Convergence rates of the Heavy-Ball method with Lojasiewicz property

J-F Aujol , Ch Dossal (1) , Aude Rondepierre

1 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse

In this paper, a joint study of the behavior of solutions of the Heavy Ball ODE and Heavy Ball type algorithms is given. Since the pioneering work of B.T. Polyak [38], it is well known that such a scheme is very efficient for C 2 strongly convex functions with Lipschitz gradient. But much less is known when only growth conditions are considered. Depending on the geometry of the function to minimize, convergence rates for convex functions, with some additional regularity such as quasi-strong convexity, or strong convexity, were recently obtained in [12]. Convergence results with much weaker assumptions are given in the present paper: namely, linear convergence rates when assuming a growth condition (which amounts to a Lojasiewicz property in the convex case). This analysis is firstly performed in continuous time for the ODE, and then transposed for discrete optimization schemes. In particular, a variant of the Heavy Ball algorithm is proposed, which converges geometrically whatever the parameters choice, and which has the best state of the art convergence rate for first order methods to minimize composite non smooth convex functions satisfying a Lojasiewicz property.

Mots clés: Lyapunov function; rate of convergence; ODEs; optimization; Lojasiewicz prop- erty; Heavy Ball method

Domaines: Optimisation et contrôle [math.OC]

J-F Aujol, Ch Dossal, Aude Rondepierre. Convergence rates of the Heavy-Ball method with Lojasiewicz property. [Research Report] IMB - Institut de Mathématiques de Bordeaux; INSA Toulouse; UPS Toulouse. 2020. ⟨hal-02928958⟩