Bellman Policy Optimization
Bellman Policy Optimization, a critic-free RLVR method derived from Policy Mirror Descent, improves LLM mathematical reasoning without intermediate state-value estimation.
The paper introduces Bellman Policy Optimization (BPO), a critic-free reinforcement learning method for LLMs with verifiable rewards, derived from Policy Mirror Descent. BPO uses the Bellman equations to reformulate PMD as a trajectory-level objective for autoregressive generation with terminal rewards, avoiding state-value estimation at intermediate states. The authors prove BPO shares the same unique optimal solution as PMD and validate it on mathematical reasoning benchmarks.
- Critic-free method derived from Policy Mirror Descent for RLVR training
- Reformulates PMD as trajectory-level objective via Bellman equations
- Proves same unique optimal solution as original PMD objective
- Mismatch-correction weight is smoothed ratio of complementary token probabilities
Full article103 words · extracted from arxiv.org · click to collapse
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.
Text extracted automatically; images, tables and formatting may be missing. Original: https://arxiv.org/abs/2609.15987