Optimal Static Mutation Strength Distributions for the $(1+\lambda)$ Evolutionary Algorithm on OneMax

TL;DR

This paper derives optimal static mutation distributions for the $(1+\lambda)$ evolutionary algorithm on OneMax, revealing complex, counter-intuitive solutions and analyzing their impact on algorithm performance.

Contribution

It introduces a novel algorithm to compute the best static mutation distributions for the $(1+\lambda)$ EA on OneMax, linking static and dynamic parameter strategies.

Findings

Optimal distributions can be complex and counter-intuitive.

Performance varies significantly with different mutation distributions.

Common distributions may not be optimal for large populations.

Abstract

Most evolutionary algorithms have parameters, which allow a great flexibility in controlling their behavior and adapting them to new problems. To achieve the best performance, it is often needed to control some of the parameters during optimization, which gave rise to various parameter control methods. In recent works, however, similar advantages have been shown, and even proven, for sampling parameter values from certain, often heavy-tailed, fixed distributions. This produced a family of algorithms currently known as "fast evolution strategies" and "fast genetic algorithms". However, only little is known so far about the influence of these distributions on the performance of evolutionary algorithms, and about the relationships between (dynamic) parameter control and (static) parameter sampling. We contribute to the body of knowledge by presenting, for the first time, an algorithm that computes the optimal static distributions, which describe the mutation operator used in the well-known simple $(1+\lambda)$ evolutionary algorithm on a classic benchmark problem OneMax. We show that, for large enough population sizes, such optimal distributions may be surprisingly complicated and counter-intuitive. We investigate certain properties of these distributions, and also evaluate the performance regrets of the $(1+\lambda)$ evolutionary algorithm using commonly used mutation distributions.