Adversarial robustness of sparse local Lipschitz predictors
This work addresses adversarial robustness for machine learning models, offering theoretical insights and practical regularization strategies, though it is incremental as it builds on existing Lipschitz-based methods.
The paper tackles the problem of certifying adversarial robustness for parametric functions by introducing sparse local Lipschitzness (SLL), which extends local Lipschitz continuity to better capture stability and reduced dimensionality. It provides tighter robustness certificates on adversarial example energy and data-dependent bounds on robust generalization error, with numerical evidence for deep neural networks.
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.