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Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions

hal.structure.identifierInstitut de Mathématiques de Bordeaux [IMB]
dc.contributor.authorAUJOL, Jean-François
hal.structure.identifierInstitut National des Sciences Appliquées - Toulouse [INSA Toulouse]
dc.contributor.authorDOSSAL, Charles
hal.structure.identifierInstitut National des Sciences Appliquées - Toulouse [INSA Toulouse]
dc.contributor.authorHOÀNG, Văn Hào
hal.structure.identifierInstitut National des Sciences Appliquées - Toulouse [INSA Toulouse]
dc.contributor.authorLABARRIÈRE, Hippolyte
hal.structure.identifierInstitut National des Sciences Appliquées - Toulouse [INSA Toulouse]
hal.structure.identifierÉquipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes [LAAS-ROC]
dc.contributor.authorRONDEPIERRE, Aude
dc.date.accessioned2024-04-04T02:41:11Z
dc.date.available2024-04-04T02:41:11Z
dc.identifier.urihttps://oskar-bordeaux.fr/handle/20.500.12278/191128
dc.description.abstractEnFirst-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.
dc.description.sponsorshipMathématiques de l'optimisation déterministe et stochastique liées à l'apprentissage profond - ANR-19-CE23-0017
dc.language.isoen
dc.subject.enConvex optimization
dc.subject.enHessian-driven damping
dc.subject.enLyapunov analysis
dc.subject.enLojasiewicz property
dc.subject.enODEs
dc.title.enFast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions
dc.typeDocument de travail - Pré-publication
dc.subject.halMathématiques [math]/Optimisation et contrôle [math.OC]
dc.identifier.arxiv2206.06853
bordeaux.hal.laboratoriesInstitut de Mathématiques de Bordeaux (IMB) - UMR 5251*
bordeaux.institutionUniversité de Bordeaux
bordeaux.institutionBordeaux INP
bordeaux.institutionCNRS
hal.identifierhal-03693218
hal.version1
hal.origin.linkhttps://hal.archives-ouvertes.fr//hal-03693218v1
bordeaux.COinSctx_ver=Z39.88-2004&rft_val_fmt=info:ofi/fmt:kev:mtx:journal&rft.au=AUJOL,%20Jean-Fran%C3%A7ois&DOSSAL,%20Charles&HO%C3%80NG,%20V%C4%83n%20H%C3%A0o&LABARRI%C3%88RE,%20Hippolyte&RONDEPIERRE,%20Aude&rft.genre=preprint


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