitive to outlier rewards than the mean, mitigating the sign flips under small rollout size (G). We generate one additional rollout for median reference (G+1), and compute advantages by using the group median. With an odd-sized group, exactly one completion is the median and receives zero advantage, we exclude this pivot rollout from backpropagation so the number of gradient-contributing samples per prompt remains G, preserving the core update cost of standard G-rollout training. Across various GRPO-family methods and a wide range of models and scales, this median-centered training consistently improves stability and final accuracy in the low-rollout regime, reducing the gap between G=2 and G=8 to within 1%. Code is available at https://github.com/lotusroot-kim/MC-GRPO

Language models as intelligent agents push the boundaries of sequential decision-making agents but struggle with limited knowledge of environmental dynamics and exponentially huge action space. Recent efforts like GLAM and TWOSOME manually constrain the action space to a restricted subset and employ reinforcement learning to align agents' knowledge with specific environments. However, they overlook fine-grained credit assignments for intra-action tokens, which is essential for efficient language agent optimization, and rely on human's prior knowledge to restrict action space. This paper proposes decomposing language agent optimization from the action level to the token level, offering finer supervision for each intra-action token and manageable optimization complexity in environments with unrestricted action spaces. Beginning with the simplification of flattening all actions, we theoretically explore the discrepancies between action-level optimization and this naive token-level optimization. We then derive the Bellman backup with Action Decomposition (BAD) to integrate credit assignments for both intra-action and inter-action tokens, effectively eliminating the discrepancies. Implementing BAD within the PPO algorithm, we introduce Policy Optimization with Action Decomposition (POAD). POAD benefits from a finer-grained credit assignment process and lower optimization complexity, leading to enhanced learning efficiency and generalization abilities in aligning language agents with interactive environments. We validate POAD across diverse testbeds, with results affirming the advantages of our approach and the correctness of our theoretical analysis.

Reinforcement Learning (RL) has significantly improved large language model reasoning, but existing RL fine-tuning methods rely heavily on heuristic techniques such as entropy regularization and reweighting to maintain stability. In practice, they often experience late-stage performance collapse, leading to degraded reasoning quality and unstable training. We derive that the magnitude of token-wise policy gradients in RL is negatively correlated with token probability and local policy entropy. Building on this result, we prove that training instability is driven by a tiny fraction of tokens, approximately 0.01\%, which we term spurious tokens. When such tokens appear in correct responses, they contribute little to the reasoning outcome but inherit the full sequence-level reward, leading to abnormally amplified gradient updates. Motivated by this observation, we propose Spurious-Token-Aware Policy Optimization (STAPO) for large-scale model refining, which selectively masks such updates and renormalizes the loss over valid tokens. Across six mathematical reasoning benchmarks using Qwen 1.7B, 8B, and 14B base models, STAPO consistently demonstrates superior entropy stability and achieves an average performance improvement of 7.13\% over GRPO, 20-Entropy and JustRL.

Reinforcement learning with verifiable rewards (RLVR) has emerged as a promising paradigm for enhancing the reasoning capabilities of large language models (LLMs). In this context, models explore reasoning trajectories and exploit rollouts with correct answers as positive signals for policy optimization. However, these rollouts might involve flawed patterns such as answer-guessing and jump-in-reasoning. Such flawed-positive rollouts are rewarded identically to fully correct ones, causing policy models to internalize these unreliable reasoning patterns. In this work, we first conduct a systematic study of flawed-positive rollouts in RL and find that they enable rapid capability gains during the early optimization stage, while constraining reasoning capability later by reinforcing unreliable patterns. Building on these insights, we propose Flawed-Aware Policy Optimization (FAPO), which presents a parameter-free reward penalty for flawed-positive rollouts, enabling the policy to leverage them as useful shortcuts in the warm-up stage, securing stable early gains, while gradually shifting optimization toward reliable reasoning in the later refinement stage. To accurately and comprehensively detect flawed-positive rollouts, we introduce a generative reward model (GenRM) with a process-level reward that precisely localizes reasoning errors. Experiments show that FAPO is effective in broad domains, improving outcome correctness, process reliability, and training stability without increasing the token budget.

We propose a new family of policy gradient methods for reinforcement learning, which alternate between sampling data through interaction with the environment, and optimizing a "surrogate" objective function using stochastic gradient ascent. Whereas standard policy gradient methods perform one gradient update per data sample, we propose a novel objective function that enables multiple epochs of minibatch updates. The new methods, which we call proximal policy optimization (PPO), have some of the benefits of trust region policy optimization (TRPO), but they are much simpler to implement, more general, and have better sample complexity (empirically). Our experiments test PPO on a collection of benchmark tasks, including simulated robotic locomotion and Atari game playing, and we show that PPO outperforms other online policy gradient methods, and overall strikes a favorable balance between sample complexity, simplicity, and wall-time.

Reinforcement learning has become a cornerstone technique for developing reasoning models in complex tasks, ranging from mathematical problem-solving to imaginary reasoning. The optimization of these models typically relies on policy gradient methods, whose efficacy hinges on the accurate estimation of an advantage function. However, prevailing methods typically employ static advantage estimation, a practice that leads to inefficient credit assignment by neglecting the dynamic utility of training samples over time. This limitation results in suboptimal policy updates, which in turn manifest as slower convergence rates and increased learning instability, as models fail to adapt to evolving sample utilities effectively. To address this problem, we introduce ADORA (Advantage Dynamics via Online Rollout Adaptation), a novel framework for policy optimization. ADORA dynamically adjusts the advantage function's weighting by adaptively categorizing training data into temporarily advantageous and disadvantageous samples, based on their evolving utility during online model rollouts. This tailored data differentiation strategy allows ADORA to be seamlessly integrated into existing policy optimization algorithms without significant architectural modifications, enabling the policy to prioritize learning from more informative experiences and thereby achieve more efficient policy updates. Extensive evaluations across diverse model families and varying data scales demonstrate that ADORA is a robust and efficient framework. It significantly enhances long reasoning in both geometric and mathematical tasks, consistently achieving notable performance gains without requiring sensitive hyperparameter tuning.

The higher-order correlation clustering problem is an expressive model, and recently, local search heuristics have been proposed for several applications. Certifying optimality, however, is NP-hard and practically hampered already by the complexity of the problem statement. Here, we focus on establishing partial optimality conditions for the special case of complete graphs and cubic objective functions. In addition, we define and implement algorithms for testing these conditions and examine their effect numerically, on two datasets.

The application of Reinforcement Learning with Verifiable Rewards (RLVR) to mathematical and coding domains has demonstrated significant improvements in the reasoning and problem-solving abilities of Large Language Models. Despite its success in single generation problem solving, the reinforcement learning fine-tuning process may harm the model's exploration ability, as reflected in decreased diversity of generations and a resulting degradation of performance during Best-of-N sampling for large N values. In this work, we focus on optimizing the max@k metric, a continuous generalization of pass@k. We derive an unbiased on-policy gradient estimate for direct optimization of this metric. Furthermore, we extend our derivations to the off-policy updates, a common element in modern RLVR algorithms, that allows better sample efficiency. Empirically, we show that our objective effectively optimizes max@k metric in off-policy scenarios, aligning the model with the Best-of-N inference strategy.

Reinforcement learning (RL) is a pivotal task for enhancing Large Language Model (LLM) reasoning. Conventional algorithms, however, typically adhere to a coarse-grained credit assignment paradigm, applying a uniform reward to all tokens in a sequence, a critical flaw in long-chain reasoning tasks. In this paper, we address this challenge and propose Dynamic Entropy Weighting, a novel mechanism that facilitates fine-grained rewards through two new algorithms: Group Token Policy Optimization (GTPO), which assigns an entropy-weighted reward to each token, and the analogous algorithm Sequence-Level GRPO (GRPO-S). Our approach is founded on the hypothesis that high policy entropy within a reasoning path is a powerful heuristic for cognitive effort at pivotal junctures, which can be repurposed into a learning signal. By repurposing policy entropy for reward shaping, we achieve true per-token credit assignment. Experimental results across challenging reasoning benchmarks validate the superiority of our approach, showing our methods significantly outperform a strong DAPO baseline and confirming our entropy-weighting mechanism as the key driver of this performance boost.

Policy-based optimizations are widely adopted today for the training and alignment of language models, where one of the most recent and effective approaches is Group-relative Policy Optimization (GRPO). In this paper, we reveals and analyze two major limitations of GRPO: (i) tokens frequently appear in completions with both positive and negative rewards, leading to conflicting gradient updates that can reduce their output probability, even though can be essential for maintaining proper structure; (ii) negatively rewarded completions may penalize confident responses and shift model decisions toward unlikely tokens, progressively flattening the output distribution and degrading learning. To address these issues and provide a more stable and effective policy optimization strategy, we introduce GTPO (Group-relative Trajectory-based Policy Optimization), which identifies conflict tokens, tokens appearing in the same position across completions with opposite rewards, protects them by skipping negative updates, while amplifying positive ones. To further prevent policy collapse, GTPO filters out completions whose entropy exceeds a provable threshold. Unlike GRPO, GTPO does not rely on KL-divergence regularization, eliminating the need for a reference model during training, while still ensuring greater training stability and improved performance, validated through multiple experiments on GSM8K, MATH and AIME 2024 benchmarks.

We revisit Group Relative Policy Optimization (GRPO) in both on-policy and off-policy optimization regimes. Our motivation comes from recent work on off-policy Proximal Policy Optimization (PPO), which improves training stability, sampling efficiency, and memory usage. In addition, a recent analysis of GRPO suggests that estimating the advantage function with off-policy samples could be beneficial. Building on these observations, we adapt GRPO to the off-policy setting. We show that both on-policy and off-policy GRPO objectives yield an improvement in the reward. This result motivates the use of clipped surrogate objectives in the off-policy version of GRPO. We then compare the empirical performance of reinforcement learning with verifiable rewards in post-training using both GRPO variants. Our results show that off-policy GRPO either significantly outperforms or performs on par with its on-policy counterpart.

We study the problem of training and fine-tuning expressive policies with online reinforcement learning (RL) given an offline dataset. Training expressive policy classes with online RL present a unique challenge of stable value maximization. Unlike simpler Gaussian policies commonly used in online RL, expressive policies like diffusion and flow-matching policies are parameterized by a long denoising chain, which hinders stable gradient propagation from actions to policy parameters when optimizing against some value function. Our key insight is that we can address stable value maximization by avoiding direct optimization over value with the expressive policy and instead construct an on-the-fly RL policy to maximize Q-value. We propose Expressive Policy Optimization (EXPO), a sample-efficient online RL algorithm that utilizes an on-the-fly policy to maximize value with two parameterized policies -- a larger expressive base policy trained with a stable imitation learning objective and a light-weight Gaussian edit policy that edits the actions sampled from the base policy toward a higher value distribution. The on-the-fly policy optimizes the actions from the base policy with the learned edit policy and chooses the value maximizing action from the base and edited actions for both sampling and temporal-difference (TD) backup. Our approach yields up to 2-3x improvement in sample efficiency on average over prior methods both in the setting of fine-tuning a pretrained policy given offline data and in leveraging offline data to train online.

The study of provable adversarial robustness for deep neural networks (DNNs) has mainly focused on static supervised learning tasks such as image classification. However, DNNs have been used extensively in real-world adaptive tasks such as reinforcement learning (RL), making such systems vulnerable to adversarial attacks as well. Prior works in provable robustness in RL seek to certify the behaviour of the victim policy at every time-step against a non-adaptive adversary using methods developed for the static setting. But in the real world, an RL adversary can infer the defense strategy used by the victim agent by observing the states, actions, etc., from previous time-steps and adapt itself to produce stronger attacks in future steps. We present an efficient procedure, designed specifically to defend against an adaptive RL adversary, that can directly certify the total reward without requiring the policy to be robust at each time-step. Our main theoretical contribution is to prove an adaptive version of the Neyman-Pearson Lemma -- a key lemma for smoothing-based certificates -- where the adversarial perturbation at a particular time can be a stochastic function of current and previous observations and states as well as previous actions. Building on this result, we propose policy smoothing where the agent adds a Gaussian noise to its observation at each time-step before passing it through the policy function. Our robustness certificates guarantee that the final total reward obtained by policy smoothing remains above a certain threshold, even though the actions at intermediate time-steps may change under the attack. Our experiments on various environments like Cartpole, Pong, Freeway and Mountain Car show that our method can yield meaningful robustness guarantees in practice.

Reinforcement learning has significantly enhanced the reasoning capabilities of Large Language Models (LLMs) in complex problem-solving tasks. Recently, the introduction of DeepSeek R1 has inspired a surge of interest in leveraging rule-based rewards as a low-cost alternative for computing advantage functions and guiding policy optimization. However, a common challenge observed across many replication and extension efforts is that when multiple sampled responses under a single prompt converge to identical outcomes, whether correct or incorrect, the group-based advantage degenerates to zero. This leads to vanishing gradients and renders the corresponding samples ineffective for learning, ultimately limiting training efficiency and downstream performance. To address this issue, we propose a consistency-aware policy optimization framework that introduces a structured global reward based on outcome consistency, the global loss based on it ensures that, even when model outputs show high intra-group consistency, the training process still receives meaningful learning signals, which encourages the generation of correct and self-consistent reasoning paths from a global perspective. Furthermore, we incorporate an entropy-based soft blending mechanism that adaptively balances local advantage estimation with global optimization, enabling dynamic transitions between exploration and convergence throughout training. Our method introduces several key innovations in both reward design and optimization strategy. We validate its effectiveness through substantial performance gains on multiple mathematical reasoning benchmarks, highlighting the proposed framework's robustness and general applicability. Code of this work has been released at https://github.com/hijih/copo-code.git.

Reinforcement learning (RL) has emerged as a powerful tool for fine-tuning large language models (LLMs) to improve complex reasoning abilities. However, state-of-the-art policy optimization methods often suffer from high computational overhead and memory consumption, primarily due to the need for multiple generations per prompt and the reliance on critic networks or advantage estimates of the current policy. In this paper, we propose A*-PO, a novel two-stage policy optimization framework that directly approximates the optimal advantage function and enables efficient training of LLMs for reasoning tasks. In the first stage, we leverage offline sampling from a reference policy to estimate the optimal value function V*, eliminating the need for costly online value estimation. In the second stage, we perform on-policy updates using a simple least-squares regression loss with only a single generation per prompt. Theoretically, we establish performance guarantees and prove that the KL-regularized RL objective can be optimized without requiring complex exploration strategies. Empirically, A*-PO achieves competitive performance across a wide range of mathematical reasoning benchmarks, while reducing training time by up to 2times and peak memory usage by over 30% compared to PPO, GRPO, and REBEL. Implementation of A*-PO can be found at https://github.com/ZhaolinGao/A-PO.

On-policy reinforcement learning (RL), particularly Proximal Policy Optimization (PPO) and Group Relative Policy Optimization (GRPO), has become the dominant paradigm for fine-tuning large language models (LLMs). While policy ratio clipping stabilizes training, this heuristic hard constraint incurs a fundamental cost: it indiscriminately truncates gradients from high-return yet high-divergence actions, suppressing rare but highly informative "eureka moments" in complex reasoning. Moreover, once data becomes slightly stale, hard clipping renders it unusable, leading to severe sample inefficiency. In this work, we revisit the trust-region objective in policy optimization and show that explicitly constraining the variance (second central moment) of the policy ratio provides a principled and smooth relaxation of hard clipping. This distributional constraint stabilizes policy updates while preserving gradient signals from valuable trajectories. Building on this insight, we propose R^2VPO (Ratio-Variance Regularized Policy Optimization), a novel primal-dual framework that supports stable on-policy learning and enables principled off-policy data reuse by dynamically reweighting stale samples rather than discarding them. We extensively evaluate R^2VPO on fine-tuning state-of-the-art LLMs, including DeepSeek-Distill-Qwen-1.5B and the openPangu-Embedded series (1B and 7B), across challenging mathematical reasoning benchmarks. Experimental results show that R^2VPO consistently achieves superior asymptotic performance, with average relative gains of up to 17% over strong clipping-based baselines, while requiring approximately 50% fewer rollouts to reach convergence. These findings establish ratio-variance control as a promising direction for improving both stability and data efficiency in RL-based LLM alignment.

Large Language Models (LLMs) have demonstrated remarkable capabilities in natural language generation and reasoning. However, their integration into automated software ecosystems is often hindered by the "Structure Gap" - the inherent tension between the probabilistic nature of token generation and the deterministic requirements of structured data formats (e.g., JSON, XML). Traditional Supervised Fine-Tuning (SFT) often fails to enforce strict syntactic constraints, leading to "hallucinated" keys or malformed structures, while constrained decoding methods impose significant inference latency. In this paper, we propose a lightweight, efficient Reinforcement Learning (RL) framework to bridge this gap. We introduce a novel Multi-dimensional Reward Function that decomposes the structured output task into a hierarchy of constraints: structural integrity, format correctness, content accuracy, and validity. Leveraging Gradient Regularized Policy Optimization (GRPO), we enable the model to internalize these constraints without the need for a separate critic network, reducing peak VRAM usage by 40% compared to PPO. We validate our approach on multiple tasks, including complex recipe generation and structured math reasoning (GSM8K-JSON). Experimental results demonstrate that our method achieves 89.7% structural accuracy and 92.1% JSON validity, significantly outperforming both zero-shot baselines (e.g., GPT-3.5) and SFT on larger models like LLaMA-3-8B. Furthermore, we provide a detailed analysis of training dynamics, revealing a distinct self-paced curriculum where the model sequentially acquires syntactic proficiency before semantic accuracy. Our model is publicly available at https://huggingface.co/Freakz3z/Qwen-JSON.

Safety and trustworthiness are indispensable requirements for real-world applications of AI systems using large language models (LLMs). This paper formulates human value alignment as an optimization problem of the language model policy to maximize reward under a safety constraint, and then proposes an algorithm, Stepwise Alignment for Constrained Policy Optimization (SACPO). One key idea behind SACPO, supported by theory, is that the optimal policy incorporating reward and safety can be directly obtained from a reward-aligned policy. Building on this key idea, SACPO aligns LLMs step-wise with each metric while leveraging simple yet powerful alignment algorithms such as direct preference optimization (DPO). SACPO offers several advantages, including simplicity, stability, computational efficiency, and flexibility of algorithms and datasets. Under mild assumptions, our theoretical analysis provides the upper bounds on optimality and safety constraint violation. Our experimental results show that SACPO can fine-tune Alpaca-7B better than the state-of-the-art method in terms of both helpfulness and harmlessness.

Reinforcement learning with verifiable rewards has emerged as a powerful paradigm for training intelligent agents. However, existing methods typically employ binary rewards that fail to capture quality differences among trajectories achieving identical outcomes, thereby overlooking potential diversity within the solution space. Inspired by the ``sweet spot'' concept in tennis-the racket's core region that produces optimal hitting effects, we introduce Sweet Spot Learning (SSL), a novel framework that provides differentiated guidance for agent optimization. SSL follows a simple yet effective principle: progressively amplified, tiered rewards guide policies toward the sweet-spot region of the solution space. This principle naturally adapts across diverse tasks: visual perception tasks leverage distance-tiered modeling to reward proximity, while complex reasoning tasks reward incremental progress toward promising solutions. We theoretically demonstrate that SSL preserves optimal solution ordering and enhances the gradient signal-to-noise ratio, thereby fostering more directed optimization. Extensive experiments across GUI perception, short/long-term planning, and complex reasoning tasks show consistent improvements over strong baselines on 12 benchmarks, achieving up to 2.5X sample efficiency gains and effective cross-task transferability. Our work establishes SSL as a general principle for training capable and robust agents.

Although Large Reasoning Models (LRMs) have progressed in solving complex problems, their chain-of-thought (CoT) reasoning often contains harmful content that can persist even when the final responses appear safe. We show that this issue still remains in existing methods which overlook the unique significance of safe reasoning, undermining their trustworthiness and posing potential risks in applications if unsafe reasoning is accessible for and exploited by malicious users. We therefore shift our focus to aligning the safety of reasoning itself in this paper and explore process supervision as the solution. However, simply rewarding safe reasoning proves inadequate due to low rollout diversity and limited training signals. To tackle this challenge, we first delve into the characteristics of safe reasoning and uncover several critical insights that 1) safe reasoning is often consolidated by a few critical steps of safety triggers; 2) compliance cues strongly correlate with unsafe continuations; and 3) corrective interventions reliably steer unsafe trajectories towards safer traces. Motivated by these, we propose Intervened Preference Optimization (IPO), an alignment method that enforces safe reasoning by substituting compliance steps with safety triggers and constructing pairs for preference learning with strong signals. Experiments on jailbreak and adversarial safety benchmarks demonstrate that IPO remarkably improves overall safety regarding both reasoning and responses, outperforming SFT-based and RL-based baselines with a relative reduction of over 30% in harmfulness, while preserving excellent performance across diverse reasoning tasks. The results highlight the importance of explicit alignment for reasoning and provide a practical path to safer LRMs.

Certified defenses based on convex relaxations are an established technique for training provably robust models. The key component is the choice of relaxation, varying from simple intervals to tight polyhedra. Counterintuitively, loose interval-based training often leads to higher certified robustness than what can be achieved with tighter relaxations, which is a well-known but poorly understood paradox. While recent works introduced various improvements aiming to circumvent this issue in practice, the fundamental problem of training models with high certified robustness remains unsolved. In this work, we investigate the underlying reasons behind the paradox and identify two key properties of relaxations, beyond tightness, that impact certified training dynamics: continuity and sensitivity. Our extensive experimental evaluation with a number of popular convex relaxations provides strong evidence that these factors can explain the drop in certified robustness observed for tighter relaxations. We also systematically explore modifications of existing relaxations and discover that improving unfavorable properties is challenging, as such attempts often harm other properties, revealing a complex tradeoff. Our findings represent an important first step towards understanding the intricate optimization challenges involved in certified training.

Large reasoning models (LRMs) exhibit diverse high-level reasoning patterns (e.g., direct solution, reflection-and-verification, and exploring multiple solutions), yet prevailing training recipes implicitly bias models toward a limited set of dominant patterns. Through a systematic analysis, we identify substantial accuracy variance across these patterns on mathematics and science benchmarks, revealing that a model's default reasoning pattern is often sub-optimal for a given problem. To address this, we introduce Group Pattern Selection Optimization (GPSO), a reinforcement learning framework that extends GRPO by incorporating multi-pattern rollouts, verifier-guided optimal pattern selection per problem, and attention masking during optimization to prevent the leakage of explicit pattern suffixes into the learned policy. By exploring a portfolio of diverse reasoning strategies and optimizing the policy on the most effective ones, GPSO enables the model to internalize the mapping from problem characteristics to optimal reasoning patterns. Extensive experiments demonstrate that GPSO delivers consistent and substantial performance gains across various model backbones and benchmarks, effectively mitigating pattern sub-optimality and fostering more robust, adaptable reasoning. All data and codes are available at https://github.com/wanghanbinpanda/GPSO.

Large language models are increasingly trained as heterogeneous systems spanning multiple domains, expert partitions, and agentic pipelines, yet prevalent proximal objectives operate at a single scale and lack a principled mechanism for coupling token-level, trajectory-level, and higher-level hierarchical stability control. To bridge this gap, we derive the Aggregational Policy Censoring Objective (APC-Obj), the first exact unconstrained reformulation of sample-based TV-TRPO, establishing that clipping-based surrogate design and trust-region optimization are dual formulations of the same problem. Building on this foundation, we develop Fiber Bundle Gating (FBG), an algebraic framework that organizes sampled RL data as a fiber bundle and decomposes ratio gating into a base-level gate on trajectory aggregates and a fiber-level gate on per-token residuals, with provable first-order agreement with the true RL objective near on-policy. From APC-Obj and FBG we derive Fibration Policy Optimization (or simply, FiberPO), a concrete objective whose Jacobian is block-diagonal over trajectories, reduces to identity at on-policy, and provides better update direction thus improving token efficiency. The compositional nature of the framework extends beyond the trajectory-token case: fibrations compose algebraically into a Fibration Gating Hierarchy (FGH) that scales the same gating mechanism to arbitrary hierarchical depth without new primitives, as demonstrated by FiberPO-Domain, a four-level instantiation with independent trust-region budgets at the domain, prompt group, trajectory, and token levels. Together, these results connect the trust-region theory, a compositional algebraic structure, and practical multi-scale stability control into a unified framework for LLM policy optimization.

Recently, test-time scaling Large Language Models (LLMs) have demonstrated exceptional reasoning capabilities across scientific and professional tasks by generating long chains-of-thought (CoT). As a crucial component for developing these reasoning models, reinforcement learning (RL), exemplified by Proximal Policy Optimization (PPO) and its variants, allows models to learn through trial and error. However, PPO can be time-consuming due to its inherent on-policy nature, which is further exacerbated by increasing response lengths. In this work, we propose Truncated Proximal Policy Optimization (T-PPO), a novel extension to PPO that improves training efficiency by streamlining policy update and length-restricted response generation. T-PPO mitigates the issue of low hardware utilization, an inherent drawback of fully synchronized long-generation procedures, where resources often sit idle during the waiting periods for complete rollouts. Our contributions are two-folds. First, we propose Extended Generalized Advantage Estimation (EGAE) for advantage estimation derived from incomplete responses while maintaining the integrity of policy learning. Second, we devise a computationally optimized mechanism that allows for the independent optimization of the policy and value models. By selectively filtering prompt and truncated tokens, this mechanism reduces redundant computations and accelerates the training process without sacrificing convergence performance. We demonstrate the effectiveness and efficacy of T-PPO on AIME 2024 with a 32B base model. The experimental results show that T-PPO improves the training efficiency of reasoning LLMs by up to 2.5x and outperforms its existing competitors.

While Reinforcement Learning with Verifiable Rewards (RLVR) has advanced the reasoning capabilities of Large Vision-Language Models (LVLMs), most existing methods in multimodal reasoning neglect the critical role of visual perception within the RLVR optimization process. In this paper, we undertake a pioneering exploration of multimodal RLVR through the novel perspective of token perception, which measures the visual dependency of each generated token. With a granular analysis of Chain-of-Thought (CoT) processes, we uncover two key insights: first, token perception in a rollout trajectory is sparsely distributed, where only a small fraction of tokens have high visual dependency for visually-grounded reasoning; second, different trajectories exhibit significant divergence in their overall visual dependency. Based on these observations, we propose Visually-Perceptive Policy Optimization (VPPO), a novel policy gradient algorithm that explicitly leverages token perception to refine the learning signal. Specifically, VPPO achieves this through a dual mechanism: it reweights a trajectory's advantage by its overall visual dependency, and focuses policy updates exclusively on perceptually pivotal tokens. On a comprehensive suite of eight perception and reasoning benchmarks, VPPO demonstrates substantial gains over leading open-source RL-tuned models, with its effectiveness consistently validated across 7B and 32B model scales. Our findings not only establish a new token-level perceptual perspective for analyzing multimodal RLVR but also present a novel and effective optimization strategy to significantly enhance the multimodal reasoning capabilities of LVLMs.

Large language models (LLMs) have recently advanced in reasoning when optimized with reinforcement learning (RL) under verifiable rewards. Existing methods primarily rely on outcome-based supervision to strengthen internal LLM reasoning, often leading to inefficient exploration and sparse rewards. To mitigate this issue, we propose Expert-Assisted Policy Optimization (EAPO), a novel RL framework that enhances exploration by incorporating multi-turn interactions with external experts during training. Unlike prior methods, where policies reason in isolation, EAPO incentivizes the policy to adaptively determine when and how to consult experts, yielding richer reward signals and more reliable reasoning trajectories. External assistance ultimately internalizes expert knowledge into the policy model, amplifying the model's inherent reasoning capabilities. During evaluation, the policy model has been well-optimized to solve questions independently, producing improved reasoning paths and more accurate solutions. Experiments on mathematical reasoning benchmarks, including AIME 2024, AIME 2025, and AIMO 2025, show that EAPO consistently outperforms expert-assisted workflow, expert-distilled models, and RL baselines, with an average gain of 5 points over self-exploratory models.

Policy optimization methods are powerful algorithms in Reinforcement Learning (RL) for their flexibility to deal with policy parameterization and ability to handle model misspecification. However, these methods usually suffer from slow convergence rates and poor sample complexity. Hence it is important to design provably sample efficient algorithms for policy optimization. Yet, recent advances for this problems have only been successful in tabular and linear setting, whose benign structures cannot be generalized to non-linearly parameterized policies. In this paper, we address this problem by leveraging recent advances in value-based algorithms, including bounded eluder-dimension and online sensitivity sampling, to design a low-switching sample-efficient policy optimization algorithm, LPO, with general non-linear function approximation. We show that, our algorithm obtains an varepsilon-optimal policy with only O(text{poly(d)}{varepsilon^3}) samples, where varepsilon is the suboptimality gap and d is a complexity measure of the function class approximating the policy. This drastically improves previously best-known sample bound for policy optimization algorithms, O(text{poly(d)}{varepsilon^8}). Moreover, we empirically test our theory with deep neural nets to show the benefits of the theoretical inspiration.

Optimization modeling is fundamental to decision-making across diverse domains.Despite progress in automating optimization formulation from natural language descriptions, Large Language Models (LLMs) often struggle to generate formally correct and usable models due to hallucinations, posing a challenge for reliable automation. Inspired by the success of Reinforcement Learning (RL) in enhancing Large Reasoning Models, we present Solver-Informed Reinforcement Learning (SIRL).This novel framework leverages external optimization solvers as verifiable reward mechanisms to significantly improve the authenticity of LLMs for optimization modeling.Acting as precise verifiers, these solvers automatically assess the executable code and the instance-level mathematical model represented by the associated LP file, yielding precise and comprehensive feedback signals -- including syntax, feasibility, and solution quality that directly inform the RL process. This automated verification process, powered by classic optimization solvers, also underpins our instance-enhanced self-consistency method to synthesize high-quality training data. Extensive experiments on diverse public benchmarks demonstrate that SIRL achieves state-of-the-art performance, substantially outperforming existing methods in generating accurate and executable optimization models.

Reinforcement Learning (RL) has played a central role in the recent surge of LLMs' math abilities by enabling self-improvement through binary verifier signals. In contrast, Supervised Learning (SL) is rarely considered for such verification-driven training, largely due to its heavy reliance on reference answers and inability to reflect on mistakes. In this work, we challenge the prevailing notion that self-improvement is exclusive to RL and propose Negative-aware Fine-Tuning (NFT) -- a supervised approach that enables LLMs to reflect on their failures and improve autonomously with no external teachers. In online training, instead of throwing away self-generated negative answers, NFT constructs an implicit negative policy to model them. This implicit policy is parameterized with the same positive LLM we target to optimize on positive data, enabling direct policy optimization on all LLMs' generations. We conduct experiments on 7B and 32B models in math reasoning tasks. Results consistently show that through the additional leverage of negative feedback, NFT significantly improves over SL baselines like Rejection sampling Fine-Tuning, matching or even surpassing leading RL algorithms like GRPO and DAPO. Furthermore, we demonstrate that NFT and GRPO are actually equivalent in strict-on-policy training, even though they originate from entirely different theoretical foundations. Our experiments and theoretical findings bridge the gap between SL and RL methods in binary-feedback learning systems.

Hybrid training methods for large language models combine supervised fine tuning (SFT) on expert demonstrations with reinforcement learning (RL) on model rollouts, typically at the sample level. We propose Entropy Gated Selective Policy Optimization (EGSPO), a three stage framework that extends sample level mixing with token level gradient modulation. Stage 1, SFT expert learning, establishes a reliable warm up policy using expert demonstrations with a pure SFT loss. Stage 2, RL rollout generation, samples trajectories from the current policy and computes per token predictive entropy. Stage 3, the EGSPO mechanism, applies entropy gated gradient allocation: a predictive entropy module routes high entropy tokens to full PPO updates to encourage exploration, and low entropy tokens to attenuated PPO updates to reduce variance and preserve knowledge. Critically, both branches incorporate the advantage function A_t, ensuring that incorrect trajectories receive consistent negative learning signals and preventing reinforcement of confident errors. EGSPO achieves consistent improvements on mathematical reasoning benchmarks, with gains of 3.8 percent on AIME and 2.9 percent on MATH over the CHORD phi baseline, while incurring only 3.4 percent additional computational overhead.

Large multimodal reasoning models have achieved rapid progress, but their advancement is constrained by two major limitations: the absence of open, large-scale, high-quality long chain-of-thought (CoT) data, and the instability of reinforcement learning (RL) algorithms in post-training. Group Relative Policy Optimization (GRPO), the standard framework for RL fine-tuning, is prone to gradient vanishing when reward variance is low, which weakens optimization signals and impairs convergence. This work makes three contributions: (1) We propose Variance-Aware Sampling (VAS), a data selection strategy guided by Variance Promotion Score (VPS) that combines outcome variance and trajectory diversity to promote reward variance and stabilize policy optimization. (2) We release large-scale, carefully curated resources containing ~1.6M long CoT cold-start data and ~15k RL QA pairs, designed to ensure quality, difficulty, and diversity, along with a fully reproducible end-to-end training codebase. (3) We open-source a family of multimodal reasoning models in multiple scales, establishing standardized baselines for the community. Experiments across mathematical reasoning benchmarks demonstrate the effectiveness of both the curated data and the proposed VAS. Comprehensive ablation studies and analyses provide further insight into the contributions of each component. In addition, we theoretically establish that reward variance lower-bounds the expected policy gradient magnitude, with VAS serving as a practical mechanism to realize this guarantee. Our code, data, and checkpoints are available at https://github.com/LengSicong/MMR1.

We present a method to formulate algorithm discovery as program search, and apply it to discover optimization algorithms for deep neural network training. We leverage efficient search techniques to explore an infinite and sparse program space. To bridge the large generalization gap between proxy and target tasks, we also introduce program selection and simplification strategies. Our method discovers a simple and effective optimization algorithm, Lion (Evo\textbf{Lved Sign Momentum}). It is more memory-efficient than Adam as it only keeps track of the momentum. Different from adaptive optimizers, its update has the same magnitude for each parameter calculated through the sign operation. We compare Lion with widely used optimizers, such as Adam and Adafactor, for training a variety of models on different tasks. On image classification, Lion boosts the accuracy of ViT by up to 2% on ImageNet and saves up to 5x the pre-training compute on JFT. On vision-language contrastive learning, we achieve 88.3% zero-shot and 91.1% fine-tuning accuracy on ImageNet, surpassing the previous best results by 2% and 0.1%, respectively. On diffusion models, Lion outperforms Adam by achieving a better FID score and reducing the training compute by up to 2.3x. For autoregressive, masked language modeling, and fine-tuning, Lion exhibits a similar or better performance compared to Adam. Our analysis of Lion reveals that its performance gain grows with the training batch size. It also requires a smaller learning rate than Adam due to the larger norm of the update produced by the sign function. Additionally, we examine the limitations of Lion and identify scenarios where its improvements are small or not statistically significant. The implementation of Lion is publicly available.

Reinforcement Learning with Verifiable Rewards (RLVR) has proven to be a highly effective strategy for endowing Large Language Models (LLMs) with robust multi-step reasoning abilities. However, its design and optimizations remain tailored to purely textual domains, resulting in suboptimal performance when applied to multimodal reasoning tasks. In particular, we observe that a major source of error in current multimodal reasoning lies in the perception of visual inputs. To address this bottleneck, we propose Perception-Aware Policy Optimization (PAPO), a simple yet effective extension of GRPO that encourages the model to learn to perceive while learning to reason, entirely from internal supervision signals. Notably, PAPO does not rely on additional data curation, external reward models, or proprietary models. Specifically, we introduce the Implicit Perception Loss in the form of a KL divergence term to the GRPO objective, which, despite its simplicity, yields significant overall improvements (4.4%) on diverse multimodal benchmarks. The improvements are more pronounced, approaching 8.0%, on tasks with high vision dependency. We also observe a substantial reduction (30.5%) in perception errors, indicating improved perceptual capabilities with PAPO. We conduct comprehensive analysis of PAPO and identify a unique loss hacking issue, which we rigorously analyze and mitigate through a Double Entropy Loss. Overall, our work introduces a deeper integration of perception-aware supervision into RLVR learning objectives and lays the groundwork for a new RL framework that encourages visually grounded reasoning. Project page: https://mikewangwzhl.github.io/PAPO.

Tool-integrated reasoning (TIR) augments large language models (LLMs) with the ability to invoke external tools, such as search engines and code interpreters, to solve tasks beyond the capabilities of language-only reasoning. While reinforcement learning (RL) has shown promise in improving TIR by optimizing final answer correctness, existing approaches often overlook the efficiency and cost associated with tool usage. This can lead to suboptimal behavior, including excessive tool calls that increase computational and financial overhead, or insufficient tool use that compromises answer quality. In this work, we propose Optimal Tool Call-controlled Policy Optimization (OTC-PO), a simple yet effective RL-based framework that encourages models to produce accurate answers with minimal tool calls. Our method introduces a tool-integrated reward that jointly considers correctness and tool efficiency, promoting high tool productivity. We instantiate this framework within both Proximal Policy Optimization (PPO) and Group Relative Preference Optimization (GRPO), resulting in OTC-PPO and OTC-GRPO. Experiments with Qwen-2.5 and Qwen-Math across multiple QA benchmarks show that our approach reduces tool calls by up to 73.1\% and improves tool productivity by up to 229.4\%, while maintaining comparable answer accuracy. To the best of our knowledge, this is the first RL-based framework that explicitly optimizes tool-use efficiency in TIR.

Recent studies, including DeepSeek-R1 and Kimi-k1.5, have demonstrated that reinforcement learning with rule-based, binary-valued reward functions can significantly enhance the reasoning capabilities of large language models. These models primarily utilize REINFORCE-based policy optimization techniques, such as REINFORCE with baseline and group relative policy optimization (GRPO). However, a key limitation remains: current policy optimization methods either neglect reward normalization or employ static normalization strategies, which fail to adapt to the dynamic nature of policy updates during training. This may result in unstable gradient estimates and hinder training stability. To address this issue, we propose Beta Normalization Policy Optimization (BNPO), a novel policy optimization method that adaptively normalizes rewards using a Beta distribution with dynamically updated parameters. BNPO aligns the normalization with the changing policy distribution, enabling more precise and lower-variance gradient estimation, which in turn promotes stable training dynamics. We provide theoretical analysis demonstrating BNPO's variance-reducing properties and show that it generalizes both REINFORCE and GRPO under binary-valued reward settings. Furthermore, we introduce an advantage decomposition mechanism to extend BNPO's applicability to more complex reward systems. Experimental results confirm that BNPO achieves state-of-the-art performance among policy optimization methods on reasoning tasks. The code is available at https://github.com/changyi7231/BNPO.

Direct preference optimization (DPO), a widely adopted offline preference optimization algorithm, aims to align large language models (LLMs) with human-desired behaviors using pairwise preference data. However, the winning response and the losing response within pairwise data are generated isolatedly, leading to weak correlations between them as well as suboptimal alignment performance. To address this issue, we propose an effective framework named BMC, for bridging and modeling correlations in pairwise data. Firstly, we increase the consistency and informativeness of the pairwise preference signals by targeted modifications, synthesizing a pseudo winning response through improving the losing response based on the winning response. Secondly, we identify that DPO alone is insufficient to model these correlations and capture nuanced variations. Therefore, we propose learning token-level correlations by dynamically leveraging the policy model's confidence during training. Comprehensive experiments on QA, math, and instruction-following tasks demonstrate the effectiveness of our approach, significantly surpassing competitive baselines, including DPO. Additionally, our in-depth quantitative analysis reveals the reasons behind our method's superior performance over DPO and showcases its versatility to other DPO variants.

Reinforcement Learning with Verifiable Rewards (RLVR) has recently emerged as a powerful paradigm for facilitating the self-improvement of large language models (LLMs), particularly in the domain of complex reasoning tasks. However, prevailing on-policy RL methods often contend with significant training instability and inefficiency. This is primarily due to a capacity-difficulty mismatch, where the complexity of training data frequently outpaces the model's current capabilities, leading to critically sparse reward signals and stalled learning progress. This challenge is particularly acute for smaller, more resource-efficient LLMs. To overcome this, we introduce the Guided Hybrid Policy Optimization (GHPO), a novel difficulty-aware reinforcement learning framework. GHPO dynamically calibrates task difficulty by employing adaptive prompt refinement to provide targeted guidance. This unique approach adaptively balances direct imitation learning for problems currently beyond the model's reach with exploration-based reinforcement learning for more manageable tasks, effectively creating a smooth and optimized learning curriculum. Extensive experiments demonstrate that GHPO achieves an average performance gain of approximately 5% across six challenging mathematics benchmarks, consistently outperforming strong on-policy reinforcement learning and curriculum learning baselines. Further analysis confirms that our framework significantly enhances both training stability and final reasoning performance, thus offering a scalable and efficient solution for developing powerful and robust reasoning models.

Automated alignment develops alignment systems with minimal human intervention. The key to automated alignment lies in providing learnable and accurate preference signals for preference learning without human annotation. In this paper, we introduce Self-Steering Optimization (SSO), an algorithm that autonomously generates high-quality preference signals based on predefined principles during iterative training, eliminating the need for manual annotation. SSO maintains the accuracy of signals by ensuring a consistent gap between chosen and rejected responses while keeping them both on-policy to suit the current policy model's learning capacity. SSO can benefit the online and offline training of the policy model, as well as enhance the training of reward models. We validate the effectiveness of SSO with two foundation models, Qwen2 and Llama3.1, indicating that it provides accurate, on-policy preference signals throughout iterative training. Without any manual annotation or external models, SSO leads to significant performance improvements across six subjective or objective benchmarks. Besides, the preference data generated by SSO significantly enhanced the performance of the reward model on Rewardbench. Our work presents a scalable approach to preference optimization, paving the way for more efficient and effective automated alignment.

Large reasoning models improve with more test-time computation, but often overthink, producing unnecessarily long chains-of-thought that raise cost without improving accuracy. Prior reinforcement learning approaches typically rely on a single outcome reward with trajectory-level length penalties, which cannot distinguish essential from redundant reasoning steps and therefore yield blunt compression. Although recent work incorporates step-level signals, such as offline pruning, supervised data construction, or verifier-based intermediate rewards, reasoning length is rarely treated as an explicit step-level optimization objective during RL. We propose Step-wise Adaptive Penalization (SWAP), a fine-grained framework that allocates length reduction across steps based on intrinsic contribution. We estimate step importance from the model's on-policy log-probability improvement toward the correct answer, then treat excess length as a penalty mass redistributed to penalize low-importance steps more heavily while preserving high-importance reasoning. We optimize with a unified outcome-process advantage within group-relative policy optimization. Extensive experiments demonstrate that SWAP reduces reasoning length by 64.3% on average while improving accuracy by 5.7% relative to the base model.

Large Language Models (LLMs) increasingly rely on chain-of-thought (CoT) reasoning to improve accuracy on complex tasks. However, always generating lengthy reasoning traces is inefficient, leading to excessive token usage and higher inference costs. This paper introduces the Hybrid Policy Optimization (i.e., HiPO), a framework for adaptive reasoning control that enables LLMs to selectively decide when to engage in detailed reasoning (Think-on) and when to respond directly (Think-off). Specifically, HiPO combines a hybrid data pipelineproviding paired Think-on and Think-off responseswith a hybrid reinforcement learning reward system that balances accuracy and efficiency while avoiding over-reliance on detailed reasoning. Experiments across mathematics and coding benchmarks demonstrate that HiPO can substantially reduce token length while maintaining or improving accuracy. Finally, we hope HiPO a can be a principled approach for efficient adaptive reasoning, advancing the deployment of reasoning-oriented LLMs in real-world, resource-sensitive settings.

Recent advances in large language models (LLMs) have demonstrated significant potential in hardware design automation, particularly in using natural language to synthesize Register-Transfer Level (RTL) code. Despite this progress, a gap remains between model capability and the demands of real-world RTL design, including syntax errors, functional hallucinations, and weak alignment to designer intent. Reinforcement Learning with Verifiable Rewards (RLVR) offers a promising approach to bridge this gap, as hardware provides executable and formally checkable signals that can be used to further align model outputs with design intent. However, in long, structured RTL code sequences, not all tokens contribute equally to functional correctness, and naïvely spreading gradients across all tokens dilutes learning signals. A key insight from our entropy analysis in RTL generation is that only a small fraction of tokens (e.g., always, if, assign, posedge) exhibit high uncertainty and largely influence control flow and module structure. To address these challenges, we present EARL, an Entropy-Aware Reinforcement Learning framework for Verilog generation. EARL performs policy optimization using verifiable reward signals and introduces entropy-guided selective updates that gate policy gradients to high-entropy tokens. This approach preserves training stability and concentrates gradient updates on functionally important regions of code. Our experiments on VerilogEval and RTLLM show that EARL improves functional pass rates over prior LLM baselines by up to 14.7%, while reducing unnecessary updates and improving training stability. These results indicate that focusing RL on critical, high-uncertainty tokens enables more reliable and targeted policy improvement for structured RTL code generation.

Group Relative Policy Optimization (GRPO) assigns a single scalar advantage to all tokens in a completion. For structured generations with explicit segments and objectives, this couples unrelated reward signals across segments, leading to objective interference and misattributed credit. We propose Blockwise Advantage Estimation, a family of GRPO-compatible methods that assigns each objective its own advantage and applies it only to the tokens in the corresponding text block, reducing reliance on hand-designed scalar rewards and scaling naturally to additional objectives. A key challenge is estimating advantages for later blocks whose rewards are conditioned on sampled prefixes; standard unbiased approaches require expensive nested rollouts from intermediate states. Concretely, we introduce an Outcome-Conditioned Baseline that approximates intermediate state values using only within-group statistics by stratifying samples according to a prefix-derived intermediate outcome. On math tasks with uncertainty estimation, our method mitigates reward interference, is competitive with a state-of-the-art reward-designed approach, and preserves test-time gains from confidence-weighted ensembling. More broadly, it provides a modular recipe for optimizing sequential objectives in structured generations without additional rollouts.

When fine-tuning pre-trained Large Language Models (LLMs) to align with human values and intentions, maximizing the estimated reward can lead to superior performance, but it also introduces potential risks due to deviations from the reference model's intended behavior. Most existing methods typically introduce KL divergence to constrain deviations between the trained model and the reference model; however, this may not be sufficient in certain applications that require tight risk control. In this paper, we introduce Risk-aware Direct Preference Optimization (Ra-DPO), a novel approach that incorporates risk-awareness by employing a class of nested risk measures. This approach formulates a constrained risk-aware advantage function maximization problem and then converts the Bradley-Terry model into a token-level representation. The objective function maximizes the likelihood of the policy while suppressing the deviation between a trained model and the reference model using a sequential risk ratio, thereby enhancing the model's risk-awareness. Experimental results across three open-source datasets: IMDb Dataset, Anthropic HH Dataset, and AlpacaEval, demonstrate the proposed method's superior performance in balancing alignment performance and model drift. Our code is opensourced at https://github.com/zlj123-max/Ra-DPO.

Scaling test-time compute is crucial for enhancing the reasoning capabilities of large language models (LLMs). Existing approaches typically employ reinforcement learning (RL) to maximize a verifiable reward obtained at the end of reasoning traces. However, such methods optimize only the final performance under a large and fixed token budget, which hinders efficiency in both training and deployment. In this work, we present a novel framework, AnytimeReasoner, to optimize anytime reasoning performance, which aims to improve token efficiency and the flexibility of reasoning under varying token budget constraints. To achieve this, we truncate the complete thinking process to fit within sampled token budgets from a prior distribution, compelling the model to summarize the optimal answer for each truncated thinking for verification. This introduces verifiable dense rewards into the reasoning process, facilitating more effective credit assignment in RL optimization. We then optimize the thinking and summary policies in a decoupled manner to maximize the cumulative reward. Additionally, we introduce a novel variance reduction technique, Budget Relative Policy Optimization (BRPO), to enhance the robustness and efficiency of the learning process when reinforcing the thinking policy. Empirical results in mathematical reasoning tasks demonstrate that our method consistently outperforms GRPO across all thinking budgets under various prior distributions, enhancing both training and token efficiency.

The soft-thinking paradigm for Large Language Model (LLM) reasoning can outperform the conventional discrete-token Chain-of-Thought (CoT) reasoning in some scenarios, underscoring its research and application value. However, while the discrete-token CoT reasoning pattern can be reinforced through policy optimization algorithms such as group relative policy optimization (GRPO), extending the soft-thinking pattern with Reinforcement Learning (RL) remains challenging. This difficulty stems from the complexities of injecting stochasticity into soft-thinking tokens and updating soft-thinking policies accordingly. As a result, previous attempts to combine soft-thinking with GRPO typically underperform their discrete-token GRPO counterparts. To fully unlock the potential of soft-thinking, this paper presents a novel policy optimization algorithm, SofT-GRPO, to reinforce LLMs under the soft-thinking reasoning pattern. SofT-GRPO injects the Gumbel noise into logits, employs the Gumbel-Softmax technique to avoid soft-thinking tokens outside the pre-trained embedding space, and leverages the reparameterization trick in policy gradient. We conduct experiments across base LLMs ranging from 1.5B to 7B parameters, and results demonstrate that SofT-GRPO enables soft-thinking LLMs to slightly outperform discrete-token GRPO on Pass@1 (+0.13% on average accuracy), while exhibiting a substantial uplift on Pass@32 (+2.19% on average accuracy). Codes and weights are available on https://github.com/zz1358m/SofT-GRPO-master

Applying Reinforcement Learning (RL) to Video Large Language Models (Video-LLMs) shows significant promise for complex video reasoning. However, popular Reinforcement Fine-Tuning (RFT) methods, such as outcome-based Group Relative Policy Optimization (GRPO), are limited by data preparation bottlenecks (e.g., noise or high cost) and exhibit unstable improvements in the quality of long chain-of-thoughts (CoTs) and downstream performance.To address these limitations, we propose VerIPO, a Verifier-guided Iterative Policy Optimization method designed to gradually improve video LLMs' capacity for generating deep, long-term reasoning chains. The core component is Rollout-Aware Verifier, positioned between the GRPO and Direct Preference Optimization (DPO) training phases to form the GRPO-Verifier-DPO training loop. This verifier leverages small LLMs as a judge to assess the reasoning logic of rollouts, enabling the construction of high-quality contrastive data, including reflective and contextually consistent CoTs. These curated preference samples drive the efficient DPO stage (7x faster than GRPO), leading to marked improvements in reasoning chain quality, especially in terms of length and contextual consistency. This training loop benefits from GRPO's expansive search and DPO's targeted optimization. Experimental results demonstrate: 1) Significantly faster and more effective optimization compared to standard GRPO variants, yielding superior performance; 2) Our trained models exceed the direct inference of large-scale instruction-tuned Video-LLMs, producing long and contextually consistent CoTs on diverse video reasoning tasks; and 3) Our model with one iteration outperforms powerful LMMs (e.g., Kimi-VL) and long reasoning models (e.g., Video-R1), highlighting its effectiveness and stability.

Large reasoning models (LRMs) are typically trained using reinforcement learning with verifiable reward (RLVR) to enhance their reasoning abilities. In this paradigm, policies are updated using both positive and negative self-generated rollouts, which correspond to distinct sample polarities. In this paper, we provide a systematic investigation into how these sample polarities affect RLVR training dynamics and behaviors. We find that positive samples sharpen existing correct reasoning patterns, while negative samples encourage exploration of new reasoning paths. We further explore how adjusting the advantage values of positive and negative samples at both the sample level and the token level affects RLVR training. Based on these insights, we propose an Adaptive and Asymmetric token-level Advantage shaping method for Policy Optimization, namely A3PO, that more precisely allocates advantage signals to key tokens across different polarities. Experiments across five reasoning benchmarks demonstrate the effectiveness of our approach.

Reinforcement learning (RL) post-training has increasingly demonstrated strong ability to elicit reasoning behaviors in large language models (LLMs). For training efficiency, rollouts are typically generated in an off-policy manner using an older sampling policy and then used to update the current target policy. To correct the resulting discrepancy between the sampling and target policies, most existing RL objectives rely on a token-level importance sampling ratio, primarily due to its computational simplicity and numerical stability. However, we observe that token-level correction often leads to unstable training dynamics when the degree of off-policyness is large. In this paper, we revisit LLM policy optimization under off-policy conditions and show that the theoretically rigorous correction term is the prefix importance ratio, and that relaxing it to a token-level approximation can induce instability in RL post-training. To stabilize LLM optimization under large off-policy drift, we propose a simple yet effective objective, Minimum Prefix Ratio (MinPRO). MinPRO replaces the unstable cumulative prefix ratio with a non-cumulative surrogate based on the minimum token-level ratio observed in the preceding prefix. Extensive experiments on both dense and mixture-of-experts LLMs, across multiple mathematical reasoning benchmarks, demonstrate that MinPRO substantially improves training stability and peak performance in off-policy regimes.

Program synthesis aims to create accurate, executable code from natural language descriptions. This field has leveraged the power of reinforcement learning (RL) in conjunction with large language models (LLMs), significantly enhancing code generation capabilities. This integration focuses on directly optimizing functional correctness, transcending conventional supervised losses. While current literature predominantly favors policy-based algorithms, attributes of program synthesis suggest a natural compatibility with value-based methods. This stems from rich collection of off-policy programs developed by human programmers, and the straightforward verification of generated programs through automated unit testing (i.e. easily obtainable rewards in RL language). Diverging from the predominant use of policy-based algorithms, our work explores the applicability of value-based approaches, leading to the development of our B-Coder (pronounced Bellman coder). Yet, training value-based methods presents challenges due to the enormous search space inherent to program synthesis. To this end, we propose an initialization protocol for RL agents utilizing pre-trained LMs and a conservative Bellman operator to reduce training complexities. Moreover, we demonstrate how to leverage the learned value functions as a dual strategy to post-process generated programs. Our empirical evaluations demonstrated B-Coder's capability in achieving state-of-the-art performance compared with policy-based methods. Remarkably, this achievement is reached with minimal reward engineering effort, highlighting the effectiveness of value-based RL, independent of reward designs.

Mirror descent (MD), a well-known first-order method in constrained convex optimization, has recently been shown as an important tool to analyze trust-region algorithms in reinforcement learning (RL). However, there remains a considerable gap between such theoretically analyzed algorithms and the ones used in practice. Inspired by this, we propose an efficient RL algorithm, called {\em mirror descent policy optimization} (MDPO). MDPO iteratively updates the policy by {\em approximately} solving a trust-region problem, whose objective function consists of two terms: a linearization of the standard RL objective and a proximity term that restricts two consecutive policies to be close to each other. Each update performs this approximation by taking multiple gradient steps on this objective function. We derive {\em on-policy} and {\em off-policy} variants of MDPO, while emphasizing important design choices motivated by the existing theory of MD in RL. We highlight the connections between on-policy MDPO and two popular trust-region RL algorithms: TRPO and PPO, and show that explicitly enforcing the trust-region constraint is in fact {\em not} a necessity for high performance gains in TRPO. We then show how the popular soft actor-critic (SAC) algorithm can be derived by slight modifications of off-policy MDPO. Overall, MDPO is derived from the MD principles, offers a unified approach to viewing a number of popular RL algorithms, and performs better than or on-par with TRPO, PPO, and SAC in a number of continuous control tasks. Code is available at https://github.com/manantomar/Mirror-Descent-Policy-Optimization.

We study why Tool-Integrated Reasoning (TIR) makes Large Language Models (LLMs) more capable. While LLMs integrated with tools like Python code interpreters show great promise, a principled theory explaining why this paradigm is effective has been missing. This work provides the first formal proof that TIR fundamentally expands an LLM's capabilities. We demonstrate that tools enable a strict expansion of the model's empirical and feasible support, breaking the capability ceiling of pure-text models by unlocking problem-solving strategies that are otherwise impossible or intractably verbose. To guide model behavior without compromising training stability and performance, we also introduce Advantage Shaping Policy Optimization (ASPO), a novel algorithm that directly modifies the advantage function to guide the policy behavior. We conduct comprehensive experiments on challenging mathematical benchmarks, leveraging a Python interpreter as the external tool. Our results show that the TIR model decisively outperforms its pure-text counterpart on the pass@k metric. Crucially, this advantage is not confined to computationally-intensive problems but extends to those requiring significant abstract insight. We further identify the emergent cognitive patterns that illustrate how models learn to think with tools. Finally, we report improved tool usage behavior with early code invocation and much more interactive turns with ASPO. Overall, our work provides the first principled explanation for TIR's success, shifting the focus from the mere fact that tools work to why and how they enable more powerful reasoning.

Recent Meta-learning for Black-Box Optimization (MetaBBO) methods harness neural networks to meta-learn configurations of traditional black-box optimizers. Despite their success, they are inevitably restricted by the limitations of predefined hand-crafted optimizers. In this paper, we present Symbol, a novel framework that promotes the automated discovery of black-box optimizers through symbolic equation learning. Specifically, we propose a Symbolic Equation Generator (SEG) that allows closed-form optimization rules to be dynamically generated for specific tasks and optimization steps. Within Symbol, we then develop three distinct strategies based on reinforcement learning, so as to meta-learn the SEG efficiently. Extensive experiments reveal that the optimizers generated by Symbol not only surpass the state-of-the-art BBO and MetaBBO baselines, but also exhibit exceptional zero-shot generalization abilities across entirely unseen tasks with different problem dimensions, population sizes, and optimization horizons. Furthermore, we conduct in-depth analyses of our Symbol framework and the optimization rules that it generates, underscoring its desirable flexibility and interpretability.

Group relative policy optimization (GRPO), a core methodological component of DeepSeekMath and DeepSeek-R1, has emerged as a cornerstone for scaling reasoning capabilities of large language models. Despite its widespread adoption and the proliferation of follow-up works, the theoretical properties of GRPO remain less studied. This paper provides a unified framework to understand GRPO through the lens of classical U-statistics. We demonstrate that the GRPO policy gradient is inherently a U-statistic, allowing us to characterize its mean squared error (MSE), derive the finite-sample error bound and asymptotic distribution of the suboptimality gap for its learned policy. Our findings reveal that GRPO is asymptotically equivalent to an oracle policy gradient algorithm -- one with access to a value function that quantifies the goodness of its learning policy at each training iteration -- and achieves asymptotically optimal performance within a broad class of policy gradient algorithms. Furthermore, we establish a universal scaling law that offers principled guidance for selecting the optimal group size. Empirical experiments further validate our theoretical findings, demonstrating that the optimal group size is universal, and verify the oracle property of GRPO.

Video large language models have demonstrated remarkable capabilities in video understanding tasks. However, the redundancy of video tokens introduces significant computational overhead during inference, limiting their practical deployment. Many compression algorithms are proposed to prioritize retaining features with the highest attention scores to minimize perturbations in attention computations. However, the correlation between attention scores and their actual contribution to correct answers remains ambiguous. To address the above limitation, we propose a novel Contribution-aware token Compression algorithm for VIDeo understanding (CaCoVID) that explicitly optimizes the token selection policy based on the contribution of tokens to correct predictions. First, we introduce a reinforcement learning-based framework that optimizes a policy network to select video token combinations with the greatest contribution to correct predictions. This paradigm shifts the focus from passive token preservation to active discovery of optimal compressed token combinations. Secondly, we propose a combinatorial policy optimization algorithm with online combination space sampling, which dramatically reduces the exploration space for video token combinations and accelerates the convergence speed of policy optimization. Extensive experiments on diverse video understanding benchmarks demonstrate the effectiveness of CaCoVID. Codes will be released.

Reinforcement Learning (RL) has emerged as a powerful tool for neural combinatorial optimization, enabling models to learn heuristics that solve complex problems without requiring expert knowledge. Despite significant progress, existing RL approaches face challenges such as diminishing reward signals and inefficient exploration in vast combinatorial action spaces, leading to inefficiency. In this paper, we propose Preference Optimization, a novel method that transforms quantitative reward signals into qualitative preference signals via statistical comparison modeling, emphasizing the superiority among sampled solutions. Methodologically, by reparameterizing the reward function in terms of policy and utilizing preference models, we formulate an entropy-regularized RL objective that aligns the policy directly with preferences while avoiding intractable computations. Furthermore, we integrate local search techniques into the fine-tuning rather than post-processing to generate high-quality preference pairs, helping the policy escape local optima. Empirical results on various benchmarks, such as the Traveling Salesman Problem (TSP), the Capacitated Vehicle Routing Problem (CVRP) and the Flexible Flow Shop Problem (FFSP), demonstrate that our method significantly outperforms existing RL algorithms, achieving superior convergence efficiency and solution quality.

Multi-turn GUI agents enable complex task completion through sequential decision-making, but suffer from severe context inflation as interaction history accumulates. Existing strategies either sacrifice long-term context via truncation or compromise spatial structure through token pruning. In this paper, we propose Coordinate Compression Policy Optimization (CCPO), an efficient policy optimization framework that couples visual compression with policy optimization for multi-turn GUI agents. CCPO introduces Coordinate-Aware Spatial Compression (CASC), which aggregates coordinates from multiple rollouts to capture target-relevant regions and progressively narrow historical attention around key visual areas. From interactions across rollouts, CASC adaptively constructs attention boundaries that concentrate computation on the most informative regions of the scene. We further design a Distance-Based Advantage that provides fine-grained learning signals based on distance rather than binary correctness, improving both grounding accuracy and compression quality. Extensive experiments demonstrate that CCPO achieves SOTA performance across four benchmarks with up to 55% token compression and 3.8times training speedup.

Reinforcement learning with verifiable rewards (RLVR) has become a standard paradigm for post-training large language models. While Group Relative Policy Optimization (GRPO) is widely adopted, its coarse credit assignment uniformly penalizes failed rollouts, lacking the token-level focus needed to efficiently address specific deviations. Self-Distillation Policy Optimization (SDPO) addresses this by providing denser, more targeted logit-level supervision that facilitates rapid early improvement, yet it frequently collapses during prolonged training. We trace this late-stage instability to two intrinsic flaws: self-distillation on already-correct samples introduces optimization ambiguity, and the self-teacher's signal reliability progressively degrades. To resolve these issues, we propose Sample-Routed Policy Optimization (SRPO), a unified on-policy framework that routes correct samples to GRPO's reward-aligned reinforcement and failed samples to SDPO's targeted logit-level correction. SRPO further incorporates an entropy-aware dynamic weighting mechanism to suppress high-entropy, unreliable distillation targets while emphasizing confident ones. Evaluated across five benchmarks and two model scales, SRPO achieves both the rapid early improvement of SDPO and the long-horizon stability of GRPO. It consistently surpasses the peak performance of both baselines, raising the five-benchmark average on Qwen3-8B by 3.4% over GRPO and 6.3% over SDPO, while simultaneously yielding moderate response lengths and lowering per-step compute cost by up to 17.2%.

Tool-integrated reasoning (TIR) enables LLM agents to solve tasks through planning, tool use, and iterative revision, but outcome-only reinforcement learning in this setting suffers from sparse, delayed rewards and weak step-level credit assignment. In long-horizon TIR trajectories, an early irrecoverable mistake can determine success or failure, making it crucial to localize the first irrecoverable step and leverage it for fine-grained credit assignment. We propose Error-Localized Policy Optimization (ELPO), which localizes the first irrecoverable step via binary-search rollout trees under a fixed rollout budget, converts the resulting tree into stable learning signals through hierarchical advantage attribution, and applies error-localized adaptive clipping to strengthen corrective updates on the critical step and its suffix. Across TIR benchmarks in math, science QA, and code execution, ELPO consistently outperforms strong Agentic RL baselines under comparable sampling budgets, with additional gains in Pass@K and Major@K scaling, rollout ranking quality, and tool-call efficiency. Our code will be publicly released soon.

Recently, Large Language Models (LLMs) have rapidly evolved, approaching Artificial General Intelligence (AGI) while benefiting from large-scale reinforcement learning to enhance Human Alignment (HA) and Reasoning. Recent reward-based optimization algorithms, such as Proximal Policy Optimization (PPO) and Group Relative Policy Optimization (GRPO) have achieved significant performance on reasoning tasks, whereas preference-based optimization algorithms such as Direct Preference Optimization (DPO) significantly improve the performance of LLMs on human alignment. However, despite the strong performance of reward-based optimization methods in alignment tasks , they remain vulnerable to reward hacking. Furthermore, preference-based algorithms (such as Online DPO) haven't yet matched the performance of reward-based optimization algorithms (like PPO) on reasoning tasks, making their exploration in this specific area still a worthwhile pursuit. Motivated by these challenges, we propose the Trust Region Preference Approximation (TRPA) algorithm, which integrates rule-based optimization with preference-based optimization for reasoning tasks. As a preference-based algorithm, TRPA naturally eliminates the reward hacking issue. TRPA constructs preference levels using predefined rules, forms corresponding preference pairs, and leverages a novel optimization algorithm for RL training with a theoretical monotonic improvement guarantee. Experimental results demonstrate that TRPA not only achieves competitive performance on reasoning tasks but also exhibits robust stability. The code of this paper are released and updating on https://github.com/XueruiSu/Trust-Region-Preference-Approximation.git.

The design of safety-critical agents based on large language models (LLMs) requires more than simple prompt engineering. This paper presents a comprehensive information-theoretic analysis of how rule encodings in system prompts influence attention mechanisms and compliance behaviour. We demonstrate that rule formats with low syntactic entropy and highly concentrated anchors reduce attention entropy and improve pointer fidelity, but reveal a fundamental trade-off between anchor redundancy and attention entropy that previous work failed to recognize. Through formal analysis of multiple attention architectures including causal, bidirectional, local sparse, kernelized, and cross-attention mechanisms, we establish bounds on pointer fidelity and show how anchor placement strategies must account for competing fidelity and entropy objectives. Combining these insights with a dynamic rule verification architecture, we provide a formal proof that hot reloading of verified rule sets increases the asymptotic probability of compliant outputs. These findings underscore the necessity of principled anchor design and dual enforcement mechanisms to protect LLM-based agents against prompt injection attacks while maintaining compliance in evolving domains.

Reinforcement Learning (RL) has been used to finetune Large Language Models (LLMs) using a reward model trained from preference data, to better align with human judgment. The recently introduced direct alignment methods, which are often simpler, more stable, and computationally lighter, can more directly achieve this. However, these approaches cannot optimize arbitrary rewards, and the preference-based ones are not the only rewards of interest for LLMs (eg., unit tests for code generation or textual entailment for summarization, among others). RL-finetuning is usually done with a variation of policy gradient, which calls for on-policy or near-on-policy samples, requiring costly generations. We introduce Contrastive Policy Gradient, or CoPG, a simple and mathematically principled new RL algorithm that can estimate the optimal policy even from off-policy data. It can be seen as an off-policy policy gradient approach that does not rely on important sampling techniques and highlights the importance of using (the right) state baseline. We show this approach to generalize the direct alignment method IPO (identity preference optimization) and classic policy gradient. We experiment with the proposed CoPG on a toy bandit problem to illustrate its properties, as well as for finetuning LLMs on a summarization task, using a learned reward function considered as ground truth for the purpose of the experiments.

While reinforcement learning (RL) shows promise in training tool-use large language models (LLMs) using verifiable outcome rewards, existing methods largely overlook the potential of explicit reasoning rewards to bolster reasoning and tool utilization. Furthermore, natively combining reasoning and outcome rewards may yield suboptimal performance or conflict with the primary optimization objective. To address this, we propose advantage-weighted policy optimization (AWPO) -- a principled RL framework that effectively integrates explicit reasoning rewards to enhance tool-use capability. AWPO incorporates variance-aware gating and difficulty-aware weighting to adaptively modulate advantages from reasoning signals based on group-relative statistics, alongside a tailored clipping mechanism for stable optimization. Extensive experiments demonstrate that AWPO achieves state-of-the-art performance across standard tool-use benchmarks, significantly outperforming strong baselines and leading closed-source models in challenging multi-turn scenarios. Notably, with exceptional parameter efficiency, our 4B model surpasses Grok-4 by 16.0 percent in multi-turn accuracy while preserving generalization capability on the out-of-distribution MMLU-Pro benchmark.

Reinforcement Learning with Verifiable Rewards (RLVR) offers a robust mechanism for enhancing mathematical reasoning in large models. However, we identify a systematic lack of emphasis on more challenging questions in existing methods from both algorithmic and data perspectives, despite their importance for refining underdeveloped capabilities. Algorithmically, widely used Group Relative Policy Optimization (GRPO) suffers from an implicit imbalance where the magnitude of policy updates is lower for harder questions. Data-wise, augmentation approaches primarily rephrase questions to enhance diversity without systematically increasing intrinsic difficulty. To address these issues, we propose a two-dual MathForge framework to improve mathematical reasoning by targeting harder questions from both perspectives, which comprises a Difficulty-Aware Group Policy Optimization (DGPO) algorithm and a Multi-Aspect Question Reformulation (MQR) strategy. Specifically, DGPO first rectifies the implicit imbalance in GRPO via difficulty-balanced group advantage estimation, and further prioritizes harder questions by difficulty-aware question-level weighting. Meanwhile, MQR reformulates questions across multiple aspects to increase difficulty while maintaining the original gold answer. Overall, MathForge forms a synergistic loop: MQR expands the data frontier, and DGPO effectively learns from the augmented data. Extensive experiments show that MathForge significantly outperforms existing methods on various mathematical reasoning tasks. The code and augmented data are all available at https://github.com/AMAP-ML/MathForge.

Large language models (LLMs) excel at general mathematical reasoning but fail catastrophically on specialized technical mathematics. In wireless communications, where problems require precise manipulation of information-theoretic bounds, optimization constraints, and signal processing formulations, even state-of-the-art models struggle to achieve competent performance. We present WirelessMathLM, demonstrating that compact models (0.5B-7B parameters) can match or exceed much larger models through domain-specific reinforcement learning with verifiable rewards. Our key insight is that wireless mathematics problems possess a unique property--verifiable correctness--that enables effective reinforcement learning without human feedback. We construct WirelessMathBench-XL, a comprehensive benchmark of 4,027 problems from 970 papers. Using Group Relative Policy Optimization (GRPO) with binary verification rewards, we train models directly from base checkpoints without supervised warm-start. Our 7B model achieves 39.5% accuracy on WirelessMathBench-XL, approaching GPT-4o (40.4%) while using about 100 times fewer parameters than DeepSeek-R1 (671B, 57.4%). Remarkably, GRPO training nearly doubles performance across all model scales (0.5B +11%, 3B +103%, 7B +81%), with positive transfer to general mathematics benchmarks--our models gain +8.4 points on average across MATH, Minerva-Math, OlympiadBench, AMC, and AIME without any training on these tasks.

Verification and validation of autonomous driving (AD) systems and components is of increasing importance, as such technology increases in real-world prevalence. Safety-critical scenario generation is a key approach to robustify AD policies through closed-loop training. However, existing approaches for scenario generation rely on simplistic objectives, resulting in overly-aggressive or non-reactive adversarial behaviors. To generate diverse adversarial yet realistic scenarios, we propose SEAL, a scenario perturbation approach which leverages learned objective functions and adversarial, human-like skills. SEAL-perturbed scenarios are more realistic than SOTA baselines, leading to improved ego task success across real-world, in-distribution, and out-of-distribution scenarios, of more than 20%. To facilitate future research, we release our code and tools: https://github.com/cmubig/SEAL

Recent research shows that Preference Alignment (PA) objectives act as divergence estimators between aligned (chosen) and unaligned (rejected) response distributions. In this work, we extend this divergence-based perspective to general alignment settings, such as reinforcement learning with verifiable rewards (RLVR), where only environmental rewards are available. Within this unified framework, we propose f-Group Relative Policy Optimization (f-GRPO), a class of on-policy reinforcement learning, and f-Hybrid Alignment Loss (f-HAL), a hybrid on/off policy objectives, for general LLM alignment based on variational representation of f-divergences. We provide theoretical guarantees that these classes of objectives improve the average reward after alignment. Empirically, we validate our framework on both RLVR (Math Reasoning) and PA tasks (Safety Alignment), demonstrating superior performance and flexibility compared to current methods.

With the proliferation of red-teaming strategies for Large Language Models (LLMs), the deficiency in the literature about improving the safety and robustness of LLM defense strategies is becoming increasingly pronounced. This paper introduces the LLM-based sentinel model as a plug-and-play prefix module designed to reconstruct the input prompt with just a few (<30) additional tokens, effectively reducing toxicity in responses from target LLMs. The sentinel model naturally overcomes the parameter inefficiency and limited model accessibility for fine-tuning large target models. We employ an interleaved training regimen using Proximal Policy Optimization (PPO) to optimize both red team and sentinel models dynamically, incorporating a value head-sharing mechanism inspired by the multi-agent centralized critic to manage the complex interplay between agents. Our extensive experiments across text-to-text and text-to-image demonstrate the effectiveness of our approach in mitigating toxic outputs, even when dealing with larger models like Llama-2, GPT-3.5 and Stable-Diffusion, highlighting the potential of our framework in enhancing safety and robustness in various applications.

Reinforcement learning with verifiable rewards (RLVR) has become a practical route to improve large language model reasoning, and Group Relative Policy Optimization (GRPO) is a widely used optimizer in this setting. However, RLVR training is typically performed with limited control over generalization. We revisit GRPO through a robustness-based generalization view, where the generalization loss is upper bounded by a combination of the empirical loss and a sharpness surrogate measured by the gradient norm. Building on this perspective, we propose Sharpness-Guided GRPO (GRPO-SG), a simple token-weighted variant of GRPO that downweights tokens likely to cause overly large gradients, reducing sharp updates and stabilizing optimization, thereby improving generalization. Experiments across mathematical reasoning, logic puzzles and tool-augmented question answering show consistent improvements over GRPO, along with smoother gradient-norm trajectories, supporting GRPO-SG as a simple and effective generalization-oriented upgrade to GRPO for RLVR.

Aligning AI systems with human preferences typically suffers from the infamous reward hacking problem, where optimization of an imperfect reward model leads to undesired behaviors. In this paper, we investigate reward hacking in offline preference optimization, which aims to improve an initial model using a preference dataset. We identify two types of reward hacking stemming from statistical fluctuations in the dataset: Type I Reward Hacking due to subpar choices appearing more favorable, and Type II Reward Hacking due to decent choices appearing less favorable. We prove that many (mainstream or theoretical) preference optimization methods suffer from both types of reward hacking. To mitigate Type I Reward Hacking, we propose POWER, a new preference optimization method that combines Guiasu's weighted entropy with a robust reward maximization objective. POWER enjoys finite-sample guarantees under general function approximation, competing with the best covered policy in the data. To mitigate Type II Reward Hacking, we analyze the learning dynamics of preference optimization and develop a novel technique that dynamically updates preference labels toward certain "stationary labels", resulting in diminishing gradients for untrustworthy samples. Empirically, POWER with dynamic labels (POWER-DL) consistently outperforms state-of-the-art methods on alignment benchmarks, achieving improvements of up to 13.0 points on AlpacaEval 2.0 and 11.5 points on Arena-Hard over DPO, while also improving or maintaining performance on downstream tasks such as mathematical reasoning. Strong theoretical guarantees and empirical results demonstrate the promise of POWER-DL in mitigating reward hacking.

Reinforcement learning (RL) has become central to enhancing reasoning in large language models (LLMs). Yet on-policy algorithms such as Group Relative Policy Optimization (GRPO) often suffer in early training: noisy gradients from low-quality rollouts lead to unstable updates and inefficient exploration. We introduce Slow-Fast Policy Optimization (SFPO), a simple yet efficient framework to address these limitations via decomposing each step into three stages: a short fast trajectory of inner steps on the same batch, a reposition mechanism to control off-policy drift, and a final slow correction. This reposition-before-update design preserves the objective and rollout process unchanged, making SFPO plug-compatible with existing policy-gradient pipelines. Extensive experiments demonstrate that SFPO consistently improves stability, reduces rollouts, and accelerates convergence of reasoning RL training. Specifically, it outperforms GRPO by up to 2.80 points in average on math reasoning benchmarks. It also achieves up to 4.93 fewer rollouts and a 4.19 reduction in wall-clock time to match GRPO's best accuracy.

Standard reinforcement learning from human feedback (RLHF) trains a reward model on pairwise preference data and then uses it for policy optimization. However, while reward models are optimized to capture relative preferences, existing policy optimization techniques rely on absolute reward magnitudes during training. In settings where the rewards are non-verifiable such as summarization, instruction following, and chat completion, this misalignment often leads to suboptimal performance. We introduce Group Ordinal Policy Optimization (GOPO), a policy optimization method that uses only the ranking of the rewards and discards their magnitudes. Our rank-based transformation of rewards provides several gains, compared to Group Relative Policy Optimization (GRPO), in settings with non-verifiable rewards: (1) consistently higher training/validation reward trajectories, (2) improved LLM-as-judge evaluations across most intermediate training steps, and (3) reaching a policy of comparable quality in substantially less training steps than GRPO. We demonstrate consistent improvements across a range of tasks and model sizes.

As large language models (LLMs) are increasingly deployed in diverse user facing applications, aligning them with real user preferences becomes essential. Existing methods like Reinforcement Learning from Human Feedback (RLHF) rely on expert annotators trained on manually defined guidelines, whose judgments may not reflect the priorities of everyday users. We introduce Reinforcement Learning from User Feedback (RLUF), a framework for aligning LLMs directly to implicit signals from users in production. RLUF addresses key challenges of user feedback: user feedback is often binary (e.g., emoji reactions), sparse, and occasionally adversarial. We train a reward model, P[Love], to predict the likelihood that an LLM response will receive a Love Reaction, a lightweight form of positive user feedback, and integrate P[Love] into a multi-objective policy optimization framework alongside helpfulness and safety objectives. In large-scale experiments, we show that P[Love] is predictive of increased positive feedback and serves as a reliable offline evaluator of future user behavior. Policy optimization using P[Love] significantly raises observed positive-feedback rates, including a 28% increase in Love Reactions during live A/B tests. However, optimizing for positive reactions introduces reward hacking challenges, requiring careful balancing of objectives. By directly leveraging implicit signals from users, RLUF offers a path to aligning LLMs with real-world user preferences at scale.

RL-based agentic search enables LLMs to solve complex questions via dynamic planning and external search. While this approach significantly enhances accuracy with agent policies optimized via large-scale reinforcement learning, we identify a critical gap in reliability: these agents fail to recognize their reasoning boundaries and rarely admit ``I DON'T KNOW'' (IDK) even when evidence is insufficient or reasoning reaches its limit. The lack of reliability often leads to plausible but unreliable answers, introducing significant risks in many real-world scenarios. To this end, we propose Boundary-Aware Policy Optimization (BAPO), a novel RL framework designed to cultivate reliable boundary awareness without compromising accuracy. BAPO introduces two key components: (i) a group-based boundary-aware reward that encourages an IDK response only when the reasoning reaches its limit, and (ii) an adaptive reward modulator that strategically suspends this reward during early exploration, preventing the model from exploiting IDK as a shortcut. Extensive experiments on four benchmarks demonstrate that BAPO substantially enhances the overall reliability of agentic search.

Trustworthy verifiers are essential for the success of reinforcement learning with verifiable reward (RLVR), which is the core methodology behind various large reasoning models such as DeepSeek-R1. In complex domains like mathematical reasoning, rule-based verifiers have been widely adopted in previous works to train strong reasoning models. However, the reliability of these verifiers and their impact on the RL training process remain poorly understood. In this work, we take mathematical reasoning as a case study and conduct a comprehensive analysis of various verifiers in both static evaluation and RL training scenarios. First, we find that current open-source rule-based verifiers often fail to recognize equivalent answers presented in different formats across multiple commonly used mathematical datasets, resulting in non-negligible false negative rates. This limitation adversely affects RL training performance and becomes more pronounced as the policy model gets stronger. Subsequently, we investigate model-based verifiers as a potential solution to address these limitations. While the static evaluation shows that model-based verifiers achieve significantly higher verification accuracy, further analysis and RL training results imply that they are highly susceptible to hacking, where they misclassify certain patterns in responses as correct (i.e., false positives). This vulnerability is exploited during policy model optimization, leading to artificially inflated rewards. Our findings underscore the unique risks inherent to both rule-based and model-based verifiers, aiming to offer valuable insights to develop more robust reward systems in reinforcement learning.

In recent years, various powerful policy gradient algorithms have been proposed in deep reinforcement learning. While all these algorithms build on the Policy Gradient Theorem, the specific design choices differ significantly across algorithms. We provide a holistic overview of on-policy policy gradient algorithms to facilitate the understanding of both their theoretical foundations and their practical implementations. In this overview, we include a detailed proof of the continuous version of the Policy Gradient Theorem, convergence results and a comprehensive discussion of practical algorithms. We compare the most prominent algorithms on continuous control environments and provide insights on the benefits of regularization. All code is available at https://github.com/Matt00n/PolicyGradientsJax.

Parallel test-time scaling typically trains separate generation and verification models, incurring high training and inference costs. We propose Advantage Decoupled Preference Optimization (ADPO), a unified reinforcement learning framework that jointly learns answer generation and self-verification within a single policy. ADPO introduces two innovations: a preference verification reward improving verification capability and a decoupled optimization mechanism enabling synergistic optimization of generation and verification. Specifically, the preference verification reward computes mean verification scores from positive and negative samples as decision thresholds, providing positive feedback when prediction correctness aligns with answer correctness. Meanwhile, the advantage decoupled optimization computes separate advantages for generation and verification, applies token masks to isolate gradients, and combines masked GRPO objectives, preserving generation quality while calibrating verification scores. ADPO achieves up to +34.1% higher verification AUC and -53.5% lower inference time, with significant gains of +2.8%/+1.4% accuracy on MathVista/MMMU, +1.9 cIoU on ReasonSeg, and +1.7%/+1.0% step success rate on AndroidControl/GUI Odyssey.

Policy compliance assessment is a fundamental task of evaluating whether an input case strictly complies with a set of human-defined rules, more generally known as policies. In practice, human experts follow a systematic, step-by-step process to identify violations with respect to specific stipulations outlined in the policy. However, such documentation of gold-standard, expert-level reasoning processes is costly to acquire. In this paper, we introduce Policy Reasoning Traces (PRT), a form of specialized generated reasoning chains that serve as a reasoning bridge to improve an LLM's policy compliance assessment capabilities. Our empirical evaluations demonstrate that the use of PRTs for both inference-time and training-time scenarios significantly enhances the performance of open-weight and commercial models, setting a new state-of-the-art for HIPAA and GDPR policies. Beyond accuracy gains, we also highlight how PRTs can improve an LLM's ability to accurately cite policy clauses, as well as influence compliance decisions through their high utilization from the raw chains of thought.

Large language model (LLM)-based agents are increasingly trained with reinforcement learning (RL) to enhance their ability to interact with external environments through tool use, particularly in search-based settings that require multi-turn reasoning and knowledge acquisition. However, existing approaches typically rely on outcome-based rewards that are only provided at the final answer. This reward sparsity becomes particularly problematic in multi-turn settings, where long trajectories exacerbate two critical issues: (i) advantage collapse, where all rollouts receive identical rewards and provide no useful learning signals, and (ii) lack of fine-grained credit assignment, where dependencies between turns are obscured, especially in long-horizon tasks. In this paper, we propose Information Gain-based Policy Optimization (IGPO), a simple yet effective RL framework that provides dense and intrinsic supervision for multi-turn agent training. IGPO models each interaction turn as an incremental process of acquiring information about the ground truth, and defines turn-level rewards as the marginal increase in the policy's probability of producing the correct answer. Unlike prior process-level reward approaches that depend on external reward models or costly Monte Carlo estimation, IGPO derives intrinsic rewards directly from the model's own belief updates. These intrinsic turn-level rewards are combined with outcome-level supervision to form dense reward trajectories. Extensive experiments on both in-domain and out-of-domain benchmarks demonstrate that IGPO consistently outperforms strong baselines in multi-turn scenarios, achieving higher accuracy and improved sample efficiency.

Aligning generative models with human preference via RLHF typically suffers from overoptimization, where an imperfectly learned reward model can misguide the generative model to output undesired responses. We investigate this problem in a principled manner by identifying the source of the misalignment as a form of distributional shift and uncertainty in learning human preferences. To mitigate overoptimization, we first propose a theoretical algorithm that chooses the best policy for an adversarially chosen reward model; one that simultaneously minimizes the maximum likelihood estimation of the loss and a reward penalty term. Here, the reward penalty term is introduced to prevent the policy from choosing actions with spurious high proxy rewards, resulting in provable sample efficiency of the algorithm under a partial coverage style condition. Moving from theory to practice, the proposed algorithm further enjoys an equivalent but surprisingly easy-to-implement reformulation. Using the equivalence between reward models and the corresponding optimal policy, the algorithm features a simple objective that combines: (i) a preference optimization loss that directly aligns the policy with human preference, and (ii) a supervised learning loss that explicitly imitates the policy with a (suitable) baseline distribution. In the context of aligning large language models (LLM), this objective fuses the direct preference optimization (DPO) loss with the supervised fune-tuning (SFT) loss to help mitigate the overoptimization towards undesired responses, for which we name the algorithm Regularized Preference Optimization (RPO). Experiments of aligning LLMs demonstrate the improved performance of RPO compared with DPO baselines. Our work sheds light on the interplay between preference optimization and SFT in tuning LLMs with both theoretical guarantees and empirical evidence.

Large language models split into two families: reasoning-centric LLMs, which strengthen internal chain-of-thought reasoning but cannot invoke external tools, and agentic LLMs, which learn to interact with environments and leverage tools but often lag in deep reasoning. This divide arises from fundamentally different training objectives, leading to mismatched strengths and inefficiency on simple queries, where both families tend to overthink or over-call tools. In this work, we present Adaptive Agent Foundation Model (A^2FM), a unified framework that follows a route-then-align principle: the model first learns task-aware routing and then aligns mode-specific trajectories under a shared backbone. To address the inefficiency gap, we introduce a third mode-instant-that handles simple queries directly, preventing unnecessary reasoning or tool calls while complementing the agentic and reasoning modes. To jointly enhance accuracy and efficiency, we propose Adaptive Policy Optimization (APO), which enforces adaptive sampling across modes and applies a cost-regularized reward. On the 32B scale, A^2FM achieves 13.4% on BrowseComp, 70.4% on AIME25, and 16.7% on HLE, setting new SOTA among comparable models and performing competitively with frontier LLMs across agentic, reasoning, and general benchmarks. Notably, the adaptive execution achieves a cost of pass of only $0.00487 per correct answer-cutting cost by 45.2% relative to reasoning and 33.5% relative to agentic, thus delivering substantially higher cost efficiency while maintaining comparable accuracy.

Large language models enable agents to autonomously perform tasks in open web environments. However, as hidden threats within the web evolve, web agents face the challenge of balancing task performance with emerging risks during long-sequence operations. Although this challenge is critical, current research remains limited to single-objective optimization or single-turn scenarios, lacking the capability for collaborative optimization of both safety and utility in web environments. To address this gap, we propose HarmonyGuard, a multi-agent collaborative framework that leverages policy enhancement and objective optimization to jointly improve both utility and safety. HarmonyGuard features a multi-agent architecture characterized by two fundamental capabilities: (1) Adaptive Policy Enhancement: We introduce the Policy Agent within HarmonyGuard, which automatically extracts and maintains structured security policies from unstructured external documents, while continuously updating policies in response to evolving threats. (2) Dual-Objective Optimization: Based on the dual objectives of safety and utility, the Utility Agent integrated within HarmonyGuard performs the Markovian real-time reasoning to evaluate the objectives and utilizes metacognitive capabilities for their optimization. Extensive evaluations on multiple benchmarks show that HarmonyGuard improves policy compliance by up to 38% and task completion by up to 20% over existing baselines, while achieving over 90% policy compliance across all tasks. Our project is available here: https://github.com/YurunChen/HarmonyGuard.

We revisit policy-gradient optimization for Large Language Models (LLMs) from a single-stream perspective. Prevailing group-based methods like GRPO reduce variance with on-the-fly baselines but suffer from critical flaws: frequent degenerate groups erase learning signals, and synchronization barriers hinder scalability. We introduce Single-stream Policy Optimization (SPO), which eliminates these issues by design. SPO replaces per-group baselines with a persistent, KL-adaptive value tracker and normalizes advantages globally across the batch, providing a stable, low-variance learning signal for every sample. Being group-free, SPO enables higher throughput and scales effectively in long-horizon or tool-integrated settings where generation times vary. Furthermore, the persistent value tracker naturally enables an adaptive curriculum via prioritized sampling. Experiments using Qwen3-8B show that SPO converges more smoothly and attains higher accuracy than GRPO, while eliminating computation wasted on degenerate groups. Ablation studies confirm that SPO's gains stem from its principled approach to baseline estimation and advantage normalization, offering a more robust and efficient path for LLM reasoning. Across five hard math benchmarks with Qwen3 8B, SPO improves the average maj@32 by +3.4 percentage points (pp) over GRPO, driven by substantial absolute point gains on challenging datasets, including +7.3 pp on BRUMO 25, +4.4 pp on AIME 25, +3.3 pp on HMMT 25, and achieves consistent relative gain in pass@k across the evaluated k values. SPO's success challenges the prevailing trend of adding incidental complexity to RL algorithms, highlighting a path where fundamental principles, not architectural workarounds, drive the next wave of progress in LLM reasoning.

Reinforcement Learning (RL) has demonstrated tremendous empirical success across numerous challenging domains. However, we lack a strong theoretical understanding of the statistical complexity of RL in environments with large state spaces, where function approximation is required for sample-efficient learning. This thesis addresses this gap by rigorously examining the statistical complexity of RL with function approximation from a learning theoretic perspective. Departing from a long history of prior work, we consider the weakest form of function approximation, called agnostic policy learning, in which the learner seeks to find the best policy in a given class Pi, with no guarantee that Pi contains an optimal policy for the underlying task. We systematically explore agnostic policy learning along three key axes: environment access -- how a learner collects data from the environment; coverage conditions -- intrinsic properties of the underlying MDP measuring the expansiveness of state-occupancy measures for policies in the class Pi, and representational conditions -- structural assumptions on the class Pi itself. Within this comprehensive framework, we (1) design new learning algorithms with theoretical guarantees and (2) characterize fundamental performance bounds of any algorithm. Our results reveal significant statistical separations that highlight the power and limitations of agnostic policy learning.

Reinforcement learning for LLM reasoning has rapidly emerged as a prominent research area, marked by a significant surge in related studies on both algorithmic innovations and practical applications. Despite this progress, several critical challenges remain, including the absence of standardized guidelines for employing RL techniques and a fragmented understanding of their underlying mechanisms. Additionally, inconsistent experimental settings, variations in training data, and differences in model initialization have led to conflicting conclusions, obscuring the key characteristics of these techniques and creating confusion among practitioners when selecting appropriate techniques. This paper systematically reviews widely adopted RL techniques through rigorous reproductions and isolated evaluations within a unified open-source framework. We analyze the internal mechanisms, applicable scenarios, and core principles of each technique through fine-grained experiments, including datasets of varying difficulty, model sizes, and architectures. Based on these insights, we present clear guidelines for selecting RL techniques tailored to specific setups, and provide a reliable roadmap for practitioners navigating the RL for the LLM domain. Finally, we reveal that a minimalist combination of two techniques can unlock the learning capability of critic-free policies using vanilla PPO loss. The results demonstrate that our simple combination consistently improves performance, surpassing strategies like GRPO and DAPO.

Reinforcement Learning (RL) can directly enhance the reasoning capabilities of large language models without extensive reliance on Supervised Fine-Tuning (SFT). In this work, we revisit the traditional Policy Gradient (PG) mechanism and propose a minimalist RL approach termed Group Policy Gradient (GPG). Unlike conventional methods, GPG directly optimize the original RL objective, thus obviating the need for surrogate loss functions. As illustrated in our paper, by eliminating both the critic and reference models, and avoiding KL divergence constraints, our approach significantly simplifies the training process when compared to Group Relative Policy Optimization (GRPO). Our approach achieves superior performance without relying on auxiliary techniques or adjustments. Extensive experiments demonstrate that our method not only reduces computational costs but also consistently outperforms GRPO across various unimodal and multimodal tasks. Our code is available at https://github.com/AMAP-ML/GPG.

Vision-Language Models (VLMs) have achieved remarkable success in visual question answering tasks, but their reliance on large numbers of visual tokens introduces significant computational overhead. While existing efficient VLM approaches reduce visual tokens through fixed-ratio compression, they operate passively and lack the ability to adapt to varying task requirements. This motivates a fundamental question: Can VLMs autonomously determine the minimum number of visual tokens required for each sample? Inspired by human active vision mechanisms, we introduce AdaptVision, an efficient VLM paradigm that enables adaptive visual token acquisition through a coarse-to-fine approach. Our model initially processes compressed visual tokens from low-resolution images and selectively acquires additional visual information by invoking a bounding box tool to crop key regions when necessary. We train AdaptVision using a reinforcement learning framework that carefully balances accuracy and efficiency. Central to our approach is Decoupled Turn Policy Optimization (DTPO), which decouples the learning objective into two components: (1) tool learning, which optimizes correct tool utilization, and (2) accuracy improvement, which refines the generated responses to improve answer correctness. Based on this formulation, we further decouple advantage estimation by computing separate advantages for tokens associated with each objective. This formulation enables more effective optimization for AdaptVision compared to vanilla GRPO. Comprehensive experiments across multiple VQA benchmarks demonstrate that AdaptVision achieves superior performance while consuming substantially fewer visual tokens than state-of-the-art efficient VLM methods.

Recently, online Reinforcement Learning with Verifiable Rewards (RLVR) has become a key paradigm for enhancing the reasoning capabilities of Large Language Models (LLMs). However, existing methods typically treat all training samples uniformly, overlooking the vast differences in problem difficulty relative to the model's current capabilities. This uniform training strategy leads to inefficient exploration of problems the model has already mastered, while concurrently lacking effective guidance on problems that are challenging its abilities the most, limiting both learning efficiency and upper-bound performance. To address this, we propose CLPO (Curriculum-guided Learning for Policy Optimization), a novel algorithm that creates a dynamic pedagogical feedback loop within the policy optimization process. The core of CLPO leverages the model's own rollout performance to conduct real-time difficulty assessment, thereby constructing an Online Curriculum. This curriculum then guides an Adaptive Problem Restructuring mechanism, where the model acts as its own teacher: it diversifies medium-difficulty problems to promote generalization and simplifies challenging problems to make them more attainable. Our approach transforms the static training procedure into a dynamic process that co-evolves with the model's capabilities. Experiments show that CLPO achieves state-of-the-art performance across eight challenging mathematical and general reasoning benchmarks, with an average pass@1 improvement of 6.96% over other methods, demonstrating its potential for more efficiently training more capable reasoning models.

This paper introduces Completion Pruning Policy Optimization (CPPO) to accelerate the training of reasoning models based on Group Relative Policy Optimization (GRPO). GRPO, while effective, incurs high training costs due to the need for sampling multiple completions for each question. Our experiment and theoretical analysis reveals that the number of completions impacts model accuracy yet increases training time multiplicatively, and not all completions contribute equally to policy training -- their contribution depends on their relative advantage. To address these issues, we propose CPPO, which prunes completions with low absolute advantages, significantly reducing the number needed for gradient calculation and updates. Additionally, we introduce a dynamic completion allocation strategy to maximize GPU utilization by incorporating additional questions, further enhancing training efficiency. Experimental results demonstrate that CPPO achieves up to 8.32times speedup on GSM8K and 3.51times on Math while preserving or even enhancing the accuracy compared to the original GRPO. We release our code at https://github.com/lzhxmu/CPPO.

Reinforcement Learning with Verifiable Rewards (RLVR) has propelled Large Language Models in complex reasoning, yet its scalability is often hindered by a training bottleneck where performance plateaus as policy entropy collapses, signaling a loss of exploration. Previous methods typically address this by maintaining high policy entropy, yet the precise mechanisms that govern meaningful exploration have remained underexplored. Our analysis suggests that an unselective focus on entropy risks amplifying irrelevant tokens and destabilizing training. This paper investigates the exploration dynamics within RLVR and identifies a key issue: the gradual elimination of valuable low-probability exploratory tokens, which we term \textit{reasoning sparks}. We find that while abundant in pre-trained models, these sparks are systematically extinguished during RLVR due to over-penalization, leading to a degeneracy in exploration. To address this, we introduce Low-probability Regularization (Lp-Reg). Its core mechanism regularizes the policy towards a heuristic proxy distribution. This proxy is constructed by filtering out presumed noise tokens and re-normalizing the distribution over the remaining candidates. The result is a less-noisy proxy where the probability of reasoning sparks is amplified, which then serves as a soft regularization target to shield these valuable tokens from elimination via KL divergence. Experiments show that Lp-Reg enables stable on-policy training for around 1,000 steps, a regime where baseline entropy-control methods collapse. This sustained exploration leads to state-of-the-art performance, achieving a 60.17% average accuracy on five math benchmarks, an improvement of 2.66% over prior methods. Code is available at https://github.com/CarlanLark/Lp-Reg.

Reinforcement Learning with Verifiable Rewards (RLVR) has emerged as a dominant paradigm for enhancing Large Language Models (LLMs) reasoning, yet its reliance on external verifiers limits its scalability. Recent findings suggest that RLVR primarily functions by eliciting latent capabilities, motivating the development of verifier-free algorithms. However, in such settings, standard methods like Group Relative Policy Optimization face a critical challenge: destructive gradient variance that often leads to training collapse. To address this issue, we introduceVerifier-Independent Curriculum Reinforcement Learning (VI-CuRL), a framework that leverages the model's intrinsic confidence to construct a curriculum independent from external verifiers. By prioritizing high-confidence samples, VI-CuRL effectively manages the bias-variance trade-off, specifically targeting the reduction of action and problem variance. We provide a rigorous theoretical analysis, proving that our estimator guarantees asymptotic unbiasedness. Empirically, VI-CuRL promotes stability and consistently outperforms verifier-independent baselines across six challenging benchmarks with/without verifiers.

In long-horizon tasks, recent agents based on Large Language Models (LLMs) face a significant challenge that sparse, outcome-based rewards make it difficult to assign credit to intermediate steps. Previous methods mainly focus on creating dense reward signals to guide learning, either through traditional reinforcement learning techniques like inverse reinforcement learning or by using Process Reward Models for step-by-step feedback. In this paper, we identify a fundamental problem in the learning dynamics of LLMs: the magnitude of policy gradients is inherently coupled with the entropy, which leads to inefficient small updates for confident correct actions and potentially destabilizes large updates for uncertain ones. To resolve this, we propose Entropy-Modulated Policy Gradients (EMPG), a framework that re-calibrates the learning signal based on step-wise uncertainty and the final task outcome. EMPG amplifies updates for confident correct actions, penalizes confident errors, and attenuates updates from uncertain steps to stabilize exploration. We further introduce a bonus term for future clarity that encourages agents to find more predictable solution paths. Through comprehensive experiments on three challenging agent tasks, WebShop, ALFWorld, and Deep Search, we demonstrate that EMPG achieves substantial performance gains and significantly outperforms strong policy gradient baselines. Project page is at https://empgseed-seed.github.io/

This paper provides a self-contained, from-scratch, exposition of key algorithms for instruction tuning of models: SFT, Rejection Sampling, REINFORCE, Trust Region Policy Optimization (TRPO), Proximal Policy Optimization (PPO), Group Relative Policy Optimization (GRPO), and Direct Preference Optimization (DPO). Explanations of these algorithms often assume prior knowledge, lack critical details, and/or are overly generalized and complex. Here, each method is discussed and developed step by step using simplified and explicit notation focused on LLMs, aiming to eliminate ambiguity and provide a clear and intuitive understanding of the concepts. By minimizing detours into the broader RL literature and connecting concepts to LLMs, we eliminate superfluous abstractions and reduce cognitive overhead. Following this exposition, we provide a literature review of new techniques and approaches beyond those detailed. Finally, new ideas for research and exploration in the form of GRAPE (Generalized Relative Advantage Policy Evolution) are presented.