  Batch Normalization is a commonly used trick to improve the training of deep
neural networks. These neural networks use L2 regularization, also called
weight decay, ostensibly to prevent overfitting. However, we show that L2
regularization has no regularizing effect when combined with normalization.
Instead, regularization has an influence on the scale of weights, and thereby
on the effective learning rate. We investigate this dependence, both in theory,
and experimentally. We show that popular optimization methods such as ADAM only
partially eliminate the influence of normalization on the learning rate. This
leads to a discussion on other ways to mitigate this issue.
