Gradient Descent Ascent (GDA) methods are the mainstream algorithms for minimax optimization in generative adversarial networks (GANs). Convergence properties of GDA have drawn significant interest in the recent literature. Specifically, for $\min_{\mathbf{x}} \max_{\mathbf{y}} f(\mathbf{x};\mathbf{y})$ where $f$ is strongly-concave in $\mathbf{y}$ and possibly nonconvex in $\mathbf{x}$, (Lin et al., 2020) proved the convergence of GDA with a stepsize ratio $\eta_{\mathbf{y}}/\eta_{\mathbf{x}}=\Theta(\kappa^2)$ where $\eta_{\mathbf{x}}$ and $\eta_{\mathbf{y}}$ are the stepsizes for $\mathbf{x}$ and $\mathbf{y}$ and $\kappa$ is the condition number for $\mathbf{y}$. While this stepsize ratio suggests a slow training of the min player, practical GAN algorithms typically adopt similar stepsizes for both variables, indicating a wide gap between theoretical and empirical results. In this paper, we aim to bridge this gap by analyzing the \emph{local convergence} of general \emph{nonconvex-nonconcave} minimax problems. We demonstrate that a stepsize ratio of $\Theta(\kappa)$ is necessary and sufficient for local convergence of GDA to a Stackelberg Equilibrium, where $\kappa$ is the local condition number for $\mathbf{y}$. We prove a nearly tight convergence rate with a matching lower bound. We further extend the convergence guarantees to stochastic GDA and extra-gradient methods (EG). Finally, we conduct several numerical experiments to support our theoretical findings.

本文探讨了梯度下降上升（GDA）方法在生成对抗网络中极小化最大化优化问题的收敛性质及实现方式，研究表明GDA在本地条件数为y时的步长比至少需要为θ（Kappa），并支持在随机GDA和额外梯度方法（EG）中的应用。

梯度下降上升的收敛性: 一个严格的局部分析