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

Hui Huang, Jethro Warnett

TL;DR

This work rigorously extends a derivative-free consensus-based optimization method to min-max problems by proving well-posedness for both finite-particle and mean-field dynamics and by providing a quantitative mean-field limit with explicit rate as the number of particles grows. The authors address challenges from agent-dependent consensus and coupled measures, employing a combination of stochastic Lyapunov techniques, Leray–Schauder fixed-point theory, and coupling arguments to obtain rates depending on moment and growth parameters. The results yield explicit convergence guarantees and a Monte-Carlo rate under suitable moment assumptions, strengthening the theoretical foundation of consensus-based methods for global min-max solutions. This advances the theoretical understanding and paves the way for robust large-scale implementations in settings with multiple interacting distributions.

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.

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

TL;DR

This work rigorously extends a derivative-free consensus-based optimization method to min-max problems by proving well-posedness for both finite-particle and mean-field dynamics and by providing a quantitative mean-field limit with explicit rate as the number of particles grows. The authors address challenges from agent-dependent consensus and coupled measures, employing a combination of stochastic Lyapunov techniques, Leray–Schauder fixed-point theory, and coupling arguments to obtain rates depending on moment and growth parameters. The results yield explicit convergence guarantees and a Monte-Carlo rate under suitable moment assumptions, strengthening the theoretical foundation of consensus-based methods for global min-max solutions. This advances the theoretical understanding and paves the way for robust large-scale implementations in settings with multiple interacting distributions.

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.
Paper Structure (13 sections, 16 theorems, 168 equations)

This paper contains 13 sections, 16 theorems, 168 equations.

Table of Contents

  1. Introduction
  2. Motivation
  3. Main Results
  4. Paper structure
  5. Necessary Lemmas
  6. Wasserstein stability estimate for weighted mean
  7. Moment estimates for \ref{['eq:cbo']} and \ref{['eq:mf_cbo']} dynamics
  8. Convergence of the weighted mean for i.i.d. samples
  9. Proof of Theorem \ref{['theorem:existence_and_uniqueness_for_cbo']}
  10. Proof of Theorem \ref{['theorem:existence_and_uniqueness_for_mfcbo']}
  11. Proof of Theorem \ref{['theorem:meanfield_of_cbo']}
  12. Auxillary Results
  13. Well-posedness of truncated mean-field equation

Key Result

Theorem 1.1

Let Assumptions ass:lip hold. Then the SDEs eq:cbo posses unique strong solutions $\{(X_t^i, Y_t^i)\}_{i\in [N]}$ for any initial conditions $\{(X_0^i,Y_0^i)\}_{i\in [N]}$ that are independent of the Brownian motions $\{(B_t^{X,i}, B_t^{Y,i})\}_{i\in [N]}$. The solutions are almost surely continuous

Theorems & Definitions (31)

  • Theorem 1.1: Existence and uniqueness for \ref{['eq:cbo']}
  • Theorem 1.2: Existence and uniqueness for \ref{['eq:mf_cbo']}
  • Theorem 1.3: Mean-field limit of \ref{['eq:cbo']}
  • Remark 1.4
  • Remark 1.5
  • Lemma 2.1: Estimate on weighted mean
  • proof
  • Lemma 2.2: Basic stability estimate for $\mathbb{Y}_\beta$
  • proof
  • Lemma 2.3: Basic Stability Estimate for $\mathbb{X}_{\alpha,\beta}$
  • ...and 21 more