We study tltt' problem of when to stop If'arning a class of feedforward networks - networks with linear outputs I1PUrOIl and fixed input weights - when they are trained with a gradient descent algorithm on a finite number of examples. Under general regularity conditions, it is shown that there a.re in general three distinct phases in the generalization performance in the learning process, and in particular, the network has hetter gt'neralization pPTformance when learning is stopped at a certain time before til(' global miniIl111lu of the empirical error is reachert. A notion of effective size of a machine is rtefil1e i and used to explain the tradeoff betwf'en the complexity of the marhine and the training error ill the learning process. The study leads nat.urally to a network size selection critt'rion, which turns Ol1t to be a generalization of Akaike's Information Criterioll for the It'arning process. It if; shown that stopping Iparning before tiJt' global minimum of the empirical error has the effect of network size splectioll. 1 INTRODUCTION The primary goal of learning in neural nets is to find a network that gives valid generalization. In achieving this goal, a central issue is the tradeoff between the training error and network complexity. This usually reduces to a problem of network size selection, which has drawn much research effort in recent years. Various principles, theories, and intuitions, including Occam's razor, statistical model selection criteria such as Akaike's Information Criterion (AIC) [11 and many others [5, 1, 10,3,111 all quantitatively support the following PAC prescription: between two machines which have the same empirical error, the machine with smaller VC-dimf'nsion generalizes better. However, it is noted that these methods or criteria do not npcpssarily If'ad to optimal (or llearly optimal) generalization performance.

Such decisions include which function approximator to use, how to trade smoothness for goodness of fit and which features arerelevant.

Recently, Ott, Grebogi and Yorke (OGY) [6] found an effective method to control chaotic systems to unstable fixed points by using onlysmall control forces; however, OGY's method is based on and limited to a linear theory and requires considerable knowledge of the dynamics of the system to be controlled. In this paper we use two radial basis function networks: one as a model of an unknown plant and the other as the controller. The controller is trained with a recurrent learning algorithm to minimize a novel objective function such that the controller can locate an unstable fixed point and drive the system into the fixed point with no a priori knowledge ofthe system dynamics. Our results indicate that the neural controller offers many advantages over OGY's technique.

EE., Linkoping University S-58183 Linkoping Sweden Abstract We describe a number of learning rules that can be used to train unsupervised parallelfeature extraction systems. The learning rules are derived using gradient ascent of a quality function. We consider anumber of quality functions that are rational functions of higher order moments of the extracted feature values. We show that one system learns the principle components of the correlation matrix.Principal component analysis systems are usually not optimal feature extractors for classification. Therefore we design quality functions which produce feature vectors that support unsupervised classification.The properties of the different systems are compared with the help of different artificially designed datasets and a database consisting of all Munsell color spectra. 1 Introduction There are a number of unsupervised Hebbian learning algorithms (see Oja, 1992 and references therein) that perform some version of the Karhunen-Loeve expansion.

Center positions are continuously updated through soft competitive learning. The width of the radial basis functions is derived from the distance to topological neighbors. During the training the observed error is accumulated locally and used to determine where to insert the next unit. This leads (in case of classification problems) to the placement of units near class borders rather than near frequency peaks as is done by most existing methods. The resulting networks need few training epochs and seem to generalize very well. This is demonstrated by examples.

Bumptrees are geometric data structures introduced by Omohundro (1991) to provide efficient access to a collection of functions on a Euclidean space of interest. We describe a modified bumptree structure that has been employed as a neural network classifier, and compare its performance on several classification tasks against that of radial basis function networks and the standard mutIi-Iayer perceptron. 1 INTRODUCTION A number of neural network studies have demonstrated the utility of the multi-layer perceptron (MLP) and shown it to be a highly effective paradigm. Studies have also shown, however, that the MLP is not without its problems, in particular it requires an extensive training time, is susceptible to local minima problems and its perfonnance is dependent upon its internal network architecture. In an attempt to improve upon the generalisation performance and computational efficiency a number of studies have been undertaken principally concerned with investigating the parametrisation of the MLP. It is well known, for example, that the generalisation performance of the MLP is affected by the number of hidden units in the network, which have to be determined empirically since theory provides no guidance.

Researchers often try to understand-post hoc-representations that emerge in the hidden layers of a neural net following training. Interpretation is difficult because these representations are typically highly distributed and continuous. By "continuous," wemean that if one constructed a scatterplot over the hidden unit activity space of patterns obtained in response to various inputs, examination at any scale would reveal the patterns to be broadly distributed over the space.

Several recurrent networks have been proposed as representations for the task of formal language learning. After training a recurrent network recognize aformal language or predict the next symbol of a sequence, the next logical step is to understand the information processing carried out by the network. Some researchers have begun to extracting finite state machines from the internal state trajectories of their recurrent networks. This paper describes how sensitivity to initial conditions and discrete measurements can trick these extraction methods to return illusory finite state descriptions.

Critical features of a domain (such as invariance under translation, rotation, and permu- Clustering with a Domain-Specific Distance Measure 103 tation) are captured within the clustering procedure, rather than reflected in the properties of feature sets created prior to clustering. The distance measure and learning problem are formally described as nested objective functions. We derive an efficient algorithm by using optimization techniques that allow us to divide up the objective function into parts which may be minimized in distinct phases. The algorithm has accurately recreated 10 prototypes from a randomly generated sample database of 100 images consisting of 20 points each in 120 experiments. Finally, by incorporating permutation invariance in our distance measure, we have a technique that we may be able to apply to the clustering of graphs. Our goal is to develop measures which will enable the learning of objects with shape or structure. Acknowledgements This work has been supported by AFOSR grant F49620-92-J-0465 and ONR/DARPA grant N00014-92-J-4048.

Given such a classification task in most cases it is not too difficult to devise a network architecture that is capable of learning the input-output relation as represented by a number of training examples.