We connect adversarial training for binary classification to a geometric
evolution equation for the decision boundary. Relying on a perspective that
recasts adversarial training as a regularization problem, we introduce a
modified training scheme that constitutes a minimizing movements scheme for a
nonlocal perimeter functional. We prove that the scheme is monotone and
consistent as the adversarial budget vanishes and the perimeter localizes, and
as a consequence we rigorously show that the scheme approximates a weighted
mean curvature flow. This highlights that the efficacy of adversarial training
may be due to locally minimizing the length of the decision boundary. In our
analysis, we introduce a variety of tools for working with the subdifferential
of a supremal-type nonlocal total variation and its regularity properties.

通过将对抗性训练与决策边界的几何演化方程相联系，我们通过将对抗性训练重新解释为一种正则化问题，引入了一种改进的训练方案，构成了一种非局部周长函数的最小化移动方案，证明了当对抗预算消失且周长局部化时，该方案是单调和一致的，并且由此严格证明了该方案近似于加权平均曲率流，强调了对抗性训练的有效性可能是由于局部最小化决策边界的长度。在我们的分析中，我们引入了一系列用于处理极值型非局部总变差及其正则性属性的工具。