# Rethinking Importance Sampling in LLM Policy Optimization: A Cumulative Token Perspective

arXiv: https://arxiv.org/abs/2605.07331 · HTML: https://mlsanity.com/paper/2605.07331/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 18 of its 29 references that are in the map.
- First posted on arXiv 8 May 2026.

## Placed in

- LLM reasoning / [Reinforcement learning for reasoning](https://mlsanity.com/lr/reinforcement-learning-for-reasoning/)

## Builds on (most cited first, 18 of 18)

- [Proximal Policy Optimization Algorithms](https://mlsanity.com/paper/1707.06347/) (2017)
- [DeepSeekMath: Pushing the Limits of Mathematical Reasoning in Open Language Models](https://mlsanity.com/paper/2402.03300/) (2024)
- [Qwen3 Technical Report](https://mlsanity.com/paper/2505.09388/) (2025)
- [DeepSeek-R1: Incentivizing Reasoning Capability in LLMs via Reinforcement Learning](https://mlsanity.com/paper/2501.12948/) (2025)
- [Training a Helpful and Harmless Assistant with Reinforcement Learning from Human Feedback](https://mlsanity.com/paper/2204.05862/) (2022)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [Understanding R1-Zero-Like Training: A Critical Perspective](https://mlsanity.com/paper/2503.20783/) (2025)
- [OpenAI o1 System Card](https://mlsanity.com/paper/2412.16720/) (2024)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [RLHF Workflow: From Reward Modeling to Online RLHF](https://mlsanity.com/paper/2405.07863/) (2024)
- [KTO: Model Alignment as Prospect Theoretic Optimization](https://mlsanity.com/paper/2402.01306/) (2024)
- [Self-Play Preference Optimization for Language Model Alignment](https://mlsanity.com/paper/2405.00675/) (2024)
- [Nash Learning from Human Feedback](https://mlsanity.com/paper/2312.00886/) (2023)
- [SimpleTIR: End-to-End Reinforcement Learning for Multi-Turn Tool-Integrated Reasoning](https://mlsanity.com/paper/2509.02479/) (2025)
- [Soft Adaptive Policy Optimization](https://mlsanity.com/paper/2511.20347/) (2025)
- [Iterative Nash Policy Optimization: Aligning LLMs with General Preferences via No-Regret Learning](https://mlsanity.com/paper/2407.00617/) (2024)
- [Improving Sampling Efficiency in RLVR through Adaptive Rollout and Response Reuse](https://mlsanity.com/paper/2509.25808/) (2025)
- [Improving LLM General Preference Alignment via Optimistic Online Mirror Descent](https://mlsanity.com/paper/2502.16852/) (2025)

## Built on by (most cited first, 7 of 7)

- [Reusing Rollouts under Policy Lag: Prefix-Normalized Policy Optimization for LLM Reinforcement Learning](https://mlsanity.com/paper/2608.01418/) (2026)
- [Video-HopChain: Multi-Hop Questions and Confidence-Gated Exploration for Video Reasoning Models](https://mlsanity.com/paper/2609.25773/) (2026)
- [REVO: Rollout-Efficient Off-Policy Distillation via Variance-Guided Reuse](https://mlsanity.com/paper/2609.37500/) (2026)
- [Free Everywhere, Exact on Trees: PPO's Dropped Correction Buys Sample Efficiency Under Aggressive Reuse](https://mlsanity.com/paper/2609.39634/) (2026)
- [Understanding Enrichment in Reinforcement Learning](https://mlsanity.com/paper/2610.02846/) (2026)
- [S3R: Fast and Faithful RL Rollouts via Self-Sparse Speculative Decoding](https://mlsanity.com/paper/2610.13202/) (2026)
- [Ascent Conditions for Length-Normalized Single-Rollout Policy Updates](https://mlsanity.com/paper/2610.14670/) (2026)

## Abstract (arXiv)

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 √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.