On the Role of Depth in the Expressivity of RNNs
This work addresses a foundational gap in understanding depth in RNNs for machine learning researchers, though it is incremental as it builds on known benefits of depth in feedforward networks.
The paper tackles the problem of understanding how depth affects the expressive power of recurrent neural networks (RNNs) and their generalization, 2RNNs, showing that depth increases memory capacity efficiently and enhances expressivity, with empirical validation on synthetic and real-world tasks.
The benefits of depth in feedforward neural networks 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 the number of parameters, thus enhancing expressivity both by enabling more complex input transformations and improving the retention of past information. We broaden 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.