Deep ReLU Networks Preserve Expected Length

Boris Hanin, Ryan Jeong, David Rolnick

Keywords: deep learning theory, random ReLU networks, length distortion, initialization, expressivity

Abstract: Assessing the complexity of functions computed by a neural network helps us understand how the network will learn and generalize. One natural measure of complexity is how the network distorts length - if the network takes a unit-length curve as input, what is the length of the resulting curve of outputs? It has been widely believed that this length grows exponentially in network depth. We prove that in fact this is not the case: the expected length distortion does not grow with depth, and indeed shrinks slightly, for ReLU networks with standard random initialization. We also generalize this result by proving upper bounds both for higher moments of the length distortion and for the distortion of higher-dimensional volumes. These theoretical results are corroborated by our experiments.

One-sentence Summary: This article proves that, both on average and with high probability, randomly initialized ReLU networks with width larger than depth do not distort lengths and volumes.