Uniform-in-time propagation of chaos for consensus-based minimax algorithm

TL;DR

This paper proves that a consensus-based minimax algorithm's particle system converges uniformly over time to a saddle point, with deviations decreasing as the number of particles increases, confirming its computational effectiveness.

Contribution

It establishes uniform-in-time propagation of chaos for the algorithm, providing theoretical guarantees of convergence and deviation bounds in large populations.

Findings

Deviation order is O(N_1^{-1} + N_2^{-1}) uniformly in time

Particles converge near a saddle point of the objective function

Convergence confirmed through exponential decay and variance concentration

Abstract

We study the large-population convergence of a consensus-based algorithm for the saddle point problem proposed by ArXiv: 2212.12334, establishing the uniform-in-time propagation of chaos using a coupling method. Our work shows that the $L^2$-deviation has order $O(N_1^{-1} + N_2^{-1})$ uniformly in time, where $N_1$ and $N_2$ denote the numbers of particles corresponding to the two competing players. It demonstrates the convergence of the particles to some location near a saddle point of the given objective function, which confirms the computational feasibility of the algorithm. The main idea behind the proofs is the exponential decay and the concentration of the variances of the particle system.