$σ$-zero: Gradient-based Optimization of $\ell_0$-norm Adversarial
  Examples
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 $\ell_2$- and
$\ell_\infty$-norm constraints to craft input perturbations, only a few
investigate sparse $\ell_1$- and $\ell_0$-norm attacks. In particular,
$\ell_0$-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 $\ell_2$- and $\ell_\infty$-norm
attacks. In this work, we propose a novel $\ell_0$-norm attack, called
$\sigma$-zero, which leverages an ad hoc differentiable approximation of the
$\ell_0$ 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 $\sigma$-zero finds minimum $\ell_0$-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.