# Convergence of iterates for first-order optimization algorithms with inertia and Hessian driven damping

arXiv: https://arxiv.org/abs/2107.05943 · HTML: https://mlsanity.com/paper/2107.05943/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 13 Jul 2021.

## Built on by (most cited first, 3 of 3)

- [Inertial Newton Algorithms Avoiding Strict Saddle Points](https://mlsanity.com/paper/2111.04596/) (2021)
- [A Variational Perspective on High-Resolution ODEs](https://mlsanity.com/paper/2311.02002/) (2023)
- [Fisher information dissipation for time inhomogeneous stochastic differential equations](https://mlsanity.com/paper/2402.01036/) (2024)

## Abstract (arXiv)

In a Hilbert space setting, for convex optimization, we show the convergence of the iterates to optimal solutions for a class of accelerated first-order algorithms. They can be interpreted as discrete temporal versions of an inertial dynamic involving both viscous damping and Hessian-driven damping. The asymptotically vanishing viscous damping is linked to the accelerated gradient method of Nesterov while the Hessian driven damping makes it possible to significantly attenuate the oscillations. By treating the Hessian-driven damping as the time derivative of the gradient term, this gives, in discretized form, first-order algorithms. These results complement the previous work of the authors where it was shown the fast convergence of the values, and the fast convergence towards zero of the gradients.