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Lecture 8Lecture 8 Artificial neural networks:Artificial neural networks: Unsupervised learningUnsupervised learning IntroductionIntroduction Hebbian learningHebbian learning Generalised Hebbian learning algorithmGeneralised Hebbian learning algorithm Competitive learningCompetitive learning Self-organising computational map:Self-organising computational map: Kohonen networkKohonen network SummarySummary Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 1
IntroductionIntroduction The main property of a neural network is anThe main property of a neural network is an ability to learn from its environment, and toability to learn from its environment, and to improve its performance through learning. Soimprove its performance through learning. So far we have consideredfar we have considered supervisedsupervised oror activeactive learninglearning −− learning with an external “teacher”learning with an external “teacher” or a supervisor who presents a training set to theor a supervisor who presents a training set to the network. But another type of learning alsonetwork. But another type of learning also exists:exists: unsupervised learningunsupervised learning.. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 2
In contrast to supervised learning, unsupervised orIn contrast to supervised learning, unsupervised or self-organised learningself-organised learning does not require andoes not require an external teacher. During the training session, theexternal teacher. During the training session, the neural network receives a number of differentneural network receives a number of different input patterns, discovers significant features ininput patterns, discovers significant features in these patterns and learns how to classify input datathese patterns and learns how to classify input data into appropriate categories. Unsupervisedinto appropriate categories. Unsupervised learning tends to follow thelearning tends to follow the neuroneuro-biological-biological organisation of the brain.organisation of the brain. Unsupervised learning algorithms aim to learnUnsupervised learning algorithms aim to learn rapidly and can be used in real-time.rapidly and can be used in real-time. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 3
Hebbian learningHebbian learning In 1949,In 1949, Donald Hebb proposedDonald Hebb proposed one of the keyone of the key ideas in biological learning, commonly known asideas in biological learning, commonly known as Hebb’s LawHebb’s Law. Hebb’s. Hebb’s Law states that ifLaw states that if neuronneuron ii isis near enough to excite neuronnear enough to excite neuron jj and repeatedlyand repeatedly participates in its activation, the synaptic connectionparticipates in its activation, the synaptic connection between these two neuronsbetween these two neurons is strengthened andis strengthened and neuronneuron jj becomes more sensitive to stimuli frombecomes more sensitive to stimuli from neuronneuron ii.. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 4
Hebb’s Law can be represented in the form of twoHebb’s Law can be represented in the form of two rules:rules: 1. If two neurons on either side of a connection1. If two neurons on either side of a connection are activated synchronously, then the weight ofare activated synchronously, then the weight of that connection is increased.that connection is increased. 2. If two neurons on either side of a connection2. If two neurons on either side of a connection are activated asynchronously, then the weightare activated asynchronously, then the weight of that connection is decreased.of that connection is decreased. Hebb’s Law provides the basis for learningHebb’s Law provides the basis for learning without a teacher. Learning here is awithout a teacher. Learning here is a locallocal phenomenonphenomenon occurring without feedback fromoccurring without feedback from the environment.the environment. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 5
Hebbian learningHebbian learning in ain a neuralneural networknetwork i g n a l sS S i g n a l s i j I n p u t O u t p u t Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 6
Using Hebb’sUsing Hebb’s Law we can express the adjustmentLaw we can express the adjustment applied to the weightapplied to the weight ww at iterationat iteration pp in thein the ijij following form:following form: ∆ w ij ( p ) = F [ y j ( p ), xi ( p )] As a special case, we can represent Hebb’s Law asAs a special case, we can represent Hebb’s Law as follows:follows: ∆wij (p) =α yj(p) xi(p) wherewhere αα is theis the learning ratelearning rate parameter.parameter. This equation is referred to as theThis equation is referred to as the activity productactivity product rulerule.. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 7
Hebbian learning implies that weights can onlyHebbian learning implies that weights can only increase. To resolve this problem, we mightincrease. To resolve this problem, we might impose a limit on the growth of synaptic weights.impose a limit on the growth of synaptic weights. It can be done by introducing a non-linearIt can be done by introducing a non-linear forgetting factorforgetting factor into Hebb’sinto Hebb’s Law:Law: ∆wij (p) = α yj (p) xi(p)−ϕ yj (p) wij (p) wherewhere ϕϕ is the forgetting factor.is the forgetting factor. Forgetting factor usually falls in the intervalForgetting factor usually falls in the interval between 0 and 1, typically between 0.01 and 0.1,between 0 and 1, typically between 0.01 and 0.1, to allow only a little “forgetting” while limitingto allow only a little “forgetting” while limiting the weight growth.the weight growth. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 8
Hebbian learning algorithmHebbian learning algorithm Step 1:Step 1: InitialisationInitialisation.. Set initial synaptic weights and thresholds to smallSet initial synaptic weights and thresholds to small random values, say in an interval [0, 1].random values, say in an interval [0, 1]. Step 2: Activation.Step 2: Activation. Compute the neuron output at iterationCompute the neuron output at iteration pp n yj(p) = ∑xi(p) wij(p)−θj i=1 wherewhere nn is the number of neuron inputs, andis the number of neuron inputs, and θθ is theis the jj threshold value of neuronthreshold value of neuron jj.. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 9
Step 3:Step 3: Learning.Learning. Update the weights in the network:Update the weights in the network: wij(p+1) = wij(p)+∆wij(p) wherewhere ∆∆ww ((pp) is the weight correction at iteration) is the weight correction at iteration pp.. ijij The weight correction is determined by theThe weight correction is determined by the generalised activity product rule:generalised activity product rule: ∆wij (p) = ϕ yj(p)[λ xi(p)− wij (p)] Step 4Step 4: Iteration.: Iteration. Increase iterationIncrease iteration pp by one, go back to Step 2.by one, go back to Step 2. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 10
Hebbian learning eHebbian learning examplexample To illustrate HebbianTo illustrate Hebbian learning, consider a fullylearning, consider a fully connected feedforwardconnected feedforward network with a single layernetwork with a single layer of five computation neuronsof five computation neurons. Each. Each neuronneuron isis represented by a McCullochrepresented by a McCulloch andand PittsPitts model withmodel with the sign activation function. The network is trainedthe sign activation function. The network is trained on the following set of input vectors:on the following set of input vectors: 0 0 0 0 0 0 1 0 0 1 X1 = 0 X2 = 0 X3 = 0 X4 = 1 X5 = 0 0 0 1 0 0 0 1 0 0 1 Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 11
Initial and final states of the networkInitial and final states of the network x 1 1 y x 1 0 y 1 1 1 1 1 1 1 1 x 0 2 0 y x 0 2 1 y 2 2 2 2 22 2 x 0 3 0 y x 0 3 0 y 3 3 3 3 3 3 x 0 4 0 y x 0 4 0 y 4 4 4 4 4 4 x 1 5 1 y x 1 5 1 y 5 5 5 5 5 5 Input layer Output layer Input layer Output layer Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 12
Initial and final weight matricesInitial and final weight matrices O u t p u t l a y e r O u t p u t l a y e r 1 2 3 4 5 1 2 3 4 5 1 1 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 2.0204 0 0 2.0204 2 2 3 0 0 1 0 0 3 0 0 1.0200 0 0 0 0 0 1 0 0 0.9996 0 4 4 0 0 I n p u t l a y e r5 0 0 0 0 1 I n p u t l a y e r5 0 2.0204 0 0 2.0204 Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 13
A test input vector, or probe, is definedA test input vector, or probe, is defined asas 1 0 X = 0 0 1 When this probe is presented to the network, weWhen this probe is presented to the network, we obtain:obtain: 0 0 0 0 0 1 0.4940 0 0 2.0204 0 0 2.02040 0.2661 1 Y = sign 0 0 1.0200 0 0 0 − 0.0907 = 0 0 0 0 0.9996 0 0 0.9478 0 0 2.0204 0 0 2.02041 0.0737 1 Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 14
Competitive learningCompetitive learning In competitive learning,In competitive learning, neuronsneurons compete amongcompete among themselves to be activated.themselves to be activated. While inWhile in HebbianHebbian learning, several outputlearning, several output neuronsneurons can be activated simultaneously, in competitivecan be activated simultaneously, in competitive learning, only a single output neuronlearning, only a single output neuron is active atis active at any time.any time. The outputThe output neuronneuron that wins the “competition” isthat wins the “competition” is called thecalled the winner-takes-allwinner-takes-all neuron.neuron. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 15
The basic idea of competitive learning wasThe basic idea of competitive learning was introduced in the early 1970s.introduced in the early 1970s. In the late 1980s, Teuvo Kohonen introduced aIn the late 1980s, Teuvo Kohonen introduced a special class of artificial neural networks calledspecial class of artificial neural networks called self-organising feature mapsself-organising feature maps. These maps are. These maps are based on competitive learning.based on competitive learning. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 16
What is a self-organising feature map?What is a self-organising feature map? Our brain is dominated by the cerebral cortex, aOur brain is dominated by the cerebral cortex, a very complex structure of billions of neuronsvery complex structure of billions of neurons andand hundreds of billions of synapses. The cortexhundreds of billions of synapses. The cortex includes areas that are responsible for differentincludes areas that are responsible for different human activities (motor, visual, auditory,human activities (motor, visual, auditory, somatosensory, etc.), andsomatosensory, etc.), and associatedassociated with differentwith different sensory inputs. We can say that each sensorysensory inputs. We can say that each sensory input is mapped into a corresponding area of theinput is mapped into a corresponding area of the cerebral cortex.cerebral cortex. The cortex is a self-organisingThe cortex is a self-organising computational map in the human brain.computational map in the human brain. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 17
Feature-mapping KohonenFeature-mapping Kohonen modelmodel Kohonen layer Kohonen layer Input layer Input layer 1 0 0 1 (a) (b) Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 18
The KohonenThe Kohonen networknetwork The KohonenThe Kohonen model provides a topologicalmodel provides a topological mapping. It placesmapping. It places a fixed number of inputa fixed number of input patterns from the input layer into a higher-patterns from the input layer into a higher- dimensional output or Kohonendimensional output or Kohonen layer.layer. Training in the Kohonen network begins with theTraining in the Kohonen network begins with the winner’s neighbourhood of a fairly large size.winner’s neighbourhood of a fairly large size. Then, as training proceeds, the neighbourhood sizeThen, as training proceeds, the neighbourhood size gradually decreases.gradually decreases. Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 19
Architecture of the KohonenArchitecture of the Kohonen NetworkNetwork y1 x1 y2 S i g n a l s S i g n a l s x2 y3 I n p u t O u t p u t Input Output layer layer Negnevitsky, Pearson Education, 2002Negnevitsky, Pearson Education, 2002 20
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