Combining fast inertial dynamics for convex optimization with Tikhonov regularization

TL;DR

This paper analyzes the convergence of a damped inertial system with Tikhonov regularization in Hilbert spaces, extending the understanding of accelerated convex optimization methods and their asymptotic behaviors.

Contribution

It introduces a new analysis of the convergence properties of a second-order differential equation with Tikhonov regularization, linking the rate of regularization decay to different convergence outcomes.

Findings

Rapid decay of psilon(t) leads to fast convergence of (x(t)) to min

Slow decay of psilon(t) results in strong ergodic convergence to minimal norm solutions

The damping coefficient lpha/t ensures rapid convergence of function values

Abstract

In a Hilbert space setting $\mathcal H$, we study the convergence properties as $t \to + \infty$ of the trajectories of the second-order differential equation \begin{equation*} \mbox{(AVD)}_{\alpha, \epsilon} \quad \quad \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) + \epsilon (t) x(t) =0, \end{equation*} where $\nabla\Phi$ is the gradient of a convex continuously differentiable function $\Phi: \mathcal H \to \mathbb R$, $\alpha$ is a positive parameter, and $\epsilon (t) x(t)$ is a Tikhonov regularization term, with $\lim_{t \to \infty}\epsilon (t) =0$. In this damped inertial system, the damping coefficient $\frac{\alpha}{t}$ vanishes asymptotically, but not too quickly, a key property to obtain rapid convergence of the values. In the case $\epsilon (\cdot) \equiv 0$, this dynamic has been highlighted recently by Su, Boyd, and Cand\`es as a continuous version of the Nesterov accelerated method. Depending on the speed of convergence of $\epsilon (t)$ to zero, we analyze the convergence properties of the trajectories of $\mbox{(AVD)}_{\alpha, \epsilon}$. We obtain results ranging from the rapid convergence of $\Phi (x(t))$ to $\min \Phi$ when $\epsilon (t)$ decreases rapidly to zero, up to the strong ergodic convergence of the trajectories to the element of minimal norm of the set of minimizers of $\Phi$, when $\epsilon (t)$ tends slowly to zero.