Consensus based optimization with memory effects: random selection and applications

Abstract

In this work we extend the class of Consensus-Based Optimization (CBO) metaheuristic methods by considering memory effects and a random selection strategy. The proposed algorithm iteratively updates a population of particles according to a consensus dynamics inspired by social interactions among individuals. The consensus point is computed taking into account the past positions of all particles. While sharing features with the popular Particle Swarm Optimization (PSO) method, the exploratory behavior is fundamentally different and allows better control over the convergence of the particle system. We discuss some implementation aspects which lead to an increased efficiency while preserving the success rate in the optimization process. In particular, we show how employing a random selection strategy to discard particles during the computation improves the overall performance. Several benchmark problems and applications to image segmentation and Neural Networks training are used to validate and test the proposed method. A theoretical analysis allows to recover convergence guarantees under mild assumptions of the objective function. This is done by first approximating the particles evolution with a continuous-in-time dynamics, and then by taking the mean-field limit of such dynamics. Convergence to a global minimizer is finally proved at the mean-field level.

Consensus-based optimisation with truncated noise

“…While the original CBO model [7] has been adapted to solve constrained optimisations [24][25][26], optimisations on manifolds [16,[27][28][29][30], multi-objective optimisation problems [31][32][33], saddle point problems [34] or the task of sampling [35], as well as has been extended to make use of memory mechanisms [17,36,37], gradient information [17,38], momentum [39], jump-diffusion processes [40] or localisation kernels for polarisation [41], we focus in this work on a variation of the original model, which incorporates a truncation in the noise term of the dynamics. More formally, given a time horizon T > 0, a time discretisation t 0 = 0 < t < • • • < K t = t K = T of [0, T], and user-specified parameters α, λ, σ > 0 as well as v b , R > 0, we consider the interacting particle system…”