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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 2
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.
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�� 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.
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�� 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..
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Biological neural networkBiological neural network
Soma Soma
Synapse
Synapse
Dendrites
Axon
Synapse
Dendrites
Axon
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�� 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.
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�� 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.
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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
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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
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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
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�� 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
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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=
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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. .
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Threshold
Inputs
x1
x2
Output
Y∑∑∑∑
Hard
Limiter
w2
w1
Linear
Combiner
θ
SingleSingle--layer twolayer two--input input perceptronperceptron
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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.
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�� 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
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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
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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?
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�� 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 −=
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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
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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
)( )()(
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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=∆
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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
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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. .
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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.
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Multilayer Multilayer perceptronperceptron with two hidden layerswith two hidden layers
Input
layer
First
hidden
layer
Second
hidden
layer
Output
layer
O u t p u t S i g n a l s
I n p u t S i g n a l s
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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.
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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.
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�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 30
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
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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
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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
)()()(
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 33
((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)
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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 ∆+=+
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 35
((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)
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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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 37
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
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�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 39
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 40
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 41
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 42
�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 43
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 44
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 45
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 46
((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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 47
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 48
�� 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 ⋅⋅+−∆⋅=∆
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 49
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 50
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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 51
�� 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).
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 52
Learning with adaptive learning rateLearning with adaptive learning rate
0 10 20 30 40 50 60 70 80 90 100
Epoch
Training for 103 Epochs
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 53
Learning with momentum and adaptive learning rateLearning with momentum and adaptive learning rate
0 10 20 30 40 50 60 70 80
Epoch
Training for 85 Epochs
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 54
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 55
�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 56
�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 57
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
O u t p u t S i g n a l s
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 58
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 59
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 60
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 61
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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 62
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 63
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 64
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 65
�� 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..
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 66
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 67
�� 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)
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 68
�� 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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 69
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.
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 70
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..
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 71
�� 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
Negnevitsky, Pearson Education, 2011Negnevitsky, Pearson Education, 2011 72
�� 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