Finite Sample Analysis of Minimax Offline Reinforcement Learning: Completeness, Fast Rates and First-Order Efficiency
Masatoshi Uehara, Masaaki Imaizumi, Nan Jiang, Nathan Kallus, Wen Sun, Tengyang Xie
TL;DR
This work develops a comprehensive finite-sample theory for offline off-policy evaluation using minimax estimators of the q- and w-functions. By introducing realizability, completeness, and adjoint variants, the authors derive polynomial, horizon-aware rates for both function estimation and policy value estimation, including the first finite-sample, first-order efficient bounds in general function-approximation settings. The methodology unifies several existing estimators (e.g., BRM, MWL, MQL, DICE families) under a single MIL framework and extends the results to state-based MIL and policy optimization. The findings have practical impact for offline RL by offering precise, non-asymptotic guarantees with explicit dependence on horizon and data complexity, and they open avenues for lower bounds and enhanced optimization strategies.
Abstract
We offer a theoretical characterization of off-policy evaluation (OPE) in reinforcement learning using function approximation for marginal importance weights and $q$-functions when these are estimated using recent minimax methods. Under various combinations of realizability and completeness assumptions, we show that the minimax approach enables us to achieve a fast rate of convergence for weights and quality functions, characterized by the critical inequality \citep{bartlett2005}. Based on this result, we analyze convergence rates for OPE. In particular, we introduce novel alternative completeness conditions under which OPE is feasible and we present the first finite-sample result with first-order efficiency in non-tabular environments, i.e., having the minimal coefficient in the leading term.