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

TL;DR

This paper explores how different structural properties of data, including correlations and priors, affect the learning behavior of perceptrons across various algorithms, revealing the limitations of Hebbian learning and the benefits of Bayesian methods.

Contribution

It provides a comprehensive analysis of perceptron learning with structured data, highlighting the impact of data correlations and priors on different learning algorithms using replica methods.

Findings

Hebb rule fails asymptotically in most structured scenarios

Gibbs and Bayesian learning are influenced by correlations in an intermediate regime

Enhanced prior knowledge improves Bayesian learning performance

Abstract

We investigate the influence of different kinds of structure 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 learning, Gibbs learning, 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. Contrarily, the behaviour of the more sophisticated methods of Gibbs and Bayes learning is influenced by the spatial correlations only in an intermediate regime of $\alpha$, where $\alpha$ specifies the size of the training set. Concerning the Bayesian case we show, how enhanced prior knowledge improves the performance.