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.

通过对无监督领域适应和生成对抗网络进行功能分析，本研究理论上解决了激烈优化所带来的不稳定性问题，展示了梯度下降法在连续函数和概率测度的无穷维空间中对极小极大问题的收敛特性，并讨论了生成对抗网络和无监督领域适应的相关条件及稳定化技术。

无穷维空间下极小极大优化问题的收敛性及对抗性训练的稳定性