Training Lipschitz Continuous Operators Using Reproducing Kernels

Abstract

This paper proposes that Lipschitz continuity is a natural outcome of regularized least squares in kernel-based learning. Lipschitz continuity is an important proxy for robustness of input-output operators. It is also instrumental for guaranteeing closed-loop stability of kernel-based controlllers through small incremental gain arguments. We introduce a new class of nonexpansive kernels that are shown to induce Hilbert spaces consisting of only Lipschitz continuous operators. The Lipschitz constant of estimated operators within such Hilbert spaces can be tuned by suitable selection of a regularization parameter. As is typical for kernel-based models, input-output operators are estimated from data by solving tractable systems of linear equations. The approach thus constitutes a promising alternative to Lipschitz-bounded neural networks, that have recently been investigated but are computationally expensive to train.