Global Convergence and Variance-Reduced Optimization for a Class of Nonconvex-Nonconcave Minimax Problems

TL;DR

This paper studies convergence properties of gradient-based algorithms for nonconvex-nonconcave minimax problems, showing global linear convergence under certain conditions and proposing a variance-reduced method for faster rates.

Contribution

It establishes convergence guarantees for AGDA in a subclass of nonconvex-nonconcave problems and introduces a variance-reduced algorithm with improved rates.

Findings

AGDA converges globally at a linear rate under specific conditions.

Stochastic AGDA achieves a sublinear convergence rate.

Variance reduction leads to faster convergence in finite-sum problems.

Abstract

Nonconvex minimax problems appear frequently in emerging machine learning applications, such as generative adversarial networks and adversarial learning. Simple algorithms such as the gradient descent ascent (GDA) are the common practice for solving these nonconvex games and receive lots of empirical success. Yet, it is known that these vanilla GDA algorithms with constant step size can potentially diverge even in the convex setting. In this work, we show that for a subclass of nonconvex-nonconcave objectives satisfying a so-called two-sided Polyak-{\L}ojasiewicz inequality, the alternating gradient descent ascent (AGDA) algorithm converges globally at a linear rate and the stochastic AGDA achieves a sublinear rate. We further develop a variance reduced algorithm that attains a provably faster rate than AGDA when the problem has the finite-sum structure.