We study the statistical limits of uniform convergence for offline policy evaluation (OPE) problems (uniform OPE for short) with model-based methods under episodic MDP setting. Uniform OPE $\sup_\Pi|Q^\pi-\hat{Q}^\pi|<\epsilon$ (initiated by Yin et al. 2021) is a stronger measure than the point-wise (fixed policy) OPE and ensures offline policy learning when $\Pi$ contains all policies (we call it global policy 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, where $d_m$ is the minimal state-action distribution induced by the behavior policy. The order $S/d_m\epsilon^2$ reveals global uniform OPE task is intrinsically harder than offline policy learning due to the extra $S$ factor. Next, our main result establishes an episode complexity of $\tilde{O}(H^2/d_m\epsilon^2)$ for \emph{local} uniform convergence that applies to all \emph{near-empirically optimal} policies for the MDPs with \emph{stationary} transition. The result implies the optimal sample complexity for offline learning and separates local uniform OPE from the global case. Paramountly, the model-based method combining with our new analysis technique (singleton absorbing MDP) can be adapted 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, which provides a unified framework for simultaneously solving different offline RL problems.
翻译:我们研究离线政策评价(OPE)问题统一趋同的统计限度( 简称为 OPE ) 。 统一 OPE $sup ⁇ pí ⁇ pi ⁇ pí ⁇ ⁇ epsilon$( 由 Yin 等人 2021 发起) 是比点( 固定政策) OPE 更强的衡量尺度, 当$\ Pi$ 包含所有政策( 我们称之为全球政策类) 时, 并确保离线政策学习的离线( 称之为全球政策类) 。 在本文中, 我们为全球统一OPE 设置了 $\ Omega (H2 S/ d_ m\\ eplon2) 的较低约束值( 超模式家庭) 。 离线( 离线) 离线( MID_ d) 和离线( IMD) 的离线( IMD) 和( IMD) 最优化的离线( IMD) 和( IMD) 最优化的离线( IMD) 解的( IMD) 解) 解( 和( IMD) 解) 最优化的( 解) 解) 定义的( 的) 解) 和( 解) 解的) 任务) 和( 解) 解的) 等( 的( 的) 的) 任务- c( 的) 的) 解- 和( 解的) 解的) 解的) 和( 解的) 任务- 解- 的) 任务- 规则- 的) 解- 的) 解式的( 解- 的) 的( 的) 的) 的) 解- 解- 的) 解- 解- 的) 的) 的) 解- 和( 的) 的) 的) 解- 解- 解- 解的) 解的(M- 的) 和(O- 的) 的) 和(O- 的) 解- 解- 解- 的) 解的) 解式的) 解的