Let zN denote a given set of N training examples. Let QN(zN) denote the expected squared error (the expectation taken over all possible examples) of the network after being trained on zN. This measures the quality of fit afforded by training on a given set of N examples. Let IMSEN denote the Integrated Mean Squared Error for training sets of size N. Given reasonable assumptions, it is straightforward to show that IMSEN E[Q N(ZN)] - 0"2, where the expectation is now over all training sets of size N, ZN is a random training set of size N, and 0"2 is the noise variance. Let CN CN(zN) denote the "delete-one cross-validation" squared error measure for a network trained on zN.

Learning of continuous valued functions using neural network en(cid:173) sembles (committees) can give improved accuracy, reliable estima(cid:173) tion of the generalization error, and active learning. The ambiguity is defined as the variation of the output of ensemble members aver(cid:173) aged over unlabeled data, so it quantifies the disagreement among the networks. It is discussed how to use the ambiguity in combina(cid:173) tion with cross-validation to give a reliable estimate of the ensemble generalization error, and how this type of ensemble cross-validation can sometimes improve performance. It is shown how to estimate the optimal weights of the ensemble members using unlabeled data. By a generalization of query by committee, it is finally shown how the ambiguity can be used to select new training data to be labeled in an active learning scheme.

We present a neural network-based face detection system. A retinally connected neural network examines small windows of an image, and decides whether each window contains a face. The system arbitrates between multiple networks to improve performance over a single network. We use a bootstrap algorithm for training, which adds false detections into the training set as training progresses. This eliminates the difficult task of manually selecting non-face training examples, which must be chosen to span the entire space of non-face images.

A statistical theory for overtraining is proposed. The analysis treats realizable stochastic neural networks, trained with Kullback(cid:173) Leibler loss in the asymptotic case. It is shown that the asymptotic gain in the generalization error is small if we perform early stop(cid:173) ping, even if we have access to the optimal stopping time. Consider(cid:173) ing cross-validation stopping we answer the question: In what ratio the examples should be divided into training and testing sets in or(cid:173) der to obtain the optimum performance. In the non-asymptotic region cross-validated early stopping always decreases the general(cid:173) ization error.

In this paper, we discuss regularisation in online/sequential learn(cid:173) ing algorithms. In environments where data arrives sequentially, techniques such as cross-validation to achieve regularisation or model selection are not possible. Further, bootstrapping to de(cid:173) termine a confidence level is not practical. To surmount these problems, a minimum variance estimation approach that makes use of the extended Kalman algorithm for training multi-layer percep(cid:173) trons is employed. The novel contribution of this paper is to show the theoretical links between extended Kalman filtering, Sutton's variable learning rate algorithms and Mackay's Bayesian estima(cid:173) tion framework.

This paper presents a method for obtaining class membership probability esti- mates for multiclass classification problems by coupling the probability estimates produced by binary classifiers. This is an extension for arbitrary code matrices of a method due to Hastie and Tibshirani for pairwise coupling of probability estimates. Experimental results with Boosted Naive Bayes show that our method produces calibrated class membership probability estimates, while having similar classification accuracy as loss-based decoding, a method for obtaining the most likely class that does not generate probability estimates.

We compare discriminative and generative learning as typified by logistic regression and naive Bayes. We show, contrary to a widely(cid:173) held belief that discriminative classifiers are almost always to be preferred, that there can often be two distinct regimes of per(cid:173) formance as the training set size is increased, one in which each algorithm does better. This stems from the observation- which is borne out in repeated experiments- that while discriminative learning has lower asymptotic error, a generative classifier may also approach its (higher) asymptotic error much faster.

When designing a two-alternative classifier, one ordinarily aims to maximize the classifier's ability to discriminate between members of the two classes. We describe a situation in a real-world business application of machine-learning prediction in which an additional constraint is placed on the nature of the solu- tion: that the classifier achieve a specified correct acceptance or correct rejection rate (i.e., that it achieve a fixed accuracy on members of one class or the other). Our domain is predicting churn in the telecommunications industry. Churn refers to customers who switch from one service provider to another. We pro- pose four algorithms for training a classifier subject to this domain constraint, and present results showing that each algorithm yields a reliable improvement in performance.

This paper develops a new approach for extremely fast detection in do- mains where the distribution of positive and negative examples is highly skewed (e.g. In such domains a cascade of simple classifiers each trained to achieve high detection rates and modest false positive rates can yield a final detector with many desir- able features: including high detection rates, very low false positive rates, and fast performance. Achieving extremely high detection rates, rather than low error, is not a task typically addressed by machine learning al- gorithms. We propose a new variant of AdaBoost as a mechanism for training the simple classifiers used in the cascade. The final face detection system can process 15 frames per second, achieves over 90% detection, and a false positive rate of 1 in a 1,000,000.

The area under an ROC curve (AUC) is a criterion used in many appli- cations to measure the quality of a classification algorithm. However, the objective function optimized in most of these algorithms is the error rate and not the AUC value. We give a detailed statistical analysis of the relationship between the AUC and the error rate, including the first exact expression of the expected value and the variance of the AUC for a fixed error rate. Our results show that the average AUC is monotonically in- creasing as a function of the classification accuracy, but that the standard deviation for uneven distributions and higher error rates is noticeable. Thus, algorithms designed to minimize the error rate may not lead to the best possible AUC values.