It begins with a boundary: A geometric view on probabilistically robust
  learning
Leon Bungert Nicolás García Trillos Matt Jacobs Daniel McKenzie Đorđe Nikolić Qingsong Wang
Although deep neural networks have achieved super-human performance on many
classification tasks, they often exhibit a worrying lack of robustness towards
adversarially generated examples. Thus, considerable effort has been invested
into reformulating Empirical Risk Minimization (ERM) into an adversarially
robust framework. Recently, attention has shifted towards approaches which
interpolate between the robustness offered by adversarial training and the
higher clean accuracy and faster training times of ERM. In this paper, we take
a fresh and geometric view on one such method -- Probabilistically Robust
Learning (PRL) (Robey et al., ICML, 2022). We propose a geometric framework for
understanding PRL, which allows us to identify a subtle flaw in its original
formulation and to introduce a family of probabilistic nonlocal perimeter
functionals to address this. We prove existence of solutions using novel
relaxation methods and study properties as well as local limits of the
introduced perimeters.