OPTIMAL CONVERGENCE RATES FOR NESTEROV ACCELERATION

Jean François Aujol (1) , Charles H Dossal (2) , Aude Rondepierre (2, 3)

1 IMB - Institut de Mathématiques de Bordeaux
2 IMT - Institut de Mathématiques de Toulouse UMR5219
3 LAAS-ROC - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes

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(t 2) is optimal for the class of convex functions. In this work, we show that better convergence rates can be obtained with some additional geometrical conditions, such as Lojasiewicz 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.

Mots clés: Lyapunov functions; rate of convergence; ODEs; optimization; Lojasiewicz property

Domaines: Optimisation et contrôle [math.OC]

Jean François Aujol, Charles H Dossal, Aude Rondepierre. OPTIMAL CONVERGENCE RATES FOR NESTEROV ACCELERATION. 2018. ⟨hal-01786117v2⟩