© Negnevitsky, Pearson Education, 2005 1 Lecture 7 Artificial neural networks: Supervised learning Introduction, or how the brain works Introduction, or

© Negnevitsky, Pearson Education, 2005 1

Lecture 7 Lecture 7 Artificial neural networks: Artificial neural networks: Supervised learningSupervised learning Introduction, or how the brain worksIntroduction, or how the brain works The neuron as a simple computing elementThe neuron as a simple computing element The perceptronThe perceptron Multilayer neural networksMultilayer neural networks Accelerated learning in multilayer neural networksAccelerated learning in multilayer neural networks The Hopfield networkThe Hopfield network Bidirectional associative memories (BAM)Bidirectional associative memories (BAM) SummarySummary


Introduction, or how the brain worksIntroduction, or how the brain works

Machine learning involves adaptive mechanisms Machine learning involves adaptive mechanisms that enable computers to learn from experience, that enable computers to learn from experience, learn by example and learn by analogy. Learning learn by example and learn by analogy. Learning capabilities can improve the performance of an capabilities can improve the performance of an intelligent system over time. The most popular intelligent system over time. The most popular approaches to machine learning are approaches to machine learning are artificial artificial neural networks neural networks and and genetic algorithmsgenetic algorithms. This . This lecture is dedicated to neural networks.lecture is dedicated to neural networks.


A A neural network neural network can be defined as a model of can be defined as a model of reasoning based on the human brain. The brain reasoning based on the human brain. The brain consists of a densely interconnected set of nerve consists of a densely interconnected set of nerve cells, or basic information-processing units, called cells, or basic information-processing units, called neuronsneurons..

The human brain incorporates nearly 10 billion The human brain incorporates nearly 10 billion neurons and 60 trillion connections, neurons and 60 trillion connections, synapsessynapses, , between them. By using multiple neurons between them. By using multiple neurons simultaneously, the brain can perform its functions simultaneously, the brain can perform its functions much faster than the fastest computers in existence much faster than the fastest computers in existence today.today.


Each neuron has a very simple structure, but an Each neuron has a very simple structure, but an army of such elements constitutes a tremendous army of such elements constitutes a tremendous processing power.processing power.

A neuron consists of a cell body, A neuron consists of a cell body, somasoma, a number , a number of fibers called of fibers called dendritesdendrites, and a single long fiber , and a single long fiber called the called the axonaxon..


Biological neural networkBiological neural network

Soma Soma

Synapse

Synapse

Dendrites

Axon

Synapse

Dendrites

Axon


Our brain can be considered as a highly complex, Our brain can be considered as a highly complex, non-linear and parallel information-processing non-linear and parallel information-processing system.system.

Information is stored and processed in a neural Information is stored and processed in a neural network simultaneously throughout the whole network simultaneously throughout the whole network, rather than at specific locations. In other network, rather than at specific locations. In other words, in neural networks, both data and its words, in neural networks, both data and its processing are processing are global global rather than local.rather than local.

Learning is a fundamental and essential Learning is a fundamental and essential characteristic of biological neural networks. The characteristic of biological neural networks. The ease with which they can learn led to attempts to ease with which they can learn led to attempts to emulate a biological neural network in a computer.emulate a biological neural network in a computer.


An artificial neural network consists of a number of An artificial neural network consists of a number of very simple processors, also called very simple processors, also called neuronsneurons, which , which are analogous to the biological neurons in the brain. are analogous to the biological neurons in the brain.

The neurons are connected by weighted links The neurons are connected by weighted links passing signals from one neuron to another.passing signals from one neuron to another.

The output signal is transmitted through the The output signal is transmitted through the neuron’s outgoing connection. The outgoing neuron’s outgoing connection. The outgoing connection splits into a number of branches that connection splits into a number of branches that transmit the same signal. The outgoing branches transmit the same signal. The outgoing branches terminate at the incoming connections of other terminate at the incoming connections of other neurons in the network.neurons in the network.


Architecture of a typical artificial neural networkArchitecture of a typical artificial neural network

InputLayer OutputLayer

MiddleLayer

I n

p u

t S

i g n

a l

s

O u

t p

u t

S i g

n a

l s


Analogy between biological and Analogy between biological and artificial neural networksartificial neural networks

BiologicalNeuralNetwork ArtificialNeuralNetworkSomaDendriteAxonSynapse

NeuronInputOutputWeight


The neuron as a simple computing elementThe neuron as a simple computing element

Diagram of aDiagram of a neuronneuron

Neuron Y

InputSignals

x1

x2

xn

OutputSignals

Y

Y

Y

w2

w1

wn

Weights


The neuron computes the weighted sum of the input The neuron computes the weighted sum of the input signals and compares the result with a signals and compares the result with a threshold threshold valuevalue, , . If the net input is less than the threshold, . If the net input is less than the threshold, the neuron output is –1. But if the net input is the neuron output is –1. But if the net input is greater than or equal to the threshold, the neuron greater than or equal to the threshold, the neuron becomes activated and its output attains a value +1.becomes activated and its output attains a value +1.

The neuron uses the following transfer or The neuron uses the following transfer or activation activation functionfunction::

This type of activation function is called a This type of activation function is called a sign sign functionfunction..

n

iiiwxX

1

1

1

X

XY

if,

if,


Activation functions of a neuronActivation functions of a neuron

Step function Sign function

+1

-1

0

+1

-1

0X

Y

X

Y

1 1

-1

0 X

Y

Sigmoid function

-1

0 X

Y

Linear function

0if,0

0if,1

X

XYstep

0if,1

0if,1

X

XYsign

Xsigmoid

eY

1

1XYlinear


Can a single neuron learn a task?Can a single neuron learn a task?

In 1958, In 1958, Frank Rosenblatt Frank Rosenblatt introduced a training introduced a training algorithm that provided the first procedure for algorithm that provided the first procedure for training a simple ANN: a training a simple ANN: a perceptronperceptron..

The perceptron The perceptron is the simplest form of a neural is the simplest form of a neural network. It consists of a single neuron with network. It consists of a single neuron with adjustable adjustable synaptic weights and a synaptic weights and a hard limiterhard limiter..


Single-layer two-input perceptronSingle-layer two-input perceptron

Threshold

Inputs

x1

x2

Output

Y

HardLimiter

w2

w1

LinearCombiner


TheThe PerceptronPerceptron

The operation of Rosenblatt’s perceptron is based The operation of Rosenblatt’s perceptron is based on the on the McCulloch and Pitts neuron modelMcCulloch and Pitts neuron model. The . The model consists of a linear combiner followed by a model consists of a linear combiner followed by a hard limiter.hard limiter.

The weighted sum of the inputs is applied to the The weighted sum of the inputs is applied to the hard limiter, which produces an output equal to +1 hard limiter, which produces an output equal to +1 if its input is positive and if its input is positive and 1 if it is negative.1 if it is negative.


The aim of the perceptron is to classify inputs, The aim of the perceptron is to classify inputs, xx11, , xx22, . . ., , . . ., xxnn, into one of two classes, say , into one of two classes, say

AA11 and and AA22..

In the case of an elementary perceptron, the n-In the case of an elementary perceptron, the n- dimensional space is divided by a dimensional space is divided by a hyperplane hyperplane into into two decision regions. The hyperplane is defined by two decision regions. The hyperplane is defined by the the linearly separable linearly separable functionfunction::

01

n

iiiwx


Linear separability in the perceptronsLinear separability in the perceptrons

x1

x2

Class A2

Class A1

1

2

x1w1 +x2w2 =0

(a) Two-inputperceptron. (b) Three-inputperceptron.

x2

x1

x3x1w1 +x2w2 +x3w3 =0

12


How does the perceptron learn its classification How does the perceptron learn its classification tasks?tasks?

This is done by making small adjustments in the This is done by making small adjustments in the weights to reduce the difference between the actual weights to reduce the difference between the actual and desired outputs of the perceptron. The initial and desired outputs of the perceptron. The initial weights are randomly assigned, usually in the range weights are randomly assigned, usually in the range [[0.5, 0.5], and then updated to obtain the output 0.5, 0.5], and then updated to obtain the output consistent with the training examples.consistent with the training examples.


If at iteration If at iteration pp, the actual output is , the actual output is YY((pp) and the ) and the desired output is desired output is YYdd ((pp), then the error is given by:), then the error is given by:

where where p p = 1, 2, = 1, 2, 3, . . .3, . . .

Iteration Iteration p p here refers to the here refers to the ppth training example th training example presented to the perceptron.presented to the perceptron.

If the error, If the error, ee((pp), is positive, we need to increase ), is positive, we need to increase perceptron output perceptron output YY((pp), but if it is negative, we ), but if it is negative, we need to decrease need to decrease YY((pp).).

)()()( pYpYpe d


)()()()1( pepxpwpw iii

where where p p = 1, 2, 3, . . . = 1, 2, 3, . . . is the is the learning ratelearning rate, a positive constant less than , a positive constant less than unity.unity.

The perceptron learning rule was first proposed by The perceptron learning rule was first proposed by Rosenblatt Rosenblatt in 1960. Using this rule we can derive in 1960. Using this rule we can derive the perceptron training algorithm for classification the perceptron training algorithm for classification tasks.tasks.

The perceptron learning ruleThe perceptron learning rule


Perceptron’s training algorithmPerceptron’s training algorithm

Step 1Step 1: Initialisation : Initialisation Set initial weights Set initial weights ww11, , ww22,…, ,…, wwnn and threshold and threshold

to random numbers in the range [to random numbers in the range [0.5, 0.5, 0.5].0.5].

If the error, If the error, ee((pp), is positive, we need to increase ), is positive, we need to increase perceptronperceptron output output YY((pp), but if it is negative, we ), but if it is negative, we need to decrease need to decrease YY((pp).).


Perceptron’s training algorithm (continued)Perceptron’s training algorithm (continued)

Step 2Step 2: Activation : Activation Activate the perceptron by applying inputs Activate the perceptron by applying inputs xx11((pp), ),

xx22((pp),…, ),…, xxnn((pp) and desired output ) and desired output YYdd ((pp). Calculate ). Calculate

the actual output at iteration the actual output at iteration p p = 1= 1

where where n n is the number of the perceptron inputs, is the number of the perceptron inputs, and and step step is a step activation function.is a step activation function.

n

iii pwpxsteppY

1

)()()(


Perceptron’s training algorithm (continued)Perceptron’s training algorithm (continued)

Step 3Step 3: Weight training : Weight training Update the weights of the perceptronUpdate the weights of the perceptron

where where wwii((pp) is the weight correction at iteration ) is the weight correction at iteration pp..

The weight correction is computed by the The weight correction is computed by the delta delta rulerule::

Step 4Step 4: Iteration : Iteration Increase iteration Increase iteration p p by one, go back to by one, go back to Step 2 Step 2 and and repeat the process until convergence.repeat the process until convergence.

)()()1( pwpwpw iii

)()()( pepxpw ii .


Example of perceptron learning: the logical operation Example of perceptron learning: the logical operation ANDAND Inputs

x1 x2

0011

0101

000

EpochDesiredoutputYd

1

Initialweights

w1 w2

1

0.30.30.30.2

0.1 0.1 0.1 0.1

0010

Actualoutput

Y

Error

e

00

11

Finalweights

w1 w20.30.30.20.3

0.1 0.1 0.10.0

0011

0101

000

2

1

0.30.30.30.2

0011

00 10

0.30.30.20.2

0.00.00.00.0

0011

0101

000

3

1

0.20.20.20.1

0.00.00.00.0

0.00.00.00.0

0010

00 11

0.20.20.10.2

0.00.00.00.1

0011

0101

000

4

1

0.20.20.20.1

0.10.10.10.1

0011

00

10

0.20.20.10.1

0.10.10.10.1

0011

0101

000

5

1

0.10.10.10.1

0.10.10.10.1

0001

00

0

0.10.10.10.1

0.10.10.10.1

0

Threshold: =0.2;learningrate: =0.1


Two-dimensional plots of basic logical operationsTwo-dimensional plots of basic logical operations

A perceptron can learn the operations A perceptron can learn the operations AND AND and and OROR, , but not but not Exclusive-ORExclusive-OR..

x1

x2

1

1

x1

x2

1

1

(b) OR (x1 x2)

x1

x2

1

1

(c) Exclusive-OR(x1 x2)

00 0

(a) AND (x1 x2)


MultilayerMultilayer neuralneural networksnetworks

A multilayer perceptron is a feedforward neural A multilayer perceptron is a feedforward neural network with one or more hidden layers. network with one or more hidden layers.

The network consists of an The network consists of an input layer input layer of source of source neurons, at least one middle or neurons, at least one middle or hidden layer hidden layer of of computational neurons, and an computational neurons, and an output layer output layer of of computational neurons.computational neurons.

The input signals are propagated in a forward The input signals are propagated in a forward direction on a layer-by-layer basis.direction on a layer-by-layer basis.


Multilayer perceptron with two hidden layersMultilayer perceptron with two hidden layers

Inputlayer

Firsthiddenlayer

Secondhiddenlayer

Outputlayer

I n

p u

t

S i g

n a

l

I n

p u

t

S i g

n a

l

ss O u

t p

u t

S

i g

n

p u

t

S i g

n

a l s

a l s


What does the middle layer hide?What does the middle layer hide? A hidden layer “hides” its desired output. A hidden layer “hides” its desired output.

Neurons in the hidden layer cannot be observed Neurons in the hidden layer cannot be observed through the input/output behaviour of the network. through the input/output behaviour of the network. There is no obvious way to know what the desired There is no obvious way to know what the desired output of the hidden layer should be.output of the hidden layer should be.

Commercial ANNs incorporate three and Commercial ANNs incorporate three and sometimes four layers, including one or two sometimes four layers, including one or two hidden layers. Each layer can contain from 10 to hidden layers. Each layer can contain from 10 to 1000 neurons. Experimental neural networks may 1000 neurons. Experimental neural networks may have five or even six layers, including three or four have five or even six layers, including three or four hidden layers, and utilise millions of neurons.hidden layers, and utilise millions of neurons.


Back-propagation neural networkBack-propagation neural network Learning in a multilayer network proceeds the Learning in a multilayer network proceeds the

same way as for a perceptron.same way as for a perceptron.

A training set of input patterns is presented to the A training set of input patterns is presented to the network.network.

The network computes its output pattern, and if The network computes its output pattern, and if there is an error there is an error or in other words a difference or in other words a difference between actual and desired output patterns between actual and desired output patterns the the weights are adjusted to reduce this error.weights are adjusted to reduce this error.


In a back-propagation neural network, the learning In a back-propagation neural network, the learning algorithm has two phases. algorithm has two phases.

First, a training input pattern is presented to the First, a training input pattern is presented to the network input layer. The network propagates the network input layer. The network propagates the input pattern from layer to layer until the output input pattern from layer to layer until the output pattern is generated by the output layer.pattern is generated by the output layer.

If this pattern is different from the desired output, If this pattern is different from the desired output, an error is calculated and then propagated an error is calculated and then propagated backwards through the network from the output backwards through the network from the output layer to the input layer. The weights are modified layer to the input layer. The weights are modified as the error is propagated.as the error is propagated.


Three-layer back-propagation neural networkThree-layer back-propagation neural network

Inputlayer

xi

x1

x2

xn

1

2

i

n

Outputlayer

1

2

k

l

yk

y1

y2

yl

Inputsignals

Error signals

wjk

Hiddenlayer

wij

1

2

j

m


The back-propagation training algorithmThe back-propagation training algorithm

Step 1Step 1: Initialisation : Initialisation Set all the weights and threshold levels of the Set all the weights and threshold levels of the network to random numbers uniformly network to random numbers uniformly distributed inside a small range:distributed inside a small range:

where where FFii is the total number of inputs of neuron is the total number of inputs of neuron i i

in the network. The weight initialisation is done in the network. The weight initialisation is done on a neuron-by-neuron basis.on a neuron-by-neuron basis.

ii FF4.2

,4.2


Step 2Step 2: Activation : Activation Activate the back-propagation neural Activate the back-propagation neural network by applying inputs network by applying inputs xx11((pp), ), xx22((pp),…, ),…, xxnn((pp) )

and desired outputs and desired outputs yydd,1,1((pp), ), yydd,2,2((pp),…, ),…, yydd,,nn((pp).).

((aa) Calculate the actual outputs of the neurons in ) Calculate the actual outputs of the neurons in the hidden layer:the hidden layer:

where where n n is the number of inputs of neuron is the number of inputs of neuron j j in the in the hidden layer, and hidden layer, and sigmoid sigmoid is the is the sigmoid sigmoid activation activation function.function.

j

n

iijij pwpxsigmoidpy

1

)()()(


((bb) Calculate the actual outputs of the neurons in ) Calculate the actual outputs of the neurons in the output layer:the output layer:

Step 2Step 2 : Activation (continued): Activation (continued)

where where m m is the number of inputs of neuron is the number of inputs of neuron k k in the in the output layer.output layer.

k

m

jjkjkk pwpxsigmoidpy

1

)()()(


Step 3Step 3: Weight training : Weight training Update the weights in the back-propagation Update the weights in the back-propagation network propagating backward the errors associated network propagating backward the errors associated with output neurons. with output neurons. ( (aa) Calculate the error gradient for the neurons in ) Calculate the error gradient for the neurons in the output layer:the output layer:

wherewhere

Calculate the weight corrections:Calculate the weight corrections:

Update the weights at the output neurons:Update the weights at the output neurons:

)()()1( pwpwpw jkjkjk

)()(1)()( pepypyp kkkk

)()()( , pypype kkdk

)()()( ppypw kjjk


((bb) Calculate the error gradient for the neurons in ) Calculate the error gradient for the neurons in the hidden layer:the hidden layer:

Step 3Step 3: Weight training (continued): Weight training (continued)

Calculate the weight corrections:Calculate the weight corrections:

Update the weights at the hidden neurons:Update the weights at the hidden neurons:

)()()(1)()(1

][ pwppypyp jk

l

kkjjj

)()()( ppxpw jiij

)()()1( pwpwpw ijijij


Step 4Step 4: Iteration : Iteration Increase iteration Increase iteration p p by one, go back to by one, go back to Step 2 Step 2 and and repeat the process until the selected error criterion repeat the process until the selected error criterion is satisfied.is satisfied.

As an example, we may consider the three-layer As an example, we may consider the three-layer back-propagation network. Suppose that the back-propagation network. Suppose that the network is required to perform logical operation network is required to perform logical operation Exclusive-ORExclusive-OR. Recall that a single-layer perceptron . Recall that a single-layer perceptron could not do this operation. Now we will apply the could not do this operation. Now we will apply the three-layer net.three-layer net.


Three-layer network for solving the Three-layer network for solving the Exclusive-OR operationExclusive-OR operation

y55

x1 31

x2

Inputlayer

Outputlayer

Hiddenlayer

42

3w13

w24

w23

w24

w35

w45

4

5

1

1

1


The effect of the threshold applied to a neuron in the The effect of the threshold applied to a neuron in the hidden or output layer is represented by its weight, hidden or output layer is represented by its weight, , , connected to a fixed input equal to connected to a fixed input equal to 1.1.

The initial weights and threshold levels are set The initial weights and threshold levels are set randomly as follows: randomly as follows: ww1313 = 0.5, = 0.5, ww1414 = 0.9, = 0.9, ww2323 = 0.4, = 0.4, ww2424 = 1.0, = 1.0, ww35 35 = = 1.2, 1.2,

ww4545 = 1.1, = 1.1, 33 = 0.8, = 0.8, 44 = = 0.1 and 0.1 and 55 = 0.3. = 0.3.


We consider a training set where inputs We consider a training set where inputs xx1 1 and and xx22 are are

equal to 1 and desired output equal to 1 and desired output yydd,5,5 is 0. The actual is 0. The actual

outputs of neurons 3 and 4 in the hidden layer are outputs of neurons 3 and 4 in the hidden layer are calculated ascalculated as

Now the actual output of neuron 5 in the output layer Now the actual output of neuron 5 in the output layer is determined as:is determined as:

Thus, the following error is obtained:Thus, the following error is obtained:

5250.01/1)( )8.014.015.01(32321313 ewxwxsigmoidy

8808.01/1)( )1.010.119.01(42421414 ewxwxsigmoidy

5097.01/1)( )3.011.18808.02.15250.0(54543535 ewywysigmoidy

5097.05097.0055, yye d


The next step is weight training. To update the The next step is weight training. To update the weights and threshold levels in our network, we weights and threshold levels in our network, we propagate the error, propagate the error, ee, from the output layer , from the output layer backward to the input layer.backward to the input layer.

First, we calculate the error gradient for neuron 5 in First, we calculate the error gradient for neuron 5 in the output layer:the output layer:

Then we determine the weight corrections assuming Then we determine the weight corrections assuming that the learning rate parameter, that the learning rate parameter, , is equal to 0.1:, is equal to 0.1:

1274.05097).0(0.5097)(10.5097)1( 555 eyy

0112.0)1274.0(8808.01.05445 yw0067.0)1274.0(5250.01.05335 yw

0127.0)1274.0()1(1.0)1( 55


Next we calculate the error gradients for neurons 3 Next we calculate the error gradients for neurons 3 and 4 in the hidden layer:and 4 in the hidden layer:

We then determine the weight corrections:We then determine the weight corrections:

0381.0)2.1(0.1274)(0.5250)(10.5250)1( 355333 wyy

0.0147.114)0.127(0.8808)(10.8808)1( 455444 wyy

0038.00381.011.03113 xw0038.00381.011.03223 xw

0038.00381.0)1(1.0)1( 33 0015.0)0147.0(11.04114 xw0015.0)0147.0(11.04224 xw

0015.0)0147.0()1(1.0)1( 44


At last, we update all weights and threshold:At last, we update all weights and threshold:

The training process is repeated until the sum ofThe training process is repeated until the sum of squared errors is less than 0.001.squared errors is less than 0.001.

5038.00038.05.0131313 www

8985.00015.09.0141414 www

4038.00038.04.0232323 www

9985.00015.00.1242424 www

2067.10067.02.1353535 www

0888.10112.01.1454545 www

7962.00038.08.0333

0985.00015.01.0444

3127.00127.03.0555


Learning curve for operation Learning curve for operation Exclusive-ORExclusive-OR

0 50 100 150 200

10 1

Epoch

Sum-Squared Network Error for 224 Epochs

100

10-1

10-2

10 -3

10 -4

Su

m-S

qu

ared

Err

or


Final results of three-layer network learningFinal results of three-layer network learning

Inputs

x1 x2

1010

1100

011

Desiredoutput

yd

0

0.0155

Actualoutput

y5 e

Sum ofsquarederrors

0.98490.98490.0175

0.0010


Network represented by McCulloch-Pitts model Network represented by McCulloch-Pitts model for solving the for solving the Exclusive-OR Exclusive-OR operationoperation

y55

x1 31

x2 42

+1.0

1

1

1+1.0

+1.0

+1.0

+1.5

+1.0

+0.5

+0.5 2.0


Decision boundariesDecision boundaries

(a)(a) DecisionDecision boundary constructed by hidden neuron 3; boundary constructed by hidden neuron 3; ((bb) Decision boundary constructed by hidden neuron 4; ) Decision boundary constructed by hidden neuron 4; ((cc) Decision boundaries constructed by the complete ) Decision boundaries constructed by the complete

three-layer networkthree-layer network

x1

x2

1

(a)

1

x2

1

1

(b)

00

x1 +x2 – 1.5 =0 x1 +x2 – 0.5 =0

x1 x1

x2

1

1

(c)

0


Accelerated learning in multilayer Accelerated learning in multilayer neural networksneural networks

A multilayer network learns much faster when the A multilayer network learns much faster when the sigmoidal activation function is represented by a sigmoidal activation function is represented by a hyperbolic tangenthyperbolic tangent::

where where a a and and bb are constants.are constants.

Suitable values for Suitable values for a a and and b b are: are: a a = 1.716 and = 1.716 and b b = 0.667= 0.667

ae

aY bX

htan

1

2


We also can accelerate training by including a We also can accelerate training by including a momentum term momentum term in the delta rule:in the delta rule:

where where is a positive number (0 is a positive number (0 1) called 1) called the the momentum constantmomentum constant. Typically, the . Typically, the momentum constant is set to 0.95. momentum constant is set to 0.95.

This equation is called the This equation is called the generalised delta rulegeneralised delta rule..

)()()1()( ppypwpw kjjkjk


Learning with momentum for operation Learning with momentum for operation Exclusive-ORExclusive-OR

0 20 40 60 80 100 12010 -4

10 -2

10 0

10 2

Epoch

Training for 126 Epochs

0 100 140-1

-0.5

0

0.5

1

1.5

Epoch

10 -3

10 1

10 -1

20 40 60 80 120

Lea

rnin

g R

ate


Learning with adaptive learning rateLearning with adaptive learning rate

To accelerate the convergence and yet avoid the To accelerate the convergence and yet avoid the danger of instability, we can apply two heuristics:danger of instability, we can apply two heuristics:

Heuristic 1Heuristic 1

If the change of the sum of squared errors has the If the change of the sum of squared errors has the same algebraic sign for several consequent epochs, same algebraic sign for several consequent epochs, then the learning rate parameter, then the learning rate parameter, , should be , should be increased.increased.

Heuristic 2Heuristic 2

If the algebraic sign of the change of the sum of If the algebraic sign of the change of the sum of squared errors alternates for several consequent squared errors alternates for several consequent epochs, then the learning rate parameter, epochs, then the learning rate parameter, , should be , should be decreased.decreased.


Adapting the learning rate requires some changes Adapting the learning rate requires some changes in the back-propagation algorithm. in the back-propagation algorithm.

If the sum of squared errors at the current epoch If the sum of squared errors at the current epoch exceeds the previous value by more than a exceeds the previous value by more than a predefined ratio (typically 1.04), the learning rate predefined ratio (typically 1.04), the learning rate parameter is decreased (typically by multiplying parameter is decreased (typically by multiplying by 0.7) and new weights and thresholds are by 0.7) and new weights and thresholds are calculated.calculated.

If the error is less than the previous one, the If the error is less than the previous one, the learning rate is increased (typically by multiplying learning rate is increased (typically by multiplying by 1.05).by 1.05).


Learning with adaptive learning rateLearning with adaptive learning rate

0 10 20 30 40 50 60 70 80 90 100Epoch

Training for 103Epochs

0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Epoch

10-4

10-2

100

102

10-3

101

10-1

Su

m-S

qu

are

d

Su

m-S

qu

are

d

Err

oErr

oLearn

ing

Learn

ing

R

ate

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Learning with momentum and adaptive learning rateLearning with momentum and adaptive learning rate

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The Hopfield NetworkThe Hopfield Network

Neural networks were designed on analogy with Neural networks were designed on analogy with the brain. The brain’s memory, however, works the brain. The brain’s memory, however, works by association. For example, we can recognise a by association. For example, we can recognise a familiar face even in an unfamiliar environment familiar face even in an unfamiliar environment within 100-200 ms. We can also recall a within 100-200 ms. We can also recall a complete sensory experience, including sounds complete sensory experience, including sounds and scenes, when we hear only a few bars of and scenes, when we hear only a few bars of music. The brain routinely associates one thing music. The brain routinely associates one thing with another.with another.


Multilayer neural networks trained with the back-Multilayer neural networks trained with the back-propagation algorithm are used for pattern propagation algorithm are used for pattern recognition problems. However, to emulate the recognition problems. However, to emulate the human memory’s associative characteristics we human memory’s associative characteristics we need a different type of network: a need a different type of network: a recurrent recurrent neural networkneural network..

A recurrent neural network has feedback loops A recurrent neural network has feedback loops from its outputs to its inputs. The presence of from its outputs to its inputs. The presence of such loops has a profound impact on the learning such loops has a profound impact on the learning capability of the network.capability of the network.


The stability of recurrent networks intrigued The stability of recurrent networks intrigued several researchers in the 1960s and 1970s. several researchers in the 1960s and 1970s. However, none was able to predict which network However, none was able to predict which network would be stable, and some researchers were would be stable, and some researchers were pessimistic about finding a solution at all. The pessimistic about finding a solution at all. The problem was solved only in 1982, when problem was solved only in 1982, when John John Hopfield Hopfield formulated the physical principle of formulated the physical principle of storing information in a dynamically stable storing information in a dynamically stable network.network.


Single-layer Single-layer nn-neuron Hopfield network-neuron Hopfield network

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The Hopfield network uses McCulloch and Pitts The Hopfield network uses McCulloch and Pitts neurons with the neurons with the sign activation function sign activation function as its as its computing element:computing element:

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The current state of the Hopfield network is The current state of the Hopfield network is determined by the current outputs of all neurons, determined by the current outputs of all neurons, yy11, ,

yy22, . . ., , . . ., yynn..

Thus, for a single-layer Thus, for a single-layer nn-neuron network, the state -neuron network, the state can be defined by the can be defined by the state vector state vector as:as:

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In the Hopfield network, synaptic weights between In the Hopfield network, synaptic weights between neurons are usually represented in matrix form as neurons are usually represented in matrix form as follows:follows:

where where MM is the number of states to be memorised is the number of states to be memorised by the network, Yby the network, Ymm is the n-dimensional binary is the n-dimensional binary

vector, I is vector, I is nn nn identity matrix, and superscript identity matrix, and superscript T T denotes matrix transposition.denotes matrix transposition.

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Possible states for the three-neuron Possible states for the three-neuron Hopfield networkHopfield network

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The stable state-vertex is determined by the weight The stable state-vertex is determined by the weight matrix matrix WW, the current input vector , the current input vector XX, and the , and the

threshold matrix threshold matrix . If the input vector is partially . If the input vector is partially incorrect or incomplete, the initial state will converge incorrect or incomplete, the initial state will converge into the stable state-vertex after a few iterations.into the stable state-vertex after a few iterations.

Suppose, for instance, that our network is required to Suppose, for instance, that our network is required to memorise two opposite states, (1, 1, 1) and (memorise two opposite states, (1, 1, 1) and (1, 1, 1, 1, 1). 1). Thus,Thus,

where where YY11 and and YY22 are the three-dimensional vectors. are the three-dimensional vectors.

oror

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The 3 The 3 3 identity matrix 3 identity matrix I I isis

Thus, we can now determine the weight matrix as Thus, we can now determine the weight matrix as follows:follows:

Next, the network is tested by the sequence of input Next, the network is tested by the sequence of input vectors, vectors, XX11 and and XX22, which are equal to the output (or, which are equal to the output (or

target) vectors target) vectors YY11 and and YY22, respectively., respectively.

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First, we activate the Hopfield network by applying First, we activate the Hopfield network by applying the input vector the input vector XX. Then, we calculate the actual . Then, we calculate the actual output vector output vector YY, and finally, we compare the result , and finally, we compare the result with the initial input vector with the initial input vector XX. .

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The remaining six states are all unstable. However, The remaining six states are all unstable. However, stable states (also calledstable states (also called fundamental memoriesfundamental memories) are ) are capable of attracting states that are close to them.capable of attracting states that are close to them.

The fundamental memory (1, 1, 1) attracts unstable The fundamental memory (1, 1, 1) attracts unstable states (states (1, 1, 1), (1, 1, 1, 1), (1, 1, 1) and (1, 1, 1, 1) and (1, 1, 1). Each of these 1). Each of these unstable states represents a single error, compared to unstable states represents a single error, compared to the fundamental memory (1, 1, 1). the fundamental memory (1, 1, 1).

The fundamental memory (The fundamental memory (1, 1, 1, 1, 1) attracts unstable 1) attracts unstable states (states (1, 1, 1, 1), (1, 1), (1, 1, 1, 1, 1) and (1, 1) and (1, 1, 1, 1). 1).

Thus, the Hopfield network can act as an Thus, the Hopfield network can act as an error error correction networkcorrection network..


Storage capacity of the Hopfield networkStorage capacity of the Hopfield network

Storage capacity Storage capacity is is or the largest number of or the largest number of fundamental memories that can be stored and fundamental memories that can be stored and retrieved correctly.retrieved correctly.

The maximum number of fundamental memories The maximum number of fundamental memories MMmaxmax that can be stored in the that can be stored in the nn-neuron recurrent -neuron recurrent

network is limited bynetwork is limited by

nMmax 0.15


Bidirectional associative memory (BAM)Bidirectional associative memory (BAM) The Hopfield network represents an The Hopfield network represents an autoassociative autoassociative

type of memory type of memory it can retrieve a corrupted or it can retrieve a corrupted or incomplete memory but cannot associate this memory incomplete memory but cannot associate this memory with another different memory.with another different memory.

Human memory is essentially Human memory is essentially associativeassociative. . One thing One thing may remind us of another, and that of another, and so may remind us of another, and that of another, and so on. We use a chain of mental associations to recover on. We use a chain of mental associations to recover a lost memory. If we forget where we left an a lost memory. If we forget where we left an umbrella, we try to recall where we last had it, what umbrella, we try to recall where we last had it, what we were doing, and who we were talking to. We we were doing, and who we were talking to. We attempt to establish a chain of associations, and attempt to establish a chain of associations, and thereby to restore a lost memory.thereby to restore a lost memory.


To associate one memory with another, we need a To associate one memory with another, we need a recurrent neural network capable of accepting an recurrent neural network capable of accepting an input pattern on one set of neurons and producing input pattern on one set of neurons and producing a related, but different, output pattern on another a related, but different, output pattern on another set of neurons.set of neurons.

Bidirectional associative memory (BAM)Bidirectional associative memory (BAM), first , first proposed by proposed by Bart KoskoBart Kosko, is a heteroassociative , is a heteroassociative network. It associates patterns from one set, set network. It associates patterns from one set, set AA, , to patterns from another set, set to patterns from another set, set BB, and vice versa. , and vice versa. Like a Hopfield network, the BAM can generalise Like a Hopfield network, the BAM can generalise and also produce correct outputs despite corrupted and also produce correct outputs despite corrupted or incomplete inputs.or incomplete inputs.


BAM operationBAM operation

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(a) Forward direction. (b) Backward direction.


The basic idea behind the BAM is to store The basic idea behind the BAM is to store pattern pairs so that when pattern pairs so that when nn-dimensional vector -dimensional vector X X from set from set A A is presented as input, the BAM is presented as input, the BAM recalls recalls mm-dimensional vector -dimensional vector Y Y from set from set BB, but , but when when Y Y is presented as input, the BAM recalls is presented as input, the BAM recalls XX..


To develop the BAM, we need to create a To develop the BAM, we need to create a correlation matrix for each pattern pair we want to correlation matrix for each pattern pair we want to store. The correlation matrix is the matrix product store. The correlation matrix is the matrix product of the input vector of the input vector XX, and the transpose of the , and the transpose of the output vector output vector YYTT. The BAM weight matrix is the . The BAM weight matrix is the sum of all correlation matrices, that is,sum of all correlation matrices, that is,

where where M M is the number of pattern pairs to be stored is the number of pattern pairs to be stored in the BAM.in the BAM.

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Stability and storage capacity of the BAMStability and storage capacity of the BAM The BAM is The BAM is unconditionally stableunconditionally stable. This means that . This means that

any set of associations can be learned without risk of any set of associations can be learned without risk of instability.instability.

The maximum number of associations to be stored in The maximum number of associations to be stored in the BAM should not exceed the number of the BAM should not exceed the number of neurons in the smaller layer.neurons in the smaller layer.

The more serious problem with the BAM is The more serious problem with the BAM is incorrect convergenceincorrect convergence. The BAM may not . The BAM may not always produce the closest association. In fact, a always produce the closest association. In fact, a stable association may be only slightly related to stable association may be only slightly related to the initial input vector. the initial input vector.

Documents

© Negnevitsky, Pearson Education, 2005 1 Lecture 7 Artificial neural networks: Supervised learning Introduction, or how the brain works Introduction, or