# ESPO: Entropy Importance Sampling Policy Optimization

arXiv: https://arxiv.org/abs/2512.00499 · HTML: https://mlsanity.com/paper/2512.00499/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Placed in Reinforcement learning for reasoning, one of 17 directions in LLM reasoning.
- Builds on 20 of its 22 references that are in the map.
- First posted on arXiv 29 Nov 2025.

## Placed in

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

## Builds on (most cited first, 20 of 20)

- [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)
- [Efficient Memory Management for Large Language Model Serving with PagedAttention](https://mlsanity.com/paper/2309.06180/) (2023)
- [Let's Verify Step by Step](https://mlsanity.com/paper/2305.20050/) (2023)
- [DAPO: An Open-Source LLM Reinforcement Learning System at Scale](https://mlsanity.com/paper/2503.14476/) (2025)
- [HybridFlow: A Flexible and Efficient RLHF Framework](https://mlsanity.com/paper/2409.19256/) (2024)
- [Group Sequence Policy Optimization](https://mlsanity.com/paper/2507.18071/) (2025)
- [Flow-GRPO: Training Flow Matching Models via Online RL](https://mlsanity.com/paper/2505.05470/) (2025)
- [Beyond the 80/20 Rule: High-Entropy Minority Tokens Drive Effective Reinforcement Learning for LLM Reasoning](https://mlsanity.com/paper/2506.01939/) (2025)
- [SimpleRL-Zoo: Investigating and Taming Zero Reinforcement Learning for Open Base Models in the Wild](https://mlsanity.com/paper/2503.18892/) (2025)
- [Open-Reasoner-Zero: An Open Source Approach to Scaling Up Reinforcement Learning on the Base Model](https://mlsanity.com/paper/2503.24290/) (2025)
- [MathArena: Evaluating LLMs on Uncontaminated Math Competitions](https://mlsanity.com/paper/2505.23281/) (2025)
- [MiniMax-M1: Scaling Test-Time Compute Efficiently with Lightning Attention](https://mlsanity.com/paper/2506.13585/) (2025)
- [VinePPO: Refining Credit Assignment in RL Training of LLMs](https://mlsanity.com/paper/2410.01679/) (2024)
- [Right Question is Already Half the Answer: Fully Unsupervised LLM Reasoning Incentivization](https://mlsanity.com/paper/2504.05812/) (2025)
- [Geometric-Mean Policy Optimization](https://mlsanity.com/paper/2507.20673/) (2025)
- [Segment Policy Optimization: Effective Segment-Level Credit Assignment in RL for Large Language Models](https://mlsanity.com/paper/2505.23564/) (2025)
- [DCPO: Dynamic Clipping Policy Optimization](https://mlsanity.com/paper/2509.02333/) (2025)
- [Stabilizing Knowledge, Promoting Reasoning: Dual-Token Constraints for RLVR](https://mlsanity.com/paper/2507.15778/) (2025)
- [BNPO: Beta Normalization Policy Optimization](https://mlsanity.com/paper/2506.02864/) (2025)

## Built on by (most cited first, 9 of 9)

- [InfoPO: Information-Driven Policy Optimization for User-Centric Agents](https://mlsanity.com/paper/2603.00656/) (2026)
- [DSDR: Dual-Scale Diversity Regularization for Exploration in LLM Reasoning](https://mlsanity.com/paper/2602.19895/) (2026)
- [Accelerating RL Post-Training Rollouts via System-Integrated Speculative Decoding](https://mlsanity.com/paper/2604.26779/) (2026)
- [A Step Back: Prefix Importance Ratio Stabilizes Policy Optimization](https://mlsanity.com/paper/2601.22718/) (2026)
- [DGPO: Distribution Guided Policy Optimization for Fine Grained Credit Assignment](https://mlsanity.com/paper/2605.03327/) (2026)
- [Each Prompt Matters: Scaling Reinforcement Learning Without Wasting Rollouts on Hundred-Billion-Scale MoE](https://mlsanity.com/paper/2512.07710/) (2025)
- [Co-Evolution of Policy and Internal Reward for Language Agents](https://mlsanity.com/paper/2604.03098/) (2026)
- [Skip-Connected Policy Optimization for Implicit Advantage](https://mlsanity.com/paper/2604.08690/) (2026)
- [Deconstructing Off-Policy Ratios: Entropy-Normalized Trust Regions for Asynchronous Reinforcement Learning](https://mlsanity.com/paper/2607.22186/) (2026)

## Abstract (arXiv)

Reinforcement learning (RL) has become a central component of post-training for large language models (LLMs), particularly for complex reasoning tasks that require stable optimization over long generation horizons. However, achieving performance at scale often introduces a fundamental trade-off between training stability and training efficiency. Token-level optimization applies fine-grained updates at the individual units, but is prone to high variance in gradient estimation, which can result in unstable training dynamics. In contrast, Sequence-level optimization often relies on aggressive clipping mechanisms to ensure stable updates. However, such design may discard a large fraction of valid training samples, leading to inefficient gradient utilization and reduced training efficiency. We refer to this phenomenon as gradient underutilization. In this work, we propose Entropy Importance Sampling Policy Optimization (ESPO), a novel framework that aims to combine fine-grained updates with stable training. ESPO decomposes sequences into groups based on predictive entropy, enabling (1) Entropy Grouping Importance Sampling to capture intra-sequence heterogeneity, and (2) Entropy Adaptive Clipping to dynamically allocate trust regions based on model uncertainty. Extensive experiments on mathematical reasoning benchmarks demonstrate that ESPO not only accelerates convergence but also achieves state-of-the-art performance, notably improving accuracy on the challenging mathematical benchmarks.