Near-Optimal Offline Reinforcement Learning via Double Variance Reduction

We consider the problem of offline reinforcement learning (RL) --- a well-motivated setting of RL that aims at policy optimization using only historical data. Despite its wide applicability, theoretical understandings of offline RL, such as its optimal sample complexity, remain largely open even in basic settings such as \emph{tabular} Markov Decision Processes (MDPs). In this paper, we propose \emph{Off-Policy Double Variance Reduction} (OPDVR), a new variance reduction-based algorithm for offline RL. Our main result shows that OPDVR provably identifies an $\epsilon$-optimal policy with $\widetilde{O}(H^2/d_m\epsilon^2)$ episodes of offline data in the finite-horizon \emph{stationary transition} setting, where $H$ is the horizon length and $d_m$ is the minimal marginal state-action distribution induced by the behavior policy. This improves over the best-known upper bound by a factor of $H$. Moreover, we establish an information-theoretic lower bound of $\Omega(H^2/d_m\epsilon^2)$ which certifies that OPDVR is optimal up to logarithmic factors. Lastly, we show that OPDVR also achieves rate-optimal sample complexity under alternative settings such as the finite-horizon MDPs with non-stationary transitions and the infinite horizon MDPs with discounted rewards.