Artificial Neural Networks
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
Commercial ANNs
• 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 hidden layers. Each layer can contain from
10 to 1000 neurons. Experimental neural 10 to 1000 neurons. Experimental neural
networks may have five or even six layers, networks may have five or even six layers,
including three or four hidden layers, and including three or four hidden layers, and
utilise millions of neurons.
utilise millions of neurons.
Example
• Character recognition
Problems that are not linearly separable
• Xor function is not linearly separable
• Using Multilayer networks with back propagation training algorithm
• There are hundreds of training algorithms
for multilayer neural networks
Multilayer neural networks Multilayer neural networks
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 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 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 do the middle layers hide?
What do the middle layers hide?
A hidden layer “hides” its desired output. Neurons A hidden layer “hides” its desired output. Neurons in the hidden layer cannot be observed through the in the hidden layer cannot be observed through the input/output behaviour of the network. There is no input/output behaviour of the network. There is no
obvious way to know what the desired output of the obvious way to know what the desired output of the
hidden layer should be.
hidden layer should be.
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 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 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
Three-layer network Three-layer network
y5 5
x1 1 3
x2
Input layer
Output layer Hidden layer
2 4
3 w13
w24 w23
w24
w35
w45
4
5
1
1
1
w13 = 0.5, w13 = 0.5, ww14 = 0.9, 14 = 0.9, w23 = 0.4, w23 = 0.4, ww24 = 1.0, 24 = 1.0, w35 = w35 = 1.2, 1.2, ww45 = 1.1, 45 = 1.1, 3 = 0.8, 3 = 0.8, 4 = 4 = 0.1 and 0.1 and
5 = 0.3 5 = 0.3
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 e.g., as follows:
randomly e.g., 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, 33 = 0.8, = 0.8, 44 = = 0.1 and 0.1 and 55 = 0.3. = 0.3.
Assuming the sigmoid activation Function
Step 2
Step 2: Activation: Activation
Activate the back-propagation neural network by Activate the back-propagation neural network by
applying inputs
applying inputs xx11((pp), ), xx22((pp),…, ),…, xxnn((pp) and desired ) and desired outputs
outputs yyd,1d,1((pp), ), yyd,2d,2((pp),…, ),…, yyd,d,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
j
n i
ij i
j p sigmoid x p w p y
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.
k
m j
jk jk
k p sigmoid x p w p y
1
) ( )
( )
(
Step 2
Step 2: Activation (continued): Activation (continued)
Class Exercise
If the sigmoid activation function is used the output of If the sigmoid activation function is used the output of
the hidden layer is the hidden layer is
1
0.5250/ 1 )
( 1 13 2 23 3 (10.5 10.4 10.8)
3 sigmoid x w x w e y
1
0.8808/ 1 )
( 1 14 2 24 4 (10.9 11.0 10.1)
4 sigmoid x w x w e y
And the actual output of neuron 5 in the output layer is And the actual output of neuron 5 in the output layer is
And the error is And the error is
1
0.5097/ 1 )
( 3 35 4 45 5 ( 0.52501.2 0.88081.1 10.3)
5 sigmoid y w y w e y
5097 .
0 5097
. 0
5 0
5
,
y y
e d
What learning law applies in a multilayer neural network?
) ( )
( )
(p y p p
wjk j k
Step 3
Step 3: Weight training output layer: Weight training output layer
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.
((aa) Calculate the error) Calculate the error
and then the error gradient for the neurons in the and then the error gradient for the neurons in the
output layer:
output layer:
Then the weight corrections:
Then the weight corrections:
Then the new weights at the output neurons:
Then the new weights at the output neurons:
1 ( )
( )) ( )
( p yk p yk p ek p
k
) ( )
( )
( p y , p y p
ek d k k
) ( )
( )
(p y p p
wjk j k
) ( )
( )
1
( p w p w p
wjk jk jk
Three-layer network for solving the Three-layer network for solving the
Exclusive-OR operation Exclusive-OR operation
y5 5
x1 1 3
x2
Input layer
Output layer 2 4
3 w13
w24 w23
w24
w35
w45
4
5
1
1
1
The error gradient for neuron 5 in the output layer:The error gradient for neuron 5 in the output layer:
1274 .
0 5097)
. 0 ( 0.5097) (1
0.5097 )
1
( 5
5
5 y y e
Determine the weight corrections assuming that the Determine the weight corrections assuming that the learning rate parameter,
learning rate parameter, , is equal to 0.1:, is equal to 0.1:
0112 .
0 )
1274 .
0 ( 8808 .
0 1 .
5 0
4
45
ww35 yy3 5 0.10.5250( 0.1274) 0.0067 0127 .
0 )
1274 .
0 ( ) 1 ( 1 . 0 )
1
( 5
5
Apportioning error in the hidden layer
• Error is apportioned in proportion to the weights of the connecting arcs.
• Higher weight indicates higher error
responsibility
((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
]
[
y p p w pp y
p l jk
k
k j
j
j
) ( )
( )
( p x p p
wij i j
) ( )
( )
1
( p w p w p
wij ij ij
Step 3
Step 3: Weight training hidden layer: Weight training hidden layer
The error gradients for neurons 3 and 4 in the hidden The error gradients for neurons 3 and 4 in the hidden layer:
layer:
Determine the weight corrections:Determine the weight corrections:
0381 .
0 ) 2 . 1 ( 0.1274) (
0.5250) (1
0.5250 )
1
( 3 5 35
3
3 y y w
0.0147 .1
1 4) 0.127 (
0.8808) (1
0.8808 )
1
( 4 5 45
4
4 y y w
0038 .
0 0381
. 0 1 1 .
3 0
1
13
w x
0038 .
0 0381 .
0 1 1 .
3 0
2
23
w x
0038 .
0 0381
. 0 ) 1 ( 1 . 0 )
1
( 3
3
0015 .
0 )
0147 .
0 ( 1 1 .
4 0
1
14
w x
0015 .
0 )
0147 .
0 ( 1 1 .
4 0
2
24
w x
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 .
13 0
13
13 w w
w
8985 .
0 0015
. 0 9 .
14 0
14
14 w w
w
4038 .
0 0038
. 0 4 .
23 0
23
23 w w
w
9985 .
0 0015
. 0 0 .
24 1
24
24 w w
w
2067 .
1 0067
. 0 2 .
35 1
35
35 w w
w
0888 .
1 0112
. 0 1 .
45 1
45
45 w w
w
7962 .
0 0038
. 0 8 .
3 0
3
3
0985 .
0 0015
. 0 1 .
4 0
4
4
3127 .
0 0127
. 0 3 .
5 0
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.
Step 4
Step 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 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.
Typical Learning Curve Typical Learning Curve
0 50 100 150 200
101
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
x
1x
21
0 1 0
1 1 0 0
0 1 1
Desired output
y
d0
0.0155
Actual output
y
5Y
Error
e
Sum of squared
errors
0.9849 e 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-ORExclusive-OR operation operation
y5 5
x1 1 3
x2 2 4
+1.0
1
1
1 +1.0
+1.0 +1.0
+1.5
+1.0
+1.0
+0.5
+0.5
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 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 e a
Y tanh abX
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 rulegeneralised delta rule..
) ( )
( )
1 (
)
( p w p y p p
w
jk
jk
j
k
Learning with an adaptive learning rate Learning with an 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 Heuristic
If the error is decreasing the learning rate
If the error is decreasing the learning rate , should be , should be increased.
increased.
If the error is increasing or remaining constant the If the error is increasing or remaining constant the
learning rate
learning rate , should be decreased., should be 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).
Typical Learning Curve Typical Learning Curve
101
Sum-Squared Error
Sum-Squared Network Error for 224 Epochs
100
10-1
10-2
10-3
Typical learning with adaptive learning rate Typical learning with adaptive learning rate
0 10 20 30 40 50 60 70 80 90 100
Epoch
Training for 103 Epochs
0 20 40 60 80 100 120
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
Typical Learning with Typical Learning with
adaptive learning rate plus momentum adaptive learning rate plus momentum
0 10 20 30 40 50 60 70 80
Epoch
Training for 85 Epochs
0.5 1 2.5
Learning Rate
10-4 10-2 100 102
Sum-Squared Error
10-3 101 10-1
1.5 2
