The performance of online algorithms for learning dichotomies is studied. In online learning, the number of examples P is equivalent to the learning time, since each example is presented only once. The learning curve, or generalization error as a function of P, depends on the schedule at which the learning rate is lowered.

We present here an analysis of the stochastic neurodynamics of a neural network composed of three-state neurons described by a master equation. An outer-product representation of the master equation is employed. In this representation, an extension of the analysis from two to three-state neurons is easily performed. We apply this formalism with approximation schemes to a simple three-state network and compare the results with Monte Carlo simulations.

This paper presents a rigorous characterization of how a general nonlinear learning machine generalizes during the training process when it is trained on a random sample using a gradient descent algorithm based on reduction of training error. It is shown, in particular, that best generalization performance occurs, in general, before the global minimum of the training error is achieved. The different roles played by the complexity of the machine class and the complexity of the specific machine in the class during learning are also precisely demarcated. 1 INTRODUCTION In learning machines such as neural networks, two major factors that affect the'goodness of fit' of the examples are network size (complexity) and training time. These are also the major factors that affect the generalization performance of the network. Many theoretical studies exploring the relation between generalization performance and machine complexity support the parsimony heuristics suggested by Occam's razor, to wit that amongst machines with similar training performance one should opt for the machine of least complexity.

Using a statistical mechanical formalism we calculate the evidence, generalisation error and consistency measure for a linear perceptron trained and tested on a set of examples generated by a non linear teacher. The teacher is said to be unrealisable because the student can never model it without error. Our model allows us to interpolate between the known case of a linear teacher, and an unrealisable, nonlinear teacher. A comparison of the hyperparameters which maximise the evidence with those that optimise the performance measures reveals that, in the nonlinear case, the evidence procedure is a misleading guide to optimising performance. Finally, we explore the extent to which the evidence procedure is unreliable and find that, despite being sub-optimal, in some circumstances it might be a useful method for fixing the hyperparameters. 1 INTRODUCTION The analysis of supervised learning or learning from examples is a major field of research within neural networks.

A neural network learning paradigm based on information theory is proposed as a way to perform in an unsupervised fashion, redundancy reduction among the elements of the output layer without loss of information from the sensory input. The model developed performs nonlinear decorrelation up to higher orders of the cumulant tensors and results in probabilistic ally independent components of the output layer. This means that we don't need to assume Gaussian distribution neither at the input nor at the output. The theory presented is related to the unsupervised-learning theory of Barlow, which proposes redundancy reduction as the goal of cognition. When nonlinear units are used nonlinear principal component analysis is obtained.

Random errors and insufficiencies in databases limit the performance of any classifier trained from and applied to the database. In this paper we propose a method to estimate the limiting performance of classifiers imposed by the database. We demonstrate this technique on the task of predicting failure in telecommunication paths. 1 Introduction Data collection for a classification or regression task is prone to random errors, e.g.

Ideally pattern recognition machines provide constant output when the inputs are transformed under a group 9 of desired invariances. These invariances can be achieved by enhancing the training data to include examples of inputs transformed by elements of g, while leaving the corresponding targets unchanged. Alternatively the cost function for training can include a regularization term that penalizes changes in the output when the input is transformed under the group. This paper relates the two approaches, showing precisely the sense in which the regularized cost function approximates the result of adding transformed (or distorted) examples to the training data. The cost function for the enhanced training set is equivalent to the sum of the original cost function plus a regularizer. For unbiased models, the regularizer reduces to the intuitively obvious choice - a term that penalizes changes in the output when the inputs are transformed under the group. For infinitesimal transformations, the coefficient of the regularization term reduces to the variance of the distortions introduced into the training data. This correspondence provides a simple bridge between the two approaches.

They provide in particular some theoretical bounds on the sample complexity, i.e. a minimal number of training samples assuring the desired accuracy with the desired confidence. However there are a few obvious deficiencies in these results: (i) the sample complexity bounds are unrealistically high (c.f. Section 4.), and (ii) for some networks they do not hold at all since VC-dimension is infinite, e.g.

We first rederive the known results for the'thermodynamic limit' of infinite perceptron size N and show explicitly that 9

An novel class of locally excitatory, globally inhibitory oscillator networks is proposed.