Log-linear convergence and optimal bounds for the (1 + 1)-ES

Authors:
Mohamed Jebalia;Anne Auger;Pierre Liardet
Affiliations:
INRIA Futurs, Université Paris Sud, LRI, Orsay cedex, France;INRIA Futurs, Université Paris Sud, LRI, Orsay cedex, France;Université de Provence, UMR, CNRS, Marseille cedex 13, France
Venue:
EA'07 Proceedings of the Evolution artificielle, 8th international conference on Artificial evolution
Year:
2007

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Abstract

The (1 + 1)-ES is modeled by a general stochastic processwhose asymptotic behavior is investigated. Under general assumptions, itis shown that the convergence of the related algorithm is sub-log-linear,bounded below by an explicit log-linear rate. For the specific case ofspherical functions and scale-invariant algorithm, it is proved using theLaw of Large Numbers for orthogonal variables, that the linear convergenceholds almost surely and that the best convergence rate is reached.Experimental simulations illustrate the theoretical results.