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

Adversarial robustness of sparse local Lipschitz predictors