Statistical Mechanics of Min-Max Problems

• URL: http://arxiv.org/abs/2409.06053v1
• Date: Mon, 9 Sep 2024 20:24:19 GMT
• Title: Statistical Mechanics of Min-Max Problems
• Authors: Yuma Ichikawa, Koji Hukushima,
• Abstract summary: This study introduces a statistical mechanical formalism for analyzing the equilibrium values of min-max problems in the high-dimensional limit. As a first step, we apply this formalism to bi-linear min-max games and simple GANs. This formalism provides a groundwork for a deeper theoretical analysis of the equilibrium properties in various machine learning methods.
• Score: 0.0
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: Min-max optimization problems, also known as saddle point problems, have attracted significant attention due to their applications in various fields, such as fair beamforming, generative adversarial networks (GANs), and adversarial learning. However, understanding the properties of these min-max problems has remained a substantial challenge. This study introduces a statistical mechanical formalism for analyzing the equilibrium values of min-max problems in the high-dimensional limit, while appropriately addressing the order of operations for min and max. As a first step, we apply this formalism to bilinear min-max games and simple GANs, deriving the relationship between the amount of training data and generalization error and indicating the optimal ratio of fake to real data for effective learning. This formalism provides a groundwork for a deeper theoretical analysis of the equilibrium properties in various machine learning methods based on min-max problems and encourages the development of new algorithms and architectures.

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