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A rapidly growing area of work has studied the existence of adversarialexamples, datapoints which have been perturbed to fool a classifier, but thevast majority of these works have focused primarily on threat models defined by$\ell_p$ norm-bounded perturbations. In this paper, we propose a new threatmodel for adversarial attacks based on the Wasserstein distance. In the imageclassification setting, such distances measure the cost of moving pixel mass,which naturally cover "standard" image manipulations such as scaling, rotation,translation, and distortion (and can potentially be applied to other settingsas well). To generate Wasserstein adversarial examples, we develop a procedurefor projecting onto the Wasserstein ball, based upon a modified version of theSinkhorn iteration. The resulting algorithm can successfully attack imageclassification models, bringing traditional CIFAR10 models down to 3% accuracywithin a Wasserstein ball with radius 0.1 (i.e., moving 10% of the image mass 1pixel), and we demonstrate that PGD-based adversarial training can improve thisadversarial accuracy to 76%. In total, this work opens up a new direction ofstudy in adversarial robustness, more formally considering convex metrics thataccurately capture the invariances that we typically believe should exist inclassifiers. Code for all experiments in the paper is available athttps://github.com/locuslab/projected_sinkhorn.

Wasserstein Adversarial Examples via Projected Sinkhorn Iterations