Stabilizing gradients for deep neural networks via efficient SVD parameterization

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 provably 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 controls generalization of RNN for the classification task. 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 the MNIST and Penn Tree Bank data sets.