# First order optimization methods based on Hessian-driven Nesterov accelerated gradient flow

arXiv: https://arxiv.org/abs/1912.09276 · HTML: https://mlsanity.com/paper/1912.09276/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Dec 2019.

## Built on by (most cited first, 6 of 6)

- [Understanding Riemannian Acceleration via a Proximal Extragradient Framework](https://mlsanity.com/paper/2111.02763/) (2021)
- [SHANG++: Robust Stochastic Acceleration under Multiplicative Noise](https://mlsanity.com/paper/2603.09355/) (2026)
- [Inertial Newton Algorithms Avoiding Strict Saddle Points](https://mlsanity.com/paper/2111.04596/) (2021)
- [Deterministic Adam-Inspired Methods with Accelerated Convergence Rate](https://mlsanity.com/paper/2604.08742/) (2026)
- [Adam-SHANG: A Convergent Adam-Type Method for Stochastic Smooth Convex Optimization](https://mlsanity.com/paper/2605.12878/) (2026)
- [A second-order-like optimizer with adaptive gradient scaling for deep learning](https://mlsanity.com/paper/2410.05871/) (2024)

## Abstract (arXiv)

A novel dynamical inertial Newton system, which is called Hessian-driven Nesterov accelerated gradient (H-NAG) flow is proposed. Convergence of the continuous trajectory are established via tailored Lyapunov function, and new first-order accelerated optimization methods are proposed from ODE solvers. It is shown that (semi-)implicit schemes can always achieve linear rate and explicit schemes have the optimal(accelerated) rates for convex and strongly convex objectives. In particular, Nesterov's optimal method is recovered from an explicit scheme for our H-NAG flow. Furthermore, accelerated splitting algorithms for composite optimization problems are also developed.