AUJOL, Jean François; DOSSAL, Charles; RONDEPIERRE, Aude

doi:10.1137/18M1186757

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AUJOL, Jean François
Institut de Mathématiques de Bordeaux [IMB]

DOSSAL, Charles
Institut de Mathématiques de Toulouse UMR5219 [IMT]

RONDEPIERRE, Aude
Institut de Mathématiques de Toulouse UMR5219 [IMT]
Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes [LAAS-ROC]

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Article de revue

Ce document a été publié dans

SIAM Journal on Optimization. 2019-12, vol. 29, n° 4, p. 3131–3153

Society for Industrial and Applied Mathematics

Résumé en anglais

In this paper, we study the behavior of solutions of the ODE associated to Nesterov acceleration. It is well-known since the pioneering work of Nesterov that the rate of convergence $O(1/t^2)$ is optimal for the class of convex functions with Lipschitz gradient. In this work, we show that better convergence rates can be obtained with some additional geometrical conditions, such as \L ojasiewicz property. More precisely, we prove the optimal convergence rates that can be obtained depending on the geometry of the function $F$ to minimize. The convergence rates are new, and they shed new light on the behavior of Nesterov acceleration schemes. We prove in particular that the classical Nesterov scheme may provide convergence rates that are worse than the classical gradient descent scheme on sharp functions: for instance, the convergence rate for strongly convex functions is not geometric for the classical Nesterov scheme (while it is the case for the gradient descent algorithm). This shows that applying the classical Nesterov acceleration on convex functions without looking more at the geometrical properties of the objective functions may lead to sub-optimal algorithms.< Réduire

Mots clés en anglais

Lyapunov functions

rate of convergence

Lojasiewicz property

ODEs

optimization

URI

https://oskar-bordeaux.fr/handle/20.500.12278/192830

DOI

http://dx.doi.org/10.1137/18M1186757

Project ANR

Centre International de Mathématiques et d'Informatique (de Toulouse) - ANR-11-LABX-0040
Université Fédérale de Toulouse - ANR-11-IDEX-0002

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251