The effect of the forget gate on bifurcation boundaries and dynamics in Recurrent Neural Networks and its implications for gradient-based optimization

Recurrent Neural Networks (RNNs) are an internal dynamics approach to identify models from time series data. They have been successfully applied, e.g. in natural language, speech and video processing [1] and the identification of nonlinear state space models [2]. Conventional RNNs, such as the Elman-RNN, are notoriously hard to optimize, since they are highly initialization dependent, prone to slow convergence, and tend to converge to poor local minima. In recent years, the vanishing/exploding gradient phenomenon, which arises when employing gradient-based optimization techniques such as Backpropagation Through Time (BPTT), has been identified as the root cause of these difficulties. This led to the development of several new RNN-architectures, such as the Long Short-Term Memory (LSTM) and the Gated Recurrent Unit (GRU), which were intended to prevent the vanishing-gradient problem and surpassed conventional RNNs in all areas of application. However, it has been shown that the gradient also vanishes in Gated Units [3] and there is no work showing, that the rate of decay is lower than in Elman-RNN. This suggests that the underlying mechanisms responsible for their success are at least in part not yet fully understood. The purpose of this paper is to provide an alternative explanation for the superior performance of Gated Units by viewing them as nonlinear dynamical systems and studying the stability of their fixed points. This work expands on the work of Doya et al. [4] and Pascanu et al. [5], who studied the effects of bifurcation boundaries in the parameter space of Elman-RNNs with one internal state on gradient-based learning.