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Algorithmica

pp 1–39 | Cite as

The (\(1+\lambda \)) Evolutionary Algorithm with Self-Adjusting Mutation Rate

  • Benjamin Doerr
  • Christian Gießen
  • Carsten Witt
  • Jing Yang
Article
  • 26 Downloads
Part of the following topical collections:
  1. Special Issue on Theory of Genetic and Evolutionary Computation

Abstract

We propose a new way to self-adjust the mutation rate in population-based evolutionary algorithms in discrete search spaces. Roughly speaking, it consists of creating half the offspring with a mutation rate that is twice the current mutation rate and the other half with half the current rate. The mutation rate is then updated to the rate used in that subpopulation which contains the best offspring. We analyze how the \((1+\lambda )\) evolutionary algorithm with this self-adjusting mutation rate optimizes the OneMax test function. We prove that this dynamic version of the \((1+\lambda )\) EA finds the optimum in an expected optimization time (number of fitness evaluations) of \(O(n\lambda /\log \lambda +n\log n)\). This time is asymptotically smaller than the optimization time of the classic \((1+\lambda )\) EA. Previous work shows that this performance is best-possible among all \(\lambda \)-parallel mutation-based unbiased black-box algorithms. This result shows that the new way of adjusting the mutation rate can find optimal dynamic parameter values on the fly. Since our adjustment mechanism is simpler than the ones previously used for adjusting the mutation rate and does not have parameters itself, we are optimistic that it will find other applications.

Keywords

Evolutionary computation Runtime analysis Mutation Self-adaptation 

Notes

Acknowledgements

This work was supported by a public Grant as part of the Investissement d’avenir Project, reference ANR-11-LABX-0056-LMH, LabEx LMH, and by a Grant by the Danish Council for Independent Research (DFF-FNU 4002–00542).

References

  1. 1.
    Alanazi, F., Lehre, P.K.: Runtime analysis of selection hyper-heuristics with classical learning mechanisms. In: Proceedings of CEC ’14, pp. 2515–2523. IEEE (2014)Google Scholar
  2. 2.
    Antipov, D., Doerr, B., Fang, J., Hetet, T.: Runtime analysis for the \((\mu +\lambda )\) EA optimizing OneMax. In: Proceedings of GECCO ’18, pp. 1459–1466. ACM (2018)Google Scholar
  3. 3.
    Auger, A., Doerr, B. (eds.): Theory of Randomized Search Heuristics. World Scientific Publishing, Singapore (2011)Google Scholar
  4. 4.
    Badkobeh, G., Lehre, P.K., Sudholt, D.: Unbiased black-box complexity of parallel search. In: Proceedings of PPSN ’14, pp. 892–901. Springer (2014)Google Scholar
  5. 5.
    Böttcher, S., Doerr, B., Neumann, F.: Optimal fixed and adaptive mutation rates for the LeadingOnes problem. In: Proceedings of PPSN ’10, pp. 1–10. Springer (2010)Google Scholar
  6. 6.
    Buzdalov, M., Doerr, B.: Runtime analysis of the \((1+(\lambda ,\lambda ))\) genetic algorithm on random satisfiable 3-CNF formulas. In: Proceedings of GECCO ’17, pp. 1343–1350. ACM (2017)Google Scholar
  7. 7.
    Cathabard, S., Lehre, P.K., Yao, X.: Non-uniform mutation rates for problems with unknown solution lengths. In: Proceedings of FOGA ’11, pp. 173–180. ACM (2011)Google Scholar
  8. 8.
    Cervantes, J., Stephens, C.R.: Rank based variation operators for genetic algorithms. In: Proceedings of GECCO ’08, pp. 905–912. ACM (2008)Google Scholar
  9. 9.
    Dang, D-C., Lehre, P.K.: Self-adaptation of mutation rates in non-elitist populations. In: Proceedings of PPSN ’16, pp. 803–813. Springer (2016)Google Scholar
  10. 10.
    Dietzfelbinger, M., Rowe, J.E., Wegener, I., Woelfel, P.: Tight bounds for blind search on the integers and the reals. Comb. Probab. Comput. 19, 711–728 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Doerr, B.: Analyzing randomized search heuristics: tools from probability theory. In: Auger, A., Doerr, B. (eds.) Theory of Randomized Search Heuristics, pp. 1–20. World Scientific Publishing, Singapore (2011)Google Scholar
  12. 12.
    Doerr, B.: Optimal parameter settings for the \((1+(\lambda , \lambda ))\) genetic algorithm. In: Proceedings of GECCO ’16, pp. 1107–1114. ACM (2016)Google Scholar
  13. 13.
    Doerr, B.: An elementary analysis of the probability that a binomial random variable exceeds its expectation. Stat. Probab. Lett. 139, 67–74 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Doerr, B., Doerr, C.: Optimal parameter choices through self-adjustment: applying the 1/5-th rule in discrete settings. In: Proceedings of GECCO ’15, pp. 1335–1342. ACM (2015)Google Scholar
  15. 15.
    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)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Doerr, B., Fouz, M., Witt, C.: Quasirandom evolutionary algorithms. In: Proceedings of GECCO ’10, pp. 1457–1464. ACM (2010)Google Scholar
  17. 17.
    Doerr, B., Fouz, M., Witt, C.: Sharp bounds by probability-generating functions and variable drift. In: Proceedings of GECCO ’11, pp. 2083–2090. ACM (2011)Google Scholar
  18. 18.
    Doerr, B., Johannsen, D., Winzen, C.: Multiplicative drift analysis. Algorithmica 64, 673–697 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Doerr, B., Doerr, C., Ebel, F.: From black-box complexity to designing new genetic algorithms. Theor. Comput. Sci. 567, 87–104 (2015a)MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Doerr, B., Doerr, C., Kötzing, T.: Solving problems with unknown solution length at (almost) no extra cost. In: Proceedings of GECCO ’15, pp. 831–838. ACM (2015b)Google Scholar
  21. 21.
    Doerr, B., Doerr, C., Kötzing, T.: The right mutation strength for multi-valued decision variables. In: Proceedings of GECCO ’16, pp. 1115–1122. ACM (2016a)Google Scholar
  22. 22.
    Doerr, B., Doerr, C., Kötzing, T.: Provably optimal self-adjusting step sizes for multi-valued decision variables. In: Proceedings of PPSN ’16, pp. 782–791. Springer (2016b)Google Scholar
  23. 23.
    Doerr, B., Doerr, C., Yang, J.: Optimal parameter choices via precise black-box analysis. In: Proceedings of GECCO ’16, pp. 1123–1130. ACM (2016c)Google Scholar
  24. 24.
    Doerr, B., Doerr, C., Yang, J.: \(k\)-bit mutation with self-adjusting \(k\) outperforms standard bit mutation. In: Proceedings of PPSN ’16, pp. 824–834. Springer (2016d)Google Scholar
  25. 25.
    Doerr, B., Doerr, C., Kötzing, T.: Unknown solution length problems with no asymptotically optimal run time. In: Proceedings of GECCO ’17, pp. 1367–1374. ACM (2017a)Google Scholar
  26. 26.
    Doerr, B., Gießen, C., Witt, C., Yang, J.: The (1+\(\lambda \)) evolutionary algorithm with self-adjusting mutation rate. In: Proceedings of GECCO ’17, pp. 1351–1358. ACM (2017b)Google Scholar
  27. 27.
    Doerr, B., Le, H.P., Makhmara, R., Nguyen, T.D.: Fast genetic algorithms. In: Proceedings of GECCO ’17, pp. 777–784. ACM (2017c)Google Scholar
  28. 28.
    Doerr, B., Lissovoi, A., Oliveto, P.S., Warwicker, J.A.: On the runtime analysis of selection hyper-heuristics with adaptive learning periods. In: Proceedings of GECCO ’18, pp. 1015–1022. ACM (2018a)Google Scholar
  29. 29.
    Doerr, B., Witt, C., Yang, J.: Runtime analysis for self-adaptive mutation rates. In: Proc. GECCO ’18, pp. 1475–1482. ACM (2018b)Google Scholar
  30. 30.
    Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theor. Comput. Sci. 276, 51–81 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Eiben, A.E., Hinterding, R., Michalewicz, Z.: Parameter control in evolutionary algorithms. IEEE Trans. Evolut. Comput. 3, 124–141 (1999)CrossRefGoogle Scholar
  32. 32.
    Garnier, J., Kallel, L., Schoenauer, M.: Rigorous hitting times for binary mutations. Evolut. Comput. 7, 173–203 (1999)CrossRefGoogle Scholar
  33. 33.
    Gießen, C., Witt, C.: The interplay of population size and mutation probability in the (1+\(\lambda \)) EA on OneMax. Algorithmica 78, 587–609 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Giel, O., Wegener, I.: Evolutionary algorithms and the maximum matching problem. In: Proceedings of STACS ’03, pp. 415–426. Springer (2003)Google Scholar
  35. 35.
    Hwang, H.-K., Panholzer, A., Rolin, N., Tsai, T.-H., Chen, W.-M.: Probabilistic analysis of the (1+1)-evolutionary algorithm. Evolut. Comput. 26, 299–345 (2018)CrossRefGoogle Scholar
  36. 36.
    Jansen, T.: Analyzing Evolutionary Algorithms—The Computer Science Perspective. Springer, Berlin (2013)CrossRefzbMATHGoogle Scholar
  37. 37.
    Jansen, T., Wegener, I.: On the analysis of a dynamic evolutionary algorithm. J. Discrete Algorithms 4, 181–199 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evolut. Comput. 13, 413–440 (2005)CrossRefGoogle Scholar
  39. 39.
    Johannsen, D.: Random combinatorial structures and randomized search heuristics. Ph.D. thesis, Saarland University (2010)Google Scholar
  40. 40.
    Kaas, R., Buhrman, J.M.: Mean, median and mode in binomial distributions. Stat. Neerl. 34, 13–18 (1980)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Kötzing, T., Lissovoi, A., Witt, C.: (1+1) EA on generalized dynamic OneMax. In: Proceedings of FOGA ’15, pp. 40–51. ACM (2015)Google Scholar
  42. 42.
    Lässig, J., Sudholt, D.: Adaptive population models for offspring populations and parallel evolutionary algorithms. In: Proceedings of FOGA ’11, pp. 181–192. ACM (2011)Google Scholar
  43. 43.
    Lehre, P.K., Özcan, E.: A runtime analysis of simple hyper-heuristics: to mix or not to mix operators. In: Proceedings of FOGA ’13, pp. 97–104. ACM (2013)Google Scholar
  44. 44.
    Lehre, P.K., Witt, C.: Concentrated hitting times of randomized search heuristics with variable drift. In: Proceedings of ISAAC ’14, pp. 686–697. Springer (2014)Google Scholar
  45. 45.
    Lissovoi, A., Oliveto, P.S., Warwicker, J.A.: On the runtime analysis of generalised selection hyper-heuristics for pseudo-Boolean optimisation. In: Proceedings of GECCO ’17, pp. 849–856. ACM (2017)Google Scholar
  46. 46.
    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, 243–284 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Mühlenbein, H.: How genetic algorithms really work: Mutation and hillclimbing. In: Proceedings of PPSN ’92, pp. 15–26. Elsevier (1992)Google Scholar
  48. 48.
    Neumann, F., Wegener, I.: Randomized local search, evolutionary algorithms, and the minimum spanning tree problem. Theor. Comput. Sci. 378, 32–40 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  49. 49.
    Neumann, F., Witt, C.: Bioinspired Computation in Combinatorial Optimization—Algorithms and Their Computational Complexity. Springer, Berlin (2010)zbMATHGoogle Scholar
  50. 50.
    Oliveto, P.S., Lehre, P.K., Neumann, F.: Theoretical analysis of rank-based mutation-combining exploration and exploitation. In: Proceedings of CEC ’09, pp. 1455–1462. IEEE (2009)Google Scholar
  51. 51.
    Qian, C., Tang, K., Zhou, Z-H.: Selection hyper-heuristics can provably be helpful in evolutionary multi-objective optimization. In: Proceedings of PPSN ’16, pp. 835–846. Springer (2016)Google Scholar
  52. 52.
    Robbins, H.: A remark on Stirling’s formula. Am. Math. Mon. 62, 26–29 (1955)MathSciNetzbMATHGoogle Scholar
  53. 53.
    Sudholt, D.: A new method for lower bounds on the running time of evolutionary algorithms. IEEE Trans. Evolut. Comput. 17, 418–435 (2013)CrossRefGoogle Scholar
  54. 54.
    Wegener, I.: Simulated annealing beats Metropolis in combinatorial optimization. In: Proceedings of ICALP ’05, pp. 589–601. Springer (2005)Google Scholar
  55. 55.
    Zarges, C.: Rigorous runtime analysis of inversely fitness proportional mutation rates. In: Proceedings of PPSN ’08, pp. 112–122. Springer (2008)Google Scholar
  56. 56.
    Zarges, C.: On the utility of the population size for inversely fitness proportional mutation rates. In: Proceedings of FOGA ’09, pp. 39–46. ACM (2009)Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Benjamin Doerr
    • 1
  • Christian Gießen
    • 2
  • Carsten Witt
    • 3
  • Jing Yang
    • 1
  1. 1.École Polytechnique, CNRSLaboratoire d’Informatique (LIX)PalaiseauFrance
  2. 2.Division Chassis & Safety, Advanced EngineeringContinentalLindau am BodenseeGermany
  3. 3.DTU ComputeTechnical University of DenmarkKgs. LyngbyDenmark

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