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We investigate the geometry of the empirical risk minimization problem for k-layer neural networks. We will provide examples showing that for the classical activation functions σ(x)=1/(1+exp(−x)) and σ(x)=tanh(x), there exists a positive-measured subset of target functions that do not have best approximations by a fixed number of layers of neural networks. In addition, we study in detail the properties of shallow networks, classifying cases when a best k-layer neural network approximation always exists or does not exist for the ReLU activation σ=max(0,x). We also determine the dimensions of shallow ReLU-activated networks.
