Generative Adversarial Networks (GANs) are a popular formulation to train generative models for complex high dimensional data. The standard method for training GANs involves a gradient descent-ascent (GDA) procedure on a minimax optimization problem. This procedure is hard to analyze in general due to the nonlinear nature of the dynamics. We study the local dynamics of GDA for training a GAN with a kernel-based discriminator. This convergence analysis is based on a linearization of a non-linear dynamical system that describes the GDA iterations, under an \textit{isolated points model} assumption from [Becker et al. 2022]. Our analysis brings out the effect of the learning rates, regularization, and the bandwidth of the kernel discriminator, on the local convergence rate of GDA. Importantly, we show phase transitions that indicate when the system converges, oscillates, or diverges. We also provide numerical simulations that verify our claims.

翻译：生成对抗网络（GAN）是训练复杂高维数据生成模型的一种主流框架。训练GAN的标准方法是针对极小极大优化问题采用梯度下降-上升（GDA）过程。由于动力学的非线性本质，这一过程通常难以分析。我们研究了基于核判别器的GAN训练中GDA的局部动力学特性。该收敛性分析基于对描述GDA迭代的非线性动力系统进行线性化处理，并采用了[Becker et al. 2022]中的**孤立点模型**假设。我们的分析揭示了学习率、正则化项以及核判别器带宽对GDA局部收敛速率的影响。重要的是，我们展示了系统收敛、振荡或发散时的相变现象，并通过数值模拟验证了我们的结论。