A speed restart scheme for a dynamics with Hessian driven damping

Abstract

In this paper, we analyze a speed restarting scheme for the dynamical system given by $$\ddot{x}(t)+\frac{\alpha }{t}\dot{x}(t)+\nabla \varphi (x(t))+\beta {\nabla }^2\varphi (x(t))\dot{x}(t)=0,$$ where $\alpha$ and $\beta$ are positive parameters, and $\varphi :{\mathbb{R}}^n\to \mathbb{R}$ is a smooth convex function. If $\varphi$ has quadratic growth, we establish a linear convergence rate for the function values along the restarted trajectories. As a byproduct, we improve the results obtained by Su, Boyd and Candès \cite{JMLR:v17:15-084}, obtained in the strongly convex case for $\alpha =3$ and $\beta =0$. Preliminary numerical experiments suggest that both adding a positive Hessian driven damping parameter $\beta$, and implementing the restart scheme help improve the performance of the dynamics and corresponding iterative algorithms as means to approximate minimizers of $\varphi$.