Hardness results for neural network approximation problems

We consider the problem of efficiently learning in two-layer neural networks. We investigate the computational complexity of agnostically learning with simple families of neural networks as the hypothesis classes. We show that it is NP-hard to find a linear threshold network of a fixed size that approximately minimizes the proportion of misclassified examples in a training set, even if there is a network that correctly classifies all of the training examples. In particular, for a training set that is correctly classified by some two-layer linear threshold network with k hidden units, it is NP-hard to find such a network that makes mistakes on a proportion smaller than c/k² of the examples, for some constant c. We prove a similar result for the problem of approximately minimizing the quadratic loss of a two-layer network with a sigmoid output unit.