On the Role of Depth in the Expressivity of RNNs

Maude Lizaire, Michael Rizvi-Martel, Éric Dupuis, Guillaume Rabusseau

TL;DR: Characterization of the effect of depth on the expressivity of RNNs: memory capacity, higher-order interactions and separation between multiplicative interactions and depth-wise nonlinearities.

Abstract: The benefits of depth in feedforward neural networks (FNNs) are well known: composing multiple layers of linear transformations with nonlinear activations enables complex computations. While similar effects are expected in recurrent neural networks (RNNs), it remains unclear how depth interacts with recurrence to shape expressive power. Here, we formally show that depth increases RNNs’ memory capacity efficiently with respect to parameters, enhancing expressivity both by enabling more complex input transformations and improving the retention of past information. We extend our analysis to 2RNNs, a generalization of RNNs with multiplicative interactions between inputs and hidden states. Unlike RNNs, which remain linear without nonlinear activations, 2RNNs perform polynomial transformations whose maximal degree grows with depth. We further show that multiplicative interactions cannot, in general, be replaced by layerwise nonlinearities. Finally, we validate these insights empirically on synthetic and real-world tasks.

Submission Number: 1018