  Behavioral Cloning (BC) aims at learning a policy that mimics the behavior
demonstrated by an expert. The current theoretical understanding of BC is
limited to the case of finite actions. In this paper, we study BC with the goal
of providing theoretical guarantees on the performance of the imitator policy
in the case of continuous actions. We start by deriving a novel bound on the
performance gap based on Wasserstein distance, applicable for continuous-action
experts, holding under the assumption that the value function is Lipschitz
continuous. Since this latter condition is hardy fulfilled in practice, even
for Lipschitz Markov Decision Processes and policies, we propose a relaxed
setting, proving that value function is always Holder continuous. This result
is of independent interest and allows obtaining in BC a general bound for the
performance of the imitator policy. Finally, we analyze noise injection, a
common practice in which the expert action is executed in the environment after
the application of a noise kernel. We show that this practice allows deriving
stronger performance guarantees, at the price of a bias due to the noise
addition.
