Robustness of classifiers to universal perturbations: a geometric perspective
This addresses the problem of adversarial robustness in machine learning for security-critical applications, offering a foundational geometric perspective.
The paper tackles the vulnerability of deep networks to universal perturbations by providing a quantitative analysis linking robustness to the geometry of the decision boundary, establishing theoretical bounds under flat and curved models and proving the existence of small universal perturbations due to systematic positive curvature.
Deep networks have recently been shown to be vulnerable to universal perturbations: there exist very small image-agnostic perturbations that cause most natural images to be misclassified by such classifiers. In this paper, we propose the first quantitative analysis of the robustness of classifiers to universal perturbations, and draw a formal link between the robustness to universal perturbations, and the geometry of the decision boundary. Specifically, we establish theoretical bounds on the robustness of classifiers under two decision boundary models (flat and curved models). We show in particular that the robustness of deep networks to universal perturbations is driven by a key property of their curvature: there exists shared directions along which the decision boundary of deep networks is systematically positively curved. Under such conditions, we prove the existence of small universal perturbations. Our analysis further provides a novel geometric method for computing universal perturbations, in addition to explaining their properties.