Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions - Inria - Institut national de recherche en sciences et technologies du numérique Accéder directement au contenu
Pré-Publication, Document De Travail Année : 2021

Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions

Résumé

Evolution Strategies (ES) are stochastic derivative-free optimization algorithms whose most prominent representative, the CMA-ES algorithm, is widely used to solve difficult numerical optimization problems. We provide the first rigorous investigation of the linear convergence of step-size adaptive ES involving a population and recombination, two ingredients crucially important in practice to be robust to local irregularities or multimodality. Our methodology relies on investigating the stability of a Markov chain associated to the algorithm. Our stability study is crucially based on recent developments connecting the stability of deterministic control models to the stability of associated Markov chains. We investigate convergence on composites of strictly increasing functions with continuously differentiable scaling-invariant functions with a global optimum. This function class includes functions with non-convex sublevel sets and discontinuous functions. We prove the existence of a constant r such that the logarithm of the distance to the optimum divided by the number of iterations of step-size adaptive ES with weighted recombination converges to r. The constant is given as an expectation with respect to the stationary distribution of a Markov chain-its sign allows to infer linear convergence or divergence of the ES and is found numerically. Our main condition for convergence is the increase of the expected log stepsize on linear functions. In contrast to previous results, our condition is equivalent to the almost sure geometric divergence of the step-size.
Fichier principal
Vignette du fichier
saes.pdf (3.86 Mo) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)

Dates et versions

hal-03286037 , version 1 (13-07-2021)
hal-03286037 , version 2 (24-06-2022)
hal-03286037 , version 3 (22-10-2022)

Identifiants

  • HAL Id : hal-03286037 , version 1

Citer

Cheikh Touré, Anne Auger, Nikolaus Hansen. Global linear convergence of Evolution Strategies with recombination on scaling-invariant functions. 2021. ⟨hal-03286037v1⟩
130 Consultations
161 Téléchargements

Partager

Gmail Facebook X LinkedIn More