On the effect of the activation function on the distribution of hidden nodes in a deep network

01/07/2019

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by   Philip M. Long, et al.

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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, and the input is in { -1, 1}^N. We show that, if the activation function ϕ 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 ϕ. We also show that this convergence may fail for ϕ that violate our assumptions.