$Q\sharp$: Provably Optimal Distributional RL for LLM Post-Training

TL;DR

This paper introduces $Q\sharp$, a theoretically grounded, distributional RL algorithm for post-training LLMs that outperforms existing methods in reasoning tasks while maintaining low divergence from the reference policy.

Contribution

The paper presents $Q\sharp$, a novel value-based, distributional RL algorithm with provable optimality for KL-regularized RL, specifically tailored for LLM post-training.

Findings

$Q\sharp$ outperforms prior baselines in math reasoning benchmarks.

It maintains a smaller KL divergence to the reference policy.

Provides the first theoretical bounds for deterministic MDPs under realizability.

Abstract

Reinforcement learning (RL) post-training is crucial for LLM alignment and reasoning, but existing policy-based methods, such as PPO and DPO, can fall short of fixing shortcuts inherited from pre-training. In this work, we introduce $Q\sharp$, a value-based algorithm for KL-regularized RL that guides the reference policy using the optimal regularized $Q$ function. We propose to learn the optimal $Q$ function using distributional RL on an aggregated online dataset. Unlike prior value-based baselines that guide the model using unregularized $Q$-values, our method is theoretically principled and provably learns the optimal policy for the KL-regularized RL problem. Empirically, $Q\sharp$ outperforms prior baselines in math reasoning benchmarks while maintaining a smaller KL divergence to the reference policy. Theoretically, we establish a reduction from KL-regularized RL to no-regret online learning, providing the first bounds for deterministic MDPs under only realizability. Thanks to distributional RL, our bounds are also variance-dependent and converge faster when the reference policy has small variance. In sum, our results highlight $Q\sharp$ as an effective approach for post-training LLMs, offering both improved performance and theoretical guarantees. The code can be found at https://github.com/jinpz/q_sharp.