On the Effect of the Activation Function on the Distribution of Hidden Nodes in a Deep Network

We analyze the joint probability distribution on the lengths of the vectors of hidden variables in different layers of a fully connected deep network, when the weights and biases are chosen randomly according to gaussian distributions. We show that if the activation function $$ satisfies a minimal set of assumptions, satisfied by all activation functions that we know that are used in practice, then, as the width of the network gets large, the "length process" converges in probability to a length map that is determined as a simple function of the variances of the random weights and biases and the activation function $$. We also show that this convergence may fail for $$ that violate our assumptions. We show how to use this analysis to choose the variance of weight initialization, depending on the activation function, so that hidden variables maintain a consistent scale throughout the network.