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A Hybrid Algorithm for Monotone Variational Inequalities

Reza Rahimi Baghbadorani, Peyman Mohajerin Esfahani, Sergio Grammatico

Abstract

Inspired by the adaptive Golden Ratio Algorithm (aGRAAL), we propose two new methods for solving monotone variational inequalities. We show that by selecting the momentum parameter beyond the golden ratio in aGRAAL, the convergence speed can be improved, which motivates us to study the switching between small and large momentum parameters to accelerate convergence. We validate the performance of our proposed algorithms on several classes of variational inequality problems studied in the machine learning and control literature, including Nash equilibrium seeking, composite minimization, Markov decision processes, and zero-sum games, and compare them to that of existing methods.

A Hybrid Algorithm for Monotone Variational Inequalities

Abstract

Inspired by the adaptive Golden Ratio Algorithm (aGRAAL), we propose two new methods for solving monotone variational inequalities. We show that by selecting the momentum parameter beyond the golden ratio in aGRAAL, the convergence speed can be improved, which motivates us to study the switching between small and large momentum parameters to accelerate convergence. We validate the performance of our proposed algorithms on several classes of variational inequality problems studied in the machine learning and control literature, including Nash equilibrium seeking, composite minimization, Markov decision processes, and zero-sum games, and compare them to that of existing methods.

Paper Structure

This paper contains 5 sections, 3 theorems, 25 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries and First Algorithm
  3. An efficient switching algorithm
  4. Numerical simulations
  5. Appendix

Key Result

Theorem B.1

Let $F\colon \mathop{\mathrm{dom}}\nolimits g \to \mathcal{V}$ be locally Lipschitz and monotone operator. Then $\left(x^k\right)_{k \in \mathbb{N}}$ and $\left(\bar{x}^k\right)_{k \in \mathbb{N}}$, generated by Algorithms alg1-alg2, satisfy the following inequality: $\blacktriangleleft$$\blacktriangleleft$

Figures (1)

  • Figure B1: Residual variation induced by Algorithm \ref{['alg1']}.

Theorems & Definitions (6)

  • Theorem B.1: Variable momentum in aGRAAL
  • proof
  • Theorem C.1: Ergodic convergence
  • proof
  • Definition E.1: Cluster point
  • Lemma E.2: Bolzano–Weierstrass theorem