Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus
This work addresses runtime analysis for evolutionary algorithms on plateau problems, offering a method that extends known results from single-instance problems to a broader class, though it appears incremental in applying Fourier analysis to a specific domain.
The authors tackled the problem of analyzing runtime of evolutionary algorithms on plateau functions by introducing a discrete Fourier analysis method, which provided precise expected runtimes and asymptotically optimal mutation rates for a new benchmark problem, with optimal static mutation rate approximately 1.59/n for ℓ = o(n).
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.