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# Offline Reinforcement Learning with Realizability and Single-policy Concentrability

*Proceedings of Thirty Fifth Conference on Learning Theory*, PMLR 178:2730-2775, 2022.

#### Abstract

Sample-efficiency guarantees for offline reinforcement learning (RL) often rely on strong assumptions on both the function classes (e.g., Bellman-completeness) and the data coverage (e.g., all-policy concentrability). Despite the recent efforts on relaxing these assumptions, existing works are only able to relax one of the two factors, leaving the strong assumption on the other factor intact. As an important open problem, can we achieve sample-efficient offline RL with weak assumptions on both factors? In this paper we answer the question in the positive. We analyze a simple algorithm based on the primal-dual formulation of MDPs, where the dual variables (discounted occupancy) are modeled using a density-ratio function against offline data. With proper regularization, the algorithm enjoys polynomial sample complexity, under only realizability and single-policy concentrability. We also provide alternative analyses based on different assumptions to shed light on the nature of primal-dual algorithms for offline RL.