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Evolution Strategies under the 1/5 Success Rule

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  • Alexandru Agapie

    (Department of Applied Mathematics, Faculty of Economic Cybernetics, Statistics and Informatics, Bucharest University of Economic Studies, Calea Dorobantilor 15-17, 010552 Bucharest, Romania
    “Gheorghe Mihoc—Caius Iacob” Institute of Mathematical Statistics and Applied Mathematics of the Romanian Academy, 050711 Bucharest, Romania)

Abstract

For large space dimensions, the log-linear convergence of the elitist evolution strategy with a 1/5 success rule on the sphere fitness function has been observed, experimentally, from the very beginning. Finding a mathematical proof took considerably more time. This paper presents a review and comparison of the most consistent theories developed so far, in the critical interpretation of the author, concerning both global convergence and the estimation of convergence rates. I discuss the local theory of the one-step expected progress and success probability for the (1+1) ES with a normal/uniform distribution inside the sphere mutation, thereby minimizing the SPHERE function, but also the adjacent global convergence and convergence rate theory, essentially based on the 1/5 rule. Small digressions into complementary theories (martingale, irreducible Markov chain, drift analysis) and different types of algorithms (population based, recombination, covariance matrix adaptation and self-adaptive ES) complete the review.

Suggested Citation

  • Alexandru Agapie, 2022. "Evolution Strategies under the 1/5 Success Rule," Mathematics, MDPI, vol. 11(1), pages 1-20, December.
    • Handle: RePEc:gam:jmathe:v:11:y:2022:i:1:p:201-:d:1020377
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    References listed on IDEAS

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    1. Alexandru Agapie, 2021. "Spherical Distributions Used in Evolutionary Algorithms," Mathematics, MDPI, vol. 9(23), pages 1-15, November.
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