Riemannian Adaptive Optimization Methods

TL;DR

This paper extends popular adaptive optimization algorithms like Adam and Adagrad to Riemannian manifolds, providing theoretical guarantees and demonstrating improved convergence in embedding tasks.

Contribution

It introduces Riemannian versions of Adam, Adagrad, and Amsgrad with convergence proofs for geodesically convex functions on product manifolds.

Findings

Faster convergence observed in Riemannian adaptive methods.

Lower train loss achieved on WordNet taxonomy embedding.

Algorithms reduce to standard Euclidean methods when applied to flat spaces.

Abstract

Several first order stochastic optimization methods commonly used in the Euclidean domain such as stochastic gradient descent (SGD), accelerated gradient descent or variance reduced methods have already been adapted to certain Riemannian settings. However, some of the most popular of these optimization tools - namely Adam , Adagrad and the more recent Amsgrad - remain to be generalized to Riemannian manifolds. We discuss the difficulty of generalizing such adaptive schemes to the most agnostic Riemannian setting, and then provide algorithms and convergence proofs for geodesically convex objectives in the particular case of a product of Riemannian manifolds, in which adaptivity is implemented across manifolds in the cartesian product. Our generalization is tight in the sense that choosing the Euclidean space as Riemannian manifold yields the same algorithms and regret bounds as those that were already known for the standard algorithms. Experimentally, we show faster convergence and to a lower train loss value for Riemannian adaptive methods over their corresponding baselines on the realistic task of embedding the WordNet taxonomy in the Poincare ball.