Abstract
The (1 + 1)-ES is modeled by a general stochastic process whose asymptotic behavior is investigated. Under general assumptions, it is shown that the convergence of the related algorithm is sub-log-linear, bounded below by an explicit log-linear rate. For the specific case of spherical functions and scale-invariant algorithm, it is proved using the Law of Large Numbers for orthogonal variables, that the linear convergence holds almost surely and that the best convergence rate is reached. Experimental simulations illustrate the theoretical results.
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References
Rechenberg, I.: Evolutionstrategie: Optimierung Technischer Systeme nach Prinzipien der Biologischen Evolution. Frommann-Holzboog Verlag, Stuttgart (1973)
Kern, S., Müller, S., Hansen, N., Büche, D., Ocenasek, J., Koumoutsakos, P.: Learning Probability Distributions in Continuous Evolutionary Algorithms - A Comparative Review. Natural Computing 3, 77–112 (2004)
Hansen, N., Ostermeier, A.: Completely derandomized self-adaptation in evolution strategies. Evolutionary Computation 9, 159–195 (2001)
Beyer, H.G.: The Theory of Evolution Strategies. Springer, Heidelberg (2001)
Bienvenüe, A., François, O.: Global convergence for evolution strategies in spherical problems: some simple proofs and difficulties. Theoretical Computer Science 306, 269–289 (2003)
Auger, A.: Convergence results for (1,λ)-SA-ES using the theory of ϕ-irreducible Markov chains. Theoretical Computer Science 334, 35–69 (2005)
Auger, A., Hansen, N.: Reconsidering the progress rate theory for evolution strategies in finite dimensions. In: Proceedings of the Genetic and Evolutionary Computation Conference (GECCO 2006), pp. 445–452 (2006)
Teytaud, O., Gelly, S.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)
Jägersküpper, J.: Lower bounds for hit-and-run direct search. In: Hromkovič, J., Královič, R., Nunkesser, M., Widmayer, P. (eds.) SAGA 2007. LNCS, vol. 4665, pp. 118–129. Springer, Heidelberg (2007)
Loève, M.: Probability Theory, 3rd edn. Van Nostrand, New York (1963)
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Jebalia, M., Auger, A., Liardet, P. (2008). Log-Linear Convergence and Optimal Bounds for the (1 + 1)-ES. In: Monmarché, N., Talbi, EG., Collet, P., Schoenauer, M., Lutton, E. (eds) Artificial Evolution. EA 2007. Lecture Notes in Computer Science, vol 4926. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-79305-2_18
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DOI: https://doi.org/10.1007/978-3-540-79305-2_18
Publisher Name: Springer, Berlin, Heidelberg
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