Understanding Riemannian Acceleration via a Proximal Extragradient Framework
This work advances theoretical understanding of optimization on Riemannian manifolds, which is important for machine learning applications involving geometric constraints, though it appears incremental as it extends an existing Euclidean framework.
The authors tackled the problem of understanding Riemannian accelerated gradient methods by proposing and analyzing Riemannian Accelerated Hybrid Proximal Extragradient (A-HPE), building on a Euclidean framework. They provided new insights into the Euclidean A-HPE and controlled metric distortion in Riemannian geometry, deriving existing and new accelerated methods as special cases.
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.