# Noisy Optimization: Fast Convergence Rates with Comparison-Based Algorithms

1 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
Abstract : Derivative Free Optimization is known to be an efficient and robust method to tackle the black-box optimization problem. When it comes to noisy functions, classical comparison-based algorithms are slower than gradient-based algorithms. For quadratic functions, Evolutionary Algorithms without large mutations have a simple regret at best $O(1/ \sqrt{N})$ when $N$ is the number of function evaluations, whereas stochastic gradient descent can reach (tightly) a simple regret in $O(1/N)$. It has been conjectured that gradient approximation by finite differences (hence, not a comparison-based method) is necessary for reaching such a $O(1/N)$. We answer this conjecture in the negative, providing a comparison-based algorithm as good as gradient methods, i.e. reaching $O(1/N)$ - under the condition, however, that the noise is Gaussian. Experimental results confirm the $O(1/N)$ simple regret, i.e., squared rate compared to many published results at $O(1/\sqrt{N})$.
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Communication dans un congrès
T. Friedrich and F. Neumann. Genetic and Evolutionary Computation Conference, Jul 2016, Denver, United States. pp.1101-1106, 2016, Proc. ACM-GECCO'16. 〈http://gecco-2016.sigevo.org/index.html/HomePage#&panel1-1〉

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https://hal.archives-ouvertes.fr/hal-01306636
Contributeur : Marie-Liesse Cauwet <>
Soumis le : mercredi 27 avril 2016 - 09:39:41
Dernière modification le : jeudi 5 avril 2018 - 12:30:12
Document(s) archivé(s) le : mardi 15 novembre 2016 - 14:13:58

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• HAL Id : hal-01306636, version 2
• ARXIV : 1604.08459

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Marie-Liesse Cauwet, Olivier Teytaud. Noisy Optimization: Fast Convergence Rates with Comparison-Based Algorithms. T. Friedrich and F. Neumann. Genetic and Evolutionary Computation Conference, Jul 2016, Denver, United States. pp.1101-1106, 2016, Proc. ACM-GECCO'16. 〈http://gecco-2016.sigevo.org/index.html/HomePage#&panel1-1〉. 〈hal-01306636v2〉

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