Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping

TL;DR

This paper proves the convergence of a class of accelerated first-order optimization algorithms with inertia and Hessian-driven damping in Hilbert spaces, enhancing stability and convergence speed.

Contribution

It introduces a novel discrete algorithm inspired by inertial dynamics with Hessian damping, improving convergence and oscillation attenuation in convex optimization.

Findings

Proves convergence of iterates to optimal solutions.

Shows attenuation of oscillations via Hessian-driven damping.

Connects continuous inertial dynamics to discrete algorithms.

Abstract

In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial dynamic involving both viscous damping and Hessian-driven damping. The asymptotically vanishing viscous damping is linked to the accelerated gradient method of Nesterov while the Hessian driven damping makes it possible to significantly attenuate the oscillations. By treating the Hessian-driven damping as the time derivative of the gradient term, this gives, in discretized form, first-order algorithms. These results complement the previous work of the authors where it was shown the fast convergence of the values, and the fast convergence towards zero of the gradients.