Training a single sigmoidal neuron is hard

We first present a brief survey of hardness results for training feedforward neural networks. These results are then completed by the proof that the simplest architecture containing only a single neuron that applies a sigmoidal activation function sigma: kappa --> [alpha, beta], satisfying certain natural axioms (e.g., the standard (logistic) sigmoid or saturated-linear function), to the weighted sum of n inputs is hard to train. In particular, the problem of finding the weights of such a unit that minimize the quadratic training error within (beta - alpha)(2) or its average (over a training set) within 5(beta - alpha)(2)/ (12n) of its infimum proves to be NP-hard. Hence, the well-known backpropagation learning algorithm appears not to be efficient even for one neuron, which has negative consequences in constructive learning.