Multiplicative versus additive noise in multi-state neural networks

TL;DR

This paper investigates how random synaptic pruning, modeled as multiplicative noise, affects multi-state neural networks, revealing that it effectively introduces additive Gaussian noise into the learning process.

Contribution

It demonstrates that random dilution acts as additive Gaussian noise in Hebbian learning and explores effects of symmetric and asymmetric pruning using statistical mechanics and path integral methods.

Findings

Dilution introduces additive Gaussian noise with zero mean.

The variance of noise depends on network connectivity and symmetry.

Anti-Hebbian couplings influence the scaling factor.

Abstract

The effects of a variable amount of random dilution of the synaptic couplings in Q-Ising multi-state neural networks with Hebbian learning are examined. A fraction of the couplings is explicitly allowed to be anti-Hebbian. Random dilution represents the dying or pruning of synapses and, hence, a static disruption of the learning process which can be considered as a form of multiplicative noise in the learning rule. Both parallel and sequential updating of the neurons can be treated. Symmetric dilution in the statics of the network is studied using the mean-field theory approach of statistical mechanics. General dilution, including asymmetric pruning of the couplings, is examined using the generating functional (path integral) approach of disordered systems. It is shown that random dilution acts as additive gaussian noise in the Hebbian learning rule with a mean zero and a variance depending on the connectivity of the network and on the symmetry. Furthermore, a scaling factor appears that essentially measures the average amount of anti-Hebbian couplings.