Lecture 7 Artificial neural networks: Supervised learning

Lecture 7

Lecture 7

Artificial neural networks:

Artificial neural networks:

Supervised learning

Supervised learning

Introduction, or how the brain works Introduction, or how the brain works

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

The perceptron The perceptron The perceptron The perceptron

Multilayer neural networks Multilayer neural networks

Accelerated learning in multilayer neural networks Accelerated learning in multilayer neural networks

The Hopfield network The Hopfield network

Bidirectional associative memories (BAM) Bidirectional associative memories (BAM)

Introduction, or how the brain works

Introduction, 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 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 algorithmsgenetic algorithms. This . This lecture is dedicated to neural networks.

A

A neural networkneural network can be defined as a model of can be defined as a model of reasoning based on the human brain. The brain reasoning based on the human brain. The brain consists of a densely interconnected set of nerve consists of a densely interconnected set of nerve cells, or basic information

cells, or basic information--processing units, called processing units, called

neurons neurons. .

The human brain incorporates nearly 10 billion The human brain incorporates nearly 10 billion 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.

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 fibers called

fibers called dendritesdendrites, and a single long fiber , and a single long fiber called the

called the axonaxon.. called the

Biological neural network

Biological neural network

Synapse

Synapse Axon

Dendrites

Axon

Soma _Soma Dendrites

Our brain can be considered as a highly complex, Our brain can be considered as a highly complex, non

non--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 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

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 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.

Architecture of a typical artificial neural network

Architecture of a typical artificial neural network

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Input Layer Output Layer

Middle Layer

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Biological Neural Network Artificial Neural Network Soma

Dendrite Axon

Neuron Input Output

Analogy between biological and

Analogy between biological and

artificial neural networks

artificial neural networks

Axon Synapse

The neuron as a simple computing element

The neuron as a simple computing element

Diagram of a neuron

Diagram of a neuron

Input Signals

x₁

Output Signals

Y w₁

Weights

Neuron _Y

x₂

x_n

Y

Y w₂

w₁

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, , 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

The neuron uses the following transfer or

The neuron uses the following transfer or activationactivation 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







θ

<

−

θ

≥

+

=

X

X

Y

if

,

1

Activation functions of a neuron

Activation functions of a neuron

Step function Sign function

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 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 network. It consists of a single neuron with network. It consists of a single neuron with

adjustable

Inputs

x₁

Output Hard

Limiter

w₁

Linear Combiner

Single

Single--layer two

layer two--input perceptron

input perceptron

Threshold

x₂

Output

Y

∑

∑

∑

∑

w₂

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 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 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

The aim of the perceptron is to classify inputs, The aim of the perceptron is to classify inputs,

xx₁₁, , xx₂₂, . . ., , . . ., xx_nn, into one of two classes, say , into one of two classes, say

A

A₁₁ and and AA₂₂. .

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 hyperplanehyperplane into into two decision regions. The hyperplane is defined by two decision regions. The hyperplane is defined by two decision regions. The hyperplane is defined by two decision regions. The hyperplane is defined by the

the linearly separablelinearly separable functionfunction::

0

1

=

θ

−

∑

=

n

i

i i

w

Linear separability in the perceptrons

Linear separability in the perceptrons

x₁ x₂

Class A

Class A₁

1

x₂

1 2

1

Class A₂

2

x₁w₁ + x₂w₂ − θ = 0

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

x₁

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

How does the perceptron learn its classification

How does the perceptron learn its classification

tasks?

tasks?

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

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 YY_{d 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 ppth training example th training example

)

(

)

(

)

(

p

Y

p

Y

p

e

=

−

Iteration

Iteration pp here refers to the here refers to the ppth training example th training example presented to the perceptron.

presented to the perceptron.

If the error,

If the error, ee((pp), is positive, we need to increase ), is positive, we need to increase perceptron output

perceptron output YY((pp), but if it is negative, we ), but if it is negative, we need to decrease

The perceptron learning rule

The perceptron learning 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.

)

(

)

(

)

(

)

1

(

p

w

p

x

p

e

p

w

+

=

+

α

⋅

⋅

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

Step 1

Step 1: Initialisation: Initialisation

Set initial weights

Set initial weights ww₁₁, , ww₂₂,…, ,…, ww_nn and threshold and threshold θθ 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, ee((pp), is positive, we need to increase ), is positive, we need to increase

Perceptron’s training algorithm

Perceptron’s training algorithm

If the error,

If the error, ee((pp), is positive, we need to increase ), is positive, we need to increase perceptron output

perceptron output YY((pp), but if it is negative, we ), but if it is negative, we need to decrease

Step 2

Step 2: Activation: Activation

Activate the perceptron by applying inputs

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

xx₂₂((pp),…, ),…, xx_nn((pp) and desired output ) and desired output YY_{d d} ((pp). ). Calculate the actual output at iteration

Calculate the actual output at iteration pp = 1= 1

Perceptron’s training algorithm (continued) Perceptron’s training algorithm (continued)

 

∑

where

where nn is the number of the perceptron inputs, is the number of the perceptron inputs, and

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

   

   

θ −

=

∑

=

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 ∆∆ww_ii((pp)) isis thethe weightweight correctioncorrection atat iterationiteration pp..

The weight correction is computed by the

The weight correction is computed by the delta delta

) ( )

( )

1

( p w p w p

w_i + = _i + ∆ _i

Perceptron’s training algorithm (continued) Perceptron’s training algorithm (continued)

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 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.

)

(

)

(

)

(

p

x

p

e

p

w

=

α

⋅

⋅

Example of perceptron learning: the logical operation

Example of perceptron learning: the logical operation ANDAND

Inputs

x₁ x₂

0 0 1 1 0 1 0 1 0 0 0 Epoch Desired output Y_d 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 0

3 0.2

0.0 0.0 0.0 0.0

Two

Two--dimensional plots of basic logical operations

dimensional plots of basic logical operations

x₁ x₂

1 1

x₁ x₂

1 1

x₁ x₂

1 1

0

0 0

(a) AND (x₁ ∩ x₂) (b) OR (x₁ ∪ x₂) (c) Exclusive-OR

(x₁ ⊕ x₂)

A perceptron can learn the operations

A perceptron can learn the operations ANDAND and and OROR, , but not

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, 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

Multilayer perceptron with two hidden layers

Multilayer perceptron with two hidden layers

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Input layer

First hidden

layer

Second hidden

layer

Output layer

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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 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

Back

Back--propagation neural network

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 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.

In a back

In 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. 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.

Three

Three--layer back

layer back--propagation neural network

propagation neural network

x_i x₁

x₂

1

2

i

1

2

k y_k

y₁

y₂ Input signals

w_jk w_ij

1

2

j

Input layer x_i

x_n n

Output layer

k

l

y_k

y_l

Error signals

jk

Hidden layer

ij j

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:

The back

The back--propagation training algorithm

propagation training algorithm

  _{2.4 2.4}

where

where FF_ii 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

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

   

 

+ −

i

i F

F

4 . 2

Step 2

Step 2: Activation: Activation

Activate the back

Activate the back--propagation neural network by propagation neural network by applying inputs

applying inputs xx₁₁((pp), ), xx₂₂((pp),…, ),…, xx_nn((pp) and desired ) and desired outputs

outputs yy_dd,1,1((pp), ), yy_dd,2,2((pp),…, ),…, yy_dd,,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

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

   

   

θ − ⋅

=

∑

= k

m

j

jk jk

k p sigmoid x p w p

y

1

) ( )

( )

(

Step 2

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

where

where mm is the number of inputs of neuron is the number of inputs of neuron kk in the in the output layer.

Step 3

Step 3: Weight training: Weight training

Update the weights in the back

Update 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:

[

]

) ( )

( p y_k p y_k p e_k p

k = ⋅ − ⋅

δ

where where

Calculate the weight corrections: Calculate the weight corrections:

Update the weights at the output neurons: Update the weights at the output neurons:

[

]

) ( )

( p y_k p y_k p e_k p

k = ⋅ − ⋅

δ

) ( )

( )

( p y _, p y p e_k = _{d k} − _k

) ( )

( )

( p y p p

w_jk =α ⋅ _j ⋅δ_k

∆

) ( )

( )

1

( p w p w p

((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:

) ( )

( )

( 1

) ( )

(

1

]

[

p y

p _jk

l

k

k j

j

j

∑

=

⋅

− ⋅

= δ

δ

Step 3

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

Calculate the weight corrections: Calculate the weight corrections:

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

) ( )

( )

( p x p p

w_ij = α ⋅ _i ⋅δ _j

∆

) ( )

( )

1

( p w p w p

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

As an example, we may consider the three--layer layer back

back--propagation network. Suppose that the propagation network. Suppose that the back

back--propagation network. Suppose that the propagation network. Suppose that the

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

Exclusive

Exclusive--OROR. Recall that a single. Recall that a single--layer perceptron layer perceptron could not do this operation. Now we will apply the could not do this operation. Now we will apply the three

Three

Three--layer network for solving the

layer network for solving the

Exclusive

Exclusive--OR operation

OR operation

y

5

x₁ 1 ₃

θ₃

w₁₃

w₂₃ w35 θ5 −1

−1

y₅

5

x₂

Input layer

Output layer

4 2

w₂₄ w₂₄

w₄₅

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:

w

w₁₃₁₃ = 0.5, = 0.5, ww₁₄₁₄ = 0.9, = 0.9, ww₂₃₂₃ = 0.4, w= 0.4, w₂₄₂₄ = 1.0, = 1.0, ww₃₅₃₅ = = −−1.2, 1.2,

w

w₁₃₁₃ = 0.5, = 0.5, ww₁₄₁₄ = 0.9, = 0.9, ww₂₃₂₃ = 0.4, w= 0.4, w₂₄₂₄ = 1.0, = 1.0, ww₃₅₃₅ = = −−1.2, 1.2,

w

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 yy_dd,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

[

]

/ 1 )

( _{1 13 2 23 3} (10.5 10.4 10.8)

3 = sigmoid x w + x w − θ = + e− ⋅ + ⋅ − ⋅ =

y

[

]

/ 1 )

( _{1 14 2 24 4} (10.9 11.0 10.1)

4 = sigmoid x w + x w − θ = + e− ⋅ + ⋅ + ⋅ =

y_{4 1 14 2 24 4}

[

]

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:

[

]

/ 1 )

( _{3 35 4 45 5} ( 0.52501.2 0.88081.1 10.3)

5 = sigmoid y w + y w −θ = +e− − ⋅ + ⋅ − ⋅ =

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 . 0 5097) . 0 ( 0.5097) (1 0.5097 ) 1 ( ₅ 5

5 = y − y e = ⋅ − ⋅ − = −

δ

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 . 0 1 . 0 5 4

45 = ⋅ ⋅ = ⋅ ⋅ − = −

∆w α y δ

0067 . 0 ) 1274 . 0 ( 5250 . 0 1 . 0 5 3

35 = ⋅ ⋅ = ⋅ ⋅ − = −

∆w α y δ

0127 . 0 ) 1274 . 0 ( ) 1 ( 1 . 0 ) 1 ( ₅

5 = ⋅ − ⋅ = ⋅ − ⋅ − = −

θ

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

( _{3 5 35}

3

3 = y − y ⋅δ ⋅ w = ⋅ − ⋅ − ⋅ − =

δ 0.0147 .1 1 4) 0.127 ( 0.8808) (1 0.8808 ) 1

( _{4 5 45}

4

4 = y − y ⋅δ ⋅ w = ⋅ − ⋅ − ⋅ = −

δ 0038 . 0 0381 . 0 1 1 . 0 3 1

13 = ⋅ ⋅ = ⋅ ⋅ =

∆w α x δ

0038 . 0 0381 . 0 1 1 . 0 3 2

23 = ⋅ ⋅ = ⋅ ⋅ =

∆w α x δ

0038 . 0 0381 . 0 ) 1 ( 1 . 0 ) 1 ( ₃

3 = ⋅ − ⋅ = ⋅ − ⋅ = −

θ

∆ α δ

0015 . 0 ) 0147 . 0 ( 1 1 . 0 4 1

14 = ⋅ ⋅ = ⋅ ⋅ − = −

∆w α x δ

0015 . 0 ) 0147 . 0 ( 1 1 . 0 4 2

24 = ⋅ ⋅ = ⋅ ⋅ − = −

∆w α x δ

0015 . 0 ) 0147 . 0 ( ) 1 ( 1 . 0 ) 1 ( ₄

4 = ⋅ − ⋅ = ⋅ − ⋅ − =

θ

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 − =

= ∆

+ = w w

w₄₅ = w₄₅ + ∆w₄₅ = 1.1− 0.0112 = 1.0888

w 7962 . 0 0038 . 0 8 . 0 3 3

3 = θ + ∆θ = − =

θ 0985 . 0 0015 . 0 1 . 0 4 4

4 = θ + ∆θ = − + = −

θ 3127 . 0 0127 . 0 3 . 0 5 5

5 = 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.

Learning curve for operation

Learning curve for operation Exclusive

Exclusive--OR

OR

101

S

u

m

-S

q

u

a

re

d

E

rr

o

r

Sum-Squared Network Error for 224 Epochs

100

10-1

0 50 100 150 200

S

u

m

-S

q

u

a

re

d

E

rr

o

r

10-2

10-3

Final results of three

Final results of three--layer network learning

layer network learning

Inputs

x₁ x₂

1 0

1 1

0 1

Desired output

y_d

0.0155

Actual output

y₅ Y

Error

e

Sum of squared

errors

0.9849

−0.0155 0.0151

0.0010 0

1 0

1 0 0

1 1 0

0.9849 0.9849 0.0175

0.0151 0.0151

Network represented by McCulloch

Network represented by McCulloch--Pitts model Pitts model for solving the

for solving the ExclusiveExclusive--OROR operationoperation

x₁ 1 3

+1.0

−1

−1 +1.0

+1.5

+0.5

−2.0

y₅

5

x₂ 2 ₄

+1.0 +1.0

+1.0

x₁ x₂

1 1

x₂

1 1

0 0

x₁ + x₂ – 1.5 = 0 x₁ + x₂ – 0.5 = 0

x₁ x₁

x₂

1 1

0

Decision boundaries

Decision boundaries

((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

three

three--layer networklayer network

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::

a

h

tan

₌

2

₋

where

where aa andand bb areare constantsconstants..

Suitable values for

Suitable values for aa and and bb are: are:

a

a = 1.716 and = 1.716 and bb = 0.667= 0.667

a

e

a

Y

bX h

tan

₋

+

=

1

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

momentum constant

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

)

(

)

(

)

1

(

)

(

p

w

p

y

p

p

w

=

β

⋅

∆

−

+

α

⋅

⋅

δ

∆

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

Learning with momentum for operation

Learning with momentum for operation ExclusiveExclusive--OROR

0 20 40 60 80 100 120 10-4

10-2 100 102

Epoch

S

u

m

-S

q

u

a

re

d

E

rr

o

r

Training for 126 Epochs

10-3 101

10-1

Epoch

-1 -0.5 0 0.5 1 1.5

L

e

a

rn

in

g

R

a

Learning with adaptive learning rate

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 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 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, αα, should be , should be decreased.

Adapting the learning rate requires some changes Adapting the learning rate requires some changes in the back

in 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).

Learning with adaptive learning rate

Learning with adaptive learning rate

0 10 20 30 40 50 60 70 80 90 100

Epoch

Training for 103 Epochs

10-4 10-2 100 102

S

u

m

-S

q

u

a

re

d

E

rr

o

r

10-3 101

10-1

0 20 40 60 80 100 120

0 0.2 0.4 0.6 0.8 1

Epoch

L

e

a

rn

in

g

R

a

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

0 10 20 30 40 50 60 70 80

Epoch

Training for 85 Epochs

10-4 10-2 100 102

S

u

m

-S

q

u

a

re

d

E

rr

o

r

10-3 101

10-1

Epoch

0 10 20 30 40 50 60 70 80 90 0

0.5 1 2.5

L

e

a

rn

in

g

R

a

te

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

within 100--200 ms. We can also recall a complete 200 ms. We can also recall a complete

The Hopfield Network

The Hopfield Network

within 100

within 100--200 ms. We can also recall a complete 200 ms. We can also recall a complete sensory experience, including sounds and scenes, sensory experience, including sounds and scenes, when we hear only a few bars of music. The

when we hear only a few bars of music. The

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..

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.

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

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.

Single

Single--layer

layer n

n--neuron Hopfield network

neuron Hopfield network

x_i x₁

x₂

I

n

p

u

t

S

i

g

n

a

l

s

y_i y₁

y₂

1

2

i

O

u

t

p

u

t

S

i

g

n

a

l

s

x_n

I

n

p

u

t

S

y_n

n

O

u

t

p

u

t

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 < −

> +

= X

X

Ysign 1, if

0 if

, 1

  

0 =

X Y, if

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, . . ., , . . ., yy_nn. .

Thus, for a single

Thus, 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:

    

 

    

 

=

n

y y y

M

2 1

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:

I Y

Y

W M

M

m

T m

m −

=

∑

=1

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, YY_mm 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

Possible states for the three

Possible states for the three--neuron

neuron

Hopfield network

Hopfield network

y y₂

(1, 1, 1)

(−1, 1, 1)

(1, 1, −1)

(−1, 1, −1)

y₁

(1, −1, 1)

(−1, −1, 1)

(−1, −1, −1) (1, −1, −1)

The stable state

The 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 state

into 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). 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 YY and and YY are the threeare the three--dimensional vectors.dimensional vectors.

    

    

=

1 1 1 1

Y

  

 

  

 

− − − =

1 1 1

2

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:           = 1 0 0 0 1 0 0 0 1 I

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

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..

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, 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 network

Storage capacity

Storage 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

M

M_maxmax that can be stored in the that can be stored in the nn--neuron recurrent neuron recurrent

Storage capacity of the Hopfield network

Storage capacity of the Hopfield network

M

M_maxmax that can be stored in the that can be stored in the nn--neuron recurrent neuron recurrent network is limited by

network is limited by

n

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

Bidirectional associative memory (BAM)

Bidirectional associative memory (BAM)

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

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

Bidirectional associative memory (BAM)(BAM), first , first proposed by

proposed by Bart KoskoBart Kosko, is a heteroassociative , is a heteroassociative proposed by

proposed by Bart KoskoBart Kosko, is a heteroassociative , is a heteroassociative 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.

BAM operation

BAM operation

y_j(p)

y₁(p)

y₂(p)

1

2

j

x_i(p)

x₁(p)

x₂(p) ₂

i

1

x (p+1) x₁(p+1)

x₂(p+1)

y_j(p)

y₁(p)

y₂(p)

1

2

j

2

i

1

y_j(p)

y_m(p)

j

m

Output layer Input

layer x_i(p)

x_n(p)

i

n

x_i(p+1)

x_n(p+1)

y_j(p)

y_m(p)

j

m

Output layer Input

layer

i

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..

when

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.

T m M

m

m Y

X

W

∑

=

=

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

Stability and storage capacity of the BAM

Stability and storage capacity of the BAM

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.