A Robust Framework for Analyzing Gradient-Based Dynamics in Bilinear Games

TL;DR

This paper introduces a frequency-domain framework using the Z-transform to analyze the stability of gradient-based algorithms in bilinear games, providing the first tight analysis of OGDA and extending to a broad class of algorithms with convergence guarantees.

Contribution

It develops a simplified, concise frequency-domain analysis for gradient algorithms in bilinear games and introduces a general family of algorithms with proven last-iterate convergence.

Findings

First tight stability analysis of OGDA in bilinear games

A broad class of algorithms with convergence guarantees

Reduction of convergence analysis to polynomial stability

Abstract

In this work, we establish a frequency-domain framework for analyzing gradient-based algorithms in linear minimax optimization problems; specifically, our approach is based on the Z-transform, a powerful tool applied in Control Theory and Signal Processing in order to characterize linear discrete-time systems. We employ our framework to obtain the first tight analysis of stability of Optimistic Gradient Descent/Ascent (OGDA), a natural variant of Gradient Descent/Ascent that was shown to exhibit last-iterate convergence in bilinear games by Daskalakis et al. \cite{DBLP:journals/corr/abs-1711-00141}. Importantly, our analysis is considerably simpler and more concise than the existing ones. Moreover, building on the intuition of OGDA, we consider a general family of gradient-based algorithms that augment the memory of the optimization through multiple historical steps. We reduce the convergence -- to a saddle-point -- of the dynamics in bilinear games to the stability of a polynomial, for which efficient algorithmic schemes are well-established. As an immediate corollary, we obtain a broad class of algorithms -- that contains OGDA as a special case -- with a last-iterate convergence guarantee to the space of Nash equilibria of the game.