Convergence rate analysis of the gradient descent-ascent method for convex-concave saddle-point problems

TL;DR

This paper provides a detailed analysis of the convergence rates of the gradient descent-ascent method for convex-concave saddle-point problems, including conditions for linear convergence and a new non-asymptotic rate.

Contribution

It introduces a new non-asymptotic convergence rate using semidefinite programming and explores convergence conditions without strong convexity.

Findings

Derived a new exact convergence rate for strongly convex-strongly concave problems.

Established necessary and sufficient conditions for linear convergence without strong convexity.

Provided insights into the parameters affecting convergence speed.

Abstract

In this paper, we study the gradient descent-ascent method for convex-concave saddle-point problems. We derive a new non-asymptotic global convergence rate in terms of distance to the solution set by using the semidefinite programming performance estimation method. The given convergence rate incorporates most parameters of the problem and it is exact for a large class of strongly convex-strongly concave saddle-point problems for one iteration. We also investigate the algorithm without strong convexity and we provide some necessary and sufficient conditions under which the gradient descent-ascent enjoys linear convergence.