CWY Parametrization: a Solution for Parallelized Optimization of Orthogonal and Stiefel Matrices
This work addresses the bottleneck of parallelizing orthogonal matrix optimization for machine learning practitioners, though it is incremental as it builds on existing Householder reflection methods.
The paper tackles the problem of optimizing orthogonal and Stiefel matrices on parallel hardware like GPUs/TPUs by introducing CWY and T-CWY parametrizations, which improve parallelization and achieve competitive performance in tasks such as neural machine translation and video prediction.
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.