Trajectory Bellman Residual Minimization: A Simple Value-Based Method for LLM Reasoning

TL;DR

This paper introduces TBRM, a simple value-based off-policy RL method for LLM reasoning that outperforms policy-based methods on mathematical benchmarks with less complexity.

Contribution

We propose TBRM, a novel trajectory Bellman residual minimization algorithm for LLMs, demonstrating its convergence and superior performance over policy-based methods.

Findings

TBRM outperforms PPO and GRPO on reasoning benchmarks.

TBRM requires only one rollout per prompt, reducing computational overhead.

The method converges to near-optimal policies from arbitrary off-policy data.

Abstract

Policy-based methods currently dominate reinforcement learning (RL) pipelines for large language model (LLM) reasoning, leaving value-based approaches largely unexplored. We revisit the classical paradigm of Bellman Residual Minimization and introduce Trajectory Bellman Residual Minimization (TBRM), an algorithm that naturally adapts this idea to LLMs, yielding a simple yet effective off-policy algorithm that optimizes a single trajectory-level Bellman objective using the model's own logits as $Q$-values. TBRM removes the need for critics, importance-sampling ratios, or clipping, and operates with only one rollout per prompt. We prove convergence to the near-optimal KL-regularized policy from arbitrary off-policy data via an improved change-of-trajectory-measure analysis. Experiments on standard mathematical-reasoning benchmarks show that TBRM consistently outperforms policy-based baselines, like PPO and GRPO, with comparable or lower computational and memory overhead. Our results indicate that value-based RL might be a principled and efficient alternative for enhancing reasoning capabilities in LLMs.