# A speed restart scheme for a dynamics with Hessian driven damping

arXiv: https://arxiv.org/abs/2301.12240 · HTML: https://mlsanity.com/paper/2301.12240/

## Key facts

- Cited by 1 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 28 Jan 2023.

## Built on by (most cited first, 1 of 1)

- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)

## Abstract (arXiv)

In this paper, we analyze a speed restarting scheme for the dynamical system given by  ddotx(t) + dfracαtdotx(t) + nabla φ(x(t)) + β nabla2 φ(x(t))dotx(t)=0,  where α and β are positive parameters, and φ:ℝn → 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 Cand`es citeJMLR: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.