AUJOL, Jean-François; DOSSAL, Charles; HOÀNG, Văn Hào; LABARRIÈRE, Hippolyte; RONDEPIERRE, Aude

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

DOSSAL, Charles
Institut National des Sciences Appliquées - Toulouse [INSA Toulouse]

HOÀNG, Văn Hào
Institut National des Sciences Appliquées - Toulouse [INSA Toulouse]

Langue

Document de travail - Pré-publication

Résumé en anglais

First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs). In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al.: ∀t ⩾ t0, ẍ(t) + α t ẋ(t) + βHF (x(t)) ẋ(t) + ∇F (x(t)) = 0, where α > 0, β > 0 and HF denotes the Hessian of F. This ODE can be derived to build numerical schemes which do not require F to be twice differentiable as shown by Attouch et al. We provide strong convergence results on the error F (x(t)) − F * and integrability properties on ∥∇F (x(t))∥ under some geometry assumptions on F such as quadratic growth around the set of minimizers. In particular, we show that the decay rate of the error for a strongly convex function is O(t −α−ε) for any ε > 0. These results are briefly illustrated at the end of the paper.< Réduire

Mots clés en anglais

Convex optimization

Hessian-driven damping

Lyapunov analysis

Lojasiewicz property

ODEs

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191128

Project ANR

Mathématiques de l'optimisation déterministe et stochastique liées à l'apprentissage profond - ANR-19-CE23-0017

Origine

Importé de hal

Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251