Abstract. One of the surprising recurring phenomena observed in experiments with boosting is that the test error of the generated classifier usually does not increase as its size becomes very large, and often is observed to decrease even after the training error reaches zero. In this paper, we show that this phenomenon is related to the distribution of margins of the training examples with respect to the generated voting classification rule, where the margin of an example is simply the difference between the number of correct votes and the maximum number of votes received by any incorrect label. We show that techniques used in the analysis of Vapnik’s support vector classifiers and of neural networks with small weights can be applied to voting methods to relate the margin distribution to the test error. We also show theoretically and experimentally that boosting is especially effective at increasing the margins of the training examples. Finally, we compare our explanation to those based on the bias-variance decomposition. 1

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We had previously shown that regularization principles lead to approximation schemes which are equivalent to networks with one layer of hidden units, called Regularization Networks. In particular, standard smoothness functionals lead to a subclass of regularization networks, the well known Radial Basis Functions approximation schemes. This paper shows that regularization networks encompass a much broader range of approximation schemes, including many of the popular general additive models and some of the neural networks. In particular, we introduce new classes of smoothness functionals that lead to different classes of basis functions. Additive splines as well as some tensor product splines can be obtained from appropriate classes of smoothness functionals. Furthermore, the same generalization that extends Radial Basis Functions (RBF) to Hyper Basis Functions (HBF) also leads from additive models to ridge approximation models, containing as special cases Breiman's hinge functions, som...

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 p (log n)=m (ignori...

In dimensions two and higher, wavelets can efficiently represent only a small range of the full diversity of interesting behavior. In effect, wavelets are welladapted for pointlike phenomena, whereas in dimensions greater than one, interesting phenomena can be organized along lines, hyperplanes, and other nonpointlike structures, for which wavelets are poorly adapted. We discuss in this paper a new subject, ridgelet analysis, which can effectively deal with linelike phenomena in dimension 2, planelike phenomena in dimension 3 and so on. It encompasses a collection of tools which all begin from the idea of analysis by ridge functions ψ(u1x1+...+unxn) whose ridge profiles ψ are wavelets, or alternatively from performing a wavelet analysis in the Radon domain. The paper reviews recent work on the continuous ridgelet transform (CRT), ridgelet frames, ridgelet orthonormal bases, ridgelets and edges and describes a new notion of smoothness naturally attached to this new representation.

In this paper, we present some general results determining minimax bounds on statistical risk for density estimation based on certain information-theoretic considerations. These bounds depend only on metric entropy conditions and are used to identify the minimax rates of convergence.

We show that the class of two layer neural networks with bounded fan-in is efficiently learnable in a realistic extension to the Probably Approximately Correct (PAC) learning model. In this model, a joint probability distribution is assumed to exist on the observations and the learner is required to approximate the neural network which minimizes the expected quadratic error. As special cases, the model allows learning real-valued functions with bounded noise, learning probabilistic concepts and learning the best approximation to a target function that cannot be well approximated by the neural network. The networks we consider have real-valued inputs and outputs, an unlimited number of threshold hidden units with bounded fan-in, and a bound on the sum of the absolute values of the output weights. The number of computation This work was supported by the Australian Research Council and the Australian Telecommunications and Electronics Research Board. The material in this paper was pres...

Gaussian mixtures (or so-called radial basis function networks) for density estimation provide a natural counterpart to sigmoidal neural networks for function tting and approximation. In both cases, it is possible to give simple expressions for the iterative improvement of performance as components of the network are introduced one at a time. In particular, for mixture density estimation we show that a k-component mixture estimated by maximum likelihood (or by an iterative likelihood improvement that we introduce) achieves log-likelihood within order 1=k of the log-likelihood achievable by any convex combination. Consequences for approximation and estimation using Kullback-Leibler risk are also given. A Minimum Description Length principle selects the optimal number of components k that minimizes the risk bound. 1 Introduction In density estimation, Gaussian mixtures provide exible-basis representations for densities that can be used to model heterogeneous data in high dimensions....

In this survey paper, we review the constructive algorithms for structure learning in feedforward neural networks for regression problems. The basic idea is to start with a small network, then add hidden units and weights incrementally until a satisfactory solution is found. By formulating the whole problem as a state space search, we first describe the general issues in constructive algorithms, with special emphasis on the search strategy. A taxonomy, based on the differences in the state transition mapping, the training algorithm and the network architecture, is then presented. Keywords--- Constructive algorithm, structure learning, state space search, dynamic node creation, projection pursuit regression, cascade-correlation, resource-allocating network, group method of data handling. I. Introduction A. Problems with Fixed Size Networks I N recent years, many neural network models have been proposed for pattern classification, function approximation and regression problems. Among...

Feedforward networks are a class of regression techniques that can be used to learn to perform some task from a set of examples. The question of generalization of network performance from a finite training set to unseen data is clearly of crucial importance. In this article we first show that the generalization error can be decomposed in two terms: the approximation error, due to the insufficient representational capacity of a finite sized network, and the estimation error, due to insufficient information about the target function because of the finite number of samples. We then consider the problem of approximating functions belonging to certain Sobolev spaces with Gaussian Radial Basis Functions. Using the above mentioned decomposition we bound the generalization error in terms of the number of basis functions and number of examples. While the bound that we derive is specific for Radial Basis Functions, a number of observations deriving from it apply to any approximation t...