On approximate learning by multi-layered feedforward circuits

Afterwards, we consider the objective to minimize the failure ratio in the presence of misclassification errors. We show that it is NP-hard to approximate the failure ratio within any positive constant for a multilayered threshold network with varying input dimension and a fixed number of neurons in the hidden layer if the thresholds of the neurons in the first hidden layer are zero. Furthermore, even obtaining weak approximations is almost NP-hard in the same situation.