Generalizing with perceptrons in the case of structured phase- and pattern-spaces

Abstract

We investigate the influence of different kinds of structures on the learning behaviour of a perceptron performing a classification task defined by a teacher rule. The underlying pattern distribution is permitted to have spatial correlations. The prior distribution for the teacher coupling vectors itself is assumed to be nonuniform. Thus, classification tasks of quite different difficulty are included. As learning algorithms we discuss Hebbian, Gibbs, and Bayesian learning with different priors, using methods from statistics and the replica formalism. We find that the Hebb rule is quite sensitive to the structure of the actual learning problem, failing asymptotically in most cases. In contrast, the behaviour of the more sophisticated methods of Gibbs and Bayes learning is influenced by the spatial correlations only in an intermediate regime of , where specifies the size of the training set. In view of the Bayesian case, we show how enhanced prior knowledge improves the performance.