Reinforcement learning typically optimizes average reward. For generative policies, the average can hide an important distinction: two policies can achieve the same mean reward while having very different chances of producing a rare but high-reward rollout. This matters as sampling increases during training and inference, since its benefit depends on retaining probability mass on high-reward outcomes. We propose to optimize this coverage directly. Rather than considering only expected reward, we consider all of its upper tails: for each reward threshold, how likely is the policy to exceed it? This turns a continuous reward into a family of binary success events. We introduce Tail-Likelihood Reinforcement Learning (TailRL), which maximizes the log-probability of exceeding a randomly chosen reward threshold. Its gradient gives more weight to rare, high-reward rollouts and can be interpreted as a mixture of Best-of-(k) gradients. TailRL requires only a simple modification to the advantage function, making it compatible with existing reinforcement learning pipelines. Across object localization, maze navigation, GUI grounding, and code optimization, TailRL leverages rare high-reward training samples to avoid suboptimal solutions and yields models that benefit more from additional samples at inference time.
Baha Zarrouki, Arslan Thobani, Jasper Hoffmann +6cs.RO cs.LG eess.SY
In Model Predictive Control (MPC), cost-function weights shape closed-loop behavior, yet changing conditions often make fixed parametrizations suboptimal and motivate context-dependent online adaptation. Learning such policies is difficult because behavior depends implicitly on numerical MPC solutions, producing nonlinear, potentially nonsmooth, long-horizon dependencies on policy parameters. This creates a bias-variance tradeoff: Reinforcement Learning (RL) optimizes realized closed-loop return from environment samples but is sample-inefficient, whereas Gradient-Based Policy Learning (GB-PL) uses low-variance solver gradients from differentiable MPC to optimize surrogate losses on predicted trajectories but can be biased under model mismatch. We propose Solver-Gradient Guided Reinforcement Learning (SG-RL), a solver-sensitivity augmentation for RL-based online MPC cost-weight adaptation. SG-RL keeps sampled closed-loop return as the objective and uses bounded solver-derived gradients as auxiliary guidance to improve stability and sample efficiency. We instantiate SG-RL in Proximal Policy Optimization (PPO) with four modular algorithms that inject solver-gradient guidance into actor-update scaling, policy loss, advantage estimation, and value-function learning. On two full-scale autonomous racing platforms with intentional model mismatch, SG-RL reaches PPO's best closed-loop return with up to 70.6% fewer samples, outperforms GB-PL baselines by at least 54% in closed-loop return, and generalizes zero-shot to unseen environments.
On-policy distillation (OPD) offers dense token-level supervision as an alternative to the sparse outcome-level advantages of reinforcement learning with verifiable rewards (RLVR). However, the teacher scores student-generated trajectories that are inherently off-policy for it, so the reliability of its supervision, and hence the source of the student's improvement, remains unclear. We quantitatively analyze teacher supervision during OPD training and find substantial noise whose prevalence increases with teacher scale. Surprisingly, the student policy is insensitive to such noise, converging to comparable performance regardless of whether noisy supervision is retained or removed. Does OPD distill at all? By analyzing what drives its gains, we find that learning concentrates on low log-probability tokens, and using a single fixed negative advantage matches the performance of teacher-provided ones. This suggests that OPD works largely by suppressing low log-probability tokens, which requires no teacher. These findings motivate On-Policy Self-Adaptation (OPSA), a supervision-free method using entropy-adaptive negative advantages. It assigns stronger learning signals to high-entropy positions, suppressing tail tokens, and evenly redistributing probability mass among head tokens. Compared with the base \texttt{Qwen3-1.7B}, OPSA improves Avg@32 by 35.41 points on AIME24, corresponding to a 263\% relative gain, and more than doubles Pass@32 across all three benchmarks. It also outperforms OPD by 16.77 points in Avg@32 on AIME24. Extensive experiments and analyses across model families and tasks further demonstrate its effectiveness and generalizability.
Expected-cost constraints can still permit rare, high-cost events. Monte Carlo conditional value at risk (CVaR) gradients can be noisy at high confidence, whereas critics that model an outcome distribution add complexity. We propose BCPPO (Bachelier-Inspired Constrained Proximal Policy Optimization), a proximal policy optimization (PPO) method. Separately initialized cost-prediction networks (critics), trained with random sample masks, produce disagreement that marks predictions sensitive to which state-action regions occur in the training data and to critic training. A Bachelier formula for the expected amount above a reference level converts this disagreement into a smooth policy-update penalty. Gradients from this penalty do not alter the critics, so temporal-difference (TD) critic learning is unchanged. A saturation-aware controller adjusts the mean-cost penalty and stops accumulated error from growing while that penalty is clipped. Deployment retains only the policy network. The disagreement penalty is neither a tail-event probability nor a guaranteed error bound, and it provides no safety guarantee. Across 175 runs with shared tasks, costs, budgets, training steps, and evaluation seeds, no comparator attains both higher mean return and lower mean CVaR than BCPPO in any task. On Push1, BCPPO has no lower return and no higher CVaR than every comparator, with at least one strict gain. These results support a practical balance among reward, caution around cost predictions that vary across trained critics, and policy-only deployment.
Decoupled Clip and Dynamic Sampling Policy Optimization (DAPO) is a prominent variant of Group Relative Policy Optimization (GRPO). DAPO introduces several improvements over GRPO. Among these, Dynamic Sampling contributes the most to DAPO's accuracy gains relative to GRPO. To improve accuracy, Dynamic Sampling enhances training stability by eliminating zero policy gradients from zero advantages. Specifically, it avoids such zero gradients by filtering out prompts where sampled responses are either entirely correct or incorrect. However, our theoretical analysis shows that Dynamic Sampling decrease training efficiency as it cannot effectively utilize hard-to-sample correct responses on hard prompts. Formally, it asymmetrically amplifies the advantages of distinct responses to the same prompts. On hard prompts, incorrect responses undergo greater amplification than correct ones. This leads the model to avoid generating the observed incorrect responses rather than capitalizing on the hard-to-sample correct ones on hard prompts, resulting in low training efficiency. To improve training efficiency, we propose Direct Advantage Amplification (DAA), which amplifies the advantages of hard-to-sample correct responses on hard prompts, as obtained by Dynamic Sampling. This ensures that, when Dynamic Sampling is used, these hard-to-sample responses can be effectively capitalized on, implying higher training efficiency. By integrating DAA into DAPO, we obtain Difficulty-aware Advantage Amplification Policy Optimization (DA3PO), which is implemented with fewer than 30 lines of code from DAPO. Experiments show that DA3PO significantly outperforms GRPO and other classical GRPO variants.
Szymon Miłosz, Piotr Duch, Szymon Grabowskics.LG cs.AI stat.ML
Searchless chess networks reach human master strength from a single forward pass by imitating a stronger teacher: the strongest, Leela Chess Zero's (Lc0) released Chessformer, distills the visit counts of an AlphaZero-style Monte Carlo Tree Search (MCTS). Imitating a search is a poor proxy for playing without one, so we fine-tune for single-pass strength with self-play reinforcement learning (RL). Its exploration is usually supplied by an entropy bonus, the reverse Kullback-Leibler (KL) divergence to uniform. We replace it with a forward, mass-covering KL toward the network's own MCTS prior (prior-directed exploration), so exploration covers the moves the prior judges promising, and pair it with an entropy-adaptive sampling temperature, set by the value head's outcome uncertainty, that sharpens once a position is decided. In about two thousand steps it raises puzzle accuracy from 93.9% to 94.9% on a 100,000-puzzle suite and mate-in-four accuracy from 77% to 81% while holding searchless strength at or slightly above the base. Measuring tactical accuracy and playing strength together across a matched-compute sweep, we find the two dissociate: accuracy gains fall in a one-point band while ratings straddle the base, and a control fine-tuned on puzzles alone posts the study's largest tactical gains while shedding roughly 260 Elo; a better puzzle-solver is not thereby a stronger player. Distribution-level measurements show what anchoring buys: without a regularizer self-play collapses onto a single line of play, and the puzzles newly solved are the near misses whose winning move the prior kept alive. The forward-KL prior tops the rating ladder, statistically tied with a reverse-KL anchor that concentrates twice as hard and drops the hardest solutions the mass-covering prior keeps in support.
Recent prominent post-training methods, such as Reinforcement Learning (RL) and On-Policy Self-Distillation (OPSD), have driven rapid progress in mathematical reasoning for large language models, yet their reliance on ground-truth labels precludes test-time training (TTT). Replacing ground truth with majority-vote pseudo-labels is a natural alternative, yet it is fragile: an incorrect vote corrupts the teacher and misleads every token. We observe that this failure mode is asymmetric: rollouts that disagree with the pseudo-label are typically wrong regardless of whether the vote itself is correct. Building on this observation, we propose Test-Time Policy Optimization (TTPO), an asymmetric objective that distills agreeing rollouts via OPSD and penalizes disagreeing rollouts with Grouped RL. Token-level selection further refines both branches: distillation down-weights already-converged positions, while RL penalizes only confident errors. Both updates remain well-grounded even under frequent pseudo-label errors, and majority-vote routing yields tighter self-supervision as the model improves. Without any labels, TTPO matches label-supervised OPSD on five competition-level benchmarks, raises Qwen3-1.7B from 38.0% to 45.2% in TTT, yields +25.2% to +36.4% without thinking, and shows strong cross-task generalization.
Group-relative reinforcement learning waits for sibling rollouts of the same prompt, which is costly for long and variable tool-use trajectories. Single-stream Policy Optimization (SPO) removes this dependency with a persistent prompt-level value estimate, but its recipe whitens one advantage per trajectory before optimizing a token-mean actor loss. We show that trajectory centering generally does not center the token-weighted quantity consumed by the actor, and fix the mismatch by standardizing terminal-outcome advantages under the action-token measure. We additionally organize prompt evidence by the policy event that generated it rather than learner receipt order. Across matched runs on ALFWorld at two model scales and on Math-TIR, SPO++ improves online learning efficiency over SPO. A paired ablation identifies action-token-measure normalization as the strongest tested component.
Reinforcement learning with verifiable rewards (RLVR) enables language models to learn multi-turn interaction with external tools, yet its sparse outcome rewards provide no signal for identifying which intermediate decisions are responsible for success. Branch sampling induces local comparisons among alternative continuations, but existing methods tend to conflate two distinct problems: allocating a fixed rollout budget and translating branch outcomes into token-level credit. We introduce Contrastive Branch Policy Optimization (CBPO), which disentangles these two problems and assigns a dedicated mechanism to each. Generation entropy screens candidate branch positions across the entire response, while path-level and node-level decay distribute a fixed budget across trajectories and positions to prevent exploration from collapsing onto a few paths or adjacent tokens. A parent trajectory together with the branches that share an identical token prefix forms an exact-prefix group, and the reward variation within this controlled group defines the Contrastive Branch Value (CBV), an outcome-based estimate of local decision sensitivity that rescales continuation advantages without altering their sign. When multiple nodes are selected along the same trajectory, CBPO partitions it into non-overlapping credit segments, thereby avoiding duplicated gradients on shared tokens. Requiring only outcome rewards and no process-level annotation, CBPO provides a practical solution for fine-grained credit assignment in tool-integrated agent training. Extensive experiments on ten benchmarks, including five for mathematical reasoning and five for knowledge-intensive search, show that CBPO consistently outperforms state-of-the-art policy-optimization and branch-based methods, attaining the highest macro-average accuracy in both domains and across two model scales.
Self-reflection is a powerful mechanism for credit assignment in human learning, converting sparse outcome feedback into actionable guidance. However, its potential for post-training Large Language Models (LLMs) remains underexplored. We propose Self-Reflective Policy Optimization (SRPO), a framework that internalizes this capability. SRPO enables LLMs to analyze their own completed trajectories, synthesize errors into concise "reflection patches," and use reflection-conditioned teacher scores on student on-policy rollouts as dense token-level training signals. This process effectively transforms sparse terminal supervision into dense, token-level learning signals without requiring external critics, separate reward models, or larger teacher models. We demonstrate that SRPO achieves state-of-the-art performance across mathematical reasoning and long-horizon agentic benchmarks with exceptional data efficiency. Using a Qwen3-8B base model, SRPO attains 73.3% on AIME'24 using only 8% (0.08x) of the training FLOPs required by scaled supervised fine-tuning, while significantly improving success rates on WebShop (64.7%), ALFWorld (76.8%), and SWE-Bench-Lite (31.2%). Code is available at https://github.com/Galleons2029/SRPO
Policy optimization (PO) for Large Language Models faces a stability--exploration trade-off, currently mediated by an action-side Policy-KL regularizer. This puts practitioners in a double bind: keeping Policy-KL constrains response behavior and consumes the action-side exploration budget, while dropping it leaves the optimization without an explicit drift control. We argue for an alternative that breaks the dilemma by moving regularization to the input side. As training progresses, the distribution over training queries induced by the current policy drifts unchecked from its pre-RL reference distribution. Concretely, Environment-Regularized Policy Optimization (ERPO) introduces a Query-KL (QKL) term that bounds this query distribution shift, together with a dataset-static reference-derived per-query weight that biases each per-query update toward queries typical under the reference. The QKL gradient flows strictly through the query likelihood; the response score function used by policy-gradient estimators does not appear in the QKL term, so QKL exerts no direct gradient pressure on the response distribution---exploration is preserved. ERPO plugs into GRPO/PPO/REINFORCE-style pipelines without additional forward passes. On six mathematical reasoning benchmarks, ERPO replaces the standard Policy-KL regularizer while achieving effective control over query distribution drift, delivering stronger accuracy and substantially more stable behavior under high-temperature decoding and long-horizon training.Our source code are available at https://github.com/alibaba/ERPO
Agentic reinforcement learning (RL) has become a critical stage in the post-training of large language models. Existing critic-free, group-relative methods estimate policy advantages from multiple rollouts, avoiding the substantial memory overhead of conventional proximal policy optimization (PPO) and achieving strong performance on long-horizon interactive tasks. Despite their success, recent studies revealed three limitations: (1) Lack explicit value generalization and effective temporal credit assignment; (2) Suffer from potential advantage collapse in long-horizon complex tasks; (3) Require a costly trade-off between sampling budget and policy performance. In this work, we propose Single-rollout Autoregressive Policy Optimization (SAPO), a low-memory and compute-efficient framework in which the policy and value functions share a single autoregressive backbone. SAPO exploits the autoregressive structure of LLMs to produce policy and value predictions at distinct causal boundaries with shared parameters, while independently optimizing the PPO objectives and auxiliary on-policy SARSA objectives. To robustly estimate the contribution of each turn, we further introduce a trajectory-level generalized advantage estimator that combines lambda-returns with batch normalization. Experiments across ALFWorld and WebShop with Qwen2.5-1.5B/7B show that SAPO trains stably and outperforms PPO and GRPO by mean +15.1 and +12.1 percentage points, respectively, while eliminating the memory cost of a separate critic model and reducing per-iteration runtime by 33.2% over PPO.
Group Relative Policy Optimization (GRPO) has become a widely used approach for post-training Large Language Models (LLMs) for reasoning. In GRPO, the group gradients induced by different queries within the same mini-batch are directly averaged to form the policy update. However, these group gradients can point in conflicting directions. Our empirical analysis suggests that group-gradient conflicts tend to be associated with less effective policy updates, motivating the need for a reliable aggregated update direction under such conflicts. Standard GRPO aggregation treats the realized group gradients as deterministic contributions and does not account for differences in their reliability during aggregation. To address this issue, we propose Gradient Uncertainty-Aware Policy Optimization (GUPO), which models each group gradient as a random variable under a Bayesian formulation and estimates its probability distribution. GUPO then derives gradient uncertainty using a Dirichlet-based formulation and uses it to calibrate the contribution of each group gradient during aggregation. Extensive experiments on multiple benchmarks demonstrate the effectiveness of GUPO.
Reinforcement learning (RL) with group-relative advantages has become the de facto standard for post-training language model reasoners. However, when optimizing multiple reward objectives, existing methods typically scalarize the reward vector with a fixed weighted sum before group-wise standardization. We show that this design leads to two fundamental problems: rollouts with distinct reward profiles can receive identical advantages, and all objectives are optimized with fixed relative weights regardless of their current level of saturation. As a result, training continues to allocate gradient budget to already-solved objectives instead of focusing on those with greater remaining headroom. We introduce \textbf{Saturation Aware Advantage Reweighting for Multi-Reward Policy Optimization} (SA-MRPO), which standardizes each reward objective independently and adaptively discounts its contribution according to a batch-level estimate of objective saturation. This dynamically reallocates optimization effort toward under-optimized objectives while empirically maintaining performance on those that are already well satisfied. We further show that saturation-aware reweighting can reverse the sign of an update, rather than merely rescale its magnitude. Across mathematical reasoning with two- and three-objective reward combinations, SA-MRPO improves the harder correctness objective over GDPO in 12 of 15 benchmark comparisons, with gains of up to $5\%$ on AIME24. On adaptive reasoning it improves accuracy on all five benchmarks, by $3.8\%$ on average and up to $9.2 \%$ on AMC23, and on coding benchmarks it improves pass rate by up to $2.3\%$, while in all settings maintaining the easier objectives near their already satisfied levels.
Group Relative Policy Optimization (GRPO) learns from reward differences within a rollout group, but receives no useful relative signal when every sampled response is incorrect. Privileged self-distillation can fill this gap with dense token supervision, yet applying it throughout training creates a different failure mode: the teacher is a biased, low-variance surrogate for the reward objective, so persistent imitation can oppose reward-improving updates after the policy becomes capable of producing successful trajectories. We introduce I-SDPO (Instance-Level Adaptive Self-Distillation Policy Optimization), which treats teacher reliance as capability-dependent. I-SDPO makes one routing decision per input instance and shares it across that instance's rollout group: all-incorrect groups use a privileged self-distillation objective, whereas any-success groups remain intact for GRPO. This design uses imitation only where group-relative rewards are uninformative. A local analysis characterizes when teacher and reward directions align and shows that a non-vanishing biased distillation weight induces an optimization bias floor. The routing rule automatically reduces the expected distillation rate as success probability rises, withdrawing teacher influence without a hand-designed schedule. On SciKnowEval, I-SDPO obtains the best result in all four scientific domains and improves average mean@16 accuracy from 56.67% with GRPO to 70.31%, with a maximum domain gain of 18.24 points.
On-policy rollout methods such as GRPO are central to post-training of large language models, yet they frequently suffer from training instabilities, cross-task capability degradation, and response-length inflation. Although prior work has characterized the subspace geometry of aggregate updates, the stepwise variation of this geometry and its relationship to model performance remain unclear. We introduce Principal-Subspace Overlap, a dimension-corrected measure of individual rollout updates relative to the dominant singular subspaces of pretrained weights. Despite low average overlap, transient spikes often precede performance degradation. To address this, we propose GCPO (Geometrically Constrained Policy Optimization), which applies hard bilateral orthogonal projections to constrain updates to the complementary subspaces, preventing such excursions by construction. Across mathematical reasoning, code generation, and tool-use tasks on Qwen3-8B and GLM4-9B, GCPO consistently outperforms GRPO and recent variants, including DAPO and GSPO, improving over the base models and the strongest baseline by up to 27.69 and 2.37 points, respectively. Furthermore, GCPO preserves general capabilities, eliminates response-length inflation, and stabilizes policy entropy. Our findings provide a new diagnostic lens and a principled design perspective for stable reinforcement learning post-training.
Group-based reinforcement learning methods for multimodal large language models typically rely on trajectory-level credit assignment that applies a single advantage to all tokens in a response. However, multimodal reasoning involves substantially higher perceptual uncertainty than text-only settings, where the model must repeatedly re-examine visual information to verify intermediate interpretations, and different visual groundings can lead to divergent reasoning paths, making such uniform credit assignment particularly inadequate and causing relative advantages to progressively degenerate toward zero. To address these challenges, we propose Multi-Branch Policy Optimization (MBPO), a tree-based framework that constructs reasoning trees at vision-language decision boundaries, enabling sibling branches to explore diverse visual hypotheses and assigning segment-level credit through branch-relative advantages. We further introduce a temporal replay buffer to reuse informative segments while controlling policy staleness. Experiments on several multimodal reasoning benchmarks show that MBPO outperforms representative baselines, improving both learning signal quality and optimization efficiency. The code is publicly available at https://github.com/ShuaiLyu0110/MBPO.
Designing effective reward functions remains a major bottleneck in Reinforcement Learning (RL). Recent work uses large foundation Vision-Language Models (VLMs) as reward models, computing text-observation similarity to bypass manual reward engineering. Although promising, these rewards are often noisy and unreliable, limiting their direct utility during deployment. We present Structure-Aware Fine-Tuning (SAFT), a simple, self-supervised method that refines these imperfect reward signals online without access to ground-truth supervision. SAFT leverages intrinsic structural priors to regularize the VLM's latent space via LoRA adapters. We rigorously evaluate SAFT across a spectrum of base model capabilities to demonstrate its versatility. Our results show that SAFT consistently denoises the reward landscape, yielding faster policy convergence and substantially improved alignment (EPIC distance) relative to the underlying base model, suggesting that failures can often be attributed to structural brittleness rather than semantic misunderstanding. By replacing extensive human preference annotation with structural inductive biases inherent to the task, SAFT offers a scalable path for stabilizing text-conditioned RL and underscores the broader value of incorporating task structure as a general inductive bias.
Ziqi Jia, Yalu Ouyang, Bo Pang +5cs.CL cs.AI cs.LG
Reinforcement learning (RL) has emerged as an effective method for enhancing the reasoning capabilities of large language models (LLMs). However, existing methods suffer from insufficient precision in feedback on generated answer trajectories and exhibit the phenomenon of problem difficulty drift. To address these challenges, we propose CVPO - Curriculum-guided Value-Variance Policy Optimization. At the response trajectory level, we find that token-level value-variance correlates with exploration intensity. Our theoretical analysis shows this variance bounds policy update magnitude. We then use the estimated trajectory value-variance to quantify the intrinsic randomness in generation. Based on this, we design a variance-aware advantage adjustment mechanism for different reward types. At the question level, we introduce a dynamic curriculum weighting method that adapts to question difficulty. This helps the model focus on tasks matched to its current ability during each training stage. Experimental results show our method outperforms strong value-based baselines like VAPO. It achieves better performance and stronger exploration, enabling more accurate and robust reasoning in language models across various math tasks.
Counterfactual inference approaches for sequential decision-making typically assume deterministic causal models, where all randomness stems from latent variables. However, Markov Decision Processes (MDPs) are inherently stochastic. We address this by formalising counterfactual policy optimisation under probabilistic nondeterministic causal models, which properly separates latent confounding from irreducible stochasticity, and here propose a first practical optimisation problem for identifying robust counterfactual policies under a sensitivity analysis framework. We validate our approach on a sepsis treatment simulator, where diabetes status acts as a hidden global confounder.
Reinforcement learning has become a central paradigm for improving the reasoning capabilities of large language models. Existing methods generally aim to reduce the failure probabilities induced across problems. In this paper, we introduce a moment-based perspective on policy optimization for LLM reasoning by treating the failure probability of a randomly sampled problem as a random variable and characterizing optimization objectives through its moments. Under this perspective, many existing methods optimize only a single moment of the failure-probability distribution, leaving its broader distributional structure largely uncharacterized. We propose \textbf{M}ulti-\textbf{M}oment \textbf{P}olicy \textbf{O}ptimization (MMPO), a novel policy optimization framework that jointly minimizes multiple moments of the failure-probability distribution. MMPO admits a direct operational interpretation as minimizing the expected truncated time required to obtain the first successful response. Beyond MMPO, we further develop a general moment-transformation framework that systematically induces different moment profiles and provides a unified view of a broader family of policy optimization objectives. Experiments across five mathematical reasoning benchmarks and models of different scales demonstrate that MMPO consistently outperforms strong baselines. We hope this moment-based perspective offers new insights into the design of policy optimization objectives for LLM reasoning.
Reinforcement learning (RL) has become a central paradigm for large language model (LLM) post-training, but optimization toward new objectives can degrade capabilities already present in the base model. KL regularization is widely used to mitigate such forgetting by constraining policy drift toward a reference model. However, standard full-policy KL regularization constrains the entire response distribution and may unnecessarily restrict exploration and target-task learning. This raises a natural question: can a more precise constraint preserve existing capabilities while minimizing interference with learning new tasks? To this end, we propose \underline{Co}rrectness-Conditioned \underline{KL} Regularization (CoKL), a conditional regularization framework that narrows the preservation constraint from the full output distribution to correctness-conditioned response distributions. We instantiate CoKL with forward KL divergence and derive a practical finite-group training objective for RL-based LLM post-training. At the population level, CoKL decouples the total probability assigned to correct responses from their correctness-conditioned distribution, thereby regularizing the relative probability allocation among reference-supported correct responses without directly anchoring incorrect outputs or total correctness mass. We further show that full-policy forward and reverse KL regularization induce a strict optimal correctness gap when the reference policy is imperfect, whereas CoKL avoids this limitation. Experiments in controlled multi-solution environments and continual post-training settings across multiple model scales demonstrate that CoKL achieves a more favorable balance between target-task improvement and prior-capability retention than existing regularization methods. Our code is available at https://github.com/Lumina04/CoKL.
Autoregressive rollout generation is a major computational cost in reinforcement learning for large language models. Reusing each rollout batch for additional learner updates amortizes this cost, but later updates become increasingly off-policy as the learner departs from the behavior policy. At a token position, exact off-policy correction must account for both the current action and the probability of reaching its prefix. The cumulative importance ratio provides this correction, but its product form can produce an unwieldy dynamic range. We study Prefix-Normalized Policy Optimization (PNPO), which replaces the cumulative ratio with the geometric mean of likelihood ratios along each causal prefix, preserving causal-prefix dependence at each position while compressing the log-weight scale. In controlled long-context mathematical reasoning experiments, we induce two off-policy regimes by using one or four policy-update epochs per rollout batch. PNPO does not consistently outperform GSPO with one epoch. With four epochs, it attains the highest observed Avg@32 on each benchmark; the unweighted mean of the three independently selected benchmark peaks is 50.24, 3.00 percentage points above GSPO. Under a matched 2,400-update budget, four-epoch PNPO reaches a final macro Avg@32 of 49.66 after 150 rollout batches, comparable to the 49.56 reached after 600 batches with one epoch. These results provide preliminary evidence that PNPO can be advantageous as training moves further off-policy.
On-policy self-distillation (OPSD) is a promising approach to improve reasoning language models, but it remains brittle in practice: making it work reliably often requires substantial engineering effort. We identify a structural source of this difficulty: vanilla OPSD is precisely the $β=1$ member of a broader policy-optimization family, where $β$ weights the KL penalty anchoring the student to a reference policy. This equivalence turns $β$ from an implicit value fixed at one into a controllable regularization parameter, yielding a more general formulation that trades off proximity to a reference policy against privileged teacher guidance. We introduce $β$-OPSD and derive its optimal policy as a geometric interpolation between the reference policy and the privileged teacher. Directly optimizing this objective with reinforcement learning, however, would be costly and high-variance. Rather than optimize the RL objective directly, we turn its closed-form solution into a distillation target. Each value of $β$ selects a target along the reference-to-teacher path, which we implement efficiently by mixing their token-level logits. In this way, inexpensive distillation approximates the solution of expensive policy optimization. Return-to-go credit assignment further aligns token updates with the sequence-level objective while retaining the simplicity of OPSD. Experiments on mathematical reasoning benchmarks show that $β$-OPSD consistently outperforms vanilla OPSD, improving optimization stability and downstream reasoning performance. Our results provide a principled route from self-distillation to policy optimization and back without sacrificing the efficiency that makes OPSD practical.
Reinforcement learning from verifiable rewards (RLVR) for mathematical reasoning suffers from a structural blind spot: on "cliff" prompts-those on which every sampled rollout in a group fails-the group-normalized advantage is identically zero, so GRPO produces no gradient on precisely the prompts at the frontier of the model's capability. We introduce LoRA Scaffolded Policy Optimization (LSPO), a sampling-time mechanism that recovers this lost gradient. Each RL step, LSPO detects cliff prompts, fits a small low-rank (LoRA) adapter by a brief supervised step on their ground-truth solutions, re-rolls the cliffs with the base-plus-adapter model, splices the now-successful completions back into the RL batch with an importance-sampling correction, and takes a GRPO step on the base alone; the adapter receives only the supervised gradient and is discarded at checkpoint, yielding a base-only model. On DeepMath-103K with DeepSeek-R1-Distill-Qwen-1.5B, evaluated over n=5 paired seeds per arm at a matched 1000-step reporting horizon, LSPO's 5-seed mean matches or beats a DAPO baseline on all 16 (benchmark, pass@k) cells (15 strict wins and one exact tie), with gains of up to +10.7 points on AIME24/pass@4, +6.7 points on AIME24 and AIME26 at pass@16, and +2.4 points on MATH500/pass@1; averaged over the 16 cells the improvement is +3.8 points.
Reinforcement learning has emerged as an effective paradigm for enhancing the mathematical reasoning capabilities of large language models. Among existing policy optimization methods, Proximal Policy Optimization (PPO) remains particularly appealing because its learned critic can, in principle, provide token-level credit assignment. However, in mathematical reasoning tasks characterized by long reasoning horizons and sparse outcome rewards, reliable token-level credit assignment remains challenging. The standard critic often fails to accurately evaluate intermediate reasoning states, resulting in noisy advantage estimates and suboptimal policy updates. In this paper, we propose ReDiPPO, a Reference-guided and Discrepancy-aware PPO framework for mathematical reasoning. ReDiPPO introduces a reference-guided critic that uses reference answers as training-time privileged signals to provide more accurate value estimation. Meanwhile, it retains a standard critic and quantifies the token-level reference-standard discrepancy between the standard value estimate and the reference-guided value estimate. This discrepancy serves as an indicator of difficult reasoning states and is used to reweight the corresponding token-level advantages during PPO optimization. Extensive experiments on diverse mathematical reasoning benchmarks demonstrate that ReDiPPO improves value-estimation accuracy and consistently outperforms strong policy optimization baselines, including PPO, DAPO, and GSPO, in final reasoning performance. Our code is available on https://github.com/cii030/ReDiPPO.
Test-time reinforcement learning (TTRL) enables language models to self-evolve at inference time without labeled feedback. Existing methods rely on answer voting and therefore do not extend naturally to open-ended generation, where valid responses cannot be mapped to a shared canonical answer. Without external reward models or stronger judges, adaptation must instead construct reliable rewards from the model's own outputs. We introduce SERPO (Self-Evolving Rubric Policy Optimization), which replaces answer voting with a closed loop that co-evolves response evidence, query-specific rubrics, and policy parameters. Good-Normal-Bad (G-N-B) response evolution organizes maximally separated rollouts into ordered archives; rubric evolution retains criteria that discriminate these archives; probabilistic criterion scoring converts verdict-token likelihoods into reward signals; and policy evolution optimizes the actor with the resulting signals. New actor rollouts then refresh both the archives and rubrics, closing the three-way evolution loop. Across two model configurations, two in-domain benchmarks, and four OOD benchmarks, SERPO improves HealthBench and ResearchQA by up to 20.63 and 20.31 points over the corresponding base models, raises the six-benchmark macro-average by up to 8.06 points, and supports OOD transfer and continued cross-benchmark evolution.
Junoh Park, Junseo Hwang, Wonguk Cho +1cs.LG cs.AI
Group Relative Policy Optimization (GRPO) has become a standard reinforcement learning method for post-training language models. Recent work shows that GRPO can reduce the base model's reasoning capacity and underperform it in Pass@k when k is large, indicating reduced coverage of reasoning paths. We find that this reduction is associated with GRPO concentrating on responses that the base model already generates with high probability. We trace this concentration to two mechanisms in the GRPO update. At the response level, high-probability responses dominate the group gradient through repeated occurrence. At the token level, GRPO's importance ratio scales gradients, further reinforcing tokens that become more likely under the current policy. We propose ReCo, a reweighting method that addresses both effects. Response contributions are normalized by their expected occurrence within the rollout group, and the token-level importance ratio is replaced with a variance-based ratio that gives larger update scale to non-saturated decision points where alternative token choices remain plausible. Across Qwen2.5-Math-1.5B/7B and Llama-3.1-8B-Instruct on five mathematical reasoning benchmarks, ReCo improves Pass@k for large values of k and is comparable to GRPO for small values of k.
Group Relative Policy Optimization (GRPO) is the dominant reinforcement learning algorithm for training reasoning capabilities in large language models, notably adopted by DeepSeek-R1. The recent improvement Dr. GRPO (COLM 2025) identifies the response-level length bias caused by per-trajectory length normalization in GRPO and proposes removing this normalization, claiming the resulting optimizer is "unbiased." We show that this claim is incomplete. Specifically, we establish an impossibility theorem: under the standard outcome reward + GRPO setting, no length-based weighting scheme can simultaneously achieve the following two properties. (P1) Gradient unbiasedness: the gradient estimator is an unbiased estimate of the true policy gradient. (P2) Length invariance: each trajectory's effective contribution to the gradient is independent of its token length. GRPO approximately satisfies P2 but violates P1; Dr. GRPO satisfies P1 but violates P2. We characterize the complete tradeoff spectrum via the parametric family f_alpha(L) = L^{alpha - 1}, where alpha = 0 recovers GRPO, alpha = 1 recovers Dr. GRPO, and provide quantitative analysis showing that Dr. GRPO's length bias can cause longer trajectories to dominate gradient updates by a factor proportional to the length ratio. Our results reveal that neither algorithm is universally "done right"; they occupy opposite ends of a fundamental and unavoidable tradeoff.
Nikola Milosevic, Nicolás Hinrichs, Nico Scherfcs.LG cs.AI stat.ML
Active Inference (AIF) frames adaptive behavior as the minimization of expected free energy (EFE), combining epistemic and pragmatic objectives within a single variational principle. We frame AIF as policy optimization and show that, for closed-loop control policies, EFE minimization can be formulated as a convex Markov decision process (MDP). In this formulation, the pragmatic terms are linear in the predictive state marginals and therefore equivalent to reward maximization in a latent MDP, while the epistemic value introduces a nonlinear component that distinguishes EFE minimization from standard reinforcement learning. This perspective further reveals the epistemic drive of active inference as a policy-dependent (performative) reward. We analyze finite-horizon, discounted, and average-reward formulations of EFE and derive a mirror descent (MD) algorithm that locally linearizes the objective around the current state marginals, yielding a policy-dependent reward that is compatible with actor-critic methods and dynamic programming. Finally, we argue that coupling world-model learning with policy optimization gives active inference the structure of performative reinforcement learning, providing a route toward grounding active inference within modern reinforcement learning and optimization theory, including convergence analysis and principled policy improvement guarantees.