AI 中文总结
本论文针对共识优化算法的三个变体（δ-CBO、共识冻结和共识跳跃），严格推导了定量混沌传播结果，证明粒子系统经验测度在大粒子极限下收敛到平均场方程解，确立了粒子系统对理想化平均场动力学的忠实逼近。
AI 中文摘要
共识优化（CBO）是一种稳健且灵活的零阶多粒子方法类，旨在为高维全局优化问题（包括真正的非凸和/或非光滑情形）提供可证明的解决方案。该算法框架基于搜索空间的随机探索与向共识点确定性收缩之间的原则性权衡。本论文关注CBO的三个近期建立的变体。δ-CBO方法与经典CBO方法类似，区别在于扩散函数为常数。共识冻结（CF）方案可理解为一种分段CBO方法，整个时间范围被划分为多个区间，在每个区间内共识点固定。共识跳跃（CH）方案可理解为一种确定性迭代方案，每次迭代跳跃到以先前分布的共识点为中心的高斯分布。本论文的核心贡献是对所有三种方案严格推导了混沌传播结果。对于每个变体，我们建立了当粒子数N趋于无穷大时，相互作用粒子系统的经验测度收敛到相应平均场方程的解。处理了两种情形：首先，在经验测度一致有界的高概率集合上；其次，在无限制设置中，其中矩界和对粒子间依赖性的仔细处理使得收敛无需依赖于有利事件的条件。混沌传播估计通过耦合方法推导。综合这些结果，为所有三种CBO变体提供了定量的混沌传播估计，并确立了粒子系统在大粒子极限下忠实逼近理想化平均场动力学。
英文摘要
Consensus-based optimization (CBO) is a robust and flexible zero-order, multi-particle class of methods devised to provide provable solutions to high-dimensional global optimization problems, including truly nonconvex and/or nonsmooth cases. The algorithmic framework is grounded in a principled trade-off between stochastic exploration of the search space and deterministic contraction toward a consensus point. This thesis is concerned with three recently established variants of CBO. The $δ$-CBO method is similar to the classic CBO method, with the difference that the diffusion function is constant. The consensus freezing (CF) scheme can be understood as a piecewise CBO method. The overall time horizon is divided into intervals in which the consensus point is fixed. The consensus hopping (CH) scheme can be understood as a deterministic iterative scheme that jumps at each iteration to a Gaussian distribution centered at the consensus point of the previous distribution. The central contribution of this thesis is the rigorous derivation of propagation of chaos results for all three schemes. For each variant, we establish that the empirical measure of the interacting particle system converges, as the number of particles $N$ approaches infinity, to the solution of the corresponding mean-field equation. Two regimes are treated: first, on a high-probability set on which the empirical measures are uniformly bounded; second, in the unrestricted setting, where moment bounds and a careful treatment of the inter-particle dependence yield convergence without imposing conditioning on favorable events. The propagation of chaos estimates are derived by the coupling method. Together, these results provide quantitative propagation of chaos estimates for all three CBO variants and establish that the particle systems faithfully approximate the idealized mean-field dynamics in the large-particle limit.
