Normalization effects on deep neural networks
This provides a mathematically informed method for selecting learning rates to ensure statistical robustness as network width increases, but it is incremental as it builds on existing normalization techniques.
The study investigated how normalization scaling factors affect deep neural networks, finding that mean-field scaling (γ_i=1) optimizes output variance and test accuracy on MNIST, with the outer layer being most sensitive.
We study the effect of normalization on the layers of deep neural networks of feed-forward type. A given layer $i$ with $N_{i}$ hidden units is allowed to be normalized by $1/N_{i}^{γ_{i}}$ with $γ_{i}\in[1/2,1]$ and we study the effect of the choice of the $γ_{i}$ on the statistical behavior of the neural network's output (such as variance) as well as on the test accuracy on the MNIST data set. We find that in terms of variance of the neural network's output and test accuracy the best choice is to choose the $γ_{i}$'s to be equal to one, which is the mean-field scaling. We also find that this is particularly true for the outer layer, in that the neural network's behavior is more sensitive in the scaling of the outer layer as opposed to the scaling of the inner layers. The mechanism for the mathematical analysis is an asymptotic expansion for the neural network's output. An important practical consequence of the analysis is that it provides a systematic and mathematically informed way to choose the learning rate hyperparameters. Such a choice guarantees that the neural network behaves in a statistically robust way as the $N_i$ grow to infinity.