  The exploding and vanishing gradient problem has been the major conceptual
principle behind most architecture and training improvements in recurrent
neural networks (RNNs) during the last decade. In this paper, we argue that
this principle, while powerful, might need some refinement to explain recent
developments. We refine the concept of exploding gradients by reformulating the
problem in terms of the cost function smoothness, which gives insight into
higher-order derivatives and the existence of regions with many close local
minima. We also clarify the distinction between vanishing gradients and the
need for the RNN to learn attractors to fully use its expressive power. Through
the lens of these refinements, we shed new light on recent developments in the
RNN field, namely stable RNN and unitary (or orthogonal) RNNs.
