On the Expressive Power of Neural Networks

In 1989 George Cybenko proved in a landmark paper that wide shallow neural networks can approximate arbitrary continuous functions on a compact set. This universal approximation theorem sparked a lot of follow-up research. Shen, Yang and Zhang determined optimal approximation rates for ReLU-networks in Lp-norms with p ∈ [1,∞). Kidger and Lyons proved a universal approximation theorem for deep narrow ReLU-networks. Telgarsky gave an example of a deep narrow ReLU-network that cannot be approximated by a wide shallow ReLU-network unless it has exponentially many neurons. However, there are even more questions that still remain unresolved. Are there any wide shallow ReLU-networks that cannot be approximated well by deep narrow ReLU-networks? Is the universal approximation theorem still true for other norms like the Sobolev norm W1,1? Do these results hold for activation functions other than ReLU? We will answer all of those questions and more with a framework of two expressive powers. The first one is well-known and counts the maximal number of linear regions of a function calculated by a ReLU-network. We will improve the best known bounds for this expressive power. The second one is entirely new.