Convergence Rates of Two-Time-Scale Gradient Descent-Ascent Dynamics for Solving Nonconvex Min-Max Problems

TL;DR

This paper analyzes the convergence rates of a continuous-time gradient descent-ascent method for nonconvex min-max problems, providing improved theoretical guarantees under various conditions.

Contribution

It characterizes finite-time convergence rates of the continuous-time gradient descent-ascent algorithm under multiple objective function conditions, using singular perturbation and Lyapunov analysis.

Findings

Convergence rates are derived for different objective function conditions.

Results improve upon previous convergence guarantees.

Analysis techniques include singular perturbation theory and Lyapunov functions.

Abstract

There are much recent interests in solving noncovnex min-max optimization problems due to its broad applications in many areas including machine learning, networked resource allocations, and distributed optimization. Perhaps, the most popular first-order method in solving min-max optimization is the so-called simultaneous (or single-loop) gradient descent-ascent algorithm due to its simplicity in implementation. However, theoretical guarantees on the convergence of this algorithm is very sparse since it can diverge even in a simple bilinear problem. In this paper, our focus is to characterize the finite-time performance (or convergence rates) of the continuous-time variant of simultaneous gradient descent-ascent algorithm. In particular, we derive the rates of convergence of this method under a number of different conditions on the underlying objective function, namely, two-sided Polyak-L ojasiewicz (PL), one-sided PL, nonconvex-strongly concave, and strongly convex-nonconcave conditions. Our convergence results improve the ones in prior works under the same conditions of objective functions. The key idea in our analysis is to use the classic singular perturbation theory and coupling Lyapunov functions to address the time-scale difference and interactions between the gradient descent and ascent dynamics. Our results on the behavior of continuous-time algorithm may be used to enhance the convergence properties of its discrete-time counterpart.