证明了粒子优化方法在任意时间下误差随粒子数呈1/N衰减
Uniform-in-time weak propagation of chaos for consensus-based optimization
- 通过线性化福克-普朗克方程分解弱误差
- 弱误差保持O(N⁻¹)且对时间均匀成立
- 适用于需要稳定收敛的全局优化场景
研究共识优化（CBO）方法在有界搜索域上的一致时间弱传播混沌。采用Delarue与Tse（ArXiv:2104.14973）提出的交互粒子系统长期行为分析方法，证明弱误差在任意时间下均为O(N⁻¹)，其中N为粒子数。证明核心为利用线性化福克-普朗克方程分解弱误差，并结合其Sobolev范数的指数衰减特性。由此结果可得：随着粒子数量和运行时间增加，CBO粒子系统的经验分布以Wasserstein型度量收敛至全局最小值处的狄拉克-δ分布。
原文摘要 · Abstract (English)
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.
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