Normalization effects on deep neural networks

TL;DR

This paper investigates how different normalization schemes, especially mean-field scaling, affect the statistical behavior and test accuracy of deep neural networks, providing insights for hyperparameter tuning.

Contribution

It offers a mathematical analysis of normalization effects, demonstrating that mean-field scaling optimizes variance and accuracy, and guides hyperparameter selection.

Findings

Mean-field scaling (γ=1) yields optimal variance and accuracy.

Outer layer scaling has a greater impact on network behavior.

Mathematical analysis informs systematic hyperparameter choice.

Abstract

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 $1/N_{i}^{\gamma_{i}}$ with $\gamma_{i}\in[1/2,1]$ and we study the effect of the choice of the $\gamma_{i}$ 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 $\gamma_{i}$'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.