The smooth output assumption, and why deep networks are better than wide ones

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.