UC Berkeley

The theory of mean-field games (MFGs) belongs to a branch of game theory that studies a large population of (weakly) interacting players. It serves as an analytically feasible framework to approximate stochastic differential games when the number of players is large and detailed characterization of the interactions is computationally expensive. As an effective modeling tool, the theory of MFGs attracts the attention of a variety of application fields in economics, finance and engineering. On the computation front, the development of machine learning provides abundant computational methods of solving for MFGs, which is remarkably meaningful in practice. At the same time, people may also wonder if the theory of MFGs, or stochastic analysis in general, could benefit the machine learning community.

This thesis starts with two MFG models, with singular and impulse types of controls, respectively. Theses two control types allow certain degrees of discontinuity, making them better mathematical models compared with regular controls where the interventions must be absolute continuous. However, due to the theoretical challenges brought by the discontinuous nature of the controls, these two models are less explored in existing literature compared with MFGs with regular controls. Both models are motivated by real-world problems. Explicit solutions to the MFGs are presented and shown to approximate Nash equilibria of the corresponding N-player games with an error of the order O(1⁄√N). Further analysis of the solutions reveals the game effect from interacting with the mean-field.

Obtaining analytical solutions of MFGs is difficult in general. The thesis then turns to the computation side of MFGs and establish the connection with generative adversarial networks, a celebrated deep learning tool that enjoys tremendous empirical success since its introduction to the machine learning community. 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.

Finally, the thesis explores the possibility of enriching the theoretical understanding of the training of GANs from the perspective of stochastic analysis. It establishes approximations, with precise error bound analysis, for the training of GANs under stochastic gradient algorithms (SGAs). The approximations are in the form of coupled stochastic differential equations (SDEs). The analysis of the SDEs and the associated invariant measures yields conditions for the stability and the convergence of GANs training. Further analysis of the invariant measure for the coupled SDEs gives rise to a fluctuation-dissipation relations (FDRs) for GANs, revealing the trade-off of the loss landscape between the generator and the discriminator and providing guidance for learning rate scheduling.