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（GANs）是一种用于训练高维复杂数据生成模型的流行框架。训练GANs的标准方法涉及在极小极大优化问题上执行梯度下降-上升（GDA）过程。由于动力学的非线性特性，该过程通常难以分析。我们研究了基于核判别器的GAN训练中GDA的局部动力学。该收敛性分析基于对描述GDA迭代的非线性动力系统的线性化，并采用了[Becker等人，2022]中的**孤立点模型**假设。我们的分析揭示了学习率、正则化以及核判别器带宽对GDA局部收敛速率的影响。重要的是，我们展示了系统在收敛、振荡或发散时的相变现象。我们还提供了数值模拟来验证我们的结论。