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

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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]

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Document de travail - Pré-publication

Resumen en inglés

In this paper, we study the behavior of solutions of the ODE associated to the Heavy Ball method. Since the pioneering work of B.T. Polyak [25], it is well known that such a scheme is very efficient for C2 strongly convex functions with Lipschitz gradient. But much less is known when the C2 assumption is dropped. Depending on the geometry of the function to minimize, we obtain optimal convergence rates for the class of convex functions with some additional regularity such as quasi-strong convexity or strong convexity. We perform this analysis in continuous time for the ODE, and then we transpose these results for discrete optimization schemes. In particular, we propose a variant of the Heavy Ball algorithm which has the best state of the art convergence rate for first order methods to minimize strongly, composite non smooth convex functions.< Leer menos

Palabras clave en inglés

Lyapunov function

rate of convergence

ODEs

optimization

strong convexity

Heavy Ball method

URI

https://oskar-bordeaux.fr/handle/20.500.12278/191610

Proyecto ANR

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

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Centros de investigación

Institut de Mathématiques de Bordeaux (IMB) - UMR 5251