Bellman Policy Optimization

2026-09-14

AI Summary

该研究通过引入Bellman Policy Optimization方法，解决大语言模型在带有可验证奖励的强化学习中的推理能力提升问题，无需估计中间状态值。

Abstract

Reinforcement learning with verifiable rewards (RLVR) improves the reasoning capabilities of large language models (LLMs). We introduce Bellman Policy Optimization (BPO), a critic-free method derived from Policy Mirror Descent (PMD). For autoregressive generation with terminal rewards, BPO uses the Bellman equations to reformulate PMD as a trajectory-level objective. The reformulation avoids estimating state values at intermediate states. We prove that it has the same unique optimal solution as the original PMD objective. We derive the practical BPO loss by approximating this objective. Its mismatch-correction weight is a smoothed ratio of complementary token probabilities. Experiments on mathematical reasoning benchmarks demonstrate the effectiveness of BPO.

Problem

• Reinforcement learning
• verifiable rewards
• large language models
• reasoning capabilities

Innovation

• Bellman Policy Optimization
• autoregressive generation
• terminal rewards
• trajectory-level objective
• mismatch-correction weight