Algorithmica

pp 1–22 | Cite as

Optimal Mutation Rates for the (1+\(\lambda \)) EA on OneMax Through Asymptotically Tight Drift Analysis

Article
Part of the following topical collections:
  1. Special Issue on Genetic and Evolutionary Computation

Abstract

We study the (1+\(\lambda \)) EA, a classical population-based evolutionary algorithm, with mutation probability c / n, where \(c>0\) and \(\lambda \) are constant, on the benchmark function OneMax, which counts the number of 1-bits in a bitstring. We improve a well-established result that allows to determine the first hitting time from the expected progress (drift) of a stochastic process, known as the variable drift theorem. Using our improved result, we show that upper and lower bounds on the expected runtime of the (1+\(\lambda \)) EA obtained from variable drift theorems are at most apart by a small lower order term if the exact drift is known. This reduces the analysis of expected optimization time to finding an exact expression for the drift. We then give an exact closed-form expression for the drift and develop a method to approximate it very efficiently, enabling us to determine approximate optimal mutation rates for the (1+\(\lambda \)) EA for various parameter settings of c and \(\lambda \) and also for moderate sizes of n. This makes the need for potentially lengthy and costly experiments in order to optimize c for fixed n and \(\lambda \) for the optimization of OneMax unnecessary. Interestingly, even for moderate n and not too small \(\lambda \) it turns out that mutation rates up to 10% larger than the asymptotically optimal rate 1 / n minimize the expected runtime. However, in absolute terms the expected runtime does not change by much when replacing 1 / n with the optimal mutation rate.

Keywords

Runtime analysis Populations Mutation 

References

  1. 1.
    Auger, A., Doerr, B. (ed.): Theory of Randomized Search Heuristics: Foundations and Recent Developments. World Scientific Publishing, (2011)Google Scholar
  2. 2.
    Abramowitz, M., Stegun, I.A.: Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards, Gaithersburg (1964)MATHGoogle Scholar
  3. 3.
    Böttcher, S., Doerr, B., Neumann, F.: Optimal fixed and adaptive mutation rates for the leadingones problem. In Proceedings of Parallel Problem Solving from Nature (PPSN 2010), vol. 6238, pp. 1–10. Springer (2010)Google Scholar
  4. 4.
    Badkobeh, G., Lehre, P.K., Sudholt, D.: Unbiased black-box complexity of parallel search. In: Parallel Problem Solving from Nature—PPSN XIII—13th International Conference, Ljubljana, Slovenia, September 13–17, 2014. Proceedings, pp. 892–901. (2014)Google Scholar
  5. 5.
    Chicano, F., Sutton, A.M., Whitley, L.D., Alba, E.: Fitness probability distribution of bit-flip mutation. Evolut. Comput. 23(2), 217–248 (2015)CrossRefGoogle Scholar
  6. 6.
    Doerr, B., Doerr, C., Yang, J .: Optimal parameter choices via precise black-box analysis. In: Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, Denver, CO, USA, July 20–24, 2016, pp. 1123–1130. (2016)Google Scholar
  7. 7.
    Doerr, B., Fouz, M., Witt, C.: Sharp bounds by probability-generating functions and variable drift. In: Procedings of the Genetic and Evolutionary Computation Conference (GECCO 2011), pp. 2083–2090. ACM Press (2011)Google Scholar
  8. 8.
    Doerr, B., Goldberg, L.A.: Adaptive drift analysis. Algorithmica 65(1), 224–250 (2013)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Doerr, B., Künnemann, M.: Royal road functions and the (1+\(\lambda \)) evolutionary algorithm: almost no speed-up from larger offspring populations. In: Proceedings of the IEEE Congress on Evolutionary Computation (CEC 2013), pp. 424–431. IEEE Press, (2013)Google Scholar
  11. 11.
    Doerr, B., Künnemann, M.: Optimizing linear functions with the (1+\(\lambda \)) evolutionary algorithm - different asymptotic runtimes for different instances. Theor. Comput. Sci. 561, 3–23 (2015)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Gießen, C., Witt, C.: Population size versus mutation strength for the (1+\(\lambda \)) EA on OneMax. In: Proceedings of GECCO ’15, pp. 1439–1446. ACM Press, (2015)Google Scholar
  13. 13.
    Gießen, C., Witt, C.: Optimal mutation rates for the (1+\(\lambda \)) EA on onemax. In: Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, Denver, CO, USA, July 20–24, 2016, pp. 1147–1154, (2016)Google Scholar
  14. 14.
    Hwang, H.-K., Panholzer, A., Rolin, N., Tsai, T.-H., Chen, W.-M.: Probabilistic analysis of the (1+1)-evolutionary algorithm. Evol. Comput. (2017). doi:10.1162/EVCO_a_00212
  15. 15.
    Jansen, T.: Analyzing Evolutionary Algorithms—The Computer Science Perspective. Natural Computing Series. Springer, Berlin (2013)CrossRefMATHGoogle Scholar
  16. 16.
    Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evolut. Comput. 13(4), 413–440 (2005)CrossRefGoogle Scholar
  17. 17.
    Johannsen, D.: Random combinatorial structures and randomized search heuristics. Ph.D. thesis, Universität des Saarlandes, Germany, (2010)Google Scholar
  18. 18.
    Lehre, P.K., Witt, C.: Concentrated hitting times of randomized search heuristics with variable drift. In: Proceedings of ISAAC ’14, Volume 8889 of Lecture Notes in Computer Science, pp. 686–697. Springer, 2014. Full technical report at http://arxiv.org/abs/1307.2559
  19. 19.
    Mitavskiy, B., Rowe, J.E., Cannings, C.: Theoretical analysis of local search strategies to optimize network communication subject to preserving the total number of links. Int. J. Intell. Comput. Cybern. 2(2), 243–284 (2009)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization—Algorithms and Their Computational Complexity. Natural Computing Series. Springer, Berlin (2010)MATHGoogle Scholar
  21. 21.
    Jonathan, E.: Rowe and Dirk Sudholt. The choice of the offspring population size in the (1, \(\lambda \)) evolutionary algorithm. Theoretical Computer Science, 545:20–38, 2014. Preliminary version in Proceedings of GECCO 2012Google Scholar
  22. 22.
    Skellam, J.G.: The frequency distribution of the difference between two poisson variates belonging to different populations. J. R. Stat. Soc. 109(3), 296–296 (1946)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Teerapabolarn, K.: A bound on the poisson-binomial relative error. Stat. Methodol. 4(4), 407–415 (2007)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Witt, C.: Tight bounds on the optimization time of a randomized search heuristic on linear functions. Comb Prob. Comput. 22(2):294–318, 2013. Preliminary version in Proceedings of STACS ’12Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2017

Authors and Affiliations

  1. 1.DTU ComputeTechnical University of DenmarkLyngbyDenmark

Personalised recommendations