Connecting GANs, MFGs, and OT

Generative adversarial networks (GANs) have enjoyed tremendous success in image generation and processing, and have recently attracted growing interests in financial modelings. This paper analyzes GANs from the perspective of mean field games (MFGs) and optimal transport (OT). It first shows a conceptual connection between GANs and MFGs: MFGs have the structure of GANs, and GANs are MFGs under the Pareto Optimality criterion. Interpreting MFGs as GANs, on one hand, enables a GANs-based algorithm (MFGANs) to solve MFGs: one neural network (NN) for the backward HJB equation and one NN for the forward FP equation, with the two NNs trained in an adversarial way. Viewing GANs as MFGs, on the other hand, reveals a new and probabilistic aspect of GANs. This new perspective, moreover, leads to an analytical connection between GANs and Optimal Transport (OT) problems, and sufficient conditions for the minimax games of GANs to be reformulated in the framework of OT. Numerical experiments demonstrate superior performance of this proposed algorithm, especially in higher dimensional case, when compared with existing NN approaches.