# Agnostic Reinforcement Learning with Low-Rank MDPs and Rich Observations

There have been many recent advances on provably efficient Reinforcement Learning (RL) in problems with rich observation spaces. However, all these works share a strong realizability assumption about the optimal value function of the true MDP. Such realizability assumptions are often too strong to hold in practice. In this work, we consider the more realistic setting of agnostic RL with rich observation spaces and a fixed class of policies $\Pi$ that may not contain any near-optimal policy. We provide an algorithm for this setting whose error is bounded in terms of the rank $d$ of the underlying MDP. Specifically, our algorithm enjoys a sample complexity bound of $\widetilde{O}\left((H^{4d} K^{3d} \log |\Pi|)/\epsilon^2\right)$ where $H$ is the length of episodes, $K$ is the number of actions and $\epsilon>0$ is the desired sub-optimality. We also provide a nearly matching lower bound for this agnostic setting that shows that the exponential dependence on rank is unavoidable, without further assumptions.

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