When is Momentum Extragradient Optimal? A Polynomial-Based Analysis

TL;DR

This paper analyzes the convergence of the momentum extragradient method in differentiable games, identifying scenarios where it accelerates convergence based on eigenvalue configurations and deriving optimal hyperparameters.

Contribution

It introduces a polynomial-based analysis to determine when the momentum extragradient method is optimally accelerated in different eigenvalue scenarios.

Findings

Identifies three eigenvalue scenarios with accelerated convergence.

Derives optimal hyperparameters for each scenario.

Provides conditions for the method's optimality.

Abstract

The extragradient method has gained popularity due to its robust convergence properties for differentiable games. Unlike single-objective optimization, game dynamics involve complex interactions reflected by the eigenvalues of the game vector field's Jacobian scattered across the complex plane. This complexity can cause the simple gradient method to diverge, even for bilinear games, while the extragradient method achieves convergence. Building on the recently proven accelerated convergence of the momentum extragradient method for bilinear games \citep{azizian2020accelerating}, we use a polynomial-based analysis to identify three distinct scenarios where this method exhibits further accelerated convergence. These scenarios encompass situations where the eigenvalues reside on the (positive) real line, lie on the real line alongside complex conjugates, or exist solely as complex conjugates. Furthermore, we derive the hyperparameters for each scenario that achieve the fastest convergence rate.