Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization
This addresses convergence issues in nonconvex minimax optimization for machine learning applications such as GANs, offering a systematic analysis that is incremental but provides new theoretical guarantees.
The paper tackles nonconvex minimax optimization problems by designing two-timescale gradient descent ascent (TTGDA) algorithms, establishing theoretical bounds for finding stationary points in both smooth and nonsmooth cases, and demonstrating superior performance in applications like GAN training.
We provide a unified analysis of two-timescale gradient descent ascent (TTGDA) for solving structured nonconvex minimax optimization problems in the form of $\min_\textbf{x} \max_{\textbf{y} \in Y} f(\textbf{x}, \textbf{y})$, where the objective function $f(\textbf{x}, \textbf{y})$ is nonconvex in $\textbf{x}$ and concave in $\textbf{y}$, and the constraint set $Y \subseteq \mathbb{R}^n$ is convex and bounded. In the convex-concave setting, the single-timescale gradient descent ascent (GDA) algorithm is widely used in applications and has been shown to have strong convergence guarantees. In more general settings, however, it can fail to converge. Our contribution is to design TTGDA algorithms that are effective beyond the convex-concave setting, efficiently finding a stationary point of the function $Φ(\cdot) := \max_{\textbf{y} \in Y} f(\cdot, \textbf{y})$. We also establish theoretical bounds on the complexity of solving both smooth and nonsmooth nonconvex-concave minimax optimization problems. To the best of our knowledge, this is the first systematic analysis of TTGDA for nonconvex minimax optimization, shedding light on its superior performance in training generative adversarial networks (GANs) and in other real-world application problems.