Averaging Stochastic Gradient Descent on Riemannian Manifolds

TL;DR

This paper introduces a geometric framework for averaging stochastic gradient descent on Riemannian manifolds, achieving faster convergence rates and improving algorithms like streaming k-PCA without prior spectral gap knowledge.

Contribution

The paper develops a novel geometric averaging method for SGD on Riemannian manifolds, enhancing convergence speed and applying it to problems like streaming k-PCA.

Findings

Achieves $O(1/n)$ convergence rate for averaged iterates on manifolds.

Accelerates streaming k-PCA to optimal convergence rate.

Provides a robust framework applicable to geodesically-strongly-convex problems.

Abstract

We consider the minimization of a function defined on a Riemannian manifold $\mathcal{M}$ accessible only through unbiased estimates of its gradients. We develop a geometric framework to transform a sequence of slowly converging iterates generated from stochastic gradient descent (SGD) on $\mathcal{M}$ to an averaged iterate sequence with a robust and fast $O(1/n)$ convergence rate. We then present an application of our framework to geodesically-strongly-convex (and possibly Euclidean non-convex) problems. Finally, we demonstrate how these ideas apply to the case of streaming $k$-PCA, where we show how to accelerate the slow rate of the randomized power method (without requiring knowledge of the eigengap) into a robust algorithm achieving the optimal rate of convergence.