Finite First Hitting Time versus Stochastic Convergence in Particle Swarm Optimisation

TL;DR

This paper analyzes the stochastic convergence of particle swarm optimization, highlighting limitations of traditional convergence criteria and proposing the expected first hitting time as a better measure, with a new algorithm demonstrating finite expected FHT.

Contribution

It introduces the expected first hitting time as a new metric for optimization ability and proposes the Noisy PSO algorithm, which achieves finite expected FHT on some functions.

Findings

Standard PSO can have infinite expected FHT.

Noisy PSO achieves finite expected FHT on certain functions.

Stagnation can occur even in simple functions like SPHERE.

Abstract

We reconsider stochastic convergence analyses of particle swarm optimisation, and point out that previously obtained parameter conditions are not always sufficient to guarantee mean square convergence to a local optimum. We show that stagnation can in fact occur for non-trivial configurations in non-optimal parts of the search space, even for simple functions like SPHERE. The convergence properties of the basic PSO may in these situations be detrimental to the goal of optimisation, to discover a sufficiently good solution within reasonable time. To characterise optimisation ability of algorithms, we suggest the expected first hitting time (FHT), i.e., the time until a search point in the vicinity of the optimum is visited. It is shown that a basic PSO may have infinite expected FHT, while an algorithm introduced here, the Noisy PSO, has finite expected FHT on some functions.