Regularity of adversarial classifiers

Determine whether minimizers of the adversarial classification problem can be assumed a priori to possess any regularity, in order to justify arguments based on regularity of the adversarial classifier boundary.

Background

The paper studies binary adversarial classification, where an adversarial classifier minimizes the robust risk associated with perturbations of size ε. The main geometric estimates compare the adversarial classifier A_ε with the Bayes classifier A_0 in a localized Hausdorff metric.

A heuristic first-variation argument would be straightforward if the boundary of A_ε were known to be regular. However, the general regularity theory for adversarial classifiers is not established. The paper therefore develops energetic competitor and barrier constructions that avoid assuming regularity, leaving the general question of whether such regularity can be assumed a priori unresolved.

References

The proof is conditional as it is not known whether we can a priori assume any regularity for $A_$; the presence of non-smooth $\sigma$ determining the adversary further complicates the matter.

— On the tradeoff between robustness and geometric accuracy in adversarial training  (2610.22955 - Morris et al., 16 Oct 2026) in Section 1, Introduction, paragraph beginning “Outline of the strategy”