Near-Optimal Distributionally Robust Reinforcement Learning with General $L_p$ Norms

To address the challenges of sim-to-real gap and sample efficiency in reinforcement learning (RL), this work studies distributionally robust Markov decision processes (RMDPs) --- optimize the worst-case performance when the deployed environment is within an uncertainty set around some nominal MDP. Despite recent efforts, the sample complexity of RMDPs has remained largely undetermined. While the statistical implications of distributional robustness in RL have been explored in some specific cases, the generalizability of the existing findings remains unclear, especially in comparison to standard RL. Assuming access to a generative model that samples from the nominal MDP, we examine the sample complexity of RMDPs using a class of generalized $L_p$ norms as the 'distance' function for the uncertainty set, under two commonly adopted $sa$-rectangular and $s$-rectangular conditions. Our results imply that RMDPs can be more sample-efficient to solve than standard MDPs using generalized $L_p$ norms in both $sa$- and $s$-rectangular cases, potentially inspiring more empirical research. We provide a near-optimal upper bound and a matching minimax lower bound for the $sa$-rectangular scenarios. For $s$-rectangular cases, we improve the state-of-the-art upper bound and also derive a lower bound using $L_\infty$ norm that verifies the tightness.