# On the Curved Geometry of Accelerated Optimization

## TL;DR

This paper introduces a differential geometric perspective on Nesterov's accelerated gradient method, modeling it as a proximal point method on a Riemannian manifold, and analyzes its convergence for quadratic problems.

## Contribution

It offers a novel geometric interpretation of AGM using Riemannian manifolds and extends the analysis to continuous-time ODEs on curved spaces.

## Key findings

- AGM can be viewed as a proximal point method on a Riemannian manifold.
- The continuous-time limit of AGM corresponds to an ODE on the manifold.
- Convergence rate analysis is provided for quadratic objectives.

## Abstract

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.