Sample-efficient offline reinforcement learning (RL) with linear function approximation has recently been studied extensively. Much of prior work has yielded the minimax-optimal bound of $\tilde{\mathcal{O}}(\frac{1}{\sqrt{K}})$, with $K$ being the number of episodes in the offline data. In this work, we seek to understand instance-dependent bounds for offline RL with function approximation. We present an algorithm called Bootstrapped and Constrained Pessimistic Value Iteration (BCP-VI), which leverages data bootstrapping and constrained optimization on top of pessimism. We show that under a partial data coverage assumption, that of \emph{concentrability} with respect to an optimal policy, the proposed algorithm yields a fast rate of $\tilde{\mathcal{O}}(\frac{1}{K})$ for offline RL when there is a positive gap in the optimal Q-value functions, even when the offline data were adaptively collected. Moreover, when the linear features of the optimal actions in the states reachable by an optimal policy span those reachable by the behavior policy and the optimal actions are unique, offline RL achieves absolute zero sub-optimality error when $K$ exceeds a (finite) instance-dependent threshold. To the best of our knowledge, these are the first $\tilde{\mathcal{O}}(\frac{1}{K})$ bound and absolute zero sub-optimality bound respectively for offline RL with linear function approximation from adaptive data with partial coverage. We also provide instance-agnostic and instance-dependent information-theoretical lower bounds to complement our upper bounds.

本研究提出了一种基于Bootstrapped and Constrained Pessimistic Value Iteration算法的离线强化学习方法，该算法结合数据自举、约束优化和悲观主义。在局部数据覆盖的假设下，该算法提供了一个快速率，即使在自适应采集的离线数据中，也能够实现绝对零的次优误差和O（1 / K）的较低界限。

线性函数逼近离线强化学习的基于实例的上界