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

Jean-François Aujol (1) , Charles Dossal (2) , Văn Hào Hoàng (2) , Hippolyte Labarrière (2) , Aude Rondepierre (2, 3)

1 IMB - Institut de Mathématiques de Bordeaux
2 INSA Toulouse - Institut National des Sciences Appliquées - Toulouse
3 LAAS-ROC - Équipe Recherche Opérationnelle, Optimisation Combinatoire et Contraintes

First-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.

Mots clés: Convex optimization; Hessian-driven damping; Lyapunov analysis; Lojasiewicz property; ODEs

Domaines: Optimisation et contrôle [math.OC]

Jean-François Aujol, Charles Dossal, Văn Hào Hoàng, Hippolyte Labarrière, Aude Rondepierre. Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions. 2022. ⟨hal-03693218v2⟩