Optimal Sample Complexity of Reinforcement Learning for Uniformly Ergodic Discounted Markov Decision Processes
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We consider the optimal sample complexity theory of tabular reinforcement learning (RL) for controlling the infinite horizon discounted reward in a Markov decision process (MDP). Optimal min-max complexity results have been developed for tabular RL in this setting, leading to a sample complexity dependence on γ and ϵ of the form Θ̃((1-γ)^-3ϵ^-2), where γ is the discount factor and ϵ is the tolerance solution error. However, in many applications of interest, the optimal policy (or all policies) will induce mixing. We show that in these settings the optimal min-max complexity is Θ̃(t_minorize(1-γ)^-2ϵ^-2), where t_minorize is a measure of mixing that is within an equivalent factor of the total variation mixing time. Our analysis is based on regeneration-type ideas, that, we believe are of independent interest since they can be used to study related problems for general state space MDPs.
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