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Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems

Amir Ali Farzin, Yuen-Man Pun, Philipp Braun, Iman Shames

TL;DR

This paper addresses the convergence gaps of conventional gradient methods in min–max problems by introducing the Generalised Extra Gradient (GEG) method, which nests multiple EG variants through tunable step-sizes. By formulating GEG as a discrete-time dynamical system and performing a stability analysis around equilibria, the authors show that, under mild smoothness and Lipschitz assumptions, saddle points of the objective $f(x,y)$ are contained within the set of asymptotically stable fixed points of the GEG dynamics for appropriate parameter choices. They demonstrate, via theoretical results and numerical experiments, that unstable fixed points attract almost no initializations and that the saddle points are captured as stable equilibria, offering convergence guarantees to meaningful equilibria in min–max settings. The work further illustrates the framework on simple and large-scale problems, highlighting practical implications for stable training in adversarial and multi-agent settings and outlining directions to connect with local min–max notions and continuous-time dynamics.

Abstract

This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of saddle points (Nash equilibria) is a subset of stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results and benefits when compared to existing methods are illustrated through numerical examples.

Properties of Fixed Points of Generalised Extra Gradient Methods Applied to Min-Max Problems

TL;DR

This paper addresses the convergence gaps of conventional gradient methods in min–max problems by introducing the Generalised Extra Gradient (GEG) method, which nests multiple EG variants through tunable step-sizes. By formulating GEG as a discrete-time dynamical system and performing a stability analysis around equilibria, the authors show that, under mild smoothness and Lipschitz assumptions, saddle points of the objective are contained within the set of asymptotically stable fixed points of the GEG dynamics for appropriate parameter choices. They demonstrate, via theoretical results and numerical experiments, that unstable fixed points attract almost no initializations and that the saddle points are captured as stable equilibria, offering convergence guarantees to meaningful equilibria in min–max settings. The work further illustrates the framework on simple and large-scale problems, highlighting practical implications for stable training in adversarial and multi-agent settings and outlining directions to connect with local min–max notions and continuous-time dynamics.

Abstract

This paper studies properties of fixed points of generalised Extra-gradient (GEG) algorithms applied to min-max problems. We discuss connections between saddle points of the objective function of the min-max problem and GEG fixed points. We show that, under appropriate step-size selections, the set of saddle points (Nash equilibria) is a subset of stable fixed points of GEG. Convergence properties of the GEG algorithm are obtained through a stability analysis of a discrete-time dynamical system. The results and benefits when compared to existing methods are illustrated through numerical examples.

Paper Structure

This paper contains 13 sections, 8 theorems, 24 equations, 2 figures, 3 tables.

Table of Contents

  1. Introduction
  2. Preliminaries & Problem Formulation
  3. Analysis of generalised Extra-gradient
  4. Derivation of preparatory and auxiliary results
  5. Saddle points and GEG stable points
  6. Numerical example
  7. Fixed points of GEG applied to $f(x,y) = xy$
  8. Fixed points of GEG applied to a function with multiple critical points
  9. Large Scale Robust Optimisation
  10. Conclusions and Future Works
  11. Mathematica Code used in the proof of Theorem \ref{['th:general']}
  12. Mathematica Code for bounding $\gamma$ such that $|\lambda|\leq1$
  13. Mathematica Code for finding a feasible nonzero $\gamma$

Key Result

Theorem 1

Let $z^e \in \mathbb{R}^n$ denote an equilibrium of eq:discrete_time_sys with $C^2$ function $w(\cdot)$. If the spectral radius of $J(z^e)$ satisfies $|\rho(J(z^e))| <1$, then $z^e$ is locally asymptotically stable. If $|\rho(J(z^e))| >1$, then $z^e$ is unstable.

Figures (2)

  • Figure 1: Numerical estimates of regions of attraction of equilibria of the dynamics \ref{['dy2']} for $f$ in \ref{['eq:multiple']}.
  • Figure 2: Evolution of gradient norm over \ref{['dy']}

Theorems & Definitions (22)

  • Definition 1
  • Definition 2: kellett2023introduction
  • Theorem 1: hespanha2018linear
  • Remark 1
  • Definition 3: Critical point daskalakis2018limit
  • Definition 4: bertsekas2009convex
  • Proposition 1: jin2020local
  • Lemma 1
  • proof
  • Remark 2
  • ...and 12 more