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

Erhan Bayraktar, Zhiyan Ding, Ibrahim Ekren, Hongyi Zhou

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.

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

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 -deviation has order uniformly in time, where and 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.
Paper Structure (12 sections, 11 theorems, 149 equations)

This paper contains 12 sections, 11 theorems, 149 equations.

Table of Contents

  1. Introduction
  2. Main problem and our contributions
  3. Organization of this paper
  4. Main results
  5. Notation and setting of the algorithm
  6. Propagation of chaos
  7. Strategies of the proof
  8. Estimates on variance
  9. Proof of Theorem \ref{['th:main']}
  10. Proof of auxiliary results
  11. Lemmata on generalized variance
  12. Technical lemmata

Key Result

proposition 1

Assume that Conditions cd:cutoff and cd:efn hold. Let $N_1, N_2 \in \N_+$, and parameters $\lambda_1, \lambda_2, \sigma_1, \sigma_2, \alpha, \beta > 0$. Also, let $\bar{\nu}^X_0 \in \prob(B_{d_1}(0, R_\cut))$ and $\bar{\nu}^Y_0 \in \prob(B_{d_2}(0, R_\cut))$. The system of SDEs eq:algorithm admits a

Theorems & Definitions (23)

  • proposition 1
  • proposition 2
  • theorem 1
  • corollary 1
  • proof
  • remark 1
  • lemma 1
  • remark 2
  • lemma 2
  • corollary 2
  • ...and 13 more