Complexity Results on Learning by Neural Nets

J.-H. Lin and J. S. Vitter. ``Complexity Results on Learning by Neural Nets,'' Machine Learning, 6, 1991, 211-230. A shortened version appears in Proceedings of the 2nd Annual ACM Workshop on Computational Learning Theory (COLT '89), Santa Cruz, CA, July-August 1989, published by Morgan Kaufmann, San Mateo, CA, 118-133.

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We consider the computational complexity of learning by neural nets. We are interested in how hard it is to design appropriate neural net architectures and to train neural nets for general and specialized learning tasks. Our main result shows that the training problem for 2-cascade neural nets (which have only two non-input nodes, one of which is hidden) is $\cal NP$-complete, which implies that finding an optimal net (in terms of the number of non-input units) that is consistent with a set of examples is also $\cal NP$-complete. This result also demonstrates a surprising gap between the computational complexities of one-node (perceptron) and two-node neural net training problems, since the perceptron training problem can be solved in polynomial time by linear programming techniques. We conjecture that training a k-cascade neural net, which is a classical threshold network training problem, is also $\cal NP$-complete, for each fixed $k \geq 2$. We also show that the problem of finding an optimal perceptron (in terms of the number of non-zero weights) consistent with a set of training examples is $\cal NP$-hard.

Our neural net learning model encapsulates the idea of modular neural nets, which is a popular approach to overcoming the scaling problem in training neural nets. We investigate how much easier the training problem becomes if the class of concepts to be learned is known a priori and the net architecture is allowed to be sufficiently non-optimal. Finally, we classify several neural net optimization problems within the polynomial-time hierarchy.