#### Optimal Uniform OPE and Model-based Offline Reinforcement Learning in Time-Homogeneous, Reward-Free and Task-Agnostic Settings

This work studies the statistical limits of uniform convergence for offline policy evaluation (OPE) problems with model-based methods (for episodic MDP) and provides a unified framework towards optimal learning for several well-motivated offline tasks. Uniform OPE $\sup_\Pi|Q^\pi-\hat{Q}^\pi|<\epsilon$ is a stronger measure than the point-wise OPE and ensures offline learning when $\Pi$ contains all policies (the global class). In this paper, we establish an $\Omega(H^2 S/d_m\epsilon^2)$ lower bound (over model-based family) for the global uniform OPE and our main result establishes an upper bound of $\tilde{O}(H^2/d_m\epsilon^2)$ for the \emph{local} uniform convergence that applies to all \emph{near-empirically optimal} policies for the MDPs with \emph{stationary} transition. Here $d_m$ is the minimal marginal state-action probability. Critically, the highlight in achieving the optimal rate $\tilde{O}(H^2/d_m\epsilon^2)$ is our design of \emph{singleton absorbing MDP}, which is a new sharp analysis tool that works with the model-based approach. We generalize such a model-based framework to the new settings: offline task-agnostic and the offline reward-free with optimal complexity $\tilde{O}(H^2\log(K)/d_m\epsilon^2)$ ($K$ is the number of tasks) and $\tilde{O}(H^2S/d_m\epsilon^2)$ respectively. These results provide a unified solution for simultaneously solving different offline RL problems.