All Constant Mutation Rates for the $(1+1)$ Evolutionary Algorithm

TL;DR

This paper demonstrates that for any mutation rate in (0,1), there exists a fitness function where this rate is nearly optimal, showing the set of optimal rates is dense in [0,1], using a specially constructed fitness landscape.

Contribution

It introduces DistantSteppingStones, a fitness function with large plateaus and valleys, to prove the density of optimal mutation rates for the (1+1) EA.

Findings

Optimal mutation rates are dense in [0,1] for the (1+1) EA.

A new fitness function, DistantSteppingStones, is constructed for the proof.

Any mutation rate in (0,1) can be nearly optimal for some fitness function.

Abstract

For every mutation rate $p \in (0, 1)$, and for all $\varepsilon > 0$, there is a fitness function $f : \{0,1\}^n \to \mathbb{R}$ with a unique maximum for which the optimal mutation rate for the $(1+1)$ evolutionary algorithm on $f$ is in $(p-\varepsilon, p+\varepsilon)$. In other words, the set of optimal mutation rates for the $(1+1)$ EA is dense in the interval $[0, 1]$. To show that, this paper introduces DistantSteppingStones, a fitness function which consists of large plateaus separated by large fitness valleys.