Adversarial robustness of sparse local Lipschitz predictors

TL;DR

This paper introduces sparse local Lipschitzness (SLL) to analyze the adversarial robustness of various models, providing tighter robustness certificates and insights into regularization strategies for improving model stability.

Contribution

It extends local Lipschitz continuity to SLL, encompassing models like Lasso and ReLU networks, and offers new robustness bounds and empirical validation.

Findings

Tighter robustness certificates for adversarial examples.

SLL captures stability and sparsity in model predictions.

Numerical evidence supports improved robustness strategies.

Abstract

This work studies the adversarial robustness of parametric functions composed of a linear predictor and a non-linear representation map. % that satisfies certain stability condition. Our analysis relies on \emph{sparse local Lipschitzness} (SLL), an extension of local Lipschitz continuity that better captures the stability and reduced effective dimensionality of predictors upon local perturbations. SLL functions preserve a certain degree of structure, given by the sparsity pattern in the representation map, and include several popular hypothesis classes, such as piece-wise linear models, Lasso and its variants, and deep feed-forward \relu networks. % are sparse local Lipschitz. We provide a tighter robustness certificate on the minimal energy of an adversarial example, as well as tighter data-dependent non-uniform bounds on the robust generalization error of these predictors. We instantiate these results for the case of deep neural networks and provide numerical evidence that supports our results, shedding new insights into natural regularization strategies to increase the robustness of these models.