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A new non-linear variant of a quantitative extension of the uniform boundedness principle is used to show sharpness of error bounds for univariate approximation by sums of sigmoid and ReLU functions. Single hidden layer feedforward neural networks with one input node perform such operations. Errors of best approximation can be expressed using moduli of smoothness of the function to be approximated (i.e., to be learned). In this context, the quantitative extension of the uniform boundedness principle indeed allows to construct counterexamples that show approximation rates to be best possible. Approximation errors do not belong to the little-o class of given bounds. By choosing piecewise linear activation functions, the discussed problem becomes free knot spline approximation. Results of the present paper also hold for non-polynomial (and not piecewise defined) activation functions like inverse tangent. Based on Vapnik–Chervonenkis dimension, first results are shown for the logistic function.

On Sharpness of Error Bounds for Univariate Approximation by Single Hidden Layer Feedforward Neural Networks