  Recurrent neural networks (RNNs) are powerful dynamical models for data with
complex temporal structure. However, training RNNs has traditionally proved
challenging due to exploding or vanishing of gradients. RNN models such as
LSTMs and GRUs (and their variants) significantly mitigate these issues
associated with training by introducing various types of gating units into the
architecture. While these gates empirically improve performance, how the
addition of gates influences the dynamics and trainability of GRUs and LSTMs is
not well understood. Here, we take the perspective of studying randomly
initialized LSTMs and GRUs as dynamical systems, and ask how the salient
dynamical properties are shaped by the gates. We leverage tools from random
matrix theory and mean-field theory to study the state-to-state Jacobians of
GRUs and LSTMs. We show that the update gate in the GRU and the forget gate in
the LSTM can lead to an accumulation of slow modes in the dynamics. Moreover,
the GRU update gate can poise the system at a marginally stable point. The
reset gate in the GRU and the output and input gates in the LSTM control the
spectral radius of the Jacobian, and the GRU reset gate also modulates the
complexity of the landscape of fixed-points. Furthermore, for the GRU we obtain
a phase diagram describing the statistical properties of fixed-points. We also
provide a preliminary comparison of training performance to the various
dynamical regimes realized by varying hyperparameters. Looking to the future,
we have introduced a powerful set of techniques which can be adapted to a broad
class of RNNs, to study the influence of various architectural choices on
dynamics, and potentially motivate the principled discovery of novel
architectures.
