A general perspective on CBO methods with stochastic rate of information

TL;DR

This paper introduces a generalized class of Consensus-Based Optimization models incorporating stochastic information rates, proving their well-posedness, convergence, and the influence of initial knowledge on consensus formation.

Contribution

It extends existing CBO models by including stochastic information rates and provides rigorous analysis of their well-posedness, convergence, and the role of initial knowledge.

Findings

Particles concentrate around the consensus point under mild conditions.

A positive initial knowledge level guarantees convergence to consensus.

The framework encompasses the first CBO models proposed in literature.

Abstract

This paper studies a class of Consensus-Based Optimization (CBO) models featuring an additional stochastic rate of information, modeling the agents' knowledge of the environment and energy landscape. The well-posedness of the stochastic system is proved, together with its finite-particle approximation and the mean-field convergence to a kinetic PDE. Particles are shown to concentrate around the consensus point under mild assumptions on the initial spatial distribution and initial level of knowledge. In particular, the analysis unveils that a positive, however small, initial level of knowledge is enough for convergence to consensus to happen. The framework presented is general enough to include the first instances of CBO proposed in the literature.