Offline Deep Q* Estimation with Diffusion Models

2026-08-14

AI Summary

This study addresses the challenge of estimating optimal Q-functions in offline reinforcement learning arising from unknown rewards and transition dynamics. We propose an operator-value decoupling framework that leverages conditional diffusion models to capture environment dynamics and achieves deep estimation by minimizing empirical Bellman residuals. A key theoretical contribution is the establishment of non-asymptotic convergence rate guarantees without relying on completeness assumptions. Extensive experiments across diverse benchmark tasks demonstrate the method’s significant effectiveness and superior performance. Collectively, this work presents a novel paradigm for offline RL that successfully integrates theoretical rigor with strong empirical results, offering a robust solution to fundamental estimation difficulties in data-constrained settings.

Abstract

In offline RL, estimating the optimal action-value function $Q^*$ can be formulated as solving the optimal Bellman equation based solely on offline observations. A fundamental challenge is that the reward function and transition kernel are unknown, so the optimal Bellman operator is not directly observable from data. To address this issue, we propose a novel framework that decouples operator estimation from value function learning. In this approach, we first formulate conditional diffusion models to estimate the reward law and transition kernel, which induces a data-driven approximation of the optimal Bellman operator. We then plug these estimators into the Bellman equation and obtain a deep estimator of $Q^*$ by minimizing the empirical Bellman residual over a neural network function class. Theoretically, we first establish sharp nonasymptotic convergence rates for learning the optimal Bellman operator through an end-to-end analysis of conditional diffusion estimation in total variation distance. We then establish the oracle value-stage rate $\widetilde{\mathcal O}\bigl(n^{-\frac{2β}{d_x+d_a+2β}}\bigr)$ for the excess Bellman residual risk. Finally, under a concentrability condition, we translate this residual bound into an $L^2$ convergence rate of $\widetilde{\mathcal O}\bigl(n^{-\fracβ{d_x+d_a+2β}}\bigr)$ for the resulting deep estimator of $Q^*$, where $d_x$ and $d_a$ denote the dimensions of the state and action spaces, respectively, and $β$ denotes the Hölder smoothness index of $Q^*$. Importantly, our theoretical analysis does not rely on completeness assumptions commonly used in deep RL theory. Extensive numerical experiments demonstrate the effectiveness of the proposed method and its strong empirical performance.

Problem

• Offline Reinforcement Learning
• Optimal Action-Value Function
• Optimal Bellman Operator
• Unknown Reward and Transition

Innovation

• Offline RL
• Diffusion Models
• Bellman Operator Estimation
• Decoupled Framework
• Non-asymptotic Convergence