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Robustness of Populations in Stochastic Environments

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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.

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References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  3. Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64(4), 673–697 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  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)

  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)

  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)

  8. Gutjahr, W.J., Pflug, GCh.: Simulated annealing for noisy cost functions. J. Glob. Optim. 8(1), 1–13 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  10. He, J., Yao, X.: A study of drift analysis for estimating computation time of evolutionary algorithms. Nat. Comput. 3(1), 21–35 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Jin, Y., Branke, J.: Evolutionary optimization in uncertain environments—a survey. IEEE Trans. Evol. Comput. 9(3), 303–317 (2005)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  13. Johannsen, D.: Random combinatorial structures and randomized search heuristics. PhD thesis, Saarland University (2010)

  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)

  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)

  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 correct

    Article  MathSciNet  MATH  Google Scholar 

  17. Oliveto, P.S., Witt, C.: Erratum: Simplified Drift Analysis for Proving Lower Bounds in Evolutionary Computation. arXiv:1211.7184 (2012)

  18. Prügel-Bennett, A.: Benefits of a population: five mechanisms that advantage population-based algorithms. IEEE Trans. Evol. Comput. 14(4), 500–517 (2010)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  20. Sudholt, D., Thyssen, C.: A simple ant colony optimizer for stochastic shortest path problems. Algorithmica 64(4), 643–672 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  21. Weisstein, E.W.: Erfc, 2015. From MathWorld—A Wolfram Web Resource. http://mathworld.wolfram.com/Erfc.html

  22. Witt, C.: Runtime analysis of the (\(\mu + 1\)) EA on simple pseudo-boolean functions. Evol. Comput. 14(1), 65–86 (2006)

    Google Scholar 

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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.

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Authors and Affiliations

  1. Technical University of Denmark, 2800, Kongens Lyngby, Denmark

    Christian Gießen

  2. Friedrich-Schiller-Universität Jena, 07743, Jena, Germany

    Timo Kötzing

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  1. Christian Gießen
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  2. Timo Kötzing
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Correspondence to Christian Gießen.

Additional information

This work started when the first author was still affiliated with Christian-Albrechts-Universität zu Kiel.

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Gießen, C., Kötzing, T. Robustness of Populations in Stochastic Environments. Algorithmica 75, 462–489 (2016). https://doi.org/10.1007/s00453-015-0072-0

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