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CWY Parametrization: a Solution for Parallelized Optimization of Orthogonal and Stiefel Matrices

Likhosherstov, V and Davis, J and Choromanski, K and Weller, A CWY Parametrization: a Solution for Parallelized Optimization of Orthogonal and Stiefel Matrices. (Unpublished)

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Abstract

We introduce an efficient approach for optimization over orthogonal groups on highly parallel computation units such as GPUs or TPUs. As in earlier work, we parametrize an orthogonal matrix as a product of Householder reflections. However, to overcome low parallelization capabilities of computing Householder reflections sequentially, we propose employing an accumulation scheme called the compact WY (or CWY) transform -- a compact parallelization-friendly matrix representation for the series of Householder reflections. We further develop a novel Truncated CWY (or T-CWY) approach for Stiefel manifold parametrization which has a competitive complexity and, again, yields benefits when computed on GPUs and TPUs. We prove that our CWY and T-CWY methods lead to convergence to a stationary point of the training objective when coupled with stochastic gradient descent. We apply our methods to train recurrent neural network architectures in the tasks of neural machine translation and video prediction.

Item Type: Article
Uncontrolled Keywords: cs.LG cs.LG stat.ML
Subjects: UNSPECIFIED
Divisions: Div F > Computational and Biological Learning
Depositing User: Cron Job
Date Deposited: 01 May 2020 20:36
Last Modified: 25 Feb 2021 05:49
DOI: