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.

Stochastic optimization algorithms typically use learning rate schedules that behave asymptotically as J.t(t)

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

The biological network is small and relatively well understood, and the silicon model can therefore span three levels of organization in the leech nervous system (neuron, ganglion, system); it represents one of the first comprehensive models of leech swimming operating in real-time. The circuit employs biophysically motivated analog neurons networked to form multiple biologically inspired silicon ganglia. These ganglia are coupled using known interganglionic connections. Thus the model retains the flavor of its biological counterpart, and though simplified, the output of the silicon circuit is similar to the output of the leech swim central pattern generator. The model operates on the same time-and spatial-scale as the leech nervous system and will provide an excellent platform with which to explore real-time adaptive locomotion in the leech and other "simple" invertebrate nervous systems.

H. Schwarze Dept. of Theoretical Physics Lund University Solvegatan 14A 223 62 Lund Sweden J.Hertz Nordita Blegdamsvej 17 2100 Copenhagen 0 Denmark Abstract The problem of learning from examples in multilayer networks is studied within the framework of statistical mechanics. Using the replica formalism we calculate the average generalization error of a fully connected committee machine in the limit of a large number of hidden units. If the number of training examples is proportional to the number of inputs in the network, the generalization error as a function of the training set size approaches a finite value. If the number of training examples is proportional to the number of weights in the network we find first-order phase transitions with a discontinuous drop in the generalization error for both binary and continuous weights. 1 INTRODUCTION Feedforward neural networks are widely used as nonlinear, parametric models for the solution of classification tasks and function approximation. Trained from examples of a given task, they are able to generalize, i.e. to compute the correct output for new, unknown inputs.

We study the complexity problem in artificial feedforward neural networks designed to approximate real valued functions of several real variables; i.e., we estimate the number of neurons in a network required to ensure a given degree of approximation to every function in a given function class. We indicate how to construct networks with the indicated number of neurons evaluating standard activation functions. Our general theorem shows that the smoother the activation function, the better the rate of approximation. 1 INTRODUCTION The approximation capabilities of feedforward neural networks with a single hidden layer has been studied by many authors, e.g., [1, 2, 5]. In [10], we have shown that such a network using practically any nonlinear activation function can approximate any continuous function of any number of real variables on any compact set to any desired degree of accuracy. A central question in this theory is the following.

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.

Automated speech recognition is notoriously hard, and thus any predictive source of information and constraints that could be incorporated into a computer speech recognition system would be desirable. Humans, especially the hearing impaired, can utilize visual information - "speech reading" - for improved accuracy (Dodd & Campbell, 1987, Sanders & Goodrich, 1971). Speech reading can provide direct information about segments, phonemes, rate, speaker gender and identity, and subtle informationfor segmenting speech from background noise or multiple speakers (De Filippo & Sims, 1988, Green & Miller, 1985). Fundamental support for the use of visual information comes from the complementary natureof the visual and acoustic speech signals. Utterances that are difficult to distinguish acoustically are the easiest to distinguish.

Scope This workshop reviewed and classified the various models which have emerged from the general concept of selective attention and context dependency, and sought to identify their commonalities. It was concluded that the motivation and mechanism ofthese functional models are "efficiency" and ''factoring'', respectively. The workshop focused on computational models of selective attention and context dependency withinthe realm of neural networks. We treated only ''functional'' models; computational models of biological neural systems, and symbolic or rule-based systems were omitted from the discussion. Presentations Thomas H. Hildebrandt presented the results of his recent survey of the literature onfunctional models of selective attention and context dependency.

We developed a system for finding address blocks on mail pieces that can process four images per second. Besides locating the address block, our system also determines the writing style, handwritten or machine printed, and moreover, it measures the skew angle of the text lines and cleans noisy images. A layout analysis of all the elements present in the image is performed in order to distinguish drawings and dirt from text and to separate text of advertisement from that of the destination address. A speed of more than four images per second is obtained on a modular hardware platform, containing a board with two of the NET32K neural net chips, a SPARC2 processor board, and a board with 2 digital signal processors. The system has been tested with more than 100,000 images. Its performance depends on the quality of the images, and lies between 85% correct location in very noisy images to over 98% in cleaner images.