Sample Complexity of Distributionally Robust Average-Reward Reinforcement Learning

Zijun Chen (Hong Kong University of Science and Technology) · Shengbo Wang (University of Southern California) · Nian Si (Hong Kong University of Science and Technology)

anchoring stateaverage-rewardcontrolled transition kernelsconvergence ratesdistributionally robust reinforcement learningf_k divergencefinite-sample convergencekl divergence markov decision processmixing timenumerical experiments optimal policyrobust average rewardsample complexity uncertainty setuniformly ergodic

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.