  Deep neural networks are known to be vulnerable to adversarial attacks (AA).
For an image recognition task, this means that a small perturbation of the
original can result in the image being misclassified. Design of such attacks as
well as methods of adversarial training against them are subject of intense
research. We re-cast the problem using techniques of Wasserstein
distributionally robust optimization (DRO) and obtain novel contributions
leveraging recent insights from DRO sensitivity analysis. We consider a set of
distributional threat models. Unlike the traditional pointwise attacks, which
assume a uniform bound on perturbation of each input data point, distributional
threat models allow attackers to perturb inputs in a non-uniform way. We link
these more general attacks with questions of out-of-sample performance and
Knightian uncertainty. To evaluate the distributional robustness of neural
networks, we propose a first-order AA algorithm and its multi-step version. Our
attack algorithms include Fast Gradient Sign Method (FGSM) and Projected
Gradient Descent (PGD) as special cases. Furthermore, we provide a new
asymptotic estimate of the adversarial accuracy against distributional threat
models. The bound is fast to compute and first-order accurate, offering new
insights even for the pointwise AA. It also naturally yields out-of-sample
performance guarantees. We conduct numerical experiments on the CIFAR-10
dataset using DNNs on RobustBench to illustrate our theoretical results. Our
code is available at https://github.com/JanObloj/W-DRO-Adversarial-Methods.
