Optimal Sample Complexity of Reinforcement Learning for Mixing Discounted Markov Decision Processes

Abstract: We consider the optimal sample complexity theory of tabular reinforcement learning (RL) for maximizing the infinite horizon discounted reward in a Markov decision process (MDP). Optimal worst-case complexity results have been developed for tabular RL problems in this setting, leading to a sample complexity dependence on $\gamma$ and $\epsilon$ of the form $\tilde \Theta((1-\gamma)^{-3}\epsilon^{-2})$, where $\gamma$ denotes the discount factor and $\epsilon$ is the solution error tolerance. However, in many applications of interest, the optimal policy (or all policies) induces mixing. We establish that in such settings, the optimal sample complexity dependence is $\tilde \Theta(t_{\text{mix}}(1-\gamma)^{-2}\epsilon^{-2})$, where $t_{\text{mix}}$ is the total variation mixing time. Our analysis is grounded in regeneration-type ideas, which we believe are of independent interest, as they can be used to study RL problems for general state space MDPs.