Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions

Youhei Akimoto 1 Anne Auger 2, 3, 4 Nikolaus Hansen 2, 3, 4
2 TAO - Machine Learning and Optimisation
LRI - Laboratoire de Recherche en Informatique, UP11 - Université Paris-Sud - Paris 11, Inria Saclay - Ile de France, CNRS - Centre National de la Recherche Scientifique : UMR8623
3 RANDOPT - Randomized Optimisation
Inria Saclay - Ile de France
Abstract : We investigate evolution strategies with weighted recombi-nation on general convex quadratic functions. We derive the asymptotic quality gain in the limit of the dimension to infinity, and derive the optimal recombination weights and the optimal step-size. This work is an extension of previous works where the asymptotic quality gain of evolution strategies with weighted recombination was derived on the infinite dimensional sphere function. Moreover, for a finite dimensional search space, we derive rigorous bounds for the quality gain on a general quadratic function. They reveal the dependency of the quality gain both in the eigenvalue distribution of the Hessian matrix and on the recombina-tion weights. Taking the search space dimension to infinity , it turns out that the optimal recombination weights are independent of the Hessian matrix, i.e., the recombination weights optimal for the sphere function are optimal for convex quadratic functions.
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Youhei Akimoto, Anne Auger, Nikolaus Hansen. Quality Gain Analysis of the Weighted Recombination Evolution Strategy on General Convex Quadratic Functions. Proceedings of the 14th ACM/SIGEVO Conference on Foundations of Genetic Algorithms, FOGA, Jan 2017, Copenhagen, Denmark. pp.111-126, ⟨10.1145/3040718.3040720⟩. ⟨hal-01516326⟩

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