Deep, Skinny Neural Networks are not Universal Approximators

Neural networks have become the model of choice in a variety of machine learning applications, due to their flexibility and generality. However, selecting network architectures and other hyperparameters is typically a matter of trial and error. To make the choice of neural network architecture more straightforward, we need to understand the limits of each architecture, both in terms of what kinds of functions any given network architecture can approximate and how those limitations impact its ability to learn functions within those limits. A number of papers [3, 6, 11, 13] have shown that neural networks with a single hidden layer are a universal approximator, i.e. that they can approximate any continuous function on a compact domain to arbitrary accuracy if the hidden layer is allowed to have an arbitrarily high dimension. In practice, however, the neural networks that have proved most effective tend to have a large number of relatively low-dimensional hidden layers. This raises the question of whether neural networks with an arbitrary number of hidden layers of bounded dimension are also a universal approximator.