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Article Dans Une Revue ACM Transactions on Evolutionary Learning and Optimization Année : 2021

Precise Runtime Analysis for Plateau Functions

Résumé

To gain a better theoretical understanding of how evolutionary algorithms (EAs) cope with plateaus of constant fitness, we propose the n -dimensional \textsc {Plateau} _k function as natural benchmark and analyze how different variants of the (1 + 1) EA optimize it. The \textsc {Plateau} _k function has a plateau of second-best fitness in a ball of radius k around the optimum. As evolutionary algorithm, we regard the (1 + 1) EA using an arbitrary unbiased mutation operator. Denoting by \alpha the random number of bits flipped in an application of this operator and assuming that \Pr [\alpha = 1] has at least some small sub-constant value, we show the surprising result that for all constant k \ge 2 , the runtime T follows a distribution close to the geometric one with success probability equal to the probability to flip between 1 and k bits divided by the size of the plateau. Consequently, the expected runtime is the inverse of this number, and thus only depends on the probability to flip between 1 and k bits, but not on other characteristics of the mutation operator. Our result also implies that the optimal mutation rate for standard bit mutation here is approximately k/(en) . Our main analysis tool is a combined analysis of the Markov chains on the search point space and on the Hamming level space, an approach that promises to be useful also for other plateau problems.
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Dates et versions

hal-03418304 , version 1 (07-11-2021)

Identifiants

Citer

Denis Antipov, Benjamin Doerr. Precise Runtime Analysis for Plateau Functions. ACM Transactions on Evolutionary Learning and Optimization, 2021, 1 (4), pp.1-28. ⟨10.1145/3469800⟩. ⟨hal-03418304⟩
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