Rep. Germany Abstract Many techniques for model selection in the field of neural networks correspond to well established statistical methods. The method of'stopped training', on the other hand, in which an oversized network is trained until the error on a further validation set of examples deteriorates,then training is stopped, is a true innovation, since model selection doesn't require convergence of the training process. Inthis paper we show that this performance can be significantly enhanced by extending the'nonconvergent model selection method' of stopped training to include dynamic topology modifications (dynamic weight pruning) and modified complexity penalty term methods in which the weighting of the penalty term is adjusted during the training process. 1 INTRODUCTION One of the central topics in the field of neural networks is that of model selection. Both the theoretical and practical side of this have been intensively investigated and a vast array of methods have been suggested to perform this task. A widely used class of techniques starts by choosing an'oversized' network architecture then either removing redundant elements based on some measure of saliency (pruning), adding a further term to the cost function penalizing complexity (penalty terms), and finally, observing the error on a further validation set of examples, then stopping training as soon as this performance begins to deteriorate (stopped training).

Inst., 19600 NW vonNeumann Dr, Beaverton, OR 97006 Abstract We propose a very simple, and well principled way of computing the optimal step size in gradient descent algorithms. The online version is very efficient computationally, and is applicable to large backpropagation networks trained on large data sets. The main ingredient is a technique for estimating the principal eigenvalue(s) and eigenvector(s) of the objective function's second derivative matrix (Hessian),which does not require to even calculate the Hessian. Severalother applications of this technique are proposed for speeding up learning, or for eliminating useless parameters. 1 INTRODUCTION Choosing the appropriate learning rate, or step size, in a gradient descent procedure such as backpropagation, is simultaneously one of the most crucial and expertintensive partof neural-network learning. We propose a method for computing the best step size which is both well-principled, simple, very cheap computationally, and, most of all, applicable to online training with large networks and data sets.

A new boundary hunting radial basis function (BH-RBF) classifier which allocates RBF centers constructively near class boundaries is described. This classifier creates complex decision boundaries only in regions where confusions occur and corresponding RBF outputs are similar. A predicted square error measure is used to determine how many centers to add and to determine when to stop adding centers. Two experiments are presented which demonstrate the advantages of the BH RBF classifier. One uses artificial data with two classes and two input features where each class contains four clusters but only one cluster is near a decision region boundary.

Given a set oftraining examples, determining the appropriate number offree parameters is a challenging problem. Constructive learning algorithms attempt to solve this problem automatically by adding hidden units, and therefore free parameters, during learning. Weexplore an alternative class of algorithms-called metamorphosis algorithms-inwhich the number of units is fixed, but the number of free parameters gradually increases during learning. The architecture we investigate is composed of RBF units on a lattice, whichimposes flexible constraints on the parameters of the network. Virtues of this approach include variable subset selection, robustparameter selection, multiresolution processing, and interpolation of sparse training data.

Memory-based classification algorithms such as radial basis functions orK-nearest neighbors typically rely on simple distances (Euclidean, dotproduct ...), which are not particularly meaningful on pattern vectors. More complex, better suited distance measures are often expensive and rather ad-hoc (elastic matching, deformable templates). We propose a new distance measure which (a) can be made locally invariant to any set of transformations of the input and (b) can be computed efficiently. We tested the method on large handwritten character databases provided by the Post Office and the NIST. Using invariances with respect to translation, rotation, scaling,shearing and line thickness, the method consistently outperformed all other systems tested on the same databases.

Holographic Recurrent Networks (HRNs) are recurrent networks which incorporate associative memory techniques for storing sequential structure.HRNs can be easily and quickly trained using gradient descent techniques to generate sequences of discrete outputs andtrajectories through continuous spaee. The performance of HRNs is found to be superior to that of ordinary recurrent networks onthese sequence generation tasks. 1 INTRODUCTION The representation and processing of data with complex structure in neural networks remains a challenge. In a previous paper [Plate, 1991b] I described Holographic Reduced Representations(HRRs) which use circular-convolution associative-memory to embody sequential and recursive structure in fixed-width distributed representations. Thispaper introduces Holographic Recurrent Networks (HRNs), which are recurrent nets that incorporate these techniques for generating sequences of symbols or trajectories through continuous space.

"Best-first model merging" is a general technique for dynamically choosing the structure of a neural or related architecture while avoiding overfitting. It is applicable to both leaming and recognition tasks and often generalizes significantly better than fixed structures. We demonstrate the approach applied to the tasks of choosing radial basis functions for function learning, choosing local affine models for curve and constraint surface modelling, and choosing the structure of a balltree or bumptree to maximize efficiency of access.

Three methods for improving the performance of (gaussian) radial basis function (RBF) networks were tested on the NETtaik task. In RBF, a new example is classified by computing its Euclidean distance to a set of centers chosen by unsupervised methods. The application of supervised learning to learn a non-Euclidean distance metric was found to reduce the error rate of RBF networks, while supervised learning of each center's variance resulted in inferior performance. The best improvement in accuracy was achieved by networks called generalized radial basis function (GRBF) networks. In GRBF, the center locations are determined by supervised learning. After training on 1000 words, RBF classifies 56.5% of letters correct, while GRBF scores 73.4% letters correct (on a separate test set). From these and other experiments, we conclude that supervised learning of center locations can be very important for radial basis function learning.

The method of Structural Risk Minimization refers to tuning the capacity of the classifier to the available amount of training data. This capacity is influenced by several factors, including: (1) properties of the input space, (2) nature and structure of the classifier, and (3) learning algorithm. Actions based on these three factors are combined here to control the capacity of linear classifiers and improve generalization on the problem of handwritten digit recognition.

Three methods for improving the performance of (gaussian) radial basis function (RBF) networks were tested on the NETtaik task. In RBF, a new example is classified by computing its Euclidean distance to a set of centers chosen by unsupervised methods. The application of supervised learning to learn a non-Euclidean distance metric was found to reduce the error rate of RBF networks, while supervised learning of each center's variance resulted in inferior performance. The best improvement in accuracy was achieved by networks called generalized radial basis function (GRBF) networks. In GRBF, the center locations are determined by supervised learning. After training on 1000 words, RBF classifies 56.5% of letters correct, while GRBF scores 73.4% letters correct (on a separate test set). From these and other experiments, we conclude that supervised learning of center locations can be very important for radial basis function learning.