  We study the behavior of untrained neural networks whose weights and biases
are randomly distributed using mean field theory. We show the existence of
depth scales that naturally limit the maximum depth of signal propagation
through these random networks. Our main practical result is to show that random
networks may be trained precisely when information can travel through them.
Thus, the depth scales that we identify provide bounds on how deep a network
may be trained for a specific choice of hyperparameters. As a corollary to
this, we argue that in networks at the edge of chaos, one of these depth scales
diverges. Thus arbitrarily deep networks may be trained only sufficiently close
to criticality. We show that the presence of dropout destroys the
order-to-chaos critical point and therefore strongly limits the maximum
trainable depth for random networks. Finally, we develop a mean field theory
for backpropagation and we show that the ordered and chaotic phases correspond
to regions of vanishing and exploding gradient respectively.
