AUJOL, Jean-François; DOSSAL, Charles; 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]
Institut de Mathématiques de Toulouse UMR5219 [IMT]

LABARRIÈRE, Hippolyte
Università degli studi di Genova = University of Genoa [UniGe]
Dipartimento di Informatica, Bioingegneria, Robotica e Ingegneria dei Sistemi [Genova] [DIBRIS]

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Document de travail - Pré-publication

Ce document a été publié dans

2024-03-11

Résumé en anglais

When considering the minimization of a quadratic or strongly convex function, it is well known that first-order methods involving an inertial term weighted by a constant-in-time parameter are particularly efficient (see Polyak [32], Nesterov [28], and references therein). By setting the inertial parameter according to the condition number of the objective function, these methods guarantee a fast exponential decay of the error. We prove that this type of schemes (which are later called Heavy Ball schemes) is relevant in a relaxed setting, i.e. for composite functions satisfying a quadratic growth condition. In particular, we adapt V-FISTA, introduced by Beck in [10] for strongly convex functions, to this broader class of functions. To the authors' knowledge, the resulting worst-case convergence rates are faster than any other in the literature, including those of FISTA restart schemes. No assumption on the set of minimizers is required and guarantees are also given in the non-optimal case, i.e. when the condition number is not exactly known. This analysis follows the study of the corresponding continuous-time dynamical system (Heavy Ball with friction system), for which new convergence results of the trajectory are shown.< Réduire

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https://oskar-bordeaux.fr/handle/20.500.12278/190207

Project ANR

Problèmes inverses aveugles et microscopie optique - ANR-21-CE48-0008
Mathématiques de l'optimisation déterministe et stochastique liées à l'apprentissage profond - ANR-19-CE23-0017
Numerical analysis, optimal control and optimal transport for AI - ANR-23-PEIA-0004

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Unités de recherche

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251