Damped inertial dynamics with vanishing Tikhonov regularization: strong asymptotic convergence towards the minimum norm solution

TL;DR

This paper introduces a damped inertial dynamic method with vanishing Tikhonov regularization in Hilbert spaces, demonstrating strong convergence to the minimum norm solution of convex minimization problems with explicit convergence rates.

Contribution

It develops a novel dynamic approach combining damping and Tikhonov regularization, ensuring strong convergence to the minimum norm solution under optimal parameter tuning.

Findings

Trajectories strongly converge to the minimum norm solution.

Convergence rates of the function values are explicitly derived.

Trade-off between fast convergence and strong convergence is characterized.

Abstract

In a Hilbert space, we provide a fast dynamic approach to the hierarchical minimization problem which consists in finding the minimum norm solution of a convex minimization problem. For this, we study the convergence properties of the trajectories generated by a damped inertial dynamic with Tikhonov regularization. When the time goes to infinity, the Tikhonov regularization parameter is supposed to tend towards zero, not too fast, which is a key property to make the trajectories strongly converge towards the minimizer of $f$ of minimum norm. According to the structure of the heavy ball method for strongly convex functions, the viscous damping coefficient is proportional to the square root of the Tikhonov regularization parameter. Therefore, it also converges to zero, which will ensure rapid convergence of values. Precisely, under a proper tuning of these parameters, based on Lyapunov's analysis, we show that the trajectories strongly converge towards the minimizer of minimum norm, and we provide the convergence rate of the values. We show a trade off between the property of fast convergence of values, and the property of strong convergence towards the minimum norm solution. This study improves several previous works where this type of results was obtained under restrictive hypotheses.