- 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