Accelerated gradient methods combining Tikhonov regularization with geometric damping driven by the Hessian

TL;DR

This paper introduces an accelerated gradient method that combines Tikhonov regularization with Hessian-driven geometric damping, achieving fast convergence and strong minimizer convergence in convex optimization.

Contribution

It develops a novel inertial dynamic with parameter tuning that leverages Hessian-driven damping to improve convergence properties in regularized convex optimization.

Findings

Achieves exponential convergence rates in strongly convex cases.

Ensures strong convergence to the minimum norm minimizer.

Attenuates oscillations through Hessian-driven geometric damping.

Abstract

In a Hilbert setting, for convex differentiable optimization, we consider accelerated gradient dynamics combining Tikhonov regularization with Hessian-driven damping. The Tikhonov regularization parameter is assumed to tend to zero as time tends to infinity, which preserves equilibria. The presence of the Tikhonov regularization term induces a strong convexity property which vanishes asymptotically. To take advantage of the exponential convergence rates attached to the heavy ball method in the strongly convex case, we consider the inertial dynamic where the viscous damping coefficient is taken proportional to the square root of the Tikhonov regularization parameter, and therefore also converges towards zero. Moreover, the dynamic involves a geometric damping which is driven by the Hessian of the function to be minimized, which induces a significant attenuation of the oscillations. Under an appropriate tuning of the parameters, based on Lyapunov's analysis, we show that the trajectories have at the same time several remarkable properties: they provide fast convergence of values, fast convergence of gradients towards zero, and strong convergence to the minimum norm minimizer. This study extends a previous paper by the authors where similar issues were examined but without the presence of Hessian driven damping.