Statistical Mechanics of Learning via Reverberation in Bidirectional Associative Memories

TL;DR

This paper uses statistical mechanics to analyze how bidirectional associative memories learn from noisy data, revealing phase diagrams and thresholds, and connecting the process to Boltzmann machines and classical conditioning.

Contribution

It provides a full analytical description of learning in bidirectional associative memories using Guerra's interpolation, including phase diagrams and a novel integral representation.

Findings

Analytical phase diagrams for learning thresholds.

Recovery of classical results in large dataset limit.

Connection to Boltzmann machines and Pavlovian conditioning.

Abstract

We study bi-directional associative neural networks that, exposed to noisy examples of an extensive number of random archetypes, learn the latter (with or without the presence of a teacher) when the supplied information is enough: in this setting, learning is heteroassociative -- involving couples of patterns -- and it is achieved by reverberating the information depicted from the examples through the layers of the network. By adapting Guerra's interpolation technique, we provide a full statistical mechanical picture of supervised and unsupervised learning processes (at the replica symmetric level of description) obtaining analytically phase diagrams, thresholds for learning, a picture of the ground-state in plain agreement with Monte Carlo simulations and signal-to-noise outcomes. In the large dataset limit, the Kosko storage prescription as well as its statistical mechanical picture provided by Kurchan, Peliti, and Saber in the eighties is fully recovered. Computational advantages in dealing with information reverberation, rather than storage, are discussed for natural test cases. In particular, we show how this network admits an integral representation in terms of two coupled restricted Boltzmann machines, whose hidden layers are entirely built of by grand-mother neurons, to prove that by coupling solely these grand-mother neurons we can correlate the patterns they are related to: it is thus possible to recover Pavlov's Classical Conditioning by adding just one synapse among the correct grand-mother neurons (hence saving an extensive number of these links for further information storage w.r.t. the classical autoassociative setting).