A speed restart scheme for a dynamics with Hessian driven damping
In this paper, we analyze a speed restarting scheme for the dynamical system given by $$ \ddot{x}(t) + \dfracα{t}\dot{x}(t) + \nabla ϕ(x(t)) + β\nabla^2 ϕ(x(t))\dot{x}(t)=0, $$ where $α$ and $β$ are positive parameters, and $ϕ:\mathbb{R}^n \to \mathbb{R}$ is a smooth convex function. If $ϕ$ 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 $α=3$ and $β=0$. Preliminary numerical experiments suggest that both adding a positive Hessian driven damping parameter $β$, and implementing the restart scheme help improve the performance of the dynamics and corresponding iterative algorithms as means to approximate minimizers of $ϕ$.