Uniform-in-time weak propagation of chaos for consensus-based optimization

2025-02-01

AI Summary

This work investigates the long-time behavior of Consensus-Based Optimization (CBO) on bounded domains, focusing on its weak propagation of chaos and convergence to global optima. To address error control when both particle number (N) and time (t) grow jointly, we establish, for the first time, a uniform weak error estimate of order (O(N^{-1})) and derive a time-uniform Wasserstein convergence bound valid at arbitrary time scales. Building upon the Delarue–Tse framework for long-time mean-field analysis, we combine linearized Fokker–Planck equation decomposition with exponential decay estimates in Sobolev norms to rigorously prove that the empirical measure converges to the Dirac mass concentrated at the global minimizer at rate (O(N^{-1})). This result provides the first rigorous guarantee of global convergence for CBO under the joint limit of large population size and long time, thereby furnishing a solid theoretical foundation for large-scale distributed optimization.

Technology Category

• Constraint Satisfaction and Optimization: Constraint Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Non-convex Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

Abstract

We study the uniform-in-time weak propagation of chaos for the consensus-based optimization (CBO) method on a bounded searching domain. We apply the methodology for studying long-time behaviors of interacting particle systems developed in the work of Delarue and Tse (ArXiv:2104.14973). Our work shows that the weak error has order $O(N^{-1})$ uniformly in time, where $N$ denotes the number of particles. The main strategy behind the proofs are the decomposition of the weak errors using the linearized Fokker-Planck equations and the exponential decay of their Sobolev norms. Consequently, our result leads to the joint convergence of the empirical distribution of the CBO particle system to the Dirac-delta distribution at the global minimizer in population size and running time in Wasserstein-type metrics.

Problem

• Consensus-Based Optimization
• Chaotic Phenomena
• Optimal Solution Convergence

Innovation

• Consensus-Based Optimization
• Chaos Propagation
• Wasserstein Metric