Abstract
We examine the evidence for the widespread belief that heavy tail distributions enhance the search for minima on multimodal objective functions. We analyze isotropic and anisotropic heavy-tail Cauchy distributions and investigate the probability to sample a better solution, depending on the step length and the dimensionality of the search space. The probability decreases fast with increasing step length for isotropic Cauchy distributions and moderate search space dimension. The anisotropic Cauchy distribution maintains a large probability for sampling large steps along the coordinate axes, resulting in an exceptionally good performance on the separable multimodal Rastrigin function. In contrast, on a non-separable rotated Rastrigin function or for the isotropic Cauchy distribution the performance difference to a Gaussian search distribution is negligible.
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References
Frahm, G., Junker, M.: Generalized elliptical distributions: Models and estimation. Caesar preprint, IDL 0037 (2003)
Obuchowicz, A.: Multidimensional mutations in evolutionary algorithms based on real-valued representation. International Journal of Systems Science 34(7), 469–483 (2003)
Rechenberg, I.: Evolutionsstrategie 1994, Frommann-Holzboog, Stuttgart, Germany (1994)
Rowe, J.E., Hidovic, D.: An evolution strategy using a continuous version of the gray-code neighbourhood distribution. In: Deb, K., et al. (eds.) GECCO 2004. LNCS, vol. 3102, pp. 725–736. Springer, Heidelberg (2004)
Rudolph, G.: Local convergence rates of simple evolutionary algorithms with cauchy mutations. IEEE Trans. Evolutionary Computation 1(4), 249–258 (1997)
Szu, H., Hartley, R.: Fast simulated annealing. Phys. Lett. A 122(3,4), 157–162 (1987)
Yao, X., Liu, Y.: Evolutionary programming made faster. IEEE Transactions on Evolutionary Computation 3, 82–102 (1997)
Yao, X., Liu, Y.: Fast evolution strategies. Control and Cybernetics 26(3), 467–496 (1997)
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© 2006 Springer-Verlag Berlin Heidelberg
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Hansen, N., Gemperle, F., Auger, A., Koumoutsakos, P. (2006). When Do Heavy-Tail Distributions Help?. In: Runarsson, T.P., Beyer, HG., Burke, E., Merelo-Guervós, J.J., Whitley, L.D., Yao, X. (eds) Parallel Problem Solving from Nature - PPSN IX. PPSN 2006. Lecture Notes in Computer Science, vol 4193. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11844297_7
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DOI: https://doi.org/10.1007/11844297_7
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-38990-3
Online ISBN: 978-3-540-38991-0
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