An adaptive online algorithm extending the learning of learning idea is proposed and theoretically motivated. Relying only on gradient flow information it can be applied to learning continuous functions or distributions, even when no explicit loss function is given and the Hessian is not available. Its efficiency is demonstrated for a non-stationary blind separation task of acoustic signals.

Furthermore it is shown that networks of noisy spiking neurons with temporal coding have a strictly larger computational power than sigmoidal neural nets with the same number of units. 1 Introduction and Definitions We consider a formal model SNN for a §piking neuron network that is basically a reformulation of the spike response model (and of the leaky integrate and fire model) without using 6-functions (see [Maass, 1996a] or [Maass, 1996b] for further backgrou nd).

To reduce the computational complexity of classification systems using tangent distance, Hastie et al. (HSS) developed an algorithm todevise rich models for representing large subsets of the data which computes automatically the "best" associated tangent subspace.Schwenk & Milgram proposed a discriminant modular classification system (Diabolo) based on several autoassociative multilayer perceptrons which use tangent distance as error reconstruction measure. We propose a gradient based constructive learning algorithm for building a tangent subspace model with discriminant capabilities which combines several of the the advantages of both HSS and Diabolo: devised tangent models hold discriminant capabilities, space requirements are improved with respect to HSS since our algorithm is discriminant and thus it needs fewer prototype models, dimension of the tangent subspace is determined automatically by the constructive algorithm, and our algorithm is able to learn new transformations.

The number of hidden layers is a crucial parameter for the architecture of multilayer neural networks. Early research, in the 60's, addressed the problem of exactly realizing Booleanfunctions with binary networks or binary multilayer networks. On the one hand, more recent work focused on approximately realizing real functions with multilayer neural networks with one hidden layer [6, 7, 11] or with two hidden units [2]. On the other hand, some authors [1, 12] were interested in finding bounds on the architecture of multilayer networks for exact realization of a finite set of points.

Neural one-unit learning rules for the problem of Independent Component Analysis(ICA) and blind source separation are introduced. In these new algorithms, every ICA neuron develops into a separator thatfinds one of the independent components. The learning rules use very simple constrained Hebbianjanti-Hebbian learning in which decorrelating feedback may be added. To speed up the convergence of these stochastic gradient descent rules, a novel computationally efficientfixed-point algorithm is introduced. 1 Introduction Independent Component Analysis (ICA) (Comon, 1994; Jutten and Herault, 1991) is a signal processing technique whose goal is to express a set of random variables aslinear combinations of statistically independent component variables. The main applications of ICA are in blind source separation, feature extraction, and blind deconvolution.

We compare different methods to combine predictions from neural networkstrained on different bootstrap samples of a regression problem. One of these methods, introduced in [6] and which we here call balancing, is based on the analysis of the ensemble generalization errorinto an ambiguity term and a term incorporating generalization performances of individual networks. We show how to estimate these individual errors from the residuals on validation patterns.Weighting factors for the different networks follow from a quadratic programming problem. On a real-world problem concerning the prediction of sales figures and on the well-known Boston housing data set, balancing clearly outperforms other recently proposedalternatives as bagging [1] and bumping [8]. 1 EARLY STOPPING AND BOOTSTRAPPING Stopped training is a popular strategy to prevent overfitting in neural networks. The complete data set is split up into a training and a validation set.

An adaptive online algorithm extending the learning of learning idea is proposed and theoretically motivated. Relying only on gradient flowinformation it can be applied to learning continuous functions or distributions, even when no explicit loss function is given andthe Hessian is not available. Its efficiency is demonstrated for a non-stationary blind separation task of acoustic signals. 1 Introduction Neural networks provide powerful tools to capture the structure in data by learning. Often the batch learning paradigm is assumed, where the learner is given all training examplessimultaneously and allowed to use them as often as desired. In large practical applications batch learning is often experienced to be rather infeasible and instead online learning is employed.

Two "exact" algorithmswere developed which lead to exactly the same results as OBS but using less time.

In the Bayes approach to statistical inference [Berger, 1985] one assumes that the prior uncertainty about parameters of an unknown data generating mechanism can be encoded in a probability distribution, the so called prior. Using the prior and the likelihood of the data given the parameters, the posterior distribution of the parameters can be derived from Bayes rule. From this posterior, various estimates for functions ofthe parameter, like predictions about unseen data, can be calculated. However, in general, those predictions cannot be realised by specific parameter values, but only by an ensemble average over parameters according to the posterior probability. Hence,exact implementations of Bayes method for neural networks require averages over network parameters which in general can be performed by time consuming 226 M.Opper and O. Winther Monte Carlo procedures.

The convergence properties of the gradient descent algorithm in the case of the linear perceptron may be obtained from the response function. We derive a general expression for the response function and apply it to the case of data with simple input correlations. It is found that correlations severely may slow down learning. This explains the success of PCA as a method for reducing training time. Motivated by this finding we furthermore propose to transform the input data by removing the mean across input variables as well as examples to decrease correlations. Numerical findings for a medical classification problem are in fine agreement with the theoretical results. 1 INTRODUCTION Learning and generalization are important areas of research within the field of neural networks.Although good generalization is the ultimate goal in feed-forward networks (perceptrons), it is of practical importance to understand the mechanism which control the amount of time required for learning, i. e. the dynamics of learning. Thisis of course particularly important in the case of a large data set. An exact analysis of this mechanism is possible for the linear perceptron and as usual it is hoped that the results to some extend may be carried over to explain the behaviour of nonlinear perceptrons.