Supervised Hebbian Learning

TL;DR

This paper introduces a supervised learning protocol for Hopfield networks that enables them to infer archetypes from datasets, relates them to restricted Boltzmann machines, and improves MNIST classification by incorporating a replica hidden layer.

Contribution

It defines a supervised learning method for Hopfield networks, establishes their equivalence to restricted Boltzmann machines for structureless data, and enhances MNIST classification using a novel quasi-ultrametric dataset organization.

Findings

Hopfield networks can infer archetypes with supervised learning.

Equivalence between supervised Hopfield and restricted Boltzmann machines for certain datasets.

Improved MNIST classification accuracy from 75% to 95%.

Abstract

In neural network's Literature, Hebbian learning traditionally refers to the procedure by which the Hopfield model and its generalizations store archetypes (i.e., definite patterns that are experienced just once to form the synaptic matrix). However, the term "Learning" in Machine Learning refers to the ability of the machine to extract features from the supplied dataset (e.g., made of blurred examples of these archetypes), in order to make its own representation of the unavailable archetypes. Here, given a sample of examples, we define a supervised learning protocol by which the Hopfield network can infer the archetypes, and we detect the correct control parameters (including size and quality of the dataset) to depict a phase diagram for the system performance. We also prove that, for structureless datasets, the Hopfield model equipped with this supervised learning rule is equivalent to a restricted Boltzmann machine and this suggests an optimal and interpretable training routine. Finally, this approach is generalized to structured datasets: we highlight a quasi-ultrametric organization (reminiscent of replica-symmetry-breaking) in the analyzed datasets and, consequently, we introduce an additional "replica hidden layer" for its (partial) disentanglement, which is shown to improve MNIST classification from 75% to 95%, and to offer a new perspective on deep architectures.