On the codimension of the set of optima: large scale optimisation with few relevant variables

Vincent Berthier 1, 2 Olivier Teytaud 1, 2
2 TAO - Machine Learning and Optimisation
CNRS - Centre National de la Recherche Scientifique : UMR8623, Inria Saclay - Ile de France, UP11 - Université Paris-Sud - Paris 11, LRI - Laboratoire de Recherche en Informatique
Abstract : The complexity of continuous optimisation by comparison-based algorithms has been developed in several recent papers.Roughly speaking, these papers conclude that a precision can be reached with cost Θ(n log(1//)) in dimension n within polylogarithmic factors for the sphere function. Compared to other (non comparison-based) algorithms, this rate is not excellent; on the other hand, it is classically considered that comparison-based algorithms have some robustness advantages, as well as scalability on parallel machines and simplicity. In the present paper we show another advantage, namely resilience to useless variables, thanks to a complexity bound Θ(m log(1//)) where m is the codimension of the set of optima, possibly m << n. In addition, experiments show that some evolutionary algorithms have a negligible computational complexity even in high dimension, making them practical for huge problems with many useless variables.
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Vincent Berthier, Olivier Teytaud. On the codimension of the set of optima: large scale optimisation with few relevant variables. Artificial Evolution 2015, 2015, Lyon, France. To appear. ⟨hal-01194519⟩

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