Consensus-based optimization with $α$-stable jump processes

arXiv (Cornell University)

Abstract

In this paper, we introduce a novel variant of the CBO method that incorporates jumps according to an $α$-stable stochastic process in a kinetic framework. This extension gives rise to nonlocal stochastic effects, which improve the exploration capabilities of the method. We formulate the method at the particle level, detailing the corresponding stochastic dynamics and its asymptotic behavior. In particular, through a Fourier-based representation, we derive the associated fractional Fokker-Planck equation, which naturally accounts for the nonlocal diffusion behaviors induced by $α$-stable processes. As a central result, we establish a rigorous convergence result for the proposed approach. Finally, we evaluate the performance of the method through a set of numerical experiments. The results demonstrate the effectiveness of the $α$-stable jump process and emphasize its potential advantages over standard diffusion-based methods, particularly in complex optimization settings.

Authors 4

1. Pedro Aceves-Sánchez

   University of Arizona

   Affiliation as printed

   Department of Mathematics , University of Arizona , Tucson , Arizona , USA

2. Giacomo Albi

   University of Verona · University of Pretoria

   Affiliation as printed

   Department of Computer Science , University of Verona , ITALY

   Department of Mathematics and Applied Mathematics , University of Pretoria , South Africa

3. Federica Ferrarese

   University of Ferrara

   Affiliation as printed

   Department of Mathematics and Computer Science & Center for Modeling, Computing and Statistics (CMCS) , University of Ferrara , via Machiavelli 30 , 44121 Ferrara , ITALY

4. Michael Herty Aachen Institute for Geometry and Applied Mathematics

   RWTH Aachen University

   Affiliation as printed

   Institute for Geometry and Applied Mathematics (IGPM) , RWTH Aachen University , GERMANY