Rate of convergence of the Nesterov accelerated gradient method in the subcritical case $α\leq 3$

Abstract: In a Hilbert space setting $\mathcal H$, given $\Phi: \mathcal H \to \mathbb R$ a convex continuously differentiable function, and $\alpha$ a positive parameter, we consider the inertial system with Asymptotic Vanishing Damping \begin{equation*} \mbox{(AVD)}{\alpha} \quad \quad \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \nabla \Phi (x(t)) =0. \end{equation*} Depending on the value of $ \alpha $ with respect to 3, we give a complete picture of the convergence properties as $t \to + \infty$ of the trajectories generated by $\mbox{(AVD)}{\alpha}$, as well as iterations of the corresponding algorithms. Our main result concerns the subcritical case $\alpha \leq 3$, where we show that $\Phi (x(t))-\min \Phi = \mathcal O (t^{-\frac{2}{3}\alpha})$. Then we examine the convergence of trajectories to optimal solutions. As a new result, in the one-dimensional framework, for the critical value $\alpha = 3 $, we prove the convergence of the trajectories without any restrictive hypothesis on the convex function $\Phi $. In the second part of this paper, we study the convergence properties of the associated forward-backward inertial algorithms. They aim to solve structured convex minimization problems of the form $\min \left\lbrace \Theta:= \Phi + \Psi \right\rbrace$, with $\Phi$ smooth and $\Psi$ nonsmooth. The continuous dynamics serves as a guideline for this study. We obtain a similar rate of convergence for the sequence of iterates $(x_k)$: for $\alpha \leq 3$ we have $\Theta (x_k)-\min \Theta = \mathcal O (k^{-p})$ for all $p <\frac{2\alpha}{3}$ , and for $\alpha > 3$ \ $\Theta (x_k)-\min \Theta = o (k^{-2})$. We conclude this study by showing that the results are robust with respect to external perturbations.