Learning Lipschitz Feedback Policies From Expert Demonstrations: Closed-Loop Guarantees, Robustness and Generalization

In this work, we propose a framework in which we use a Lipschitz-constrained loss minimization scheme to learn feedback control policies with guarantees on closed-loop stability, adversarial robustness, and generalization. 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. Our analysis exploits the Lipschitz property of the learned policies to obtain closed-loop guarantees on stability, adversarial robustness, and generalization over scenarios unexplored by the expert. In particular, first, we establish robust closed-loop stability under the learned control policy, where we provide guarantees that the closed-loop trajectory under the learned policy stays within a bounded region around the expert trajectory and converges asymptotically to a bounded region around the origin. Second, we derive bounds on the closed-loop regret with respect to the expert policy and on the deterioration of the closed-loop performance under bounded (adversarial) disturbances to the state measurements. These bounds provide certificates for closed-loop performance and adversarial robustness for learned policies. Third, we derive a (probabilistic) bound on generalization error for the learned policies. Numerical results validate our analysis and demonstrate the effectiveness of our robust feedback policy learning framework. Finally, our results support 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.