Exactly Computing the Local Lipschitz Constant of ReLU Networks

TL;DR

This paper introduces a novel method for exactly computing the local Lipschitz constant of ReLU neural networks, providing insights into robustness and generalization with strong theoretical and empirical results.

Contribution

It offers a new analytic framework relating Lipschitz constants to the generalized Jacobian and presents an algorithm for exact computation of these constants in ReLU networks.

Findings

Exact Lipschitz constants can be computed efficiently.

Regularized training affects the Lipschitz constant.

Existing estimators may be loose or inaccurate.

Abstract

The local Lipschitz constant of a neural network is a useful metric with applications in robustness, generalization, and fairness evaluation. We provide novel analytic results relating the local Lipschitz constant of nonsmooth vector-valued functions to a maximization over the norm of the generalized Jacobian. We present a sufficient condition for which backpropagation always returns an element of the generalized Jacobian, and reframe the problem over this broad class of functions. We show strong inapproximability results for estimating Lipschitz constants of ReLU networks, and then formulate an algorithm to compute these quantities exactly. We leverage this algorithm to evaluate the tightness of competing Lipschitz estimators and the effects of regularized training on the Lipschitz constant.