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An alternative approach to well-posedness of McKean-Vlasov equations arising in Consensus-Based Optimization

Alessandro Baldi

TL;DR

This paper tackles the well-posedness of the mean-field McKean–Vlasov equation arising from Consensus-Based Optimization (CBO), where the interaction fields lack global Lipschitz continuity. It introduces a truncation of the measure space to force global Lipschitzness of the drift and diffusion, enabling a Sznitman-style fixed-point construction to obtain strong solutions, and then removes the truncation via uniform moment bounds to recover the original problem. The authors prove existence of strong solutions and extend pathwise uniqueness to a class where the instantaneous consensus $ ext{M}_eta( ho_t)$ remains bounded. As an alternative to Leray–Shauder-type arguments, this truncation-based approach sharpens the theoretical understanding of CBO's mean-field dynamics and provides robust a priori estimates for uniqueness.

Abstract

In this work we study the mean-field description of Consensus-Based Optimization (CBO), a derivative-free particle optimization method. Such a description is provided by a non-local SDE of McKean-Vlasov type, whose fields lack of global Lipschitz continuity. We propose a novel approach to prove the well-posedness of the mean-field CBO equation based on a truncation argument. The latter is performed through the introduction of a cut-off function, defined on the space of probability measures, acting on the fields. This procedure allows us to study the well-posedness problem in the classical framework of Sznitman. Through this argument, we recover the established result on the existence of strong solutions, and we extend the class of solutions for which pathwise uniqueness holds.

An alternative approach to well-posedness of McKean-Vlasov equations arising in Consensus-Based Optimization

TL;DR

This paper tackles the well-posedness of the mean-field McKean–Vlasov equation arising from Consensus-Based Optimization (CBO), where the interaction fields lack global Lipschitz continuity. It introduces a truncation of the measure space to force global Lipschitzness of the drift and diffusion, enabling a Sznitman-style fixed-point construction to obtain strong solutions, and then removes the truncation via uniform moment bounds to recover the original problem. The authors prove existence of strong solutions and extend pathwise uniqueness to a class where the instantaneous consensus remains bounded. As an alternative to Leray–Shauder-type arguments, this truncation-based approach sharpens the theoretical understanding of CBO's mean-field dynamics and provides robust a priori estimates for uniqueness.

Abstract

In this work we study the mean-field description of Consensus-Based Optimization (CBO), a derivative-free particle optimization method. Such a description is provided by a non-local SDE of McKean-Vlasov type, whose fields lack of global Lipschitz continuity. We propose a novel approach to prove the well-posedness of the mean-field CBO equation based on a truncation argument. The latter is performed through the introduction of a cut-off function, defined on the space of probability measures, acting on the fields. This procedure allows us to study the well-posedness problem in the classical framework of Sznitman. Through this argument, we recover the established result on the existence of strong solutions, and we extend the class of solutions for which pathwise uniqueness holds.
Paper Structure (11 sections, 10 theorems, 69 equations)

This paper contains 11 sections, 10 theorems, 69 equations.

Table of Contents

  1. Introduction
  2. Plan of the paper.
  3. Notation and preliminaries.
  4. Strategies for the proof of well-posedness of McKean--Vlasov SDEs
  5. The classical proof of Sznitman in the case of globally Lipschitz fields.
  6. Lack of global Lipschitz continuity: the established proof for the mean-field CBO equation.
  7. An alternative approach for the mean-field CBO equation: a proof à la Sznitman through a truncation argument.
  8. Structural assumptions and their consequences
  9. Proof of existence through a truncation argument
  10. An a priori estimate and the proof of uniqueness
  11. The Sznitman's argument of well-posedness

Key Result

Theorem 1.1

Let $f \in \mathcal{O}(s,\ell)$, with $s,\ell\ge0$, and $p \ge 2 \vee p_\mathcal{M}(s,\ell)$ (see Definition def:ClassO_sl and eq:p_M below). Given a $d$-dimensional standard Brownian motion ${}\mkern3mu\overline{\mkern-3muB}$ defined on a filtered probability space $(\Omega, \mathscr{F}, (\mathscr{

Theorems & Definitions (20)

  • Theorem 1.1
  • Theorem 2.1
  • proof
  • Definition 3.1
  • Proposition 3.2: MeanFieldHoff
  • Remark 3.3
  • Lemma 4.1
  • proof
  • Proposition 4.2
  • proof
  • ...and 10 more