Riemannian Natural Gradient Methods

TL;DR

This paper introduces a Riemannian natural gradient method for large-scale optimization on manifolds, establishing its convergence properties and demonstrating its effectiveness in neural network training with batch normalization.

Contribution

It extends the natural gradient method to Riemannian manifolds, providing convergence analysis and practical validation in neural network applications.

Findings

Proposed method converges globally under standard assumptions.

Achieves local linear or quadratic convergence rates under certain conditions.

Neural network Jacobian stability is satisfied with high probability for wide networks.

Abstract

This paper studies large-scale optimization problems on Riemannian manifolds whose objective function is a finite sum of negative log-probability losses. Such problems arise in various machine learning and signal processing applications. By introducing the notion of Fisher information matrix in the manifold setting, we propose a novel Riemannian natural gradient method, which can be viewed as a natural extension of the natural gradient method from the Euclidean setting to the manifold setting. We establish the almost-sure global convergence of our proposed method under standard assumptions. Moreover, we show that if the loss function satisfies certain convexity and smoothness conditions and the input-output map satisfies a Riemannian Jacobian stability condition, then our proposed method enjoys a local linear -- or, under the Lipschitz continuity of the Riemannian Jacobian of the input-output map, even quadratic -- rate of convergence. We then prove that the Riemannian Jacobian stability condition will be satisfied by a two-layer fully connected neural network with batch normalization with high probability, provided that the width of the network is sufficiently large. This demonstrates the practical relevance of our convergence rate result. Numerical experiments on applications arising from machine learning demonstrate the advantages of the proposed method over state-of-the-art ones.