All Constant Mutation Rates for the $(1+1)$ Evolutionary Algorithm
Andrew James Kelley
TL;DR
This work proves that the set of optimal mutation rates for the (1+1) EA can be dense in (0,1) by constructing a fitness function $\textsc{DistantSteppingStones}_p$ with multiple large plateaus separated by deep valleys. The central result shows that the optimal mutation rate for this function converges to the target $p$ as the problem size grows, with a precise runtime bound: $\mathbb{E}[T(p)]=O(\alpha^n\log n)$ where $\alpha=1/(p^{p}(1-p)^{1-p})$, while any other mutation rate $q\neq p$ yields exponential runtime $\mathbb{E}[T(q)]=\Omega(\eta^n)$ for some $\eta>\alpha$. The analysis relies on a stepping-stone argument, bounding the time to reach successive plateaus and showing that there are $\Theta(\log n)$ such stones, which collectively enforce that the optimal rate tracks $p$ in the limit. The result both extends understanding of mutation-rate selection and provides a constructive means to realize dense optimal-rate sets in the (1+1) EA.
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.