lecture 7 artificial neural networks - ankara...

Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 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 The perceptronperceptron

�� 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 networksneural networks and and genetic algorithmsgenetic algorithms. This . This

lecture is dedicated to neural networks.lecture is dedicated to neural networks.


�� A A neural networkneural 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 informationcells, or basic information--processing units, called 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 of , a number of

fibersfibers called called dendritesdendrites, and a single long , and a single long fiberfiber

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,

nonnon--linear and parallel informationlinear and parallel information--processing 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 globalglobal 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

neuronneuron’’s outgoing connection. The outgoing 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

Input Layer Output Layer

Middle Layer

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


Biological Neural Network Artificial Neural Network

Soma

Dendrite

Axon

Synapse

Neuron

Input

Output

Weight

Analogy between biological and Analogy between biological and

artificial neural networksartificial neural networks


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

Diagram of a neuronDiagram of a neuron

Neuron Y

Input Signals

x1

x2

xn

Output Signals

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, , θθ. If the net input is less than the threshold, . If the net input is less than the threshold,

the neuron output is the neuron output is ––1. But if the net input is greater 1. But if the net input is greater

than or equal to the threshold, the neuron becomes than or equal to the threshold, the neuron becomes

activated and its output attains a value +1.activated and its output attains a value +1.

�� The neuron uses the following transfer or The neuron uses the following transfer or activationactivation

functionfunction::

�� This type of activation function is called a This type of activation function is called a sign sign

functionfunction..

∑=

=n

i

iiwxX

1

θ<−

θ≥+=

X

XY

if ,1

if ,1


Activation functions of a neuronActivation functions of a neuron

Step function Sign function

+1

-1

0

+1

-1

0X

Y

X

Y

+1

-1

0 X

Y

Sigmoid function

+1

-1

0 X

Y

Linear function

<

≥=

0 if ,0

0 if ,1

X

XYstep

<−

≥+=

0 if ,1

0 if ,1

X

XYsign

X

sigmoid

eY

−+=1

1XYlinear=


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

�� In 1958, In 1958, Frank RosenblattFrank 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 The perceptronperceptron 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

adjustableadjustable synaptic weights and a synaptic weights and a hard limiterhard limiter. .


Threshold

Inputs

x1

x2

Output

Y∑∑∑∑

Hard

Limiter

w2

w1

Linear

Combiner

θ

SingleSingle--layer twolayer two--input input perceptronperceptron


The The PerceptronPerceptron

�� The operation of RosenblattThe operation of Rosenblatt’’s s perceptronperceptron is based 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 The aim of the perceptronperceptron is to classify inputs, 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 In the case of an elementary perceptronperceptron, the n, the n--

dimensional space is divided by a dimensional space is divided by a hyperplanehyperplane into into

two decision regions. The two decision regions. The hyperplanehyperplane is defined by is defined by

the the linearly separablelinearly separable functionfunction::

0

1

=θ−∑=

n

i

iiwx


Linear Linear separabilityseparability in the in the perceptronsperceptrons

x1

x2

Class A2

Class A1

1

2

x1w1 + x2w2 − θ = 0

(a) Two-input perceptron. (b) Three-input perceptron.

x2

x1

x3x1w1 + x2w2 + x3w3 − θ = 0

12


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 and desired outputs of the perceptronperceptron. The initial . 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.

How does the How does the perceptronperceptron learn its classification learn its classification

tasks?tasks?


�� 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 YYd d ((pp), then the error is given by:), then the error is given by:

where where pp = 1, 2, 3, . . .= 1, 2, 3, . . .

Iteration Iteration pp here refers to the here refers to the ppthth training example training example

presented to the presented to the perceptronperceptron..

�� 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).).

)()()( pYpYpe d −=


The The perceptronperceptron learning rulelearning rule

where where pp = 1, 2, 3, . . .= 1, 2, 3, . . .

αα is the is the learning ratelearning rate, a positive constant less than, a positive constant less than

unity.unity.

The The perceptronperceptron learning rule was first proposed bylearning rule was first proposed by

Rosenblatt Rosenblatt in 1960. Using this rule we can derive in 1960. Using this rule we can derive

the the perceptronperceptron training algorithm for classification training algorithm for classification

tasks.tasks.

)()()()1( pepxpwpw iii ⋅⋅+=+ α


Step 1Step 1: Initialisation: Initialisation

Set initial weights Set initial weights ww11, , ww22,,……, , wwnn and threshold and threshold θθto random numbers in the range [to random numbers in the range [−−0.5, 0.5].0.5, 0.5].

PerceptronPerceptron’’ss training algorithmtraining algorithm

Step 2Step 2: Activation: Activation

Activate the Activate the perceptronperceptron by applying inputs by applying inputs xx11((pp), ),

xx22((pp),),……, , xxnn((pp) and desired output ) and desired output YYd d ((pp). ).

Calculate the actual output at iteration Calculate the actual output at iteration pp = 1= 1

where where nn is the number of the is the number of the perceptronperceptron inputs, inputs,

and and stepstep is a step activation function.is a step activation function.

θ−= ∑

=

n

i

ii pwpxsteppY

1

)( )()(


Step 3Step 3: Weight training: Weight training

Update the weights of the Update the weights of the perceptronperceptron

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 pp by one, go back to by one, go back to Step 2Step 2 and and

repeat the process until convergence.repeat the process until convergence.

)()()1( pwpwpw iii ∆+=+

PerceptronPerceptron’’ss training algorithm (continued)training algorithm (continued)

)()()( pepxpw ii ⋅⋅α=∆


Example of Example of perceptronperceptron learning: the logical operation learning: the logical operation ANDANDInputs

x1 x2

0

0

1

1

0

1

0

1

0

0

0

EpochDesiredoutputYd

1

Initial

weightsw1 w2

1

0.3

0.3

0.3

0.2

−0.1

−0.1

−0.1

−0.1

0

0

1

0

Actualoutput

Y

Error

e

0

0

−1

1

Final

weightsw1 w2

0.3

0.3

0.2

0.3

−0.1

−0.1

−0.1

0.0

0

0

1

1

0

1

0

1

0

0

0

2

1

0.3

0.3

0.3

0.2

0

0

1

1

0

0

−1

0

0.3

0.3

0.2

0.2

0.0

0.0

0.0

0.0

0

0

1

1

0

1

0

1

0

0

0

3

1

0.2

0.2

0.2

0.1

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0.0

0

0

1

0

0

0

−1

1

0.2

0.2

0.1

0.2

0.0

0.0

0.0

0.1

0

0

1

1

0

1

0

1

0

0

0

4

1

0.2

0.2

0.2

0.1

0.1

0.1

0.1

0.1

0

0

1

1

0

0

−1

0

0.2

0.2

0.1

0.1

0.1

0.1

0.1

0.1

0

0

1

1

0

1

0

1

0

0

0

5

1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0

0

0

1

0

0

0

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0.1

0

Threshold: θ = 0.2; learning rate: α = 0.1


TwoTwo--dimensional plots of basic logical operationsdimensional plots of basic logical operations

x1

x2

1

(a) AND (x1 ∩ x2)

1

x1

x2

1

1

(b) OR (x1 ∪ x2)

x1

x2

1

1

(c) Exclusive-OR

(x1 ⊕ x2)

00 0

A A perceptronperceptron can learn the operations can learn the operations ANDAND and and OROR, ,

but not but not ExclusiveExclusive--OROR. .


Multilayer neural networksMultilayer neural networks

�� A multilayer A multilayer perceptronperceptron is a is a feedforwardfeedforward neural 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 layerinput layer of source of source

neurons, at least one middle or neurons, at least one middle or hidden layerhidden layer of of

computational neurons, and an computational neurons, and an output layeroutput 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 layerdirection on a layer--byby--layer basis.layer basis.


Multilayer Multilayer perceptronperceptron with two hidden layerswith two hidden layers

Input

layer

First

hidden

layer

Second

hidden

layer

Output

layer


I n p u t S i g n a l s


What does the middle layer hide?What does the middle layer hide?

�� A hidden layer A hidden layer ““hideshides”” its desired output. 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 Commercial ANNsANNs incorporate three and 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 have five or even six layers, including three or

four hidden layers, and utilise millions of neurons.four hidden layers, and utilise millions of neurons.


BackBack--propagation neural networkpropagation neural network

�� Learning in a multilayer network proceeds the Learning in a multilayer network proceeds the

same way as for a same way as for a perceptronperceptron. .

�� 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 backIn a back--propagation neural network, the learning 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.


ThreeThree--layer backlayer back--propagation neural networkpropagation neural network

Input

layer

xi

x1

x2

xn

1

2

i

n

Output

layer

1

2

k

l

yk

y1

y2

yl

Input signals

Error signals

wjk

Hidden

layer

wij

1

2

j

m


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 ii

in the network. The weight initialisation is done in the network. The weight initialisation is done

on a neuronon a neuron--byby--neuron basis.neuron basis.

The backThe back--propagation training algorithmpropagation training algorithm

+−

ii FF

4.2 ,

4.2


Step 2Step 2: Activation: Activation

Activate the backActivate the back--propagation neural network by propagation neural network by

applying inputs applying inputs xx11((pp), ), xx22((pp),),……, , xxnn((pp) and desired ) and desired

outputs 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 nn is the number of inputs of neuron is the number of inputs of neuron jj in the in the

hidden layer, and hidden layer, and sigmoidsigmoid is the is the sigmoidsigmoid activation activation

function.function.

θ−⋅= ∑

=j

n

i

ijij 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:

where where mm is the number of inputs of neuron is the number of inputs of neuron kk in the in the

output layer.output layer.

θ−⋅= ∑

=k

m

j

jkjkk pwpxsigmoidpy

1

)()()(

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


Step 3Step 3: Weight training: Weight training

Update the weights in the backUpdate the weights in the back--propagation network propagation network

propagating backward the errors associated with propagating backward the errors associated with

output neurons.output neurons.

((aa) Calculate the error gradient for the neurons in the ) Calculate the error gradient for the neurons in the

output layer: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)()( pepypyp kkkk ⋅−⋅=δ

)()()( , pypype kkdk −=

)()()( ppypw kjjk δα ⋅⋅=∆

)()()1( pwpwpw jkjkjk ∆+=+


((bb) Calculate the error gradient for the neurons in ) Calculate the error gradient for the neurons in

the hidden layer:the hidden layer:

Calculate the weight corrections:Calculate the weight corrections:

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

)()()(1)()(

1

][ p wppypyp jk

l

k

kjjj ∑=

⋅−⋅= δδ

)()()( ppxpw jiij δα ⋅⋅=∆

)()()1( pwpwpw ijijij ∆+=+

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


Step 4Step 4: Iteration: Iteration

Increase iteration Increase iteration pp by one, go back to by one, go back to Step 2Step 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 threeAs an example, we may consider the three--layer layer

backback--propagation network. Suppose that the propagation network. Suppose that the

network is required to perform logical operation network is required to perform logical operation

ExclusiveExclusive--OROR. Recall that a single. Recall that a single--layer layer perceptronperceptron

could not do this operation. Now we will apply the could not do this operation. Now we will apply the

threethree--layer net.layer net.


ThreeThree--layer network for solving the layer network for solving the

ExclusiveExclusive--OR operationOR operation

y55

x1 31

x2

Input

layer

Output

layer

Hidden layer

42

θ3

w13

w24

w23

w24

w35

w45

θ4

θ5

−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, θθ, ,

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, ww3535 = = −−1.2, 1.2,

ww4545 = 1.1, = 1.1, θθ33 = 0.8, = 0.8, θθ44 = = −−0.1 and 0.1 and θθ55 = 0.3.= 0.3.


�� We consider a training set where inputs We consider a training set where inputs xx11 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

[ ] 5250.01 /1)( )8.014.015.01(32321313 =+=θ−+= ⋅−⋅+⋅−ewxwx sigmoidy

[ ] 8808.01 /1)( )1.010.119.01(42421414 =+=θ−+= ⋅+⋅+⋅−ewxwx sigmoidy

�� 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:

[ ] 5097.01 /1)( )3.011.18808.02.15250.0(54543535 =+=θ−+= ⋅−⋅+⋅−−ewywy sigmoidy

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:

1274.05097).0( 0.5097)(1 0.5097)1( 555 −=−⋅−⋅=−= e y yδ

�� Then we determine the weight corrections assuming Then we determine the weight corrections assuming

that the learning rate parameter, that the learning rate parameter, αα, is equal to 0.1:, is equal to 0.1:

0112.0)1274.0(8808.01.05445 −=−⋅⋅=⋅⋅=∆ δα yw

0067.0)1274.0(5250.01.05335 −=−⋅⋅=⋅⋅=∆ δα yw

0127.0)1274.0()1(1.0)1( 55 −=−⋅−⋅=⋅−⋅=θ∆ δα


�� 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)(1 0.5250)1( 355333 =−⋅−⋅−⋅=⋅⋅−= wyy δδ

0.0147.11 4) 0.127 ( 0.8808)(10.8808)1( 455444 −=⋅−⋅−⋅=⋅⋅−= wyy δδ

0038.00381.011.03113 =⋅⋅=⋅⋅=∆ δα xw

0038.00381.011.03223 =⋅⋅=⋅⋅=∆ δα xw

0038.00381.0)1(1.0)1( 33 −=⋅−⋅=⋅−⋅=θ∆ δα0015.0)0147.0(11.04114 −=−⋅⋅=⋅⋅=∆ δα xw

0015.0)0147.0(11.04224 −=−⋅⋅=⋅⋅=∆ δα xw

0015.0)0147.0()1(1.0)1( 44 =−⋅−⋅=⋅−⋅=θ∆ δα


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

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=−=θ∆+θ=θ

0985.00015.01.0444−=+−=θ∆+θ=θ

3127.00127.03.0555=+=θ∆+θ=θ

�� The training process is repeated until the sum of The training process is repeated until the sum of

squared errors is less than 0.001. squared errors is less than 0.001.


Learning curve for operation Learning curve for operation ExclusiveExclusive--OROR

0 50 100 150 200

101

Epoch

Sum-Squared Error

Sum-Squared Network Error for 224 Epochs

100

10-1

10-2

10-3

10-4


Final results of threeFinal results of three--layer network learninglayer network learning

Inputs

x1 x2

1

0

1

0

1

1

0

0

0

1

1

Desired

output

yd

0

0.0155

Actual

output

y5Y

Error

e

Sum of

squarederrors

0.9849

0.9849

0.0175

−0.0155 0.0151

0.0151

−0.0175

0.0010


Network represented by McCullochNetwork represented by McCulloch--Pitts model Pitts model

for solving the for solving the ExclusiveExclusive--OROR 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


((aa) Decision boundary constructed by hidden neuron 3;) Decision 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

threethree--layer networklayer 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

Decision boundariesDecision boundaries


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

sigmoidalsigmoidal activation function is represented by a activation function is represented by a

hyperbolic tangenthyperbolic tangent::

where where aa and and bb are constants.are constants.

Suitable values for Suitable values for aa and and bb are: are:

aa = 1.716 and = 1.716 and bb = 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 termmomentum term in the delta rule:in the delta rule:

where where ββ is a positive number (0 is a positive number (0 ≤≤ ββ << 1) called the 1) called the

momentum constantmomentum constant. Typically, the momentum . Typically, the momentum

constant is set to 0.95.constant is set to 0.95.

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

)()()1()( ppypwpw kjjkjk δαβ ⋅⋅+−∆⋅=∆


Learning with momentum for operation Learning with momentum for operation ExclusiveExclusive--OROR

0 20 40 60 80 100 12010

-4

10-2

100

102

Epoch

Sum-Squared Error

Training for 126 Epochs

0 100 140-1

-0.5

0

0.5

1

1.5

Epoch

Learning Rate

10-3

101

10-1

20 40 60 80 120


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 same If the change of the sum of squared errors has the same

algebraic sign for several consequent epochs, then the algebraic sign for several consequent epochs, then the

learning rate parameter, learning rate parameter, αα, should be increased., should be 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, αα, should be , should be

decreased.decreased.


�� Adapting the learning rate requires some changes Adapting the learning rate requires some changes

in the backin the back--propagation algorithm. 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 90 100

Epoch


0 20 40 60 80 100 1200

0.2

0.4

0.6

0.8

1

Epoch

Learning Rate

10-4

10-2

100

102

Sum-Squared Error

10-3

101

10-1


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

0 10 20 30 40 50 60 70 80

Epoch


0 10 20 30 40 50 60 70 80 900

0.5

1

2.5

Epoch

Learning Rate

10-4

10-2

100

102

Sum-Squared Error

10-3

101

10-1

1.5

2


�� Neural networks were designed on analogy with Neural networks were designed on analogy with

the brain. The brainthe brain. The brain’’s memory, however, works 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 100within 100--200 ms. We can also recall a 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.

The Hopfield NetworkThe Hopfield Network


�� Multilayer neural networks trained with the backMultilayer 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 memoryhuman memory’’s associative characteristics we 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

HopfieldHopfield formulated the physical principle of formulated the physical principle of

storing information in a dynamically stable storing information in a dynamically stable

network.network.


SingleSingle--layer layer nn--neuron Hopfield networkneuron Hopfield network

xi

x1

x2

xnI n p u t S i g n a l s

yi

y1

y2

yn

1

2

i

n



�� The Hopfield network uses McCulloch and Pitts The Hopfield network uses McCulloch and Pitts

neurons with the neurons with the sign activation functionsign activation function as its as its

computing element:computing element:

0=

0<−

>+

=

XY

X

X

Y sign

if ,

if ,1

0 if ,1


�� 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 singleThus, for a single--layer layer nn--neuron network, the state neuron network, the state

can be defined by the can be defined by the state vectorstate vector as:as:

=

ny

y

y

M

2

1

Y


�� 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, by the network, YYmm is the is the nn--dimensional binary dimensional binary

vector, vector, II is is nn ×× nn identity matrix, and superscript identity matrix, and superscript TT

denotes a matrix transposition.denotes a matrix transposition.

IYYW MM

m

Tmm −= ∑

=1


Possible states for the threePossible states for the three--neuron neuron

Hopfield networkHopfield network

y1

y2

y3

(1, −1, 1)(−1, −1, 1)

(−1, −1, −1) (1, −1, −1)

(1, 1, 1)(−1, 1, 1)

(1, 1, −1)(−1, 1, −1)

0


�� The stable stateThe stable state--vertex is determined by the weight 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 θθθθθθθθ. 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 stateinto the stable state--vertex after a few iterations.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,

oror

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

=

1

1

1

1Y

−

−

−

=

1

1

1

2Y [ ]1 1 11 =TY [ ]1 1 12 −−−=T

Y


�� The 3 The 3 ×× 3 identity matrix 3 identity matrix II 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.

=

1 0 0

0 1 0

0 0 1

I

[ ] [ ]

−−−−

−

−

−

+

=

1 0 0

0 1 0

0 0 1

21 1 1

1

1

1

1 1 1

1

1

1

W

=

0 2 2

2 0 2

2 2 0


�� 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..

=

−

=

1

1

1

0

0

0

1

1

1

0 2 2

2 0 2

2 2 0

1 signY

−

−

−

=

−

−

−

−

=

1

1

1

0

0

0

1

1

1

0 2 2

2 0 2

2 2 0

2 signY


�� The remaining six states are all unstable. However, The remaining six states are all unstable. However,

stable states (also called stable 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 1). Each of

these unstable states represents a single error, these unstable states represents a single error,

compared to the fundamental memory (1, 1, 1). compared to the fundamental memory (1, 1, 1).

�� The fundamental memory (The fundamental memory (−−1, 1, −−1, 1, −−1) attracts 1) attracts

unstable states (unstable 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 capacityStorage capacity isis 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

n Mmax 15.0=

Storage capacity of the Hopfield networkStorage capacity of the Hopfield network


�� The Hopfield network represents an The Hopfield network represents an autoassociativeautoassociative

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.

Bidirectional associative memory (BAM)Bidirectional associative memory (BAM)


�� 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 memoryBidirectional associative memory (BAM)(BAM), first , first

proposed by proposed by Bart Bart KoskoKosko, is a , is a heteroassociativeheteroassociative

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

yj(p)

y1(p)

y2(p)

ym(p)

1

2

j

m

Output

layer

Input

layer

xi(p)

x1(p)

x2(p)

xn(p)

2

i

n

1

xi(p+1)

x1(p+1)

x2(p+1)

xn(p+1)

yj(p)

y1(p)

y2(p)

ym(p)

1

2

j

m

Output

layer

Input

layer

2

i

n

1

(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

XX from set from set AA is presented as input, the BAM is presented as input, the BAM

recalls recalls mm--dimensional vector dimensional vector YY from set from set BB, but , but

when when YY 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 MM is the number of pattern pairs to be stored is the number of pattern pairs to be stored

in the BAM.in the BAM.

Tm

M

m

m YXW ∑=

=1


�� 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 The maximum number of associations to be stored

in the BAM should not exceed the number of in 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.

Stability and storage capacity of the BAMStability and storage capacity of the BAM

lecture 7 artificial neural networks - ankara...

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