Batch normalization is sufficient for universal function approximation in CNNs

Normalization techniques, for which Batch Normalization (BN) is a popular choice, is an integral part of many deep learning architectures and contributes significantly to the learning success. We provide a partial explanation for this phenomenon by proving that training normalization parameters alone is already sufficient for universal function approximation if the number of available, potentially random features matches or exceeds the weight parameters of the target networks that can be expressed. Our bound on the number of required features does not only improve on a recent result for fully-connected feed-forward architectures but also applies to CNNs with and without residual connections and almost arbitrary activation functions (which include ReLUs). Our explicit construction of a given target network solves a depth-width trade-off that is driven by architectural constraints and can explain why switching off entire neurons can have representational benefits, as has been observed empirically. To validate our theory, we explicitly match target networks that outperform experimentally obtained networks with trained BN parameters by utilizing a sufficient number of random features.