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Training a Single Sigmoidal Neuron Is Hard

Jií íma

sima{at}cs.cas.cz, Institute of Computer Science, Academy of Sciences of the Czech Republic, P.O. Box 5, 18207 Prague 8, Czech Republic

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 : [,ß], 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 (ß-)² or its average (over a training set) within 5(ß-)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.

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