Stabilizing Gradients for Deep Neural Networks via Efficient SVD Parameterization
Jiong Zhang · Qi Lei · Inderjit Dhillon

Wed Jul 11th 05:40 -- 05:50 PM @ Victoria

Vanishing and exploding gradients are two of the main obstacles in training deep neural networks, especially in capturing long range dependencies in recurrent neural networks (RNNs). In this paper, we present an efficient parametrization of the transition matrix of an RNN that allows us to stabilize the gradients that arise in its training. Specifically, we parameterize the transition matrix by its singular value decomposition (SVD), which allows us to explicitly track and control its singular values. We attain efficiency by using tools that are common in numerical linear algebra, namely Householder reflectors for representing the orthogonal matrices that arise in the SVD. By explicitly controlling the singular values, our proposed Spectral-RNN method allows us to easily solve the exploding gradient problem and we observe that it empirically solves the vanishing gradient issue to a large extent. We note that the SVD parameterization can be used for any rectangular weight matrix, hence it can be easily extended to any deep neural network, such as a multi-layer perceptron. Theoretically, we demonstrate that our parameterization does not lose any expressive power, and show how it potentially makes the optimization process easier. Our extensive experimental results also demonstrate that the proposed framework converges faster, and has good generalization, especially in capturing long range dependencies, as shown on the synthetic addition and copy tasks, as well as on MNIST and Penn Tree Bank data sets.

Author Information

Jiong Zhang (University of Texas at Austin)
Qi Lei (University of Texas at Austin)
Inderjit Dhillon (UT Austin & Amazon)

Inderjit Dhillon is the Gottesman Family Centennial Professor of Computer Science and Mathematics at UT Austin, where he is also the Director of the ICES Center for Big Data Analytics. His main research interests are in big data, machine learning, network analysis, linear algebra and optimization. He received his B.Tech. degree from IIT Bombay, and Ph.D. from UC Berkeley. Inderjit has received several awards, including the ICES Distinguished Research Award, the SIAM Outstanding Paper Prize, the Moncrief Grand Challenge Award, the SIAM Linear Algebra Prize, the University Research Excellence Award, and the NSF Career Award. He has published over 160 journal and conference papers, and has served on the Editorial Board of the Journal of Machine Learning Research, the IEEE Transactions of Pattern Analysis and Machine Intelligence, Foundations and Trends in Machine Learning and the SIAM Journal for Matrix Analysis and Applications. Inderjit is an ACM Fellow, an IEEE Fellow, a SIAM Fellow and an AAAS Fellow.

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