Towards a regularity theory for ReLU networks -- chain rule and global error estimates

Understand

Although for neural networks with locally Lipschitz continuous activation functions the classical derivative exists almost everywhere, the standard chain rule is in general not applicable.

• We will consider a way of introducing a derivative for neural networks that admits a chain rule, which is both rigorous and easy to work with.
• In addition we will present a method of converting approximation results on bounded domains to global (pointwise) estimates.
• This can be used to extend known neural network approximation theory to include the study of regularity properties.