  In recent years there has been significant interest in the effect of
different types of adversarial perturbations in data classification problems.
Many of these models incorporate the adversarial power, which is an important
parameter with an associated trade-off between accuracy and robustness. This
work considers a general framework for adversarially-perturbed classification
problems, in a large data or population-level limit. In such a regime, we
demonstrate that as adversarial strength goes to zero that optimal classifiers
converge to the Bayes classifier in the Hausdorff distance. This significantly
strengthens previous results, which generally focus on $L^1$-type convergence.
The main argument relies upon direct geometric comparisons and is inspired by
techniques from geometric measure theory.
