  Research in adversarial learning follows a cat and mouse game between
attackers and defenders where attacks are proposed, they are mitigated by new
defenses, and subsequently new attacks are proposed that break earlier
defenses, and so on. However, it has remained unclear as to whether there are
conditions under which no better attacks or defenses can be proposed. In this
paper, we propose a game-theoretic framework for studying attacks and defenses
which exist in equilibrium. Under a locally linear decision boundary model for
the underlying binary classifier, we prove that the Fast Gradient Method attack
and the Randomized Smoothing defense form a Nash Equilibrium. We then show how
this equilibrium defense can be approximated given finitely many samples from a
data-generating distribution, and derive a generalization bound for the
performance of our approximation.
