Best k-layer neural network approximations

We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set s1, dots, sn ∈ ℝp with corresponding responses t1,dots,tn ∈ ℝq, fitting a k-layer neural network nuθ : ℝp → ℝq involves estimation of the weights θ ∈ ℝm via an ERM: [ infθ ∈ ℝm ; sumi=1n lVert ti - nuθ(si) rVert22. ] We show that even for k = 2, this infimum is not attainable in general for common activations like ReLU, hyperbolic tangent, and sigmoid functions. A high-level explanation is like that for the nonexistence of best rank-r approximations of higher-order tensors --- the set of parameters is not a closed set --- but the geometry involved for best k-layer neural networks approximations is more subtle. In addition, we show that for smooth activations σ(x)= 1/bigl(1 + exp(-x)bigr) and σ(x)=tanh(x), such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation σ(x)=max(0,x), we completely classifying cases where the ERM for a best two-layer neural network approximation attains its infimum. As an aside, we obtain a precise description of the geometry of the space of two-layer neural networks with d neurons in the hidden layer: it is the join locus of a line and the d-secant locus of a cone.