Papers › σ-zero: Gradient-based Optimization of ℓ₀-norm Adversarial Examples

σ-zero: Gradient-based Optimization of ℓ₀-norm Adversarial Examples

2 Feb 2024arXiv:2402.01879archive 2025-07-28

Antonio Emanuele Cinà, Francesco Villani, Maura Pintor, Lea Schönherr, Battista Biggio, Marcello Pelillo

Evaluating the adversarial robustness of deep networks to gradient-based attacks is challenging. While most attacks consider ℓ₂- and ℓ_∞-norm constraints to craft input perturbations, only a few investigate sparse ℓ₁- and ℓ₀-norm attacks. In particular, ℓ₀-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 ℓ₂- and ℓ_∞-norm attacks. In this work, we propose a novel ℓ₀-norm attack, called σ-zero, which leverages a differentiable approximation of the ℓ₀ 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\texttt{-zero} finds minimum ℓ₀-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.

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