Fast convergence of inertial dynamics with Hessian-driven damping under geometry assumptions

TL;DR

This paper analyzes a Hessian-driven inertial ODE for optimization, demonstrating fast convergence rates under geometric conditions, with implications for designing efficient algorithms that do not require twice differentiability.

Contribution

The paper provides convergence analysis of a Hessian-driven inertial ODE under geometric assumptions, establishing explicit decay rates and integrability properties for optimization.

Findings

Error decay rate for strongly convex functions is O(t^{- ext{alpha}- ext{epsilon}}).

Convergence results hold under quadratic growth conditions.

The analysis supports the development of algorithms not requiring second derivatives.

Abstract

First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) \cite{su2014differential}. In this perspective, studying the properties of the corresponding trajectories may lead to convergence results which can be transfered to the numerical scheme. In this paper we analyse the following ODE introduced by Attouch et al. in \cite{attouch2016fast}: \begin{equation*} \forall t\geqslant t_0,~\ddot{x}(t)+\frac{\alpha}{t}\dot{x}(t)+\beta H_F(x(t))\dot{x}(t)+\nabla F(x(t))=0,\end{equation*} where $\alpha>0$, $\beta>0$ and $H_F$ denotes the Hessian of $F$. This ODE can be derived to build numerical schemes which do not require $F$ to be twice differentiable as shown in \cite{attouch2020first,attouch2021convergence}. We provide strong convergence results on the error $F(x(t))-F^*$ and integrability properties on $\|\nabla F(x(t))\|$ under some geometry assumptions on $F$ such as quadratic growth around the set of minimizers. In particular, we show that the decay rate of the error for a strongly convex function is $O(t^{-\alpha-\varepsilon})$ for any $\varepsilon>0$. These results are briefly illustrated at the end of the paper.