A particle consensus approach to solving nonconvex-nonconcave min-max problems

TL;DR

This paper introduces a novel zero-order particle-based method for solving complex nonconvex-nonconcave min-max problems, capable of finding global solutions without convexity assumptions.

Contribution

It presents a new particle consensus algorithm with stochastic exploration and theoretical convergence guarantees for challenging min-max optimization tasks.

Findings

Successfully identifies global min-max solutions

Outperforms gradient-based methods in nonconvex settings

Demonstrates convergence through mean-field analysis

Abstract

We propose a zero-order optimization method for sequential min-max problems based on two populations of interacting particles. The systems are coupled so that one population aims to solve the inner maximization problem, while the other aims to solve the outer minimization problem. The dynamics are characterized by a consensus-type interaction with additional stochasticity to promote exploration of the objective landscape. Without relying on convexity or concavity assumptions, we establish theoretical convergence guarantees of the algorithm via a suitable mean-field approximation of the particle systems. Numerical experiments illustrate the validity of the proposed approach. In particular, the algorithm is able to identify a global min-max solution, in contrast to gradient-based methods, which typically converge to possibly suboptimal stationary points.