Understanding Riemannian Acceleration via a Proximal Extragradient Framework

TL;DR

This paper extends the Accelerated Hybrid Proximal Extragradient framework to Riemannian manifolds, providing new insights and methods for accelerated gradient optimization in curved spaces.

Contribution

It introduces the Riemannian A-HPE framework, building on Euclidean A-HPE, and analyzes how to control metric distortion for accelerated methods on Riemannian manifolds.

Findings

Unified Riemannian accelerated gradient methods as special cases

Characterization of acceleration through the new framework

Insights into Euclidean A-HPE and metric distortion control

Abstract

We contribute to advancing the understanding of Riemannian accelerated gradient methods. In particular, we revisit Accelerated Hybrid Proximal Extragradient(A-HPE), a powerful framework for obtaining Euclidean accelerated methods \citep{monteiro2013accelerated}. Building on A-HPE, we then propose and analyze Riemannian A-HPE. The core of our analysis consists of two key components: (i) a set of new insights into Euclidean A-HPE itself; and (ii) a careful control of metric distortion caused by Riemannian geometry. We illustrate our framework by obtaining a few existing and new Riemannian accelerated gradient methods as special cases, while characterizing their acceleration as corollaries of our main results.