Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus

TL;DR

This paper introduces a novel Fourier analysis method to precisely analyze the runtime of evolutionary algorithms on plateau problems, providing exact expected runtimes and optimal mutation rates for various scenarios.

Contribution

The paper presents a new Fourier analysis technique that yields exact runtime calculations and optimal mutation strategies for plateau problems in evolutionary algorithms.

Findings

Exact expected runtime for the Needle problem using Fourier analysis.

Determination of asymptotically optimal static and fitness-dependent mutation rates.

Extension of results from LeadingOnes to broader classes of plateau problems.

Abstract

We propose a new method based on discrete Fourier analysis to analyze the time evolutionary algorithms spend on plateaus. This immediately gives a concise proof of the classic estimate of the expected runtime of the $(1+1)$ evolutionary algorithm on the Needle problem due to Garnier, Kallel, and Schoenauer (1999). We also use this method to analyze the runtime of the $(1+1)$ evolutionary algorithm on a new benchmark consisting of $n/\ell$ plateaus of effective size $2^\ell-1$ which have to be optimized sequentially in a LeadingOnes fashion. Using our new method, we determine the precise expected runtime both for static and fitness-dependent mutation rates. We also determine the asymptotically optimal static and fitness-dependent mutation rates. For $\ell = o(n)$, the optimal static mutation rate is approximately $1.59/n$. The optimal fitness dependent mutation rate, when the first $k$ fitness-relevant bits have been found, is asymptotically $1/(k+1)$. These results, so far only proven for the single-instance problem LeadingOnes, thus hold for a much broader class of problems. We expect similar extensions to be true for other important results on LeadingOnes. We are also optimistic that our Fourier analysis approach can be applied to other plateau problems as well.