research-article

Learning rate adaptation by line search in evolution strategies with recombination

Authors:

Armand Gissler,

Nikolaus HansenAuthors Info & Claims

GECCO '22: Proceedings of the Genetic and Evolutionary Computation Conference

Pages 630 - 638

https://doi.org/10.1145/3512290.3528760

Published: 08 July 2022 Publication History

Abstract

In this paper, we investigate the effect of a learning rate for the mean in Evolution Strategies with recombination. We study the effect of a half-line search after the mean shift direction is established, hence the learning rate value is conditioned to the direction. We prove convergence and study convergence rates in different dimensions and for different population sizes on the sphere function with the step-size proportional to the distance to the optimum.

We empirically find that a perfect half-line search increases the maximal convergence rate on the sphere function by up to about 70%, assuming the line search imposes no additional costs. The speedup becomes less pronounced with increasing dimension. The line search reduces---however does not eliminate---the dependency of the convergence rate on the step-size. The optimal step-size assumes considerably smaller values with line search, which is consistent with previous results for different learning rate settings. The step-size difference is more pronounced in larger dimension and with larger population size, thereby diminishing an important advantage of a large population.

References

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Youhei Akimoto, Anne Auger, Tobias Glasmachers, and Daiki Morinaga. Global linear convergence of evolution strategies on more than smooth strongly convex functions. SIAM Journal on Optimization, 2022 (accepted).

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Youhei Akimoto, Anne Auger, and Nikolaus Hansen. Quality gain analysis of the weighted recombination evolution strategy on general convex quadratic functions. Theoretical Computer Science, 832:42--67, 2020.

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Dirk V Arnold. Weighted multirecombination evolution strategies. Theoretical computer science, 361(1):18--37, 2006.

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Anne Auger, Dimo Brockhoff, and Nikolaus Hansen. Mirrored sampling in evolution strategies with weighted recombination. In Proceedings of the 13th annual conference on Genetic and evolutionary computation, pages 861--868, 2011.

Digital Library

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Anne Auger and Nikolaus Hansen. Reconsidering the progress rate theory for evolution strategies in finite dimensions. In Proceedings of the 8th annual conference on genetic and evolutionary computation GECCO, pages 445--452. ACM, 2006.

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Anne Auger and Nikolaus Hansen. Theory of evolution strategies: a new perspective. In A. Auger and B. Doerr, editors, Theory of Randomized Search Heuristics: Foundations and Recent Developments, chapter 10, pages 289--325. World Scientific Publishing, 2011.

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Anne Auger and Nikolaus Hansen. Linear convergence on positively homogeneous functions of a comparison based step-size adaptive randomized search: the (1+1) ES with generalized one-fifth success rule. arXiv:1310.8397 [cs.NA], 2013.

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Anne Auger and Nikolaus Hansen. Linear convergence of comparison-based step-size adaptive randomized search via stability of markov chains. SIAM Journal on Optimization, 26(3):1589--1624, 2016.

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Hans-Georg Beyer. Mutate large, but inherit small! On the analysis of rescaled mutations in (1,λ)-ES with noisy fitness data. In International Conference on Parallel Problem Solving from Nature, pages 109--118. Springer, 1998.

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Armand Gissler, Anne Auger, and Nikolaus Hansen. Supplementary material for Learning rate adaptation by line search in evolution strategies with recombination. hal-03626292, 2022.

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Nikolaus Hansen and Andreas Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary computation, 9(2):159--195, 2001.

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Jens Jägersküpper. Lower bounds for hit-and-run direct search. In Juraj Hromkovič, Richard Královič, Marc Nunkesser, and Peter Widmayer, editors, Stochastic Algorithms: Foundations and Applications, volume 4665, pages 118--129, Berlin, Heidelberg, 2007. Springer.

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Mohamed Jebalia and Anne Auger. Log-linear convergence of the scale-invariant (μ/μ_w, λ)-ES and optimal μ for intermediate recombination for large population sizes. In International Conference on Parallel Problem Solving from Nature, pages 52--62. Springer, 2010.

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Ingo Rechenberg. Evolutionsstrategie'94, volume 1581. Frommann-Holzboog-Verlag, Stuttgart (Germany), 1994.

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Cheikh Touré, Anne Auger, and Nikolaus Hansen. Global linear convergence of evolution strategies with recombination on scaling-invariant functions. Journal of Global Optimization, 2022 (under revision).

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Cheikh Touré, Armand Gissler, Anne Auger, and Nikolaus Hansen. Scaling-invariant functions versus positively homogeneous functions. Journal of Optimization Theory and Applications, 191:363--383, 2021.

Digital Library

Cited By

Nomura MAkimoto YOno ISilva SPaquete L(2023)CMA-ES with Learning Rate Adaptation: Can CMA-ES with Default Population Size Solve Multimodal and Noisy Problems?Proceedings of the Genetic and Evolutionary Computation Conference10.1145/3583131.3590358(839-847)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583131.3590358

Index Terms

Learning rate adaptation by line search in evolution strategies with recombination
1. Theory of computation
  1. Design and analysis of algorithms
    1. Mathematical optimization
      1. Continuous optimization
        Nonconvex optimization
      2. Mixed discrete-continuous optimization
        Bio-inspired optimization

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cover image ACM Conferences

GECCO '22: Proceedings of the Genetic and Evolutionary Computation Conference

July 2022

1472 pages

ISBN:9781450392372

DOI:10.1145/3512290

Editor:
Jonathan E. Fieldsend
University of Exeter
,
General Chair:
Markus Wagner
The University of Adelaide

Copyright © 2022 ACM.

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Published: 08 July 2022

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GECCO '22: Genetic and Evolutionary Computation Conference

July 9 - 13, 2022

Massachusetts, Boston

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Cited By

Nomura MAkimoto YOno ISilva SPaquete L(2023)CMA-ES with Learning Rate Adaptation: Can CMA-ES with Default Population Size Solve Multimodal and Noisy Problems?Proceedings of the Genetic and Evolutionary Computation Conference10.1145/3583131.3590358(839-847)Online publication date: 15-Jul-2023
https://dl.acm.org/doi/10.1145/3583131.3590358

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