Simplifying Momentum-based Positive-definite Submanifold Optimization with Applications to Deep Learning
This work addresses a specific bottleneck in optimization for deep learning with low precision, offering an incremental improvement over existing methods.
The paper tackles the computational challenge of Riemannian submanifold optimization with momentum by simplifying it for sparse or structured symmetric positive-definite matrices, resulting in matrix-inverse-free second-order optimizers for deep learning that use only matrix multiplications.
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 sparse or 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 with low precision by using only matrix multiplications. Code: https://github.com/yorkerlin/StructuredNGD-DL