Sample Complexity of Offline Reinforcement Learning with Deep ReLU Networks

TL;DR

This paper provides the first theoretical analysis of the sample complexity for offline reinforcement learning using deep ReLU networks, accounting for complex function regularities and distributional shifts.

Contribution

It establishes a novel sample complexity bound for offline RL with deep ReLU networks under Besov regularity and correlated structures, extending beyond linear models.

Findings

Sample complexity bound depends on horizon, dimension, smoothness, and distribution shift.

Introduces the Besov dynamic closure and correlated structure concepts for analysis.

First theoretical characterization of deep neural network offline RL under general Besov conditions.

Abstract

Offline reinforcement learning (RL) leverages previously collected data for policy optimization without any further active exploration. Despite the recent interest in this problem, its theoretical results in neural network function approximation settings remain elusive. In this paper, we study the statistical theory of offline RL with deep ReLU network function approximation. In particular, we establish the sample complexity of $n = \tilde{\mathcal{O}}( H^{4 + 4 \frac{d}{\alpha}} \kappa_{\mu}^{1 + \frac{d}{\alpha}} \epsilon^{-2 - 2\frac{d}{\alpha}} )$ for offline RL with deep ReLU networks, where $\kappa_{\mu}$ is a measure of distributional shift, {$H = (1-\gamma)^{-1}$ is the effective horizon length}, $d$ is the dimension of the state-action space, $\alpha$ is a (possibly fractional) smoothness parameter of the underlying Markov decision process (MDP), and $\epsilon$ is a user-specified error. Notably, our sample complexity holds under two novel considerations: the Besov dynamic closure and the correlated structure. While the Besov dynamic closure subsumes the dynamic conditions for offline RL in the prior works, the correlated structure renders the prior works of offline RL with general/neural network function approximation improper or inefficient {in long (effective) horizon problems}. To the best of our knowledge, this is the first theoretical characterization of the sample complexity of offline RL with deep neural network function approximation under the general Besov regularity condition that goes beyond {the linearity regime} in the traditional Reproducing Hilbert kernel spaces and Neural Tangent Kernels.