On the approximation of functions by tanh neural networks
This provides theoretical guarantees for function approximation in machine learning, though it is incremental as it focuses on a specific activation function and comparison.
The paper tackles the problem of approximating Sobolev-regular and analytic functions using tanh neural networks, deriving explicit error bounds in high-order Sobolev norms and showing that tanh networks with only two hidden layers achieve comparable or better rates than deeper ReLU networks.
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.