  Adversarial training, a special case of multi-objective optimization, is an
increasingly prevalent machine learning technique: some of its most notable
applications include GAN-based generative modeling and self-play techniques in
reinforcement learning which have been applied to complex games such as Go or
Poker. In practice, a \emph{single} pair of networks is typically trained in
order to find an approximate equilibrium of a highly nonconcave-nonconvex
adversarial problem. However, while a classic result in game theory states such
an equilibrium exists in concave-convex games, there is no analogous guarantee
if the payoff is nonconcave-nonconvex. Our main contribution is to provide an
approximate minimax theorem for a large class of games where the players pick
neural networks including WGAN, StarCraft II, and Blotto Game. Our findings
rely on the fact that despite being nonconcave-nonconvex with respect to the
neural networks parameters, these games are concave-convex with respect to the
actual models (e.g., functions or distributions) represented by these neural
networks.
