\textbackslash sigma-zero: Gradient-based Optimization of \textbackslash ell\_0-norm Adversarial Examples

Antonio Emanuele Cinà

Primo

;Francesco Villani;Maura Pintor;Lea Schönherr;Battista Biggio;Marcello Pelillo

2025-01-01

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Abstract

Evaluating the adversarial robustness of deep networks to gradient-based attacks is challenging. While most attacks consider ω2- and ω→-norm constraints to craft input perturbations, only a few investigate sparse ω1- and ω0-norm attacks. In particular, ω0-norm attacks remain the least studied due to the inherent complexity of optimizing over a non-convex and non-differentiable constraint. However, evaluating adversarial robustness under these attacks could reveal weaknesses otherwise left untested with more conventional ω2- and ω→-norm attacks. In this work, we propose a novel ω0-norm attack, called ε-zero, which leverages a differentiable approximation of the ω0 norm to facilitate gradient-based optimization, and an adaptive projection operator to dynamically adjust the trade-off between loss minimization and perturbation sparsity. Extensive evaluations using MNIST, CIFAR10, and ImageNet datasets, involving robust and non-robust models, show that ε-zero finds minimum ω0-norm adversarial examples without requiring any time-consuming hyperparameter tuning, and that it outperforms all competing sparse attacks in terms of success rate, perturbation size, and efficiency.