From particle swarm optimization to consensus based optimization: Stochastic modeling and mean-field limit

Abstract

In this paper, we consider a continuous description based on stochastic differential equations of the popular particle swarm optimization (PSO) process for solving global optimization problems and derive in the large particle limit the corresponding mean-field approximation based on Vlasov–Fokker–Planck-type equations. The disadvantage of memory effects induced by the need to store the local best position is overcome by the introduction of an additional differential equation describing the evolution of the local best. A regularization process for the global best permits to formally derive the respective mean-field description. Subsequently, in the small inertia limit, we compute the related macroscopic hydrodynamic equations that clarify the link with the recently introduced consensus based optimization (CBO) methods. Several numerical examples illustrate the mean field process, the small inertia limit and the potential of this general class of global optimization methods.

On the mean-field limit for the consensus-based optimization

“…In this section we extend our discussions to the model of particle swarm optimization (PSO) proposed recently by Grassi and Pareschi [19], where they only numerically verified the mean-limit result. We consider PSO based on a continuous description in the form of a system of stochastic differential equations:…”

“…which means that X α (µ N t ) is a global best location at time t. It has been proved that CBO can guarantee global convergence under suitable assumptions [16] and it is a powerful and robust method to solve many interesting non-convex high-dimensional optimization problems in machine learning [7,14]. By now, CBO methods have also been generalized to optimization over manifolds [13][14][15]22] and several variants have been explored, which use additionally, for instance, personal best information [30], binary interaction dynamics [3] or connect CBO with Particle Swarm Optimization [8,19]. The readers are referred to [31] for a comprehensive review on the recent developments of the CBO methods.…”

SiSyPHE: A Python package for the Simulation of Systems of interacting mean-field Particles with High Efficiency

“…These models are usually based on partial differential equations (PDE) derived using phenomenological considerations that are often difficult to justify mathematically (Degond et al, 2021;Degond & Motsch, 2008;Dimarco & Motsch, 2016). Finally, inspired by models in biology, there is an ever growing literature on the design of algorithms based on the simulation of artificial particle systems to solve tough optimization problems (Grassi & Pareschi, 2020;Kennedy & Eberhart, 1995;Pinnau et al, 2017;Totzeck, 2021) and to construct new more efficient Markov Chain Monte Carlo methods (Cappé et al, 2004;Clarté et al, 2021;Del Moral, 1998, 2013Doucet et al, 2001). The simulation of systems of particles is also at the core of molecular dynamics (Leimkuhler & Matthews, 2015), although the present library is not specifically written for this purpose.…”

Mean-field particle swarm optimization

“…In this work we review some recent results on the mean-field modeling of particle swarm optimization with the goal of providing a robust mathematical theory for PSO methods and their convergence to the global minimum, based on a continuous description of their dynamics [67][68][69][70][71][72]. A major difficulty in the mathematical description of PSO methods, and other metaheuristic algorithms, is the presence of memory mechanisms that make their interpretation in terms of differential equations particularly challenging.…”

“…Adopting the same regularization process for the global best as in CBO methods [50,51], it is then possible to pass to the mean-field limit and derive the corresponding Vlasov-Fokker-Planck equation that characterizes the behavior of the system in the limit of a large number of particles [68,70].…”