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Runtime Analysis of a Heavy-Tailed \((1+(\lambda ,\lambda ))\) Genetic Algorithm on Jump Functions

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Parallel Problem Solving from Nature – PPSN XVI (PPSN 2020)

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Abstract

It was recently observed that the \((1+(\lambda ,\lambda ))\) genetic algorithm can comparably easily escape the local optimum of the jump functions benchmark. Consequently, this algorithm can optimize the jump function with jump size k in an expected runtime of only \(n^{(k + 1)/2}k^{-k/2}e^{O(k)}\) fitness evaluations (Antipov, Doerr, Karavaev (GECCO 2020)). This performance, however, was obtained with non-standard parameter setting depending on the jump size k.

To overcome this difficulty, we propose to choose two parameters of the \((1+(\lambda ,\lambda ))\) genetic algorithm randomly from a power-law distribution. Via a mathematical runtime analysis, we show that this algorithm with natural instance-independent choices of the power-law parameters on all jump functions with jump size at most n/4 has a performance close to what the best instance-specific parameters in the previous work obtained. This price for instance-independence can be made as small as an \(O(n\log (n))\) factor. Given the difficulty of the jump problem and the runtime losses from using mildly suboptimal fixed parameters (also discussed in this work), this appears to be a fair price.

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Notes

  1. 1.

    The omitted proofs can be found in the preprint  [3].

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Acknowledgements

This study was funded by RFBR and CNRS, project number 20-51-15009.

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Authors and Affiliations

  1. ITMO University, St. Petersburg, Russia

    Denis Antipov

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

    Denis Antipov & Benjamin Doerr

Authors
  1. Denis Antipov
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  2. Benjamin Doerr
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Correspondence to Denis Antipov .

Editor information

Editors and Affiliations

  1. Leiden University, Leiden, The Netherlands

    Thomas Bäck

  2. Leiden University, Leiden, The Netherlands

    Mike Preuss

  3. Leiden University, Leiden, The Netherlands

    André Deutz

  4. Sorbonne University, Paris, France

    Hao Wang

  5. Sorbonne University, Paris, France

    Carola Doerr

  6. Leiden University, Leiden, The Netherlands

    Michael Emmerich

  7. University of Münster, Münster, Germany

    Heike Trautmann

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Antipov, D., Doerr, B. (2020). Runtime Analysis of a Heavy-Tailed \((1+(\lambda ,\lambda ))\) Genetic Algorithm on Jump Functions. In: Bäck, T., et al. Parallel Problem Solving from Nature – PPSN XVI. PPSN 2020. Lecture Notes in Computer Science(), vol 12270. Springer, Cham. https://doi.org/10.1007/978-3-030-58115-2_38

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  • DOI: https://doi.org/10.1007/978-3-030-58115-2_38

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