Offline Deep Q* Estimation with Diffusion Models

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. We establish the smoothness of $Q^*$ through the Bellman equation using the the H\"older assumptions on the conditional reward and transition densities. With suitable sample sizes, we obtain the oracle value-stage rate $\widetilde{\mathcal O}\bigl(n^{-\frac{2\alpha}{d_x+d_a+2\alpha}}\bigr)$ for the excess Bellman residual risk. We then translate this risk bound into an expected $L^2$ estimation error of order $\widetilde{\mathcal O}\bigl(n^{-\frac{\alpha}{d_x+d_a+2\alpha}}\bigr)$ under a concentrability condition. For the total offline budget $\widetilde n=n+n_r+n_x$, we obtain a value estimation rate of $\widetilde{\mathcal O}\bigl(\widetilde n^{-\frac{\alpha}{2d_x+d_a+2\alpha}}\bigr)$ and establish an information-theoretic lower bound of order $\widetilde n^{-\frac{\alpha}{d_x+d_a+2\alpha}}$. Importantly, our theoretical analysis does not rely on Bellman completeness assumptions commonly used in deep RL theory.