Consensus-based optimization with $\alpha$-stable jump processes

TL;DR

This paper introduces a new consensus-based optimization method incorporating $oldsymbol{ extalpha}$-stable jump processes, enhancing exploration in complex optimization problems through nonlocal stochastic effects and fractional Fokker-Planck equations.

Contribution

It extends the CBO method by integrating $oldsymbol{ extalpha}$-stable jumps, providing a rigorous convergence analysis and demonstrating improved performance in complex optimization tasks.

Findings

The method effectively explores complex landscapes due to nonlocal jumps.

Convergence to optimal solutions is rigorously established.

Numerical experiments show advantages over standard diffusion-based methods.

Abstract

In this paper, we introduce a novel variant of the CBO method that incorporates jumps according to an $\alpha$-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 $\alpha$-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 $\alpha$-stable jump process and emphasize its potential advantages over standard diffusion-based methods, particularly in complex optimization settings.