Accelerated Gradient Methods for Geodesically Convex Optimization: Tractable Algorithms and Convergence Analysis

TL;DR

This paper introduces the first fully accelerated first-order algorithms for geodesically convex optimization on Riemannian manifolds, matching Euclidean iteration complexities and validated through numerical experiments.

Contribution

It extends Nesterov acceleration to Riemannian settings with tractable algorithms and novel convergence analysis techniques.

Findings

Algorithms achieve Euclidean iteration complexity for geodesically convex functions.

First fully accelerated methods for Riemannian optimization.

Numerical experiments confirm theoretical convergence rates.

Abstract

We propose computationally tractable accelerated first-order methods for Riemannian optimization, extending the Nesterov accelerated gradient (NAG) method. For both geodesically convex and geodesically strongly convex objective functions, our algorithms are shown to have the same iteration complexities as those for the NAG method on Euclidean spaces, under only standard assumptions. To the best of our knowledge, the proposed scheme is the first fully accelerated method for geodesically convex optimization problems. Our convergence analysis makes use of novel metric distortion lemmas as well as carefully designed potential functions. A connection with the continuous-time dynamics for modeling Riemannian acceleration in (Alimisis et al., 2020) is also identified by letting the stepsize tend to zero. We validate our theoretical results through numerical experiments.