Statistical Mechanics of Min-Max Problems

Yuma Ichikawa, Koji Hukushima
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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.
最小-最大问题的统计力学
最小最大优化问题又称鞍点问题,因其在公平波束成形、生成式对抗网络(GAN)和对抗学习等多个领域的应用而备受关注。然而,如何理解这些最小问题的特性仍然是一个巨大的挑战。本研究引入了一种统计力学形式主义,用于分析高维极限下的最小-最大问题的均衡值,同时适当解决最小和最大的操作顺序问题。作为第一步,我们将这一形式主义应用于双线性最小-最大博弈和简单的 GAN,推导出训练数据量与泛化误差之间的关系,并指出有效学习所需的假数据与真实数据的最佳比例。这一形式主义为更深入地从理论上分析基于最小最大问题的各种机器学习方法中的均衡属性奠定了基础,并促进了新算法和新架构的开发。
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