We have calculated, both analytically and in simulations, the rate of convergence at long times in the backpropagation learning algorithm fornetworks with and without hidden units. Our basic finding for units using the standard sigmoid transfer function is lit convergence of the error for large t, with at most logarithmic corrections fornetworks with hidden units. Other transfer functions may lead to a 8lower polynomial rate of convergence. Our analytic calculations were presented in (Tesauro, He & Ahamd, 1989). Here we focus in more detail on our empirical measurements of the convergence ratein numerical simulations, which confirm our analytic results.

We present an application of back-propagation networks to handwritten digitrecognition. Minimal preprocessing of the data was required, but architecture of the network was highly constrained and specifically designed for the task. The input of the network consists of normalized images of isolated digits. The method has 1 % error rate and about a 9% reject rate on zipcode digits provided by the U.S. Postal Service. 1 INTRODUCTION The main point of this paper is to show that large back-propagation (BP) networks canbe applied to real image-recognition problems without a large, complex preprocessing stage requiring detailed engineering. Unlike most previous work on the subject (Denker et al., 1989), the learning network is directly fed with images, rather than feature vectors, thus demonstrating the ability of BP networks to deal with large amounts of low level information. Previous work performed on simple digit images (Le Cun, 1989) showed that the architecture of the network strongly influences the network's generalization ability. Good generalization can only be obtained by designing a network architecture that contains a certain amount of a priori knowledge about the problem. The basic design principleis to minimize the number of free parameters that must be determined by the learning algorithm, without overly reducing the computational power of the network.

Selective sampling

A new concept for unsupervised learning based upon examples introduced tothe neural network is proposed. Each example is considered as an interpolation node of the velocity field in the phase space. The velocities at these nodes are selected such that all the streamlines converge to an attracting set imbedded in the subspace occupied by the cluster of examples. The synaptic interconnections are found from learning procedure providing selected field. The theory is illustrated by examples. This paper is devoted to development of a new concept for unsupervised learning based upon examples introduced to an artificial neural network.

Cascade-Correlation is a new architecture and supervised learning algorithm forartificial neural networks. Instead of just adjusting the weights in a network of fixed topology. Cascade-Correlation begins with a minimal network,then automatically trains and adds new hidden units one by one, creating a multi-layer structure. Once a new hidden unit has been added to the network, its input-side weights are frozen. This unit then becomes a permanent feature-detector in the network, available for producing outputs or for creating other, more complex feature detectors. TheCascade-Correlation architecture has several advantages over existing algorithms: it learns very quickly, the network .determines

We introduce a cost function for learning in feed-forward neural networks which is an explicit function of the internal representation inaddition to the weights. The learning problem can then be formulated as two simple perceptrons and a search for internal representations. Back-propagation is recovered as a limit. The frequency of successful solutions is better for this algorithm than for back-propagation when weights and hidden units are updated on the same timescale i.e. once every learning step. 1 INTRODUCTION In their review of back-propagation in layered networks, Rumelhart et al. (1986) describe the learning process in terms of finding good "internal representations" of the input patterns on the hidden units. However, the search for these representations isan indirect one, since the variables which are adjusted in its course are the connection weights, not the activations of the hidden units themselves when specific input patterns are fed into the input layer. Rather, the internal representations are represented implicitly in the connection weight values. More recently, Grossman et al. (1988 and 1989)1 suggested a way in which the search for internal representations could be made much more explicit.

Yoshua Bengio Renato De Mori Dept Computer Science Dept Computer Science McGill University McGill University Montreal, Canada H3A2A7 RegisCardin Dept Computer Science McGill University ABSTRACT We attempt to combine neural networks with knowledge from speech science to build a speaker independent speech recognition system.This knowledge is utilized in designing the preprocessing, input coding, output coding, output supervision and architectural constraints. To handle the temporal aspect of speech we combine delays, copies of activations of hidden and output units at the input level, and Back-Propagation for Sequences (BPS), a learning algorithm for networks with local self-loops. This strategy is demonstrated in several experiments, inparticular a nasal discrimination task for which the application of a speech theory hypothesis dramatically improved generalization. 1 INTRODUCTION The strategy put forward in this research effort is to combine the flexibility and learning abilities of neural networks with as much knowledge from speech science as possible in order to build a speaker independent automatic speech recognition system. This knowledge is utilized in each of the steps in the construction ofan automated speech recognition system: preprocessing, input coding, output coding, output supervision, architectural design. Fast Fourier Transform (FFT), or compressing the frame sequence in such a way as to conserve an approximately constant rate of change.

The forward modeling approach is a methodology for learning control whendata is available in distal coordinate systems. We extend previous work by considering how this methodology can be applied to the optimization of quantities that are distal not only in space but also in time. In many learning control problems, the output variables of the controller are not the natural coordinates in which to specify tasks and evaluate performance. Tasks are generally more naturally specified in "distal" coordinate systems (e.g., endpoint coordinates for manipulator motion) than in the "proximal" coordinate system of the controller (e.g., joint angles or torques). Furthermore, the relationship between proximal coordinates and distal coordinates is often not known a priori and, if known, not easily inverted.

A method for storing analog vectors in Hopfield's continuous feedback modelis proposed. By analog vectors we mean vectors whose components are real-valued. The vectors to be stored are set as equilibria of the network. The network model consists of one layer of visible neurons and one layer of hidden neurons. We propose a learning algorithm, which results in adjusting the positions of the equilibria, as well as guaranteeing their stability.

Neu.·al Network Analysis of Distributed Representations of Dynamical Sensory-Motor rrransformations in the Leech Shawn R. LockerYt Van Fangt and Terrence J. Sejnowski Computational Neurobiology Laboratory Salk Institute for Biological Studies Box 85800, San Diego, CA 92138 ABSTRACT Interneurons in leech ganglia receive multiple sensory inputs and make synaptic contacts with many motor neurons. These "hidden" units coordinate several different behaviors. We used physiological and anatomical constraints to construct a model of the local bending reflex. Dynamical networks were trained on experimentally derived input-output patterns using recurrent back-propagation. Units in the model were modified to include electrical synapses and multiple synaptic time constants.