The sample complexity of pattern classification 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 classification 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 classification problem and the learning algorithm finds 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 feedforward 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 misclassification probability is no more than a certain error estimate (that is related to squared error on the training set) plus A3 √((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 examples is considerably smaller than the number of weights. It also supports heuristics (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 classifiers: when the input domain is a totally bounded metric space, we use the same approach to give upper bounds on misclassification probability for classifiers with decision boundaries that are far from the training examples.

A fuzzy neural network approach for contractor pre-qualification

Nonlinearity, uncertainty and subjectivity are the three predominant characteristics of contractors prequalification which cause the process more of an art than a scientific evaluation. A fuzzy neural network (FNN) model, amalgamating both the fuzzy set and neural network theories, has been developed aiming to improve the objectiveness of contractor prequalification. Through the FNN theory, the fuzzy rules as used by the prequalifiers can be identified and the corresponding membership functions can be transformed. Eighty-five cases with detailed decision criteria and rules for prequalifying Hong Kong civil engineering contractors were collected. These cases were used for training (calibrating) and testing the FNN model. The performance of the FNN model was compared with the original results produced by the prequalifiers and those generated by the general feedforward neural network (GFNN, i.e. a crisp neural network) approach. Contractor’s ranking orders, the model efficiency (R2) and the mean absolute percentage error (MAPE) were examined during the testing phase. These results indicate the applicability of the neural network approach for contractor prequalification and the benefits of the FNN model over the GFNN model. The FNN is a practical approach for modelling contractor prequalification.