A LimitedCapacity Minimax Theorem for NonConvex Games or: How I Learned to Stop Worrying about MixedNash and Love Neural Nets
Abstract
Adversarial training, a special case of multiobjective optimization, is an increasingly prevalent machine learning technique: some of its most notable applications include GANbased generative modeling and selfplay 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 nonconcavenonconvex adversarial problem. However, while a classic result in game theory states such an equilibrium exists in concaveconvex games, there is no analogous guarantee if the payoff is nonconcavenonconvex. 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 nonconcavenonconvex with respect to the neural networks parameters, these games are concaveconvex with respect to the actual models (e.g., functions or distributions) represented by these neural networks.
 Publication:

arXiv eprints
 Pub Date:
 February 2020
 DOI:
 10.48550/arXiv.2002.05820
 arXiv:
 arXiv:2002.05820
 Bibcode:
 2020arXiv200205820G
 Keywords:

 Statistics  Machine Learning;
 Computer Science  Computer Science and Game Theory;
 Computer Science  Machine Learning
 EPrint:
 Appears in: Proceedings of the 24th International Conference on Artificial Intelligence and Statistics (AISTATS 2021). 19 pages