
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 algorithms
genetic 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
neurons
neurons.
.
 The human brain incorporates nearly 10 billion
The human brain incorporates nearly 10 billion
neurons and 60 trillion connections,
neurons and 60 trillion connections, synapses
synapses,
,
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, soma
soma, a number of
, a number of
fibers called
fibers called dendrites
dendrites, and a single long fiber
, and a single long fiber
called the
called the axon
axon.
.


Biological neural network
Biological 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 neurons
neurons, 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 network
Architecture 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


Analogy between biological and
Analogy between biological and
artificial neural networks
artificial neural networks


Diagram of a neuron
Diagram 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
value
value,
, 
. 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 greater
the neuron output is –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 activation
activation
function
function:
:
 This type of activation function is called a
This type of activation function is called a sign
sign
function
function.
.



n
i
i
iw
x
X
1 









X
X
Y
if
,
1
if
,
1


Activation functions of a neuron
Activation functions of a neuron
Step function Sign function
+1
-1
0
+1
-1
0
X
Y
X
Y
+1
-1
0 X
Y
Sigm oid function
+1
-1
0 X
Y
L inear function






0
if
,
0
0
if
,
1
X
X
Y step








0
if
,
1
0
if
,
1
X
X
Y sign
X
sigmoid
e
Y



1
1
X
Y linear



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 perceptron
perceptron.
.
 The perceptron is the simplest form of a neural
The perceptron 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 limiter
hard limiter.
.


T
h
r
e
s
h
o
l
d
I
n
p
u
t
s
x
1
x
2
O
u
t
p
u
t
Y

H
a
r
d
L
i
m
i
t
e
r
w
2
w
1
L
i
n
e
a
r
C
o
m
b
i
n
e
r

Single-layer two-input perceptron
Single-layer two-input perceptron


The Perceptron
The Perceptron
 The operation of Rosenblatt’s perceptron is based
The operation of Rosenblatt’s perceptron is based
on the
on the McCulloch and Pitts neuron model
McCulloch 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,
x
x1
1,
, x
x2
2, . . .,
, . . ., x
xn
n, into one of two classes, say
, into one of two classes, say
A
A1
1 and
and A
A2
2.
.
 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 function
function:
:
0
1





n
i
i
iw
x


Linear separability in the perceptrons
Linear separability in the perceptrons
x1
x2
Class A2
Class A1
1
2
x1w1 + x2w2
  = 0
(a) Two-input perceptron. (b) Three-input perceptron.
x2
x1
x3
x1w1 + x2w2 + x3w3
  = 0
1
2

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


 If at iteration
If at iteration p
p, the actual output is
, the actual output is Y
Y(
(p
p) and the
) and the
desired output is
desired output is Y
Yd
d (
(p
p), then the error is given by:
), then the error is given by:
where
where p
p = 1, 2, 3, . . .
= 1, 2, 3, . . .
Iteration
Iteration p
p here refers to the
here refers to the p
pth training example
th training example
presented to the perceptron.
presented to the perceptron.
 If the error,
If the error, e
e(
(p
p), is positive, we need to increase
), is positive, we need to increase
perceptron output
perceptron output Y
Y(
(p
p), but if it is negative, we
), but if it is negative, we
need to decrease
need to decrease Y
Y(
(p
p).
).
)
(
)
(
)
( p
Y
p
Y
p
e d 



The perceptron learning rule
The perceptron learning rule
where
where p
p = 1, 2, 3, . . .
= 1, 2, 3, . . .

 is the
is the learning rate
learning 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.
)
(
)
(
)
(
)
1
( p
e
p
x
p
w
p
w i
i
i 



 


Step 1
Step 1: Initialisation
: Initialisation
Set initial weights
Set initial weights w
w1
1,
, w
w2
2,…,
,…, w
wn
n 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, e
e(
(p
p), is positive, we need to increase
), is positive, we need to increase
perceptron output
perceptron output Y
Y(
(p
p), but if it is negative, we
), but if it is negative, we
need to decrease
need to decrease Y
Y(
(p
p).
).
Perceptron’s training algorithm
Perceptron’s training algorithm


Step 2
Step 2: Activation
: Activation
Activate the perceptron by applying inputs
Activate the perceptron by applying inputs x
x1
1(
(p
p),
),
x
x2
2(
(p
p),…,
),…, x
xn
n(
(p
p) and desired output
) and desired output Y
Yd
d (
(p
p).
).
Calculate the actual output at iteration
Calculate 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.
Perceptron’s training algorithm (continued)










 

n
i
i
i p
w
p
x
step
p
Y
1
)
(
)
(
)
(


Step 3
Step 3: Weight training
: Weight training
Update the weights of the perceptron
Update the weights of the perceptron
where
where 
w
wi
i(
(p
p) is the weight correction at iteration
) is the weight correction at iteration p
p.
.
The weight correction is computed by the
The weight correction is computed by the delta
delta
rule
rule:
:
Step 4
Step 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
( p
w
p
w
p
w i
i
i 



)
(
)
(
)
( p
e
p
x
p
w i
i 






Example of perceptron learning: the logical operation
Example of perceptron learning: the logical operation AND
AND
Inputs
x1 x2
0
0
1
1
0
1
0
1
0
0
0
Epoch
Desired
output
Yd
1
Initial
weights
w1 w2
1
0.3
0.3
0.3
0.2
 0.1
 0.1
 0.1
 0.1
0
0
1
0
Actual
output
Y
Error
e
0
0
 1
1
Final
weights
w1 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:  = 0.2; learning rate:  = 0.1


Two-dimensional plots of basic logical operations
Two-dimensional 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)
0
0 0
A perceptron can learn the operations
A perceptron can learn the operations AND
AND and
and OR
OR,
,
but not
but not Exclusive-OR
Exclusive-OR.
.


 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 layers
Multilayer perceptron with 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


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
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.


Back-propagation neural network
Back-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 network
Three-layer back-propagation 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 1
Step 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 F
Fi
i 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.
The back-propagation training algorithm
The back-propagation training algorithm










i
i F
F
4
.
2
,
4
.
2


Step 2
Step 2: Activation
: Activation
Activate the back-propagation neural network by
Activate the back-propagation neural network by
applying inputs
applying inputs x
x1
1(
(p
p),
), x
x2
2(
(p
p),…,
),…, x
xn
n(
(p
p) and desired
) and desired
outputs
outputs y
yd
d,1
,1(
(p
p),
), y
yd
d,2
,2(
(p
p),…,
),…, y
yd
d,
,n
n(
(p
p).
).
(
(a
a) 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
i
ij
i
j p
w
p
x
sigmoid
p
y
1
)
(
)
(
)
(


(
(b
b) Calculate the actual outputs of the neurons in
) Calculate the actual outputs of the neurons in
the output layer:
the output layer:
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
j
jk
jk
k p
w
p
x
sigmoid
p
y
1
)
(
)
(
)
(
Step 2
Step 2: Activation (continued)
: Activation (continued)


Step 3
Step 3: Weight training
: Weight training
Update the weights in the back-propagation network
Update the weights in the back-propagation network
propagating backward the errors associated with
propagating backward the errors associated with
output neurons.
output neurons.
(
(a
a) Calculate the error gradient for the neurons in the
) Calculate the error gradient for the neurons in the
output layer:
output layer:
where
where
Calculate the weight corrections:
Update the weights at the output neurons:
Update the weights at the output neurons:
  )
(
)
(
1
)
(
)
( p
e
p
y
p
y
p k
k
k
k 




)
(
)
(
)
( , p
y
p
y
p
e k
k
d
k 

)
(
)
(
)
( p
p
y
p
w k
j
jk 
 



)
(
)
(
)
1
( p
w
p
w
p
w jk
jk
jk 





(
(b
b) Calculate the error gradient for the neurons in
) Calculate the error gradient for the neurons in
the hidden layer:
the hidden layer:
Update the weights at the hidden neurons:
Update the weights at the hidden neurons:
)
(
)
(
)
(
1
)
(
)
(
1
]
[ p
w
p
p
y
p
y
p jk
l
k
k
j
j
j 




 

)
(
)
(
)
( p
p
x
p
w j
i
ij 
 



)
(
)
(
)
1
( p
w
p
w
p
w ij
ij
ij 



Step 3
Step 3: Weight training (continued)
: Weight training (continued)


Step 4
Step 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-OR
Exclusive-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 operation
y5
5
x1 3
1
x2
Input
layer
Output
layer
Hidden layer
4
2

3
w13
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:
w
w13
13 = 0.5,
= 0.5, w
w14
14 = 0.9,
= 0.9, w
w23
23 = 0.4,
= 0.4, w
w24
24 = 1.0,
= 1.0, w
w35
35 =
= 
1.2,
1.2,
w
w45
45 = 1.1,
= 1.1, 
3
3 = 0.8,
= 0.8, 
4
4 =
= 
0.1 and
0.1 and 
5
5 = 0.3.
= 0.3.


 We consider a training set where inputs
We consider a training set where inputs x
x1
1 and
and x
x2
2 are
are
equal to 1 and desired output
equal to 1 and desired output y
yd
d,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 as
calculated as
  5250
.
0
1
/
1
)
( )
8
.
0
1
4
.
0
1
5
.
0
1
(
3
23
2
13
1
3 





 





e
w
x
w
x
sigmoid
y
  8808
.
0
1
/
1
)
( )
1
.
0
1
0
.
1
1
9
.
0
1
(
4
24
2
14
1
4 





 





e
w
x
w
x
sigmoid
y
 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
.
0
1
/
1
)
( )
3
.
0
1
1
.
1
8808
.
0
2
.
1
5250
.
0
(
5
45
4
35
3
5 





 






e
w
y
w
y
sigmoid
y
5097
.
0
5097
.
0
0
5
5
, 




 y
y
e 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, e
e, 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
.
0
5097)
.
0
(
0.5097)
(1
0.5097
)
1
( 5
5
5 







 e
y
y

 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:
0112
.
0
)
1274
.
0
(
8808
.
0
1
.
0
5
4
45 








 
 y
w
0067
.
0
)
1274
.
0
(
5250
.
0
1
.
0
5
3
35 








 
 y
w
0127
.
0
)
1274
.
0
(
)
1
(
1
.
0
)
1
( 5
5 











 



 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
( 35
5
3
3
3 










 w
y
y 

0.0147
.1
1
4)
0.127
(
0.8808)
(1
0.8808
)
1
( 45
5
4
4
4 










 w
y
y 

0038
.
0
0381
.
0
1
1
.
0
3
1
13 






 
 x
w
0038
.
0
0381
.
0
1
1
.
0
3
2
23 






 
 x
w
0038
.
0
0381
.
0
)
1
(
1
.
0
)
1
( 3
3 










 

0015
.
0
)
0147
.
0
(
1
1
.
0
4
1
14 








 
 x
w
0015
.
0
)
0147
.
0
(
1
1
.
0
4
2
24 








 
 x
w
0015
.
0
)
0147
.
0
(
)
1
(
1
.
0
)
1
( 4
4 










 



 At last, we update all weights and threshold:
At last, we update all weights and threshold:
5038
.
0
0038
.
0
5
.
0
13
13
13 




 w
w
w
8985
.
0
0015
.
0
9
.
0
14
14
14 




 w
w
w
4038
.
0
0038
.
0
4
.
0
23
23
23 




 w
w
w
9985
.
0
0015
.
0
0
.
1
24
24
24 




 w
w
w
2067
.
1
0067
.
0
2
.
1
35
35
35 






 w
w
w
0888
.
1
0112
.
0
1
.
1
45
45
45 




 w
w
w
7962
.
0
0038
.
0
8
.
0
3
3
3 








0985
.
0
0015
.
0
1
.
0
4
4
4 










3127
.
0
0127
.
0
3
.
0
5
5
5 








 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 Exclusive-OR
Exclusive-OR
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 three-layer network learning
Final results of three-layer network learning
Inputs
x1 x2
1
0
1
0
1
1
0
0
0
1
1
Desired
output
yd
0
0.0155
Actual
output
y5
Y
Error
e
Sum of
squared
errors
e
0.9849
0.9849
0.0175
 0.0155
0.0151
0.0151
 0.0175
0.0010


Network represented by McCulloch-Pitts model
Network represented by McCulloch-Pitts model
for solving the
for solving the Exclusive-OR
operation
y5
5
x1 3
1
x2 4
2
+1.0
 1
 1
 1
+1.0
+1.0
+1.0
+1.5
+1.0
+0.5
+0.5
 2.0


(
(a
a) Decision boundary constructed by hidden neuron 3;
) Decision boundary constructed by hidden neuron 3;
(
(b
b) Decision boundary constructed by hidden neuron 4;
) Decision boundary constructed by hidden neuron 4;
(
(c
c) Decision boundaries constructed by the complete
) Decision boundaries constructed by the complete
three-layer network
three-layer network
x1
x2
1
(a)
1
x2
1
1
(b)
0
0
x1 + x2 – 1.5 = 0 x1 + x2 – 0.5 = 0
x1 x1
x2
1
1
(c)
0
Decision boundaries
Decision boundaries


Accelerated learning in multilayer
Accelerated learning in multilayer
neural networks
neural 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 tangent
hyperbolic tangent:
:
where
where a
a and
and b
b 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
a
e
a
Y bX
h
tan


 
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 the
1) called the
momentum constant
momentum 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 rule
generalised delta rule.
.
)
(
)
(
)
1
(
)
( p
p
y
p
w
p
w k
j
jk
jk 

 









Learning with momentum for operation
Learning with momentum for operation Exclusive-OR
Exclusive-OR
0 20 40 60 80 100 120
10-4
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 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 1
Heuristic 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, 
, should be increased.
, should be increased.
Heuristic 2
Heuristic 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).


0 10 20 30 40 50 60 70 80 90 100
Epoch
0 20 40 60 80 100 120
0
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 rate
Learning 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 90
0
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 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.
The Hopfield Network
The Hopfield Network


 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 network
neural 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 n
n-neuron Hopfield network
-neuron Hopfield network
xi
x1
x2
xn
I
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


 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:













X
Y
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,
y
y1
1,
, y
y2
2, . . .,
, . . ., y
yn
n.
.
Thus, for a single-layer
Thus, for a single-layer n
n-neuron network, the state
-neuron network, the state
can be defined by the
can be defined by the state vector
state vector as:
as:















n
y
y
y

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 M
M is the number of states to be memorised
is the number of states to be memorised
by the network,
by the network, Y
Ym
m is the
is the n
n-dimensional binary
-dimensional binary
vector,
vector, I
I is
is n
n 
 n
n identity matrix, and superscript
identity matrix, and superscript T
T
denotes a matrix transposition.
denotes a matrix transposition.
I
Y
Y
W M
M
m
T
m
m 


1


Possible states for the three-neuron
Possible states for the three-neuron
Hopfield network
Hopfield 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 state-vertex is determined by the weight
The stable state-vertex is determined by the weight
matrix
matrix W
W, the current input vector
, the current input vector X
X, 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,
or
or
where
where Y
Y1
1 and
and Y
Y2
2 are the three-dimensional vectors.
are the three-dimensional vectors.











1
1
1
1
Y














1
1
1
2
Y  
1
1
1
1 
T
Y  
1
1
1
2 



T
Y


 The 3
The 3 
 3 identity matrix
3 identity matrix I
I is
is
 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, X
X1
1 and
and X
X2
2, which are equal to the output (or
, which are equal to the output (or
target) vectors
target) vectors Y
Y1
1 and
and Y
Y2
2, respectively.
, respectively.











1
0
0
0
1
0
0
0
1
I
   







































1
0
0
0
1
0
0
0
1
2
1
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 X
X. Then, we calculate the actual
. Then, we calculate the actual
output vector
output vector Y
Y, and finally, we compare the result
, and finally, we compare the result
with the initial input vector
with the initial input vector X
X.
.





















































1
1
1
0
0
0
1
1
1
0
2
2
2
0
2
2
2
0
1 sign
Y



























































1
1
1
0
0
0
1
1
1
0
2
2
2
0
2
2
2
0
2 sign
Y


 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 memories
fundamental 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 network
correction 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
M
Mmax
max that can be stored in the
that can be stored in the n
n-neuron recurrent
-neuron recurrent
network is limited by
network is limited by
n
Mmax 15
.
0

Storage capacity of the Hopfield network
Storage capacity of the Hopfield network


 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 associative
associative. 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)


 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), first
, first
proposed by
proposed by Bart Kosko
Bart Kosko, is a heteroassociative
, is a heteroassociative
network. It associates patterns from one set, set
network. It associates patterns from one set, set A
A,
,
to patterns from another set, set
to patterns from another set, set B
B, 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 operation
BAM 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 n
n-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 m
m-dimensional vector
-dimensional vector Y
Y from set
from set B
B, but
, but
when
when Y
Y is presented as input, the BAM recalls
is presented as input, the BAM recalls
X
X.
.


 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 X
X, and the transpose of the
, and the transpose of the
output vector
output vector Y
YT
T
. 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.
T
m
M
m
m Y
X
W 


1


 The BAM is
The BAM is unconditionally stable
unconditionally 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 convergence
incorrect 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 BAM
Stability and storage capacity of the BAM

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