On the Value of Interaction and Function Approximation in Imitation Learning

We study the statistical guarantees for the Imitation Learning (IL) problem in episodic MDPs.Rajaraman et al. (2020) show an information theoretic lower bound that in the worst case, a learner which can even actively query the expert policy suffers from a suboptimality growing quadratically in the length of the horizon, $H$. We study imitation learning under the $\mu$-recoverability assumption of Ross et al. (2011) which assumes that the difference in the $Q$-value under the expert policy across different actions in a state do not deviate beyond $\mu$ from the maximum. We show that the reduction proposed by Ross et al. (2010) is statistically optimal: the resulting algorithm upon interacting with the MDP for $N$ episodes results in a suboptimality bound of $\widetilde{\mathcal{O}} \left( \mu |\mathcal{S}| H / N \right)$ which we show is optimal up to log-factors. In contrast, we show that any algorithm which does not interact with the MDP and uses an offline dataset of $N$ expert trajectories must incur suboptimality growing as $\gtrsim |\mathcal{S}| H^2/N$ even under the $\mu$-recoverability assumption. This establishes a clear and provable separation of the minimax rates between the active setting and the no-interaction setting.