Lower bounds for artificial neural network approximations: A proof that shallow neural networks fail to overcome the curse of dimensionality

Artificial neural networks (ANNs) have become a very powerful tool in the approximation of high-dimensional functions. Especially, deep ANNs, consisting large number hidden layers, been successfully used series practical relevant computational problems involving input data ranging from classification tasks supervised learning to optimal decision reinforcement learning. There are also mathematical results scientific literature which study capacities ANNs context target In particular, there show that sufficiently capacity overcome curse dimensionality certain function classes sense parameters approximating grows at most polynomially dimension d∈N functions under considerations. proofs several such it is crucial involved and consist layers considered It topic this work look bit more detailed deepness main result proves exists concretely specified sequence can be approximated without by but cannot if shallow or not enough.