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A Potential Function for a Variable-Metric Evolution Strategy

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Parallel Problem Solving from Nature – PPSN XVIII (PPSN 2024)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15149))

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Abstract

This paper works towards an analysis of a variable-metric evolution strategy by means of drift analysis. Drift analysis has been effective for proving convergence and analyzing the runtime of a simple (1+1)-ES. We make a first step towards including covariance matrix adaptation (CMA). To this end, we develop a novel class of potential functions for the (1+1)-CMA-ES optimizing two-dimensional convex quadratic functions. We leverage invariances to efficiently sample a representative space of states. We use simulations to gain an empirical estimate of the expected minimal drift induced by the candidate potential function and to tune potential function parameters. Our results indicate that the tuned potential function is negative and uniformly bounded away from zero, which yields linear convergence.

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Notes

  1. 1.

    Under slight misuse of notation, we incorporate the step size into the covariance matrix at this point, writing C instead of \(\sigma ^2 C\) from now on. The parameter \(\sigma \) is re-introduced in the normal form, see equation (1).

  2. 2.

    https://github.com/RUB-INI-Theory-of-Machine-Learning/EmpiricalDriftAnalysis/blob/main/Empirical_Drift_Analysis___Supplements.pdf.

  3. 3.

    https://github.com/RUB-INI-Theory-of-Machine-Learning/EmpiricalDriftAnalysis/tree/main/plots.

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Authors and Affiliations

  1. Fakultät für Informatik, Institut für Neuroinformatik, Ruhr-Universität Bochum, Bochum, Germany

    Stephan Frank & Tobias Glasmachers

Authors
  1. Stephan Frank
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  2. Tobias Glasmachers
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Correspondence to Stephan Frank .

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Editors and Affiliations

  1. University of Applied Sciences Upper Austria, Wels, Austria

    Michael Affenzeller

  2. University of Applied Sciences Upper Austria, Hagenberg, Austria

    Stephan M. Winkler

  3. Leiden University, Leiden, The Netherlands

    Anna V. Kononova

  4. University of Paderborn, Paderborn, Germany

    Heike Trautmann

  5. Jožef Stefan Institute, Ljubljana, Slovenia

    Tea Tušar

  6. University of Coimbra, Coimbra, Portugal

    Penousal Machado

  7. Leiden University, LEIDEN, The Netherlands

    Thomas Bäck

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Frank, S., Glasmachers, T. (2024). A Potential Function for a Variable-Metric Evolution Strategy. In: Affenzeller, M., et al. Parallel Problem Solving from Nature – PPSN XVIII. PPSN 2024. Lecture Notes in Computer Science, vol 15149. Springer, Cham. https://doi.org/10.1007/978-3-031-70068-2_14

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  • DOI: https://doi.org/10.1007/978-3-031-70068-2_14

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