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Online imitation learning is the problem of how best to mimic expertdemonstrations, given access to the environment or an accurate simulator. Priorwork has shown that in the infinite sample regime, exact moment matchingachieves value equivalence to the expert policy. However, in the finite sampleregime, even if one has no optimization error, empirical variance can lead to aperformance gap that scales with $H^2 / N$ for behavioral cloning and $H /\sqrt{N}$ for online moment matching, where $H$ is the horizon and $N$ is thesize of the expert dataset. We introduce the technique of replay estimation toreduce this empirical variance: by repeatedly executing cached expert actionsin a stochastic simulator, we compute a smoother expert visitation distributionestimate to match. In the presence of general function approximation, we provea meta theorem reducing the performance gap of our approach to the parameterestimation error for offline classification (i.e. learning the expert policy).In the tabular setting or with linear function approximation, our meta theoremshows that the performance gap incurred by our approach achieves the optimal$\widetilde{O} \left( \min({H^{3/2}} / {N}, {H} / {\sqrt{N}} \right)$dependency, under significantly weaker assumptions compared to prior work. Weimplement multiple instantiations of our approach on several continuous controltasks and find that we are able to significantly improve policy performanceacross a variety of dataset sizes.

Minimax Optimal Online Imitation Learning via Replay Estimation