  Adversarial training is one of the most popular methods for training methods
robust to adversarial attacks, however, it is not well-understood from a
theoretical perspective. We prove and existence, regularity, and minimax
theorems for adversarial surrogate risks. Our results explain some empirical
observations on adversarial robustness from prior work and suggest new
directions in algorithm development. Furthermore, our results extend previously
known existence and minimax theorems for the adversarial classification risk to
surrogate risks.
