skip to main content
10.1145/2725494.2725496acmconferencesArticle/Chapter ViewAbstractPublication PagesfogaConference Proceedingsconference-collections
research-article

A More Efficient Rank-one Covariance Matrix Update for Evolution Strategies

Published: 17 January 2015 Publication History

Abstract

Learning covariance matrices of Gaussian distributions is at the heart of most variable-metric randomized algorithms for continuous optimization. If the search space dimensionality is high, updating the covariance or its factorization is computationally expensive. Therefore, we adopt an algorithm from numerical mathematics for rank-one updates of Cholesky factors. Our methods results in a quadratic time covariance matrix update scheme with minimal memory requirements. The numerically stable algorithm leads to triangular Cholesky factors. Systems of linear equations where the linear transformation is defined by a triangular matrix can be solved in quadratic time. This can be exploited to avoid the additional iterative update of the inverse Cholesky factor required in some covariance matrix adaptation algorithms proposed in the literature. When used together with the (1+1)-CMA-ES and the multi-objective CMA-ES, the new method leads to a memory reduction by a factor of almost four and a faster covariance matrix update. The numerical stability and runtime improvements are demonstrated on a set of benchmark functions.

References

[1]
Y. Akimoto, A. Auger, and N. Hansen. Comparison-based natural gradient optimization in high dimension. In Proceedings of the 16th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 373--380. ACM, 2014.
[2]
D. V. Arnold and N. Hansen. Active covariance matrix adaptation for the (1 + 1)-CMA-ES. In Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 385--392. ACM, 2010.
[3]
H.-G. Beyer and H.-P. Schwefel. Evolution strategies--A comprehensive introduction. Natural Computing, 1(1):3--52, 2002.
[4]
P. E. Gill, G. H. Golub, W. Murray, and M. A. Saunders. Methods for modifying matrix factorizations. Mathematics of Computation, 28(126):505--535, 1974.
[5]
N. Hansen. Adaptive encoding: How to render search coordinate system invariant. In G. Rudolph et al., editors, Proceedings of the 10th International Conference on Parallel Problem Solving from Nature (PPSN X), LNCS, pages 205--214, 2008.
[6]
N. Hansen. The CMA evolution strategy: A tutorial, 2011.
[7]
N. Hansen and A. Ostermeier. Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation, 9(2):159--195, 2001.
[8]
C. Igel, N. Hansen, and S. Roth. Covariance matrix adaptation for multi-objective optimization. Evolutionary Computation, 15(1):1--28, 2007.
[9]
C. Igel, V. Heidrich-Meisner, and T. Glasmachers. Shark. Journal of Machine Learning Research, 9:993--996, 2008.
[10]
C. Igel, T. Suttorp, and N. Hansen. A computational efficient covariance matrix update and a (1 + 1)-CMA for evolution strategies. In Proceedings of the 8th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 453--460. ACM, 2006.
[11]
P. Larrañaga. A review on estimation of distribution algorithms. In P. Larrañaga and J. Lozano, editors, Estimation of distribution algorithms, pages 57--100. Springer, 2002.
[12]
I. Loshchilov. A computationally efficient limited memory CMA-ES for large scale optimization. In Proceedings of the 16th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 397--404. ACM, 2014.
[13]
J. Poland and A. Zell. Main vector adaptation: A CMA variant with linear time and space complexity. In Proceedings of the 10th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 1050--1055. Morgan Kaufmann Publishers, 2001.
[14]
R. Ros and N. Hansen. A simple modification in CMA-ES achieving linear time and space complexity. In G. Rudolph et al., editors, Proceedings of the 10th International Conference on Parallel Problem Solving from Nature (PPSN X), LNCS, pages 296--305. Springer, 2008.
[15]
Y. Sun, T. Schaul, F. Gomez, and J. Schmidhuber. A linear time natural evolution strategy for non-separable functions. In 15th Annual Conference on Genetic and Evolutionary Computation Conference Companion, pages 61--62. ACM, 2013.
[16]
T. Suttorp, N. Hansen, and C. Igel. Efficient covariance matrix update for variable metric evolution strategies. Machine Learning, 75(2):167--197, 2009.
[17]
T. Voß, H. Hansen, and C. Igel. Improved step size adaptation for the MO-CMA-ES. In Proceedings of the 12th Annual Genetic and Evolutionary Computation Conference (GECCO), pages 487--494. ACM Press, 2010.

Cited By

View all
  • (2024)Distributed Evolution Strategies With Multi-Level Learning for Large-Scale Black-Box OptimizationIEEE Transactions on Parallel and Distributed Systems10.1109/TPDS.2024.343768835:11(2087-2101)Online publication date: Nov-2024
  • (2024)Robust Object Pose Tracking for Augmented Reality Guidance and TeleoperationIEEE Transactions on Instrumentation and Measurement10.1109/TIM.2024.339810873(1-15)Online publication date: 2024
  • (2024)An Efficient Differential Grouping Algorithm for Large-Scale Global OptimizationIEEE Transactions on Evolutionary Computation10.1109/TEVC.2022.323007028:1(32-46)Online publication date: Feb-2024
  • Show More Cited By

Index Terms

  1. A More Efficient Rank-one Covariance Matrix Update for Evolution Strategies

    Recommendations

    Comments

    Information & Contributors

    Information

    Published In

    cover image ACM Conferences
    FOGA '15: Proceedings of the 2015 ACM Conference on Foundations of Genetic Algorithms XIII
    January 2015
    200 pages
    ISBN:9781450334341
    DOI:10.1145/2725494
    Permission to make digital or hard copies of all or part of this work for personal or classroom use is granted without fee provided that copies are not made or distributed for profit or commercial advantage and that copies bear this notice and the full citation on the first page. Copyrights for components of this work owned by others than the author(s) must be honored. Abstracting with credit is permitted. To copy otherwise, or republish, to post on servers or to redistribute to lists, requires prior specific permission and/or a fee. Request permissions from [email protected].

    Sponsors

    Publisher

    Association for Computing Machinery

    New York, NY, United States

    Publication History

    Published: 17 January 2015

    Permissions

    Request permissions for this article.

    Check for updates

    Author Tags

    1. cholesky factorization
    2. cma-es
    3. covariance matrix adaptation
    4. rank-one update

    Qualifiers

    • Research-article

    Funding Sources

    • Danish National Advanced Technology Foundation

    Conference

    FOGA '15
    Sponsor:
    FOGA '15: Foundations of Genetic Algorithms XIII
    January 17 - 22, 2015
    Aberystwyth, United Kingdom

    Acceptance Rates

    FOGA '15 Paper Acceptance Rate 16 of 26 submissions, 62%;
    Overall Acceptance Rate 72 of 131 submissions, 55%

    Contributors

    Other Metrics

    Bibliometrics & Citations

    Bibliometrics

    Article Metrics

    • Downloads (Last 12 months)44
    • Downloads (Last 6 weeks)2
    Reflects downloads up to 22 Feb 2025

    Other Metrics

    Citations

    Cited By

    View all
    • (2024)Distributed Evolution Strategies With Multi-Level Learning for Large-Scale Black-Box OptimizationIEEE Transactions on Parallel and Distributed Systems10.1109/TPDS.2024.343768835:11(2087-2101)Online publication date: Nov-2024
    • (2024)Robust Object Pose Tracking for Augmented Reality Guidance and TeleoperationIEEE Transactions on Instrumentation and Measurement10.1109/TIM.2024.339810873(1-15)Online publication date: 2024
    • (2024)An Efficient Differential Grouping Algorithm for Large-Scale Global OptimizationIEEE Transactions on Evolutionary Computation10.1109/TEVC.2022.323007028:1(32-46)Online publication date: Feb-2024
    • (2024)Square-Root Higher-Order Unscented Estimators for Robust Orbit DeterminationIEEE Transactions on Aerospace and Electronic Systems10.1109/TAES.2024.342385160:6(7820-7837)Online publication date: Dec-2024
    • (2023)Comparison of MCMC Adaptation Schemes: A Preliminary Empirical StudyProceedings of the Companion Conference on Genetic and Evolutionary Computation10.1145/3583133.3590707(303-306)Online publication date: 15-Jul-2023
    • (2023)Clinamen2: Functional-style evolutionary optimization in Python for atomistic structure searchesComputer Physics Communications10.1016/j.cpc.2023.109065(109065)Online publication date: Dec-2023
    • (2021)Active Update of Mutation Matrix Adaptation for Variable Metric Evolution Strategy2021 17th International Conference on Computational Intelligence and Security (CIS)10.1109/CIS54983.2021.00013(21-25)Online publication date: Nov-2021
    • (2020)Fast Covariance Matrix Adaptation for Large-Scale Black-Box OptimizationIEEE Transactions on Cybernetics10.1109/TCYB.2018.287764150:5(2073-2083)Online publication date: May-2020
    • (2018)Adaptive Lattice Filters for Systems of Space-Time Processing of Non-Stationary Gaussian ProcessesRadioelectronics and Communications Systems10.3103/S073527271811001861:11(477-514)Online publication date: 27-Dec-2018
    • (2018)Адаптивные решетчатые фильтры для систем пространственно-временной обработки нестационарных гауссовых процессовИзвестия высших учебных заведений. Радиоэлектроника10.20535/S002134701811001861:11(607-644)Online publication date: 28-Nov-2018
    • Show More Cited By

    View Options

    Login options

    View options

    PDF

    View or Download as a PDF file.

    PDF

    eReader

    View online with eReader.

    eReader

    Figures

    Tables

    Media

    Share

    Share

    Share this Publication link

    Share on social media