Stability is a fundamental property of dynamical systems, yet to this date it has had little bearing on the practice of recurrent neural networks. In this work, we conduct a thorough investigation of stable recurrent models. Theoretically, we prove stable recurrent neural networks are well approximated by feed-forward networks for the purpose of both inference and training by gradient descent. Empirically, we demonstrate stable recurrent models often perform as well as their unstable counterparts on benchmark sequence tasks. Taken together, these findings shed light on the effective power of recurrent networks and suggest much of sequence learning happens, or can be made to happen, in the stable regime. Moreover, our results help to explain why in many cases practitioners succeed in replacing recurrent models by feed-forward models.
 Recurrent neural networks are a popular modeling choice for solving sequence learning problems arising in domains such as speech recognition and natural language processing. At the outset, recurrent neural networks are non-linear dynamical systems commonly trained to fit sequence data via some variant of gradient descent.Stability is of fundamental importance in the study of dynamical system. Surprisingly, however, stability has had little impact on the practice of recurrent neural networks. Recurrent models trained in practice do not satisfy stability in an obvious manner, suggesting that perhaps training happens in a chaotic regime. The difficulty of training recurrent models has compelled practitioners to successfully replace recurrent models with non-recurrent, feed-forward architectures.This state of affairs raises important unresolved questions. Is sequence modeling in practice inherently unstable? When and why are recurrent models really needed?In this work, we shed light on both of these questions through a theoretical and empirical investigation of stability in recurrent models.We first prove stable recurrent models can be approximated by feed-forward networks. In particular, not only are the models equivalent for inference, they are also equivalent for training via gradient descent. While it is easy to contrive non-linear recurrent models that on some input sequence cannot be approximated by feed-forward models, our result implies such models are inevitably unstable. This means in particular they must have exploding gradients, which is in general an impediment to learnibility via gradient descent.Second, across a variety of different sequence tasks, we show how recurrent models can often be made stable without loss in performance. We also show models that are nominally unstable often operate in the stable regime on the data distribution. Combined with our first result, these observation helps to explain why an increasingly large body of empirical research succeeds in replacing recurrent models with feed-forward models in important applications, including translation BID25 BID5 , speech synthesis BID24 , and language modeling . While stability does not always hold in practice to begin with, it is often possible to generate a high-performing stable model by imposing stability during training.Our results also shed light on the effective representational properties of recurrent networks trained in practice. In particular, stable models cannot have long-term memory. Therefore, when stable and unstable models achieve similar results, either the task does not require long-term memory, or the unstable model does not have it.
Stable recurrent models can be approximated by feed-forward networks and empirically perform as well as unstable models on benchmark tasks. Studies the stability of RNNs and investigation of spectral normalization to sequential predictions.