A mean-field games laboratory for generative modeling

• URL: http://arxiv.org/abs/2304.13534v5
• Date: Tue, 24 Oct 2023 17:21:36 GMT
• Title: A mean-field games laboratory for generative modeling
• Authors: Benjamin J. Zhang and Markos A. Katsoulakis
• Abstract summary: Mean-field games (MFGs) are a framework for explaining, enhancing, and designing generative models. We study the mathematical properties of each generative model by studying their associated MFG's optimality condition. We propose and demonstrate an HJB-regularized SGM with improved performance over standard SGMs.
• Score: 5.837881923712394
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We demonstrate the versatility of mean-field games (MFGs) as a mathematical framework for explaining, enhancing, and designing generative models. In generative flows, a Lagrangian formulation is used where each particle (generated sample) aims to minimize a loss function over its simulated path. The loss, however, is dependent on the paths of other particles, which leads to a competition among the population of particles. The asymptotic behavior of this competition yields a mean-field game. We establish connections between MFGs and major classes of generative flows and diffusions including continuous-time normalizing flows, score-based generative models (SGM), and Wasserstein gradient flows. Furthermore, we study the mathematical properties of each generative model by studying their associated MFG's optimality condition, which is a set of coupled forward-backward nonlinear partial differential equations. The mathematical structure described by the MFG optimality conditions identifies the inductive biases of generative flows. We investigate the well-posedness and structure of normalizing flows, unravel the mathematical structure of SGMs, and derive a MFG formulation of Wasserstein gradient flows. From an algorithmic perspective, the optimality conditions yields Hamilton-Jacobi-Bellman (HJB) regularizers for enhanced training of generative models. In particular, we propose and demonstrate an HJB-regularized SGM with improved performance over standard SGMs. We present this framework as an MFG laboratory which serves as a platform for revealing new avenues of experimentation and invention of generative models.

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