Towards Riemannian Accelerated Gradient Methods

TL;DR

This paper introduces a Riemannian version of Nesterov's Accelerated Gradient method, demonstrating accelerated convergence for geodesically smooth and strongly convex problems on manifolds, with a constructive and computationally feasible algorithm.

Contribution

It develops a new Riemannian accelerated gradient algorithm with provable convergence, improving upon previous methods by being constructive and computationally tractable.

Findings

Achieves accelerated convergence near the minimizer on Riemannian manifolds.

Provides a new estimate sequence and bound on nonlinear metric distortion.

Applicable to geodesically smooth and strongly convex problems.

Abstract

We propose a Riemannian version of Nesterov's Accelerated Gradient algorithm (RAGD), and show that for geodesically smooth and strongly convex problems, within a neighborhood of the minimizer whose radius depends on the condition number as well as the sectional curvature of the manifold, RAGD converges to the minimizer with acceleration. Unlike the algorithm in (Liu et al., 2017) that requires the exact solution to a nonlinear equation which in turn may be intractable, our algorithm is constructive and computationally tractable. Our proof exploits a new estimate sequence and a novel bound on the nonlinear metric distortion, both ideas may be of independent interest.