Abstract
We propose a new black-box complexity model for search algorithms evaluating λ search points in parallel. The parallel unbiased black-box complexity gives lower bounds on the number of function evaluations every parallel unbiased black-box algorithm needs to optimise a given problem. It captures the inertia caused by offspring populations in evolutionary algorithms and the total computational effort in parallel metaheuristics. Our model applies to all unary variation operators such as mutation or local search. We present lower bounds for the LeadingOnes function and general lower bound for all functions with a unique optimum that depend on the problem size and the degree of parallelism, λ. The latter is tight for OneMax; we prove that a (1+λ) EA with adaptive mutation rates is an optimal parallel unbiased black-box algorithm.
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References
Chvátal, V.: The tail of the hypergeometric distribution. Discrete Math. 25(3), 285–287 (1979)
Doerr, B., Doerr, C., Ebel, F.: Lessons from the black-box: fast crossover-based genetic algorithms. In: Proc. of GECCO 2013, pp. 781–788. ACM (2013)
Doerr, B., Johannsen, D., Kötzing, T., Lehre, P.K., Wagner, M., Winzen, C.: Faster black-box algorithms through higher arity operators. In: Proc. of FOGA 2011, pp. 163–172. ACM (2011)
Doerr, B., Winzen, C.: Towards a complexity theory of randomized search heuristics: Ranking-based black-box complexity. In: Kulikov, A., Vereshchagin, N. (eds.) CSR 2011. LNCS, vol. 6651, pp. 15–28. Springer, Heidelberg (2011)
Doerr, B., Winzen, C.: Playing Mastermind with Constant-Size Memory. Theory of Computing Systems (2012)
Droste, S., Jansen, T., Wegener, I.: Upper and lower bounds for randomized search heuristics in black-box optimization. Theory of Computing Systems 39(4), 525–544 (2006)
He, J., Chen, T., Yao, X.: Average drift analysis and its application. CoRR, abs/1308.3080 (2013)
He, J., Yao, X.: A Study of Drift Analysis for Estimating Computation Time of Evolutionary Algorithms. Natural Computing 3(1), 21–35 (2004)
Jansen, T., De Jong, K.A., Wegener, I.: On the choice of the offspring population size in evolutionary algorithms. Evolutionary Computation 13, 413–440 (2005)
Johannsen, D.: Random Combinatorial Structures and Randomized Search Heuristics. PhD thesis, Universität des Saarlandes, Saarbrücken, Germany and the Max-Planck-Institut für Informatik (2010)
Lässig, J., Sudholt, D.: General upper bounds on the running time of parallel evolutionary algorithms. Evolutionary Computation (in press), http://www.mitpressjournals.org/doi/pdf/10.1162/EVCO_a_00114
Lässig, J., Sudholt, D.: Analysis of speedups in parallel evolutionary algorithms for combinatorial optimization. In: Asano, T., Nakano, S.-i., Okamoto, Y., Watanabe, O. (eds.) ISAAC 2011. LNCS, vol. 7074, pp. 405–414. Springer, Heidelberg (2011)
Lehre, P.K., Witt, C.: Black-box search by unbiased variation. Algorithmica 64(4), 623–642 (2012)
Luque, G., Alba, E.: Parallel Genetic Algorithms–Theory and Real World Applications. Springer (2011)
Mambrini, A., Sudholt, D., Yao, X.: Homogeneous and heterogeneous island models for the set cover problem. In: Coello, C.A.C., Cutello, V., Deb, K., Forrest, S., Nicosia, G., Pavone, M. (eds.) PPSN 2012, Part I. LNCS, vol. 7491, pp. 11–20. Springer, Heidelberg (2012)
Rowe, J.E., Sudholt, D.: The choice of the offspring population size in the (1,λ) EA. In: Proc. of GECCO 2012, pp. 1349–1356 (2012)
Rowe, J.E., Vose, M.D.: Unbiased black box search algorithms. In: Proc. of GECCO 2011, p. 2035. ACM, New York (2011)
Teytaud, O., Gelly, S.: General lower bounds for evolutionary algorithms. In: Runarsson, T.P., Beyer, H.-G., Burke, E.K., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds.) PPSN 2006. LNCS, vol. 4193, pp. 21–31. Springer, Heidelberg (2006)
Zarges, C.: Rigorous runtime analysis of inversely fitness proportional mutation rates. In: Rudolph, G., Jansen, T., Lucas, S., Poloni, C., Beume, N. (eds.) PPSN 2008. LNCS, vol. 5199, pp. 112–122. Springer, Heidelberg (2008)
Zarges, C.: On the utility of the population size for inversely fitness proportional mutation rates. In: Proc. of FOGA 2009, pp. 39–46. ACM (2009)
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Badkobeh, G., Lehre, P.K., Sudholt, D. (2014). Unbiased Black-Box Complexity of Parallel Search. In: Bartz-Beielstein, T., Branke, J., Filipič, B., Smith, J. (eds) Parallel Problem Solving from Nature – PPSN XIII. PPSN 2014. Lecture Notes in Computer Science, vol 8672. Springer, Cham. https://doi.org/10.1007/978-3-319-10762-2_88
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DOI: https://doi.org/10.1007/978-3-319-10762-2_88
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