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.

这项研究针对最小-最大优化问题的理解难题，提出了一种统计力学形式以分析高维极限下的平衡值。研究表明，在双线性最小-最大博弈和简单GAN中，训练数据量与泛化误差之间的关系，为有效学习提供了假数据与真实数据的最优比例，从而为基于最小-最大问题的多种机器学习方法的理论分析奠定基础。

最小-最大问题的统计力学