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.

在对数据分类问题的不同类型的对抗扰动的影响方面，引入对抗性能力作为一个重要参数，以精确性和稳健性之间的权衡关系。本研究考虑对对抗性扰动分类问题的一般框架，在大数据或整体数据的情况下进行研究。在这样的情况下，我们证明了当对抗性强度趋近于零时，最优分类器在 Hausdorff 距离上收敛于贝叶斯分类器。这一结果显著增强了先前通常集中在 $L^1$ 型收敛上的研究成果。主要论证依赖于直接的几何比较，并受到几何测度理论技术的启发。