  When several models have similar training scores, classical model selection
heuristics follow Occam's razor and advise choosing the ones with least
capacity. Yet, modern practice with large neural networks has often led to
situations where two networks with exactly the same number of parameters score
similar on the training set, but the deeper one generalizes better to unseen
examples. With this in mind, it is well accepted that deep networks are
superior to shallow wide ones. However, theoretically there is no difference
between the two. In fact, they are both universal approximators.
  In this work we propose a new unsupervised measure that predicts how well a
model will generalize. We call it the output sharpness, and it is based on the
fact that, in reality, boundaries between concepts are generally unsharp. We
test this new measure on several neural network settings, and architectures,
and show how generally strong the correlation is between our metric, and test
set performance.
  Having established this measure, we give a mathematical probabilistic
argument that predicts network depth to be correlated with our proposed
measure. After verifying this in real data, we are able to formulate the key
argument of the work: output sharpness hampers generalization; deep networks
have an in built bias against it; therefore, deep networks beat wide ones.
  All in all the work not only provides a helpful predictor of overfitting that
can be used in practice for model selection (or even regularization), but also
provides a much needed theoretical grounding for the success of modern deep
neural networks.
