Two-Timescale Gradient Descent Ascent Algorithms for Nonconvex Minimax Optimization

Understand

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 $\Phi(\cdot) := \max_{\textbf{y} \in Y} f(\cdot, \textbf{y})$.