Fast Convergence of Inertial Dynamics with Hessian-Driven Damping Under Geometry Assumptions

First-order optimization algorithms can be considered as a discretization of ordinary differential equations (ODEs) (Su et al. in Adv Neural Inf Process Syst 27, 2014). In this perspective, studying the properties corresponding trajectories may lead to convergence results which transfered numerical scheme. paper we analyse following ODE introduced by Attouch (J Differ Equ 261(10):5734–5783, 2016): $$\begin{aligned} \forall t\geqslant t_0,~\ddot{x}(t)+\frac{\alpha }{t}{\dot{x}}(t)+\beta H_F(x(t)){\dot{x}}(t)+\nabla F(x(t))=0, \end{aligned}$$ where $$\alpha >0$$ , $$\beta and $$H_F$$ denotes Hessian F. This derived build schemes do not require F twice differentiable shown (Math Program 1–43, 2020) (Optimization 72:1–40, 2021). We provide strong on error $$F(x(t))-F^*$$ integrability $$\Vert \nabla F(x(t))\Vert $$ under some geometry assumptions such quadratic growth around set minimizers. particular, show that decay rate for strongly convex function is $$O(t^{-\alpha -\varepsilon })$$ any $$\varepsilon . These are briefly illustrated at end paper.