Exact Solutions of the Inner Optimization Problem of Adversarial Robustness

Abstract

We propose a robust framework that uses adversarially robust training to safeguard the ML models against perturbed testing data. Our contributions can be seen from both computational and statistical perspectives. Firstly, from a computational/optimization point of view, we derive the ready-to-use exact solution for several widely used loss functions with a variety of norm constraints on adversarial perturbation for various supervised and unsupervised ML problems, including regression, classification, two-layer neural networks, graphical models, and matrix completion. The solutions are either in closed-form, or an easily tractable optimization problem such as 1-D convex optimization, semidefinite programming, difference of convex programming or a sorting-based algorithm. Secondly, from statistical/generalization viewpoint, using some of these results, we derive novel bounds of the adversarial Rademacher complexity for various problems, which entails new generalization bounds. Thirdly, we perform some sanity-check experiments on real-world datasets for supervised problems such as regression and classification, as well as for unsupervised problems such as matrix completion and learning graphical models. The full-length version of this work with detailed proofs of our theoretical claims is available at https://arxiv.org/pdf/2208.09449.