Article

Optimizing monotone functions can be difficult

Authors:

Benjamin Doerr,

Christine ZargesAuthors Info & Claims

PPSN'10: Proceedings of the 11th international conference on Parallel problem solving from nature: Part I

Pages 42 - 51

Published: 11 September 2010 Publication History

Abstract

Extending previous analyses on function classes like linear functions, we analyze how the simple (1+1) evolutionary algorithm optimizes pseudo-Boolean functions that are strictly monotone. Contrary to what one would expect, not all of these functions are easy to optimize. The choice of the constant c in the mutation probability p(n) = c/n can make a decisive difference.

We show that if c > 1, then the (1+1) EA finds the optimum of every such function in Θ(n log n) iterations. For c = 1, we can still prove an upper bound of O(n^3/2). However, for c > 33, we present a strictly monotone function such that the (1+1) EA with overwhelming probability does not find the optimum within 2^Ω(n) iterations. This is the first time that we observe that a constant factor change of the mutation probability changes the run-time by more than constant factors.

References

[1]

Mühlenbein, H.: How genetic algorithms really work. Mutation and hillclimbing. In: Proceedings of PPSN II, pp. 15-25. North-Holland, Amsterdam (1992)

[2]

Droste, S., Jansen, T., Wegener, I.: On the optimization of unimodal functions with the (1+1) evolutionary algorithm. In: Eiben, A.E., Bäck, T., Schoenauer, M., Schwefel, H.-P. (eds.) PPSN 1998. LNCS, vol. 1498, pp. 13-22. Springer, Heidelberg (1998)

Digital Library

[3]

Horn, J., Goldberg, D., Deb, K.: Long path problems. In: Davidor, Y., Männer, R., Schwefel, H.-P. (eds.) PPSN 1994. LNCS, vol. 866, pp. 149-158. Springer, Heidelberg (1994)

Digital Library

[4]

Igel, C., Toussaint, M.: A no-free-lunch theorem for non-uniform distributions of target functions. J. of Mathematical Modelling and Algorithms 3, 313-322 (2004)

[5]

Oliveto, P., He, J., Yao, X.: Time complexity of evolutionary algorithms for combinatorial optimization: A decade of results. International Journal of Automation and Computing 4, 281-293

[6]

Droste, S., Jansen, T., Wegener, I.: On the analysis of the (1+1) evolutionary algorithm. Theoretical Computer Science 276, 51-81 (2002)

Digital Library

[7]

He, J., Yao, X.: Drift analysis and average time complexity of evolutionary algorithms. Artificial Intelligence 127, 57-85 (2001)

Digital Library

[8]

He, J., Yao, X.: Erratum to {7}. Artificial Intelligence 140, 245-248 (2002)

Digital Library

[9]

Jägersküpper, J.: A blend of Markov-chain and drift analysis. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 41-51. Springer, Heidelberg (2008)

Digital Library

[10]

Doerr, B., Fouz, M., Witt, C.: Quasirandom evolutionary algorithms. In: Proceedings of GECCO 2010, pp. 1457-1464. ACM, New York (2010)

Digital Library

[11]

Doerr, B., Johannsen, D., Winzen, C.: Drift analysis and linear functions revisited. In: Proceedings of CEC 2010, pp. 1967-1974. IEEE, Los Alamitos (2010)

[12]

Doerr, B., Goldberg, L.: Adaptive drift analysis. In: Schaefer, R., et al. (eds.) PPSN XI, Part I. LNCS, vol. 6238, pp. 32-41. Springer, Heidelberg (2010)

Digital Library

[13]

Jansen, T., Wegener, I.: On the choice of the mutation probability for the (1+1) EA. In: Deb, K., Rudolph, G., Lutton, E., Merelo, J.J., Schoenauer, M., Schwefel, H.- P., Yao, X. (eds.) PPSN 2000. LNCS, vol. 1917, pp. 89-98. Springer, Heidelberg (2000)

Digital Library

[14]

Bäck, T., Fogel, D., Michalewicz, Z. (eds.): Handbook of Evolutionary Computation. Oxford University Press, Oxford (1997)

[15]

Ochoa, G.: Setting the mutation rate: Scope and limitations of the 1/L heuristic. In: Proceedings of GECCO 2002, pp. 495-502. Morgan Kaufmann, San Francisco (2002)

Digital Library

[16]

Cervantes, J., Stephens, C.R.: Limitations of existing mutation rate heuristics and how a rank GA overcomes them. IEEE Trans. on Evol. Comp. 13, 369-397 (2009)

Digital Library

[17]

Jansen, T.: On the brittleness of evolutionary algorithms. In: Stephens, C.R., Toussaint, M., Whitley, L.D., Stadler, P.F. (eds.) FOGA 2007. LNCS, vol. 4436, pp. 54-69. Springer, Heidelberg (2007)

Digital Library

[18]

Alon, N., Spencer, J.H.: The Probabilistic Method, 2nd edn. Wiley, Chichester (2000)

[19]

Mitzenmacher, M., Upfal, E.: Probability and Computing: Randomized Algorithms and Probabilistic Analysis. Cambridge University Press, Cambridge (2005)

Digital Library

[20]

Oliveto, P.S., Witt, C.: Simplified drift analysis for proving lower bounds in evolutionary computation. Algorithmica (2010), http://dx.doi.org/10.1007/s00453-010-9387-z

Digital Library

Cited By

Lengler JMartinsson ASteger AAuger AStützle T(2019)The (1 + 1)-EA with mutation rate (1 + ϵ)/n is efficient on monotone functionsProceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3319619.3326767(25-26)Online publication date: 13-Jul-2019
https://dl.acm.org/doi/10.1145/3319619.3326767
Lengler JZou XFriedrich TDoerr CArnold D(2019)Exponential slowdown for larger populationsProceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms10.1145/3299904.3340309(87-101)Online publication date: 27-Aug-2019
https://dl.acm.org/doi/10.1145/3299904.3340309
Doerr BGoldberg L(2013)Adaptive Drift AnalysisAlgorithmica10.1007/s00453-011-9585-365:1(224-250)Online publication date: 1-Jan-2013
https://dl.acm.org/doi/10.1007/s00453-011-9585-3
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Published In

cover image Guide Proceedings

PPSN'10: Proceedings of the 11th international conference on Parallel problem solving from nature: Part I

September 2010

741 pages

ISBN:3642158439

Editors:
Robert Schaefer
AGH University of Science and Technology, Faculty of Electrical Engineering, Automatics, Computer Science and Electronics, Department of Computer Science, Kraków, Poland
,
Carlos Cotta
Universidad de Málaga, Departamento Lenguajes y Ciencias de la Computación, ETSI Informática, Málaga, Spain
,
Joanna Kołodziej
University of Bielsko-Biala, Department of Mathematics and Computer Science, Bielsko-Biala, Poland
,
Günter Rudolph
Technische Universität Dortmund, Fakultät für Informatik, Dortmund, Germany

Sponsors

Hewlett-Packard Polska
Microsoft: Microsoft
Intel: Intel

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Springer-Verlag

Berlin, Heidelberg

Publication History

Published: 11 September 2010

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Cited By

Lengler JMartinsson ASteger AAuger AStützle T(2019)The (1 + 1)-EA with mutation rate (1 + ϵ)/n is efficient on monotone functionsProceedings of the Genetic and Evolutionary Computation Conference Companion10.1145/3319619.3326767(25-26)Online publication date: 13-Jul-2019
https://dl.acm.org/doi/10.1145/3319619.3326767
Lengler JZou XFriedrich TDoerr CArnold D(2019)Exponential slowdown for larger populationsProceedings of the 15th ACM/SIGEVO Conference on Foundations of Genetic Algorithms10.1145/3299904.3340309(87-101)Online publication date: 27-Aug-2019
https://dl.acm.org/doi/10.1145/3299904.3340309
Doerr BGoldberg L(2013)Adaptive Drift AnalysisAlgorithmica10.1007/s00453-011-9585-365:1(224-250)Online publication date: 1-Jan-2013
https://dl.acm.org/doi/10.1007/s00453-011-9585-3
Minku LSudholt DYao XMoore J(2012)Evolutionary algorithms for the project scheduling problemProceedings of the 14th annual conference on Genetic and evolutionary computation10.1145/2330163.2330332(1221-1228)Online publication date: 7-Jul-2012
https://dl.acm.org/doi/10.1145/2330163.2330332
Doerr BJohannsen DWinzen C(2012)Multiplicative Drift AnalysisAlgorithmica10.1007/s00453-012-9622-x64:4(673-697)Online publication date: 1-Dec-2012
https://dl.acm.org/doi/10.1007/s00453-012-9622-x
Sutton AWhitley DHowe ALanzi P(2011)Mutation rates of the (1+1)-EA on pseudo-boolean functions of bounded epistasisProceedings of the 13th annual conference on Genetic and evolutionary computation10.1145/2001576.2001709(973-980)Online publication date: 12-Jul-2011
https://dl.acm.org/doi/10.1145/2001576.2001709
Cathabard SLehre PYao XBeyer HLangdon W(2011)Non-uniform mutation rates for problems with unknown solution lengthsProceedings of the 11th workshop proceedings on Foundations of genetic algorithms10.1145/1967654.1967670(173-180)Online publication date: 5-Jan-2011
https://dl.acm.org/doi/10.1145/1967654.1967670
Doerr BGoldberg L(2010)Adaptive drift analysisProceedings of the 11th international conference on Parallel problem solving from nature: Part I10.5555/1885031.1885036(32-41)Online publication date: 11-Sep-2010
https://dl.acm.org/doi/10.5555/1885031.1885036

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