Fast convex optimization via inertial dynamics with Hessian driven damping

We first study the fast minimization properties of the trajectories of the second-order evolution equation \begin{equation*} \ddot{x}(t) + \frac{\alpha}{t} \dot{x}(t) + \beta \nabla^2 \Phi (x(t))\dot{x} (t) + \nabla \Phi (x(t)) = 0, \end{equation*} where $\Phi : \mathcal H \to \mathbb R$ is a smooth convex function acting on a real Hilbert space $\mathcal H$, and $\alpha$, $\beta$ are positive parameters. This inertial system combines an isotropic viscous damping which vanishes asymptotically, and a geometrical Hessian driven damping, which makes it naturally related to Newton's and Levenberg-Marquardt methods. For $\alpha\geq 3$, and $\beta >0$, along any trajectory, fast convergence of the values \begin{align*} \Phi(x(t))- \min_{\mathcal H}\Phi =\mathcal O\left(t^{-2}\right) \end{align*} is obtained, together with rapid convergence of the gradients $\nabla \Phi (x(t))$ to zero. For $\alpha > 3$, just assuming that $\argmin \Phi \neq \emptyset,$ we show that any trajectory converges weakly to a minimizer of $\Phi$, and that $ \Phi(x(t))- \min_{\mathcal H}\Phi = o(t^{-2})$. Strong convergence is established in various practical situations. In particular, for the strongly convex case, we obtain an even faster speed of convergence which can be arbitrarily fast depending on the choice of $\alpha$. More precisely, we have $\Phi(x(t))- \min_{\mathcal H}\Phi = \mathcal O(t^{-\frac{2}{3}\alpha})$. Then, we extend the results to the case of a general proper lower-semicontinuous convex function $\Phi : \mathcal H \rightarrow \mathbb R \cup \lbrace + \infty \rbrace$. This is based on the crucial property that the inertial dynamic with Hessian driven damping can be equivalently written as a first-order system in time and space, allowing to extend it by simply replacing the gradient with the subdifferential. By explicit-implicit time discretization, this opens a gate to new $-$ possibly more rapid $-$ inertial algorithms, expanding the field of FISTA methods for convex structured optimization problems.



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