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Algorithmica

, Volume 75, Issue 3, pp 462–489 | Cite as

Robustness of Populations in Stochastic Environments

  • Christian GießenEmail author
  • Timo Kötzing
Article

Abstract

We consider stochastic versions of OneMax and LeadingOnes and analyze the performance of evolutionary algorithms with and without populations on these problems. It is known that the (\(1+1\)) EA on OneMax performs well in the presence of very small noise, but poorly for higher noise levels. We extend these results to LeadingOnes and to many different noise models, showing how the application of drift theory can significantly simplify and generalize previous analyses. Most surprisingly, even small populations (of size \(\varTheta (\log n)\)) can make evolutionary algorithms perform well for high noise levels, well outside the abilities of the (\(1+1\)) EA. Larger population sizes are even more beneficial; we consider both parent and offspring populations. In this sense, populations are robust in these stochastic settings.

Keywords

Run time analysis Stochastic fitness function Evolutionary algorithm Populations Robustness 

Notes

Acknowledgments

The research leading to these results has received funding from the European Union Seventh Framework Programme (FP7/2007–2013) under Grant Agreement No. 618091 (SAGE) and by the Danish Council for Independent Research (DFF), through Grant 4002-00542. The authors would like to thank Benjamin Doerr and Tobias Friedrich for many useful discussions on the topic. Furthermore, the reviewers of both the conference and the journal version of this paper gave a lot of helpful feedback, for which we are very grateful.

References

  1. 1.
    Bianchi, L., Dorigo, M., Gambardella, L.M., Gutjahr, W.J.: A survey on metaheuristics for stochastic combinatorial optimization. Nat. Comput. 8(2), 239–287 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Doerr, B., Hota, A., Kötzing, T.: Ants easily solve stochastic shortest path problems. In: Genetic and Evolutionary Computation Conference, GECCO ’12, Philadelphia, PA, USA, July 7–11, 2012, pp. 17–24 (2012)Google Scholar
  3. 3.
    Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Dang, D.-C., Lehre, P.K.: Evolution under partial information. In: Genetic and Evolutionary Computation Conference, GECCO ’14, Vancouver, BC, Canada, July 12–16, 2014, pp. 1359–1366 (2014)Google Scholar
  5. 5.
    Droste, S.: Analysis of the (1+1) EA for a noisy onemax. In: Genetic and Evolutionary Computation—GECCO 2004, Genetic and Evolutionary Computation Conference, Seattle, WA, USA, June 26–30, 2004, Proceedings, Part I, pp. 1088–1099 (2004)Google Scholar
  6. 6.
    Feldmann, M., Kötzing, T.: Optimizing expected path lengths with ant colony optimization using fitness proportional update. In: Foundations of Genetic Algorithms XII, FOGA ’13, Adelaide, SA, Australia, January 16–20, 2013, pp. 65–74 (2013)Google Scholar
  7. 7.
    Gießen, C., Kötzing, T.: Robustness of populations in stochastic environments. In: Genetic and Evolutionary Computation Conference, GECCO ’14, Vancouver, BC, Canada, July 12–16, 2014, pp. 1383–1390 (2014)Google Scholar
  8. 8.
    Gutjahr, W.J., Pflug, GCh.: Simulated annealing for noisy cost functions. J. Glob. Optim. 8(1), 1–13 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gutjahr, W.J.: A converging ACO algorithm for stochastic combinatorial optimization. In: Stochastic Algorithms: Foundations and Applications, Second International Symposium, SAGA 2003, Hatfield, UK, September 22–23, 2003, Proceedings, pp. 10–25 (2003)Google Scholar
  10. 10.
    He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 9(3), 303–317 (2005)CrossRefGoogle Scholar
  12. 12.
    Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evol. Comput. 13(4), 413–440 (2005)CrossRefGoogle Scholar
  13. 13.
    Johannsen, D.: Random combinatorial structures and randomized search heuristics. PhD thesis, Saarland University (2010)Google Scholar
  14. 14.
    Lehre, P.K.: Fitness-levels for non-elitist populations. In: 13th Annual Genetic and Evolutionary Computation Conference, GECCO 2011, Proceedings, Dublin, Ireland, July 12–16, 2011, pp. 2075–2082 (2011)Google Scholar
  15. 15.
    Mitavskiy, B., Rowe, J.E., Cannings, C.: Preliminary theoretical analysis of a local search algorithm to optimize network communication subject to preserving the total number of links. In: Proceedings of the IEEE Congress on Evolutionary Computation, CEC 2008, June 1–6, 2008, Hong Kong, China, pp. 1484–1491 (2008)Google Scholar
  16. 16.
    Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica 59(3), 369–386 (2011). Kindly check whether the references [16] and [17] are correctMathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Oliveto, P.S., Witt, C.: Erratum: Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation. arXiv:1211.7184 (2012)
  18. 18.
    Prügel-Bennett, A.: Benefits of a population: five mechanisms that advantage population-based algorithms. IEEE Trans. Evol. Comput. 14(4), 500–517 (2010)CrossRefGoogle Scholar
  19. 19.
    Rowe, J.E., Sudholt, D.: The choice of the offspring population size in the (1, \(\lambda \)) evolutionary algorithm. Theor. Comput. Sci. 545, 20–38 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Sudholt, D., Thyssen, C.: A simple ant colony optimizer for stochastic shortest path problems. Algorithmica 64(4), 643–672 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Weisstein, E.W.: Erfc, 2015. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Erfc.html
  22. 22.
    Witt, C.: Runtime analysis of the (\(\mu + 1\)) EA on simple pseudo-boolean functions. Evol. Comput. 14(1), 65–86 (2006)Google Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Technical University of DenmarkKongens LyngbyDenmark
  2. 2.Friedrich-Schiller-Universität JenaJenaGermany

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