Continuous-time Analysis for Variational Inequalities: An Overview and Desiderata

TL;DR

This paper reviews recent advances in continuous-time analysis for variational inequalities, emphasizing its potential to improve algorithm stability and performance in solving complex multi-objective and zero-sum game problems in machine learning.

Contribution

It provides an overview of how continuous-time perspectives can inform the design and analysis of algorithms for variational inequalities, highlighting challenges and desiderata.

Findings

Continuous-time analysis offers insights into algorithm stability.

Drawing parallels between single- and multi-objective problems aids understanding.

Understanding continuous dynamics can help achieve desirable algorithm properties.

Abstract

Algorithms that solve zero-sum games, multi-objective agent objectives, or, more generally, variational inequality (VI) problems are notoriously unstable on general problems. Owing to the increasing need for solving such problems in machine learning, this instability has been highlighted in recent years as a significant research challenge. In this paper, we provide an overview of recent progress in the use of continuous-time perspectives in the analysis and design of methods targeting the broad VI problem class. Our presentation draws parallels between single-objective problems and multi-objective problems, highlighting the challenges of the latter. We also formulate various desiderata for algorithms that apply to general VIs and we argue that achieving these desiderata may profit from an understanding of the associated continuous-time dynamics.