Unveiling and Mitigating Adversarial Vulnerabilities in Iterative Optimizers
Elad Sofer, Tomer Shaked, Caroline Chaux, Nir Shlezinger
TL;DR
This work reveals that iterative optimizers used for inference—a class traditionally not learned from data—exhibit adversarial sensitivity similar to neural networks when cast as ML models through deep unfolding. It establishes theoretical links between the learned hyperparameters of unfolded proximal gradient and ADMM methods and the Lipschitz continuity of the update mappings, showing how adversarial perturbations effectively reshape the optimization surface and shift minima. The authors demonstrate, both theoretically (via Lipschitz bounds and a least-squares proposition) and empirically (across sparse recovery, RPCA, and hybrid beamforming), that unfolding can both expose and mitigate vulnerability: conventional training can increase sensitivity, while adversarial training of unfolded optimizers reduces it with only modest losses on clean data. The work further provides practical defense insights for signal-processing and communications applications, suggesting adversarial-aware unfolding as a scalable defense that preserves the underlying optimization structure while improving robustness.
Abstract
Machine learning (ML) models are often sensitive to carefully crafted yet seemingly unnoticeable perturbations. Such adversarial examples are considered to be a property of ML models, often associated with their black-box operation and sensitivity to features learned from data. This work examines the adversarial sensitivity of non-learned decision rules, and particularly of iterative optimizers. Our analysis is inspired by the recent developments in deep unfolding, which cast such optimizers as ML models. We show that non-learned iterative optimizers share the sensitivity to adversarial examples of ML models, and that attacking iterative optimizers effectively alters the optimization objective surface in a manner that modifies the minima sought. We then leverage the ability to cast iteration-limited optimizers as ML models to enhance robustness via adversarial training. For a class of proximal gradient optimizers, we rigorously prove how their learning affects adversarial sensitivity. We numerically back our findings, showing the vulnerability of various optimizers, as well as the robustness induced by unfolding and adversarial training.