On the Sample Complexity of Imitation Learning for Smoothed Model Predictive Control

Abstract

Recent work in imitation learning has shown that having an expert controller that is both suitably smooth and stable enables much stronger guarantees on the performance of the approximating learned controller. Constructing such smoothed expert controllers for arbitrary systems remains challenging, especially in the presence of input and state constraints. We show how such a smoothed expert can be designed for a general class of systems using a log-barrier-based relaxation of a standard Model Predictive Control (MPC) optimization problem. Our principal theoretical contributions include (1) demonstrating that the Jacobian of the barrier MPC controller can be written as a convex combination of pieces arising from the explicit MPC formulation, (2) bounding the Hessian of the barrier MPC as a function of the strength of the barrier function, and (3) presenting new results in both matrix and convex analysis for computing perturbed adjugate matrices and a tight (up to constant) lower bound on the distance of a solution with a self-concordant-barrier to the constraint set. We consider randomized smoothing as a point of comparison and show empirically that, unlike randomized smoothing, barrier MPC yields better performance while guaranteeing constraint satisfaction.