# Acceleration via Symplectic Discretization of High-Resolution Differential Equations

arXiv: https://arxiv.org/abs/1902.03694 · HTML: https://mlsanity.com/paper/1902.03694/

## Key facts

- Cited by 27 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Feb 2019.

## Built on by (most cited first, 27 of 27)

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- [A More Stable Accelerated Gradient Method Inspired by Continuous-Time Perspective](https://mlsanity.com/paper/2112.04922/) (2021)
- [Whiplash Gradient Descent Dynamics](https://mlsanity.com/paper/2203.02140/) (2022)
- [Reducing Discretization Error in the Frank-Wolfe Method](https://mlsanity.com/paper/2304.01432/) (2023)
- [Accelerated Optimization Landscape of Linear-Quadratic Regulator](https://mlsanity.com/paper/2307.03590/) (2023)
- [A General Continuous-Time Formulation of Stochastic ADMM and Its Variants](https://mlsanity.com/paper/2404.14358/) (2024)
- [Symplectic Bregman divergences](https://mlsanity.com/paper/2408.12961/) (2024)
- [Generalized Continuous-Time Models for Nesterov's Accelerated Gradient Methods](https://mlsanity.com/paper/2409.00913/) (2024)

## Abstract (arXiv)

We study first-order optimization methods obtained by discretizing ordinary differential equations (ODEs) corresponding to Nesterov's accelerated gradient methods (NAGs) and Polyak's heavy-ball method. We consider three discretization schemes: an explicit Euler scheme, an implicit Euler scheme, and a symplectic scheme. We show that the optimization algorithm generated by applying the symplectic scheme to a high-resolution ODE proposed by Shi et al. [2018] achieves an accelerated rate for minimizing smooth strongly convex functions. On the other hand, the resulting algorithm either fails to achieve acceleration or is impractical when the scheme is implicit, the ODE is low-resolution, or the scheme is explicit.