Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning

Published: 24 Apr 2023, Last Modified: 21 Jun 2023ICML 2023 PosterEveryoneRevisions
Abstract: Riemannian submanifold optimization with momentum is computationally challenging because, to ensure that the iterates remain on the submanifold, we often need to solve difficult differential equations. Here, we simplify such difficulties for a class of structured symmetric positive-definite matrices with the affine-invariant metric. We do so by proposing a generalized version of the Riemannian normal coordinates that dynamically orthonormalizes the metric and locally converts the problem into an unconstrained problem in the Euclidean space. We use our approach to simplify existing approaches for structured covariances and develop matrix-inverse-free $2^\text{nd}$-order optimizers for deep learning in low precision settings.
Submission Number: 861