The Sample Complexity of Pattern Classiication with Neural Networks: the Size of the Weights Is More Important than the Size of the Network

Sample complexity results from computational learning theory, when applied to neural network learning for pattern classiication problems, suggest that for good generalization performance the number of training examples should grow at least linearly with the number of adjustable parameters in the network. Results in this paper show that if a large neural network is used for a pattern classiication problem and the learning algorithm nds a network with small weights that has small squared error on the training patterns, then the generalization performance depends on the size of the weights rather than the number of weights. For example, consider a two-layer feed-forward network of sigmoid units, in which the sum of the magnitudes of the weights associated with each unit is bounded by A and the input dimension is n. We show that the misclassiication probability is no more than a certain error estimate (that is related to squared error on the training set) plus A 3 p (log n)=m (ignoring log A and log m factors), where m is the number of training patterns. This may explain the generalization performance of neural networks, particularly when the number of training 1 examples is considerably smaller than the number of weights. It also supports heuris-tics (such as weight decay and early stopping) that attempt to keep the weights small during training. The proof techniques appear to be useful for the analysis of other pattern classiiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassiication probability for classiiers with decision boundaries that are far from the training examples.