Nearly Minimax Optimal Reinforcement Learning for Discounted MDPs

Abstract: We study the reinforcement learning problem for discounted Markov Decision Processes (MDPs) under the tabular setting. We propose a model-based algorithm named UCBVI-$\gamma$, which is based on the \emph{optimism in the face of uncertainty principle} and the Bernstein-type bonus. We show that UCBVI-$\gamma$ achieves an $$\tilde{O}\big({\sqrt{SAT}}/{(1-\gamma)<sup>{1.5}}\big)$$ regret, where $S$ is the number of states, $A$ is the number of actions, $\gamma$ is the discount factor and $T$ is the number of steps. In addition, we construct a class of hard MDPs and show that for any algorithm, the expected regret is at least $$\tilde{\Omega}\big({\sqrt{SAT}}/{(1-\gamma)<sup>{1.5}}\big)$$. Our upper bound matches the minimax lower bound up to logarithmic factors, which suggests that UCBVI-$\gamma$ is nearly minimax optimal for discounted MDPs.