On the approximation of functions by tanh neural networks

TL;DR

This paper establishes explicit error bounds for approximating Sobolev and analytic functions using shallow tanh neural networks, demonstrating their efficiency compared to deeper ReLU networks.

Contribution

It provides the first explicit Sobolev norm error bounds for shallow tanh neural networks and compares their approximation rates favorably to deeper ReLU networks.

Findings

Tanh neural networks with two hidden layers can approximate functions as well as or better than deeper ReLU networks.

Explicit error bounds are derived in high-order Sobolev norms.

Shallow tanh networks are effective for high-regularity function approximation.

Abstract

We derive bounds on the error, in high-order Sobolev norms, incurred in the approximation of Sobolev-regular as well as analytic functions by neural networks with the hyperbolic tangent activation function. These bounds provide explicit estimates on the approximation error with respect to the size of the neural networks. We show that tanh neural networks with only two hidden layers suffice to approximate functions at comparable or better rates than much deeper ReLU neural networks.