Affiliation:
1. ITMO University, St. Petersburg, Russia and École Polytechnique, CNRS, France
2. École Polytechnique, CNRS, France
Abstract
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.
Funder
National Center for Cognitive Research of ITMO University
Investissements d’avenir project
Publisher
Association for Computing Machinery (ACM)
Cited by
4 articles.
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1. Tight Runtime Bounds for Static Unary Unbiased Evolutionary Algorithms on Linear Functions;Proceedings of the Genetic and Evolutionary Computation Conference;2023-07-12
2. Fourier Analysis Meets Runtime Analysis: Precise Runtimes on Plateaus;Proceedings of the Genetic and Evolutionary Computation Conference;2023-07-12
3. The (1 + (λ, λ)) global SEMO algorithm;Proceedings of the Genetic and Evolutionary Computation Conference;2022-07-08
4. Escaping Local Optima with Local Search: A Theory-Driven Discussion;Lecture Notes in Computer Science;2022