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A Block Coordinate and Variance-Reduced Method for Generalized Variational Inequalities of Minty Type

Jelena Diakonikolas

TL;DR

This paper addresses solving Generalized Minty Variational Inequalities (GMVI) with block-decomposable structure by introducing a Randomized Extrapolated Method (REM) that leverages nonuniform block Lipschitz constants and variance reduction. The method achieves strong convergence guarantees with complexities that can scale linearly with the number of blocks $m$ under highly nonuniform Lipschitz parameters, and up to a $\\sqrt{m}$ improvement in finite-sum settings via importance-based sampling. It unifies block-coordinate and finite-sum perspectives, showing how lazy updates and extrapolated estimators yield practical runtime benefits while maintaining provable gap or distance-to-solution guarantees. The paper corroborates its theoretical results with diverse GMVI instances, including policy evaluation in reinforcement learning, matrix games, box-constrained $\ell_\infty$ regression, and least absolute deviation, highlighting REM’s potential to outperform classical full-vector methods in large-scale, nonuniform regimes.

Abstract

Block coordinate methods have been extensively studied for minimization problems, where they come with significant complexity improvements whenever the considered problems are compatible with block decomposition and, moreover, block Lipschitz parameters are highly nonuniform. For the more general class of variational inequalities with monotone operators, essentially none of the existing methods transparently shows potential complexity benefits of using block coordinate updates in such settings. Motivated by this gap, we develop a new randomized block coordinate method and study its oracle complexity and runtime. We prove that in the setting where block Lipschitz parameters are highly nonuniform -- the main setting in which block coordinate methods lead to high complexity improvements in any of the previously studied settings -- our method can lead to complexity improvements by a factor order-$m$, where $m$ is the number of coordinate blocks. The same method further applies to the more general problem with a finite-sum operator with $m$ components, where it can be interpreted as performing variance reduction. Compared to the state of the art, the method leads to complexity improvements up to a factor $\sqrt{m},$ obtained when the component Lipschitz parameters are highly nonuniform.

A Block Coordinate and Variance-Reduced Method for Generalized Variational Inequalities of Minty Type

TL;DR

This paper addresses solving Generalized Minty Variational Inequalities (GMVI) with block-decomposable structure by introducing a Randomized Extrapolated Method (REM) that leverages nonuniform block Lipschitz constants and variance reduction. The method achieves strong convergence guarantees with complexities that can scale linearly with the number of blocks under highly nonuniform Lipschitz parameters, and up to a improvement in finite-sum settings via importance-based sampling. It unifies block-coordinate and finite-sum perspectives, showing how lazy updates and extrapolated estimators yield practical runtime benefits while maintaining provable gap or distance-to-solution guarantees. The paper corroborates its theoretical results with diverse GMVI instances, including policy evaluation in reinforcement learning, matrix games, box-constrained regression, and least absolute deviation, highlighting REM’s potential to outperform classical full-vector methods in large-scale, nonuniform regimes.

Abstract

Block coordinate methods have been extensively studied for minimization problems, where they come with significant complexity improvements whenever the considered problems are compatible with block decomposition and, moreover, block Lipschitz parameters are highly nonuniform. For the more general class of variational inequalities with monotone operators, essentially none of the existing methods transparently shows potential complexity benefits of using block coordinate updates in such settings. Motivated by this gap, we develop a new randomized block coordinate method and study its oracle complexity and runtime. We prove that in the setting where block Lipschitz parameters are highly nonuniform -- the main setting in which block coordinate methods lead to high complexity improvements in any of the previously studied settings -- our method can lead to complexity improvements by a factor order-, where is the number of coordinate blocks. The same method further applies to the more general problem with a finite-sum operator with components, where it can be interpreted as performing variance reduction. Compared to the state of the art, the method leads to complexity improvements up to a factor obtained when the component Lipschitz parameters are highly nonuniform.

Paper Structure

This paper contains 20 sections, 5 theorems, 89 equations, 2 algorithms.

Table of Contents

  1. Introduction
  2. Problem Setup
  3. Main Result
  4. Related Work
  5. Randomized Extrapolated Method for GMVI: Algorithm Description
  6. Block coordinate considerations for Algorithm \ref{['alg:main']}.
  7. Convergence Analysis
  8. Gap Estimate Construction and Main Convergence Result
  9. Sampling Distributions and Resulting Complexity
  10. Uniform Sampling.
  11. Importance-Based Sampling.
  12. Combining Variance Reduction with Block Coordinate Strategies
  13. Examples
  14. Policy Evaluation in Reinforcement Learning
  15. Matrix Games
  16. ...and 5 more sections

Key Result

Lemma 1

Consider algorithm updates specified in eq:alg-iteration and let $\bar{\bm{x}}_k = \frac{1}{A_k}\sum_{i=1}^k a_i {\bm{x}}_i$ for $k \geq 1$. Then, for any $k \geq 1$ and any ${\bm{x}} \in \mathrm{dom}(g),$ where

Theorems & Definitions (9)

  • Lemma 1: A Generic Gap Bound
  • proof
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Theorem 1: Main Theorem
  • proof
  • Lemma 4