Depth Degeneracy in Neural Networks: Vanishing Angles in Fully Connected ReLU Networks on Initialization

Stacking many layers to create truly deep neural networks is arguably what has led to the recent explosion of these methods. However, many properties of deep neural networks are not yet understood. One such mystery is the depth degeneracy phenomenon: the deeper you make your network, the closer your network is to a constant function on initialization. In this paper, we examine the evolution of the angle between two inputs to a ReLU neural network as a function of the number of layers. By using combinatorial expansions, we find precise formulas for how fast this angle goes to zero as depth increases. Our formulas capture microscopic fluctuations that are not visible in the popular framework of infinite width limits, and yet have a significant effect on predicted behaviour. The formulas are given in terms of the mixed moments of correlated Gaussians passed through the ReLU function. We also find a surprising combinatorial connection between these mixed moments and the Bessel numbers.