On Sharpness of Error Bounds for Multivariate Neural Network Approximation

Sharpness of error bounds for best non-linear multivariate approximation by sums of logistic activation functions and piecewise polynomials is investigated. The error bounds are given in terms of moduli of smoothness. They describe approximation properties of single hidden layer feedforward neural networks with multiple input nodes. Sharpness with respect to Lipschitz classes is established by constructing counterexamples with a non-linear, quantitative extension of the uniform boundedness principle.