Bellman Policy Optimization
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.
Related
- The Effect of Quadrature on the Convergence of Policy Iteration for Hamilton-Jacobi-Bellman Equations
- Policy iteration for Hamilton-Jacobi-Isaacs equations with control constraints and comparison with Hamilton-Jacobi-Bellman equations
- Offline Policy Evaluation via Mixed Bellman Residuals and Adaptive Critic Representations
- Monotone Neural Policy Iteration for High-Dimensional First-Order Hamilton--Jacobi--Bellman Equations
- Model Selection for Off-Policy Evaluation via Pairwise Bellman Validation
BibTeX
@misc{song2026bellman,
title = {{Bellman Policy Optimization}},
author = {Zhuoqing Song and Haotian Xu and Xikun Zhang and Lidong Bing},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.15987},
eprint = {2609.15987},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.15987},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bellman Policy Optimization
AU - Zhuoqing Song
AU - Haotian Xu
AU - Xikun Zhang
AU - Lidong Bing
PY - 2026
DA - 2026/09/14
PB - arXiv
DO - 10.48550/arxiv.2609.15987
UR - https://arxiv.org/abs/2609.15987
AB - 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.
ER -