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# Beyond No Regret: Instance-Dependent PAC Reinforcement Learning

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:358-418, 2022.

#### Abstract

The theory of reinforcement learning has focused on two fundamental problems: achieving low regret, and identifying $\epsilon$-optimal policies. While a simple reduction allows one to apply a low-regret algorithm to obtain an $\epsilon$-optimal policy and achieve the worst-case optimal rate, it is unknown whether low-regret algorithms can obtain the instance-optimal rate for policy identification. We show this is not possible—there exists a fundamental tradeoff between achieving low regret and identifying an $\epsilon$-optimal policy at the instance-optimal rate. Motivated by our negative finding, we propose a new measure of instance-dependent sample complexity for PAC tabular reinforcement learning which explicitly accounts for the attainable state visitation distributions in the underlying MDP. We then propose and analyze a novel, planning-based algorithm which attains this sample complexity—yielding a complexity which scales with the suboptimality gaps and the “reachability” of a state. We show our algorithm is nearly minimax optimal, and on several examples that our instance-dependent sample complexity offers significant improvements over worst-case bounds.