Typical methods for gradient descent in neural network learning involve calculation of derivatives based on a detailed knowledge of the network model. This requires extensive, time consuming calculations for each pattern presentation and high precision that makes it difficult to implement in VLSI. We present here a perturbation technique that measures, not calculates, the gradient. Since the technique uses the actual network as a measuring device, errors in modeling neuron activation and synaptic weights do not cause errors in gradient descent. The method is parallel in nature and easy to implement in VLSI. We describe the theory of such an algorithm, an analysis of its domain of applicability, some simulations using it and an outline of a hardware implementation.

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. Several other 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 part of 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 method for creating a nonlinear encoder-decoder for multidimensional data with compact representations is presented. The commonly used technique of autoassociation is extended to allow nonlinear representations, and an objective function which penalizes activations of individual hidden units is shown to result in minimum dimensional encodings with respect to allowable error in reconstruction. 1 INTRODUCTION Reducing dimensionality of data with minimal information loss is important for feature extraction, compact coding and computational efficiency. The data can be tranformed into "good" representations for further processing, constraints among feature variables may be identified, and redundancy eliminated. Many algorithms are exponential in the dimensionality of the input, thus even reduction by a single dimension may provide valuable computational savings. Autoassociating feed forward networks with one hidden layer have been shown to extract the principal components of the data (Baldi & Hornik, 1988). Such networks have been used to extract features and develop compact encodings of the data (Cottrell, Munro & Zipser, 1989). Principal Components Analysis projects the data into a linear subspace -email: demers@cs.ucsd.edu

In stochastic learning, weights are random variables whose time evolution is governed by a Markov process. We summarize the theory of the time evolution of P, and give graphical examples of the time evolution that contrast the behavior of stochastic learning with true gradient descent (batch learning). Finally, we use the formalism to obtain predictions of the time required for noise-induced hopping between basins of different optima. We compare the theoretical predictions with simulations of large ensembles of networks for simple problems in supervised and unsupervised learning. Despite the recent application of convergence theorems from stochastic approximation theory to neural network learning (Oja 1982, White 1989) there remain outstanding questions about the search dynamics in stochastic learning.

Memory-based classification algorithms such as radial basis functions or K-nearest neighbors typically rely on simple distances (Euclidean, dot product...), 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.

The implementation uses noise injection and multiplicative correlation to estimate derivatives, as in [Anderson, Kerns 92]. One intended application of this technique is setting circuit parameters on-chip automatically, rather than manually [Kirk 91]. Gradient descent optimization may be used to adjust synapse weights for a backpropagation or other on-chip learning implementation. The approach combines the features of continuous multidimensional gradient descent and the potential for an annealing style of optimization. We present data measured from our analog VLSI implementation. 1 Introduction This work is similar to [Anderson, Kerns 92], but represents two advances. First, we describe the extension of the technique to multiple dimensions. Second, we demonstrate an implementation of the multidimensional technique in analog VLSI, and provide results measured from the chip. Unlike previous work using noise sources in adaptive systems, we use the noise as a means of estimating the gradient of a function f(y), rather than performing an annealing process [Alspector 88]. We also estimate gr-;:dients continuously in position and time, in contrast to [Umminger 89] and [J abri 91], which utilize discrete position gradient estimates.

We have designed, fabricated, and tested an analog VLSI chip which computes radial basis functions in parallel. We have developed a synapse circuit that approximates a quadratic function. We aggregate these circuits to form radial basis functions. These radial basis functions are then averaged together using a follower aggregator.

Our efforts proceed in the context of a problem suggested by the operational needs of a particular electric utility to make daily forecasts of short-term load or demand. Forecasts are made at midday (1 p.m.) on a weekday t ( Monday - Thursday), for the next evening peak e(t) (occuring usually about 8 p.m. in the winter), the daily minimum d(t

Channel equalization problem is an important problem in high-speed communications. The sequences of symbols transmitted are distorted by neighboring symbols. Traditionally, the channel equalization problem is considered as a channel-inversion operation. One problem of this approach is that there is no direct correspondence between error probability and residual error produced by the channel inversion operation. In this paper, the optimal equalizer design is formulated as a classification problem. The optimal classifier can be constructed by Bayes decision rule. In general it is nonlinear. An efficient hybrid linear/nonlinear equalizer approach has been proposed to train the equalizer. The error probability of new linear/nonlinear equalizer has been shown to be better than a linear equalizer in an experimental channel. 1 INTRODUCTION

Two theorems and a lemma are presented about the use of jackknife estimator and the cross-validation method for model selection. Theorem 1 gives the asymptotic form for the jackknife estimator. Combined with the model selection criterion, this asymptotic form can be used to obtain the fit of a model. The model selection criterion we used is the negative of the average predictive likehood, the choice of which is based on the idea of the cross-validation method. Lemma 1 provides a formula for further exploration of the asymptotics of the model selection criterion. Theorem 2 gives an asymptotic form of the model selection criterion for the regression case, when the parameters optimization criterion has a penalty term. Theorem 2 also proves the asymptotic equivalence of Moody's model selection criterion (Moody, 1992) and the cross-validation method, when the distance measure between response y and regression function takes the form of a squared difference. 1 INTRODUCTION Selecting a model for a specified problem is the key to generalization based on the training data set.