On the Expressive Power of Deep Neural Networks

Maithra Raghu, Ben Poole, Jon Kleinberg, Surya Ganguli, Jascha Sohl-Dickstein

Abstract:

We study the expressive power of deep neural networks before and after
training. Considering neural nets after random initialization, we show that
three natural measures of expressivity all display an exponential dependence
on the depth of the network. We prove, theoretically and experimentally,
that all of these measures are in fact related to a fourth quantity, trajectory
length. This quantity grows exponentially in the depth of the network, and
is responsible for the depth sensitivity observed. These results translate
to consequences for networks during and after training. They suggest that
parameters earlier in a network have greater influence on its expressive power
– in particular, given a layer, its influence on expressivity is determined by
the remaining depth of the network after that layer. This is verified with
experiments on MNIST and CIFAR-10. We also explore the effect of training
on the input-output map, and find that it trades off between the stability
and expressivity of the input-output map.

TL;DR: Derives and explains the exponential depth sensitivity of different expressivity measures for deep neural networks, and explores consequences during and after training.

Keywords: Theory, Deep learning

Conflicts: cornell.edu, google.com, stanford.edu