Stronger and Faster Wasserstein Adversarial Attacks

Deep models, while being extremely flexible and accurate, are surprisingly vulnerable to "small, imperceptible" perturbations known as adversarial attacks. While the majority of existing attacks focus on measuring perturbations under the $\ell_p$ metric, Wasserstein distance, which takes geometry in pixel space into account, has long been known to be a suitable metric for measuring image quality and has recently risen as a compelling alternative to the $\ell_p$ metric in adversarial attacks. However, constructing an effective attack under the Wasserstein metric is computationally much more challenging and calls for better optimization algorithms. We address this gap in two ways: (a) we develop an exact yet efficient projection operator to enable a stronger projected gradient attack; (b) we show that the Frank-Wolfe method equipped with a suitable linear minimization oracle works extremely fast under Wasserstein constraints. Our algorithms not only converge faster but also generate much stronger attacks. For instance, we decrease the accuracy of a residual network on CIFAR-10 to $3.4\%$ within a Wasserstein perturbation ball of radius $0.005$, in contrast to $65.6\%$ using the previous Wasserstein attack based on an \emph{approximate} projection operator. Furthermore, employing our stronger attacks in adversarial training significantly improves the robustness of adversarially trained models.

Complete Hierarchy of Relaxation for Constrained Signomial Positivity

In this article, we prove that the Sums-of-AM/GM Exponential (SAGE) relaxation generalized to signomial over a constrained set is complete, with a compactness assumption. The high-level structure of the proof is as follows. We first apply variable change to convert a set of rational exponents to polynomial equations. In addition, we make the observation that linear constraints of the variables may also be converted to polynomial equations after variable change. Note that any convex set may be expressed as a set of linear constraints. Further, we use redundant constraints to find reduction to Positivstellensatz. We rely on Positivstellensatz results from algebraic geometry to obtain a decomposition of positive polynomials. Lastly, we explicitly show that the decomposition is of a form certifiable by SAGE.