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On orthogonality and learning recurrent networks with long term dependencies (1702.00071v4)

Published 31 Jan 2017 in cs.LG and cs.NE

Abstract: It is well known that it is challenging to train deep neural networks and recurrent neural networks for tasks that exhibit long term dependencies. The vanishing or exploding gradient problem is a well known issue associated with these challenges. One approach to addressing vanishing and exploding gradients is to use either soft or hard constraints on weight matrices so as to encourage or enforce orthogonality. Orthogonal matrices preserve gradient norm during backpropagation and may therefore be a desirable property. This paper explores issues with optimization convergence, speed and gradient stability when encouraging or enforcing orthogonality. To perform this analysis, we propose a weight matrix factorization and parameterization strategy through which we can bound matrix norms and therein control the degree of expansivity induced during backpropagation. We find that hard constraints on orthogonality can negatively affect the speed of convergence and model performance.

Citations (230)

Summary

  • The paper shows that imposing orthogonality in weight matrices effectively mitigates vanishing and exploding gradients in RNNs.
  • The paper employs singular value decomposition to balance strict and soft orthogonality constraints, enhancing convergence and generalization.
  • The paper finds that slight relaxations in orthogonality can speed up training while maintaining accuracy, as evidenced by results on tasks like permuted sequential MNIST.

Orthogonality in Recurrent Neural Networks: Implications for Learning Long Term Dependencies

The paper "On orthogonality and learning recurrent networks with long term dependencies" explores the challenges and techniques for training Recurrent Neural Networks (RNNs) in tasks that necessitate the capturing of long-term dependencies. The underpinning issue addressed by the authors is the vanishing and exploding gradients problem, a well-acknowledged challenge in training deep networks, particularly those with recurrent architectures. Specific emphasis is placed on the role of orthogonality in mitigating these gradient-related problems.

Problem Context and Motivation

Training neural networks with long-term dependencies is notoriously difficult due to gradient instability. Exploding gradients can be managed using techniques such as gradient clipping or weight norm penalties; however, preventing vanishing gradients requires more exhaustive solutions. The stability of training is often enhanced by imposing orthogonality constraints on weight matrices. Orthogonal matrices are particularly advantageous as they preserve the norm of gradients during backpropagation, stabilizing the learning process. The authors focus on examining the effects of orthogonal and nearly-orthogonal weight matrices on the performance and optimization of RNNs.

Methodology and Approach

The paper proposes an approach involving factorization and parametrization of weight matrices to control the expansivity during backpropagation. This involves decomposing weight matrices using singular value decomposition (SVD) into orthogonal matrices, allowing the authors to modulate singular values, hence influencing network training dynamics. The research investigates both hard (strict orthogonality) and soft (regularization-based) constraints on orthogonality and their implications on convergence and model generalization.

The empirical investigations encompass synthetic tasks, like sequence copying and basic addition, to evaluate the effects on networks known for memory challenges. Further evaluations involve real datasets, such as sequential MNIST and Penn Treebank (PTB), to extend validations to complex, structured data.

Results and Implications

The inclusion of orthogonality particularly impacts convergence speed and model accuracy. The results demonstrate that while orthogonal initializations facilitate training, maintaining strict orthogonality can constrict the model's representational capabilities. A notable finding is that loosening orthogonality constraints, even marginally, can enhance training speed without substantially affecting the gradient stability.

Factorized RNNs equipped with sigmoidal constraints improved model performance across synthetic and real data tasks. For instance, in the permuted sequential MNIST task, models with strategic deviations from orthogonality nearly matched Long Short-Term Memory (LSTM) networks, despite their inherent structural simplicity and parameter efficiency.

Future Directions

This research opens pathways for future exploration in two main areas: further refinement of orthogonality and non-orthogonality balance, and the extension of these techniques to more advanced RNN architectures, including newer variants like Gated Recurrent Units (GRUs) or LSTMs with additional orthogonality-constrained layers.

Additionally, applying such techniques to unsupervised learning tasks involving RNNs could uncover new opportunities for enhancing learning efficiency. Adaptation of orthogonality concepts to other domains, such as convolutional networks, may also yield benefits in gradient management and learning dynamics.

Conclusion

The exploration of orthogonality as a mechanism for enhancing learning in RNNs presents a compelling narrative around balancing constraints and flexibility. The results advocate for a nuanced application of orthogonality constraints, emphasizing how small deviations from strict constraints can yield substantial gains in learning efficiency and model accuracy. This work not only elucidates key aspects of training stability but also sets a foundational understanding to inform future studies on orthogonality in deep learning architectures.