Search papers, labs, and topics across Lattice.
This paper theoretically analyzes the role of depth in the expressivity of RNNs and 2RNNs (RNNs with multiplicative interactions). It proves that depth increases RNNs' memory capacity more efficiently than width, and that depth in 2RNNs enables polynomial transformations of increasing degree. The analysis also demonstrates that multiplicative interactions in 2RNNs cannot be generally replaced by standard layerwise nonlinearities, highlighting their unique contribution to expressivity.
Forget simply scaling width – stacking layers unlocks exponentially more memory capacity in RNNs, and multiplicative interactions are key to their expressive power.
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.
