Well-posedness and mean-field limit estimate of a consensus-based algorithm for min-max problems

TL;DR

This paper provides a rigorous theoretical foundation for a consensus-based algorithm solving nonconvex-nonconcave min-max problems, including well-posedness and mean-field limit estimates, enhancing understanding of its convergence and stability.

Contribution

It offers the first quantitative mean-field limit estimate and proves well-posedness for both finite particle and mean-field models in this context.

Findings

Quantitative estimate of mean-field limit with respect to particle number

Proof of well-posedness for finite particle and mean-field models

Enhanced theoretical understanding of the consensus-based min-max algorithm

Abstract

The recent work arXiv:2407.17373 proposes a derivative-free consensus-based particle method that computes global solutions to nonconvex-nonconcave min-max problems and establishes global exponential convergence in the sense of the mean-field law. This paper aims to address the theoretical gaps in arXiv:2407.17373, specifically by providing a quantitative estimate of the mean-field limit with respect to the number of particles, as well as establishing the well-posedness of both the finite particle model and the corresponding mean-field dynamics.