Training neural networks that require adversarial optimization, such as generative adversarial networks (GANs) and unsupervised domain adaptations (UDAs), suffers from instability. This instability problem comes from the difficulty of the minimax optimization, and there have been various approaches in GANs and UDAs to overcome this problem. In this study, we tackle this problem theoretically through a functional analysis. Specifically, we show the convergence property of the minimax problem by the gradient descent over the infinite-dimensional spaces of continuous functions and probability measures under certain conditions. Using this setting, we can discuss GANs and UDAs comprehensively, which have been studied independently. In addition, we show that the conditions necessary for the convergence property are interpreted as stabilization techniques of adversarial training such as the spectral normalization and the gradient penalty.

翻译：需要对抗优化的神经网络训练，如生成对抗网络（GANs）和无监督领域自适应（UDAs），常面临不稳定性问题。这一不稳定性源于极小极大优化的固有困难，已有多种方法在GANs和UDAs中尝试克服此问题。本研究通过泛函分析从理论上探讨该问题。具体而言，我们证明了在特定条件下，基于连续函数空间和概率测度空间的无穷维梯度下降方法下，极小极大问题的收敛性质。借助这一框架，可以统一分析以往独立研究的GANs和UDAs。此外，我们指出，实现该收敛性质所需的条件可解释为对抗训练中的稳定化技术，例如谱归一化和梯度惩罚。