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We give a rigorous analysis of the statistical behavior of gradients in arandomly initialized fully connected network N with ReLU activations. Ourresults show that the empirical variance of the squares of the entries in theinput-output Jacobian of N is exponential in a simple architecture-dependentconstant beta, given by the sum of the reciprocals of the hidden layer widths.When beta is large, the gradients computed by N at initialization vary wildly.Our approach complements the mean field theory analysis of random networks.From this point of view, we rigorously compute finite width corrections to thestatistics of gradients at the edge of chaos.

Which Neural Net Architectures Give Rise To Exploding and Vanishing Gradients?