This paper presents a systematic study on gap-dependent sample complexity in offline reinforcement learning. Prior work showed when the density ratio between an optimal policy and the behavior policy is upper bounded (the optimal policy coverage assumption), then the agent can achieve an $O\left(\frac{1}{\epsilon^2}\right)$ rate, which is also minimax optimal. We show under the optimal policy coverage assumption, the rate can be improved to $O\left(\frac{1}{\epsilon}\right)$ when there is a positive sub-optimality gap in the optimal $Q$-function. Furthermore, we show when the visitation probabilities of the behavior policy are uniformly lower bounded for states where an optimal policy's visitation probabilities are positive (the uniform optimal policy coverage assumption), the sample complexity of identifying an optimal policy is independent of $\frac{1}{\epsilon}$. Lastly, we present nearly-matching lower bounds to complement our gap-dependent upper bounds.

翻译：本文对离线强化学习中基于差距的样本复杂性进行了系统研究。 当最佳政策和行为政策之间的密度比被上限(最佳政策覆盖面假设)时, 先前的工作显示, 当该代理商可以达到美元( left) (frac{ 1\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\