Artificial neural networks: Supervised learning

Lecture 7
Artificial neural networks:
Supervised learning
■ Introduction, or how the brain works
■ The neuron as a simple computing element
■ The perceptron
■ Multilayer neural networks
■ Accelerated learning in multilayer neural networks
■ The Hopfield network
■ Bidirectional associative memories (BAM)
■ Summary
 Negnevitsky, Pearson Education,
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Introduction, or how the brain works
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Machine learning involves adaptive mechanisms
that enable computers to learn from experience,
learn by example and learn by analogy.
Learning capabilities can improve the
performance of an intelligent system over time.
The most popular approaches to machine
learning are artificial neural networks and
genetic algorithms. This
lecture is dedicated to neural networks.

■ A neural network can be defined as a model of
reasoning based on the human brain. The brain
consists of a densely interconnected set of nerve
cells, or basic information-processing units, called
neurons.
■ The human brain incorporates nearly 10 billion
neurons and 60 trillion connections, synapses,
between them. By using multiple neurons
simultaneously, the brain can perform its functions
much faster than the fastest computers in
existence today.
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■ Each neuron has a very simple structure, but an
army of such elements constitutes a tremendous
processing power.
■ A neuron consists of a cell body, soma, a number of
fibers called dendrites, and a single long fiber
called the axon.
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■ Our brain can be considered as a highly complex,
non-linear and parallel information-processing
system.
■ Information is stored and processed in a neural
network simultaneously throughout the whole
network, rather than at specific locations. In
other words, in neural networks, both data and its
processing are global rather than local.
■ Learning is a fundamental and essential
characteristic of biological neural networks.
The ease with which they can learn led to
attempts to emulate a biological neural network in
a computer.
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■ An artificial neural network consists of a number of
very simple processors, also called neurons, which
are analogous to the biological neurons in the
brain.
■ The neurons are connected by weighted links
passing signals from one neuron to another.
■ The output signal is transmitted through the
neuron’s outgoing connection. The outgoing
connection splits into a number of branches that
transmit the same signal. The outgoing
branches terminate at the incoming connections
of other neurons in the network.
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Can a single neuron learn a task?
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■ In 1958, Frank Rosenblatt introduced a
training algorithm that provided the first
procedure for training a simple ANN: a
perceptron.
■ The perceptron is the simplest form of a neural
network. It consists of a single neuron with
adjustable synaptic weights and a hard limiter.

The Perceptron
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■ The operation of Rosenblatt’s perceptron is based
on the McCulloch and Pitts neuron model. The
model consists of a linear combiner followed by
a hard limiter.
■ The weighted sum of the inputs is applied to the
hard limiter, which produces an output equal to +1
if its input is positive and 1 if it is negative.

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

Multilayer neural networks
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■ A multilayer perceptron is a feedforward neural
network with one or more hidden layers.
■ The network consists of an input layer of source
neurons, at least one middle or hidden layer of
computational neurons, and an output layer of
computational neurons.
■ The input signals are propagated in a forward
direction on a layer-by-layer basis.

What does the middle layer hide?
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■ A hidden layer “hides” its desired output.
Neurons in the hidden layer cannot be observed
through the input/output behaviour of the network.
There is no obvious way to know what the
desired output of the hidden layer should be.
■ Commercial ANNs incorporate three and
sometimes four layers, including one or two
hidden layers. Each layer can contain from 10
to 1000 neurons.Experimental neural networks
may have five or even six layers, including three
or four hidden layers, and utilise millions of
neurons.

Back-propagation neural network
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■ Learning in a multilayer network proceeds the
same way as for a perceptron.
■ A training set of input patterns is presented to the
network.
■ The network computes its output pattern, and if
there is an error  or in other words a difference
between actual and desired output patterns  the
weights are adjusted to reduce this error.

■ In a back-propagation neural network, the learning
algorithm has two phases.
■ First, a training input pattern is presented to the
network input layer. The network propagates
the input pattern from layer to layer until the
output pattern is generated by the output layer.
■ If this pattern is different from the desired output,
an error is calculated and then propagated
backwards through the network from the output
layer to the input layer. The weights are
modified as the error is propagated.
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Step 1: Initialisation
Set all the weights and threshold levels of the
network to random numbers uniformly
distributed inside a small range:
where Fi is the total number of inputs of neuron i
in the network. The weight initialisation is done
on a neuron-by-neuron basis.
The back-propagation training algorithm
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Step 2: Activation
Activate the back-propagation neural network by
applying inputs x1(p), x2(p),…, xn(p) and desired
outputs yd,1(p), yd,2(p),…, yd,n(p).
(a) Calculate the actual outputs of the neurons in
the hidden layer:
where n is the number of inputs of neuron j in the
hidden layer, and sigmoid is the sigmoid activation
function.
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(b) Calculate the actual outputs of the neurons in
the output layer:
where m is the number of inputs of neuron k in the
output layer.
Step 2: Activation (continued)
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Step 3: Weight training
Update the weights in the back-propagation network
propagating backward the errors associated with
output neurons.
(a) Calculate the error gradient for the neurons in the
output layer:
Calculate the weight corrections:
Update the weights at the output neurons:
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Calculate the weight corrections:
Update the weights at the hidden neurons:
Step 3: Weight training (continued)
(b) Calculate the error gradient for the
neurons in the hidden layer:
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Step 4: Iteration
Increase iteration p by one, go back to Step 2 and
repeat the process until the selected error criterion
is satisfied.
As an example, we may consider the three-layer
back-propagation network. Suppose that the
network is required to perform logical operation
Exclusive-OR. Recall that a single-layer
perceptron could not do this operation.
Now we will apply the three-layer net.
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■ The effect of the threshold applied to a neuron in the
hidden or output layer is represented by its weight,
, connected to a fixed input equal to 1.
■ The initial weights and threshold levels are set
randomly as follows:
w13 = 0.5, w14 = 0.9, w23 = 0.4, w24 = 1.0, w35 =
1.2,
w45 = 1.1, 3 = 0.8, 4 = 0.1 and 5 = 0.3.
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Accelerated learning in multilayer
neural networks
■ A multilayer network learns much faster when the
sigmoidal activation function is represented by a
hyperbolic tangent:
where a and b are constants.
Suitable values for a and b are:
a = 1.716 and b = 0.667
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■ We also can accelerate training by including a
momentum term in the delta rule:
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where  is a positive number (0    1) called
the momentum constant. Typically, the
momentum constant is set to 0.95.
This equation is called the generalised delta rule.

Learning with adaptive learning rate
To accelerate the convergence and yet avoid the
danger of instability, we can apply two heuristics:
Heuristic 1
If the change of the sum of squared errors has the same
algebraic sign for several consequent epochs, then the
learning rate parameter, , should be increased.
Heuristic 2
If the algebraic sign of the change of the sum of
squared errors alternates for several consequent
epochs, then the learning rate parameter, , should
be decreased.
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■ Adapting the learning rate requires some changes
in the back-propagation algorithm.
■ If the sum of squared errors at the current epoch
exceeds the previous value by more than a
predefined ratio (typically 1.04), the learning rate
parameter is decreased (typically by multiplying
by 0.7) and new weights and thresholds are
calculated.
■ If the error is less than the previous one, the
learning rate is increased (typically by multiplying
by 1.05).
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■ Neural networks were designed on analogy with
the brain. The brain’s memory, however, works
by association. For example, we can
recognise a familiar face even in an unfamiliar
environment within 100-200 ms. We can also
recall a complete sensory experience, including
sounds and scenes, when we hear only a few
bars of music. The brain routinely associates
one thing with another.
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The Hopfield Network

■ Multilayer neural networks trained with the back-
propagation algorithm are used for pattern
recognition problems. However, to emulate
the human memory’s associative characteristics
we need a different type of network: a recurrent
neural network.
■ A recurrent neural network has feedback loops
from its outputs to its inputs. The presence of
such loops has a profound impact on the learning
capability of the network.
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■ The stability of recurrent networks intrigued
several researchers in the 1960s and 1970s.
However, none was able to predict which network
would be stable, and some researchers were
pessimistic about finding a solution at all. The
problem was solved only in 1982, when John
Hopfield formulated the physical principle of
storing information in a dynamically stable
network.
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■ In the Hopfield network, synaptic weights between
neurons are usually represented in matrix form as
follows:
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where M is the number of states to be memorised
by the network, Ym is the n-dimensional binary
vector, I is n  n identity matrix, and superscript T
denotes a matrix transposition.

■ The remaining six states are all unstable.
However, stable states (also called fundamental
memories) are capable of attracting states that are
close to them.
■ The fundamental memory (1, 1, 1) attracts unstable
states (1, 1, 1), (1, 1, 1) and (1, 1, 1). Each
of these unstable states represents a single error,
compared to the fundamental memory (1, 1, 1).
■ The fundamental memory (1, 1, 1) attracts unstable
states (1, 1, 1), (1, 1, 1) and (1, 1, 1).
■ Thus, the Hopfield network can act as an error
correction network.
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■ Storage capacity is or the largest number of
fundamental memories that can be stored and
retrieved correctly.
■ The maximum number of fundamental memories
Mmax that can be stored in the n-neuron recurrent
network is limited by
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Storage capacity of the Hopfield network

■ The Hopfield network represents an autoassociative
type of memory  it can retrieve a corrupted or
incomplete memory but cannot associate this memory
with another different memory.
■ Human memory is essentially associative. One
thing may remind us of another, and that of another,
and so on. We use a chain of mental
associations to recover a lost memory. If we
forget where we left an umbrella, we try to recall
where we last had it, what we were doing, and who
we were talking to. We
attempt to establish a chain of associations, and
thereby to restore a lost memory.
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Bidirectional associative memory (BAM)

■ To associate one memory with another, we need a
recurrent neural network capable of accepting an
input pattern on one set of neurons and producing
a related, but different, output pattern on another
set of neurons.
■ Bidirectional associative memory (BAM), first
proposed by Bart Kosko, is a heteroassociative
network. It associates patterns from one set, set
A, to patterns from another set, set B, and vice
versa. Like a Hopfield network, the BAM can
generalise and also produce correct outputs
despite corrupted or incomplete inputs.
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The basic idea behind the BAM is to store
pattern pairs so that when n-dimensional vector
X from set A is presented as input, the BAM
recalls m-dimensional vector Y from set B, but
when Y is presented as input, the BAM recalls X.
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■ To develop the BAM, we need to create a
correlation matrix for each pattern pair we want to
store. The correlation matrix is the matrix
product of the input vector X, and the transpose
of the output vector YT. The BAM weight
matrix is the sum of all correlation matrices, that
is,
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where M is the number of pattern pairs to be stored
in the BAM.

■ The BAM is unconditionally stable. This means
that any set of associations can be learned without
risk of instability.
■ The maximum number of associations to be stored
in the BAM should not exceed the number of
neurons in the smaller layer.
■ The more serious problem with the BAM is
incorrect convergence. The BAM may not
always produce the closest association. In
fact, a stable association may be only slightly
related to the initial input vector.
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Stability and storage capacity of the BAM

Artificial neural networks: Supervised learning

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