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

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.