Sample Complexity of Distributionally Robust Average-Reward Reinforcement Learning

Zijun Chen, Shengbo Wang, Nian Si

Keywords: Reinforcement Learning, Distributional Robustness, Markov Decision Process, Ergodicity, Long-run Average Reward

TL;DR: We propose priori knowledge-free algorithms for distributionally robust average-reward estimation, with both finite-sample theoretical guarantees and numerical validation.

Abstract: Motivated by practical applications where stable long-term performance is critical—such as robotics, operations research, and healthcare—we study the problem of distributionally robust (DR) average-reward reinforcement learning. We propose two algorithms that achieve near-optimal sample complexity. The first reduces the problem to a DR discounted Markov decision process (MDP), while the second, Anchored DR Average-Reward MDP, introduces an anchoring state to stabilize the controlled transition kernels within the uncertainty set. Assuming the nominal MDP is uniformly ergodic, we prove that both algorithms attain a sample complexity of $\widetilde{O}\left(|\mathbf{S}||\mathbf{A}| t_{\mathrm{mix}}^2\varepsilon^{-2}\right)$ for estimating the optimal policy as well as the robust average reward under KL and $f_k$-divergence-based uncertainty sets, provided the uncertainty radius is sufficiently small. Here, $\varepsilon$ is the target accuracy, $|\mathbf{S}|$ and $|\mathbf{A}|$ denote the sizes of the state and action spaces, and $t_{\mathrm{mix}}$ is the mixing time of the nominal MDP. This represents the first finite-sample convergence guarantee for DR average-reward reinforcement learning. We further validate the convergence rates of our algorithms through numerical experiments.

Primary Area: Reinforcement learning (e.g., decision and control, planning, hierarchical RL, robotics)

Submission Number: 11653