  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.
