  In this work, we propose a framework to learn feedback control policies with
guarantees on closed-loop generalization and adversarial robustness. These
policies are learned directly from expert demonstrations, contained in a
dataset of state-control input pairs, without any prior knowledge of the task
and system model. We use a Lipschitz-constrained loss minimization scheme to
learn feedback policies with certified closed-loop robustness, wherein the
Lipschitz constraint serves as a mechanism to tune the generalization
performance and robustness to adversarial disturbances. Our analysis exploits
the Lipschitz property to obtain closed-loop guarantees on generalization and
robustness of the learned policies. In particular, we derive a finite sample
bound on the policy learning error and establish robust closed-loop stability
under the learned control policy. We also derive bounds on the closed-loop
regret with respect to the expert policy and the deterioration of closed-loop
performance under bounded (adversarial) disturbances to the state measurements.
Numerical results validate our analysis and demonstrate the effectiveness of
our robust feedback policy learning framework. Finally, our results suggest the
existence of a potential tradeoff between nominal closed-loop performance and
adversarial robustness, and that improvements in nominal closed-loop
performance can only be made at the expense of robustness to adversarial
perturbations.
