Towards Better Training Signal: Advantage Clipped Policy Optimization

Reinforcement learning, including reinforcement learning with verifiable rewards (RLVR), has emerged as a powerful approach for LLM post-training. Central to these approaches is the design of the importance sampling (IS) ratio used in off-policy policy-gradient estimation. Existing methods face a fundamental bias-variance dilemma: token-level IS ratios, as adopted by PPO (Schulman et al., 2017) and GRPO (Shao et al., 2024), introduce bias by ignoring prefix state distribution mismatch; full sequence ratios provide exact trajectory-level correction but suffer from high variance due to the multiplicative accumulation of per-token ratios, while GSPO (Zheng et al., 2025) improves numerical stability via length normalization at the cost of deviating from the exact full-sequence IS correction. In this work, we identify the cumulative token IS ratio, the product of per-token ratios up to position $t$, as a theoretically principled solution to this dilemma. We prove that, under the token-level policy-gradient formulation, this ratio provides an unbiased prefix correction for each token-level gradient term and has strictly lower variance than the full sequence ratio. Building on this insight, we propose CTPO (Cumulative Token Policy Optimization), which combines the cumulative token IS ratio with position-adaptive clipping that scales log-space clip bounds according to the natural $\sqrt{t}$ growth of the cumulative log-ratio. This yields more consistent regularization across token positions. We implement and evaluate CTPO in the tool-integrated reasoning setting on several challenging mathematical reasoning benchmarks, achieving the best average performance across both model scales compared with strong GRPO and GSPO baselines. Code will be available at https://github.com/horizon-llm/CTPO.

Reinforcement Learning from Verifiable Rewards (RLVR) has emerged as a promising framework for enhancing the reasoning ability of large language models. However, much of the existing work is guided by heuristic intuition, leading to divergent algorithmic choices, even contradictory ones that nevertheless report empirical gains. To better understand this phenomenon, we conduct a theoretical analysis of RLVR updates. Our study reveals that differences in off-policy degree, determined by the number of gradient steps per rollout, substantially affect the distribution of importance sampling ratios and their clipping behavior, thereby altering which tokens dominate the update. Building on this insight, we characterize gradient expectation as the central quantity governing update dynamics and analyze the roles of token probability, advantage, and importance sampling ratio. Motivated by these findings, we propose Adaptive Clip Policy Optimization (ACPO), which adjusts clipping boundaries across token groups according to the empirical variance of their importance sampling ratios. Experiments on 3B and 7B models across diverse reasoning benchmarks, spanning mathematical problem solving, tabular QA, and logic puzzles, demonstrate that ACPO outperforms strong baselines such as DAPO and CISPO. These results demonstrate that principled, analysis-driven approaches yield more robust and effective RLVR methods. Code is available in: https://github.com/Control-derek/ACPO

RL with verifiable rewards can substantially improve LLM reasoning, yet standard GRPO-style training often treats easy, hard, and learnable questions alike through uniform sampling and weighting, leading to inefficient compute allocation. We study GRPO by tracking token log-probabilities, group-normalized advantages, and the induced token-level update weights. This reveals three recurring dynamics as training proceeds: (1) confidence inflation, (2) advantage contraction, and (3) hierarchical convergence. These findings suggest that the utility of each update depends strongly on both question difficulty and the model's current competence. Motivated by this, we propose Confidence and Difficulty-adaptive Policy Optimization (CoDaPO), which assigns each question a bounded value from rollout confidence and empirical difficulty. CoDaPO then uses this value to reweight policy updates and resample high-value learnable questions within mini-batches, thereby increasing discovery within the learnable band under a fixed compute budget. Across twelve benchmarks, CoDaPO consistently improves accuracy over existing RL methods. Our code is publicly available at https://github.com/tmlr-group/CoDaPO.