A Mirror Descent Framework for Adversarial Attacks over the Clipped $l_{\infty ,1}$ Ball and Group Sparsity Constraints

We introduce a Mirror Descent framework to solve optimization problems over a clipped ${\ell }_{\infty ,1}$ ball, generalizing the ${\ell }_1$-APGD algorithm to group-norm constraints. We resolve the expensive high-dimensional Bregman projection by reducing it to a continuous resource-allocation problem that is computed exactly via a efficient 1D root search. Furthermore, we propose a structurally relaxed projection that decouples within-group coordinate dependencies, reducing computational overhead while systematically enforcing group sparsity. Instantiating this framework as Group-Structured Mirror Descent (GSMD) for adversarial attack, our extensive evaluations on ImageNet reveal a clear trade-off: the exact projection accelerates objective convergence, whereas the relaxation yields strictly sparser modifications. GSMD generates localized, coherent perturbations and achieves state-of-the-art attack success rates on both standard and robust classifiers, requiring significantly fewer backward passes than existing structured attacks.