Concept Learning and Optimisation with Hopfield Networks Kevin Swingler University of Stirling Presentation to UKCI 2011, Manchester

Hopfield Networks

Example – Learning Digit Images Learned PatternsClueRecalled Pattern

How? Learning w ij = w ij + u i u j Recall u i = Σ j≠i w ij u j uiui ujuj w ij

Pattern Discovery Random PatternScore against TargetLearning rate = Score σ

Learning Concepts or Optimal Patterns σ Concept = Symmetry

How? Weight update rule is adapted to: w ij = w ij + σu i u j uiui ujuj w ij

Examples Concept = SymmetryConcept = Horizontal

Speed Simple Target MethodGAEDAHopfield Mean Search Length n =

Relation to EDA EDA Probabilities, R = r 1... r n Probability that element i=1: P(p i = 1) = r i Probability update rule: r i = r i +f(P) if p i = 1 Current pattern, P = p 1... p n f(P) = Score for pattern P

Relation to EDA Probability that element i=1: P(p i = 1) = g(W, p j≠i ) Not stochastic, but marginal probabilities are not known Settling the network from a random state samples from the learned distribution without the need for joint distribution sampling explicitly.

Practicalities 1 If P is an attractor state for the network, then so is P` Scores for target patterns need their distance metric altered accordingly Or compare both the pattern and its inverse and score the highest =

Practicalities 2 The random patterns used during training must have elements drawn from an even distribution Deviation from this impairs learning

Uses and Benefits In situations where there are many possible solutions, this provides a method of sampling random good solutions without the need for additional searching Solutions tend to be close to the seeded start point, so you can use this method to find a local optimum that is close to a start point – again, without actually searching

Limitations The storage capacity of the network The size of the search space – Can we use a sparser connected network? Inverse patterns need care Currently: – Only tested on binary patterns – No evolution of patterns – stimuli must all be random

Thank You