σ-zero: Gradient-based Optimization of ℓ_0-norm Adversarial Examples

Evaluating the adversarial robustness of deep networks to gradient-based attacks is challenging. While most attacks consider ℓ_2- and ℓ_∞-norm constraints to craft input perturbations, only a few investigate sparse ℓ_1- and ℓ_0-norm attacks. In particular, ℓ_0-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 ℓ_2- and ℓ_∞-norm attacks. In this work, we propose a novel ℓ_0-norm attack, called σ-zero, which leverages an ad hoc differentiable approximation of the ℓ_0 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 σ-zero finds minimum ℓ_0-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 scalability.