Achieving Minimax Optimal Sample Complexity of Offline Reinforcement Learning: A DRO-Based Approach

Offline reinforcement learning aims to learn from pre-collected datasets without active exploration. This problem faces significant challenges, including limited data availability and distributional shifts. Existing approaches adopt a pessimistic stance towards uncertainty by penalizing rewards of under-explored state-action pairs to estimate value functions conservatively. In this paper, we show that the distributionally robust optimization (DRO) based approach can also address these challenges and is minimax optimal. Specifically, we directly model the uncertainty in the transition kernel and construct an uncertainty set of statistically plausible transition kernels. We then find the policy that optimizes the worst-case performance over this uncertainty set. We first design a metric-based Hoeffding-style uncertainty set such that with high probability the true transition kernel is in this set. We prove that to achieve a sub-optimality gap of $\epsilon$, the sample complexity is $\mathcal{O}(SC^{\pi^*}\epsilon^{-2}(1-\gamma)^{-4})$, where $\gamma$ is the discount factor, $S$ is the number of states, and $C^{\pi^*}$ is the single-policy clipped concentrability coefficient which quantifies the distribution shift. To achieve the optimal sample complexity, we further propose a less conservative Bernstein-style uncertainty set, which, however, does not necessarily include the true transition kernel. We show that an improved sample complexity of $\mathcal{O}(SC^{\pi^*}\epsilon^{-2}(1-\gamma)^{-3})$ can be obtained, which matches with the minimax lower bound for offline reinforcement learning, and thus is minimax optimal.