# On the Curved Geometry of Accelerated Optimization

arXiv: https://arxiv.org/abs/1812.04634 · HTML: https://mlsanity.com/paper/1812.04634/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Dec 2018.

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

- [Convergence and Stability of the Stochastic Proximal Point Algorithm with Momentum](https://mlsanity.com/paper/2111.06171/) (2021)
- [Revisiting the Role of Euler Numerical Integration on Acceleration and Stability in Convex Optimization](https://mlsanity.com/paper/2102.11537/) (2021)
- [Smoothing DiLoCo with Primal Averaging for Faster Training of LLMs](https://mlsanity.com/paper/2512.17131/) (2025)
- [Practical Schemes for Finding Near-Stationary Points of Convex Finite-Sums](https://mlsanity.com/paper/2105.12062/) (2021)
- [Fast symplectic integrator for Nesterov-type acceleration method](https://mlsanity.com/paper/2106.07620/) (2021)
- [A theoretical and empirical study of new adaptive algorithms with additional momentum steps and shifted updates for stochastic non-convex optimization](https://mlsanity.com/paper/2110.08531/) (2021)

## Abstract (arXiv)

In this work we propose a differential geometric motivation for Nesterov's accelerated gradient method (AGM) for strongly-convex problems. By considering the optimization procedure as occurring on a Riemannian manifold with a natural structure, The AGM method can be seen as the proximal point method applied in this curved space. This viewpoint can also be extended to the continuous time case, where the accelerated gradient method arises from the natural block-implicit Euler discretization of an ODE on the manifold. We provide an analysis of the convergence rate of this ODE for quadratic objectives.