On Convergence of Gradient Descent Ascent: A Tight Local Analysis
This provides improved theoretical understanding for training generative adversarial networks (GANs) by bridging practical and theoretical stepsize requirements, though it is incremental as it refines existing convergence analyses.
The paper tackles the gap between theoretical and empirical stepsize ratios in Gradient Descent Ascent (GDA) for minimax optimization, showing that a stepsize ratio of Θ(κ) is necessary and sufficient for local convergence to a Stackelberg Equilibrium, with a nearly tight convergence rate and extensions to stochastic and extra-gradient methods.
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 $η_{\mathbf{y}}/η_{\mathbf{x}}=Θ(κ^2)$ where $η_{\mathbf{x}}$ and $η_{\mathbf{y}}$ are the stepsizes for $\mathbf{x}$ and $\mathbf{y}$ and $κ$ 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 $Θ(κ)$ is necessary and sufficient for local convergence of GDA to a Stackelberg Equilibrium, where $κ$ 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.