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Precise Runtime Analysis for Plateaus

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Parallel Problem Solving from Nature – PPSN XV (PPSN 2018)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11102))

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Abstract

To gain a better theoretical understanding of how evolutionary algorithms cope with plateaus of constant fitness, we analyze how the \((1 + 1)\) EA optimizes the n-dimensional \(\textsc {Plateau} _k\) function. This function has a plateau of second-best fitness in a radius of k around the optimum. As optimization 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 \(\Pr [\alpha = 1] = \varOmega (1)\), we show the surprising result that for \(k \ge 2\) the expected optimization time of this algorithm is

$$ \frac{n^k}{k!\Pr [1 \le \alpha \le k]}(1 + o(1)), $$

that is, the size of the plateau times the expected waiting time for an iteration flipping between 1 and k bits. Our result implies that the optimal mutation rate for this function 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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Notes

  1. 1.

    As common in optimization, we reserve the notion unimodal for objective functions such that each non-optimal search point has a strictly better neighbor.

  2. 2.

    In the usual asymptotic sense, that is, meaning all but a lower order fraction.

  3. 3.

    We call a mutation rate optimal when it differs from the truly optimal rate at most by lower order terms, that is, e.g. a factor of \((1 \pm o(1))\).

  4. 4.

    For reasons of space, not all mathematical proofs could fit into this extended abstract. The proofs can be found in [1].

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Authors and Affiliations

  1. ITMO University, 49 Kronverkskiy prosp., 197101, Saint-Petersburg, Russia

    Denis Antipov

  2. École Polytechnique, CNRS, Laboratoire d’Informatique (LIX), Palaiseau, France

    Benjamin Doerr

Authors
  1. Denis Antipov
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  2. Benjamin Doerr
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Corresponding author

Correspondence to Denis Antipov .

Editor information

Editors and Affiliations

  1. Inria Saclay, Palaiseau, France

    Anne Auger

  2. University of Coimbra, Coimbra, Portugal

    Carlos M. Fonseca

  3. University of Coimbra, Coimbra, Portugal

    Nuno Lourenço

  4. University of Coimbra, Coimbra, Portugal

    Penousal Machado

  5. University of Coimbra, Coimbra, Portugal

    Luís Paquete

  6. Colorado State University, Fort Collins, Colorado, USA

    Darrell Whitley

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Antipov, D., Doerr, B. (2018). Precise Runtime Analysis for Plateaus. In: Auger, A., Fonseca, C., Lourenço, N., Machado, P., Paquete, L., Whitley, D. (eds) Parallel Problem Solving from Nature – PPSN XV. PPSN 2018. Lecture Notes in Computer Science(), vol 11102. Springer, Cham. https://doi.org/10.1007/978-3-319-99259-4_10

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  • DOI: https://doi.org/10.1007/978-3-319-99259-4_10

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  • Online ISBN: 978-3-319-99259-4

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