Best k-layer neural network approximations

Abstract

We show that the empirical risk minimization (ERM) problem for neural networks has no solution in general. Given a training set $s_1,\dots ,s_n\in {\mathbb{R}}^p$ with corresponding responses $t_1,\dots ,t_n\in {\mathbb{R}}^q$, fitting a $k$-layer neural network ${\nu }_{\theta }:{\mathbb{R}}^p\to {\mathbb{R}}^q$ involves estimation of the weights $\theta \in {\mathbb{R}}^m$ via an ERM: \[ \inf_{θ\in \mathbb{R}^m} \; \sum_{i=1}^n \lVert t_i - ν_θ(s_i) \rVert_2^2. \] 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 $\sigma (x)=1/(1+\exp (-x))$ and $\sigma (x)=\tanh (x)$, such failure to attain an infimum can happen on a positive-measured subset of responses. For the ReLU activation $\sigma (x)=\max (0,x)$, 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.

19 pages