Artificial Intelligence Chapter 20.5: Neural Networks

1
Artificial IntelligenceChapter 20.5 Neural
Networks

• Michael Scherger
• Department of Computer Science
• Kent State University

2
Contents

• Introduction
• Simple Neural Networks for Pattern Classification
• Pattern Association
• Neural Networks Based on Competition
• Backpropagation Neural Network

3
Introduction

• Much of these notes come from Fundamentals of
  Neural Networks Architectures, Algorithms, and
  Applications by Laurene Fausett, Prentice Hall,
  Englewood Cliffs, NJ, 1994.

4
Introduction

• Aims
• Introduce some of the fundamental techniques and
  principles of neural network systems
• Investigate some common models and their
  applications

5
What are Neural Networks?

• Neural Networks (NNs) are networks of neurons,
  for example, as found in real (i.e. biological)
  brains.
• Artificial Neurons are crude approximations of
  the neurons found in brains. They may be physical
  devices, or purely mathematical constructs.
• Artificial Neural Networks (ANNs) are networks of
  Artificial Neurons, and hence constitute crude
  approximations to parts of real brains. They may
  be physical devices, or simulated on conventional
  computers.
• From a practical point of view, an ANN is just a
  parallel computational system consisting of many
  simple processing elements connected together in
  a specific way in order to perform a particular
  task.
• One should never lose sight of how crude the
  approximations are, and how over-simplified our
  ANNs are compared to real brains.

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Why Study Artificial Neural Networks?

• They are extremely powerful computational devices
  (Turing equivalent, universal computers)
• Massive parallelism makes them very efficient
• They can learn and generalize from training data
  so there is no need for enormous feats of
  programming
• They are particularly fault tolerant this is
  equivalent to the graceful degradation found in
  biological systems
• They are very noise tolerant so they can cope
  with situations where normal symbolic systems
  would have difficulty
• In principle, they can do anything a
  symbolic/logic system can do, and more. (In
  practice, getting them to do it can be rather
  difficult)

7
What are Artificial Neural Networks Used for?

• As with the field of AI in general, there are two
  basic goals for neural network research
• Brain modeling The scientific goal of building
  models of how real brains work
• This can potentially help us understand the
  nature of human intelligence, formulate better
  teaching strategies, or better remedial actions
  for brain damaged patients.
• Artificial System Building The engineering goal
  of building efficient systems for real world
  applications.
• This may make machines more powerful, relieve
  humans of tedious tasks, and may even improve
  upon human performance.

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What are Artificial Neural Networks Used for?

• Brain modeling
• Models of human development help children with
  developmental problems
• Simulations of adult performance aid our
  understanding of how the brain works
• Neuropsychological models suggest remedial
  actions for brain damaged patients
• Real world applications
• Financial modeling predicting stocks, shares,
  currency exchange rates
• Other time series prediction climate, weather,
  airline marketing tactician
• Computer games intelligent agents, backgammon,
  first person shooters
• Control systems autonomous adaptable robots,
  microwave controllers
• Pattern recognition speech recognition,
  hand-writing recognition, sonar signals
• Data analysis data compression, data mining
• Noise reduction function approximation, ECG
  noise reduction
• Bioinformatics protein secondary structure, DNA
  sequencing

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Learning in Neural Networks

• There are many forms of neural networks. Most
  operate by passing neural activations through a
  network of connected neurons.
• One of the most powerful features of neural
  networks is their ability to learn and generalize
  from a set of training data. They adapt the
  strengths/weights of the connections between
  neurons so that the final output activations are
  correct.

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Learning in Neural Networks

• There are three broad types of learning
• Supervised Learning (i.e. learning with a
  teacher)
• Reinforcement learning (i.e. learning with
  limited feedback)
• Unsupervised learning (i.e. learning with no help)

11
A Brief History

• 1943 McCulloch and Pitts proposed the
  McCulloch-Pitts neuron model
• 1949 Hebb published his book The Organization of
  Behavior, in which the Hebbian learning rule was
  proposed.
• 1958 Rosenblatt introduced the simple single
  layer networks now called Perceptrons.
• 1969 Minsky and Paperts book Perceptrons
  demonstrated the limitation of single layer
  perceptrons, and almost the whole field went into
  hibernation.
• 1982 Hopfield published a series of papers on
  Hopfield networks.
• 1982 Kohonen developed the Self-Organizing Maps
  that now bear his name.
• 1986 The Back-Propagation learning algorithm for
  Multi-Layer Perceptrons was re-discovered and the
  whole field took off again.
• 1990s The sub-field of Radial Basis Function
  Networks was developed.
• 2000s The power of Ensembles of Neural Networks
  and Support Vector Machines becomes apparent.

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Overview

• Artificial Neural Networks are powerful
  computational systems consisting of many simple
  processing elements connected together to perform
  tasks analogously to biological brains.
• They are massively parallel, which makes them
  efficient, robust, fault tolerant and noise
  tolerant.
• They can learn from training data and generalize
  to new situations.
• They are useful for brain modeling and real world
  applications involving pattern recognition,
  function approximation, prediction,

13
The Nervous System

• The human nervous system can be broken down into
  three stages that may be represented in block
  diagram form as
• The receptors collect information from the
  environment e.g. photons on the retina.
• The effectors generate interactions with the
  environment e.g. activate muscles.
• The flow of information/activation is represented
  by arrows feed forward and feedback.

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Levels of Brain Organization

• The brain contains both large scale and small
  scale anatomical structures and different
  functions take place at higher and lower levels.
  There is a hierarchy of interwoven levels of
  organization
• Molecules and Ions
• Synapses
• Neuronal microcircuits
• Dendritic trees
• Neurons
• Local circuits
• Inter-regional circuits
• Central nervous system
• The ANNs we study in this module are crude
  approximations to levels 5 and 6.

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Brains vs. Computers

• There are approximately 10 billion neurons in the
  human cortex, compared with 10 of thousands of
  processors in the most powerful parallel
  computers.
• Each biological neuron is connected to several
  thousands of other neurons, similar to the
  connectivity in powerful parallel computers.
• Lack of processing units can be compensated by
  speed. The typical operating speeds of biological
  neurons is measured in milliseconds (10-3 s),
  while a silicon chip can operate in nanoseconds
  (10-9 s).
• The human brain is extremely energy efficient,
  using approximately 10-16 joules per operation
  per second, whereas the best computers today use
  around 10-6 joules per operation per second.
• Brains have been evolving for tens of millions of
  years, computers have been evolving for tens of
  decades.

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Structure of a Human Brain
17
Slice Through a Real Brain
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Biological Neural Networks

• The majority of neurons encode their outputs or
  activations as a series of brief electical pulses
  (i.e. spikes or action potentials).
• Dendrites are the receptive zones that receive
  activation from other neurons.
• The cell body (soma) of the neurons processes
  the incoming activations and converts them into
  output activations.
• 4. Axons are transmission lines that send
  activation to other neurons.
• 5. Synapses allow weighted transmission of
  signals (using neurotransmitters) between axons
  and dendrites to build up large neural networks.

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The McCulloch-Pitts Neuron

• This vastly simplified model of real neurons is
  also known as a Threshold Logic Unit
• A set of synapses (i.e. connections) brings in
  activations from other neurons.
• A processing unit sums the inputs, and then
  applies a non-linear activation function (i.e.
  squashing/transfer/threshold function).
• An output line transmits the result to other
  neurons.

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Networks of McCulloch-Pitts Neurons

• Artificial neurons have the same basic components
  as biological neurons. The simplest ANNs consist
  of a set of McCulloch-Pitts neurons labeled by
  indices k, i, j and activation flows between them
  via synapses with strengths wki, wij

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Some Useful Notation

• We often need to talk about ordered sets of
  related numbers we call them vectors, e.g.
• x (x1, x2, x3, , xn) , y (y1, y2, y3, , ym)
• The components xi can be added up to give a
  scalar (number), e.g.
• s x1 x2 x3 xn SUM(i, n, xi)
• Two vectors of the same length may be added to
  give another vector, e.g.
• z x y (x1 y1, x2 y2, , xn yn)
• Two vectors of the same length may be multiplied
  to give a scalar, e.g.
• p x.y x1y1 x2 y2 xnyn SUM(i, N,
  xiyi)

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Some Useful Functions

• Common activation functions
• Identity function
• f(x) x for all x
• Binary step function (with threshold ?) (aka
  Heaviside function or threshold function)

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Some Useful Functions

• Binary sigmoid
• Bipolar sigmoid

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The McCulloch-Pitts Neuron Equation

• Using the above notation, we can now write down a
  simple equation for the output out of a
  McCulloch-Pitts neuron as a function of its n
  inputs ini

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Review

• Biological neurons, consisting of a cell body,
  axons, dendrites and synapses, are able to
  process and transmit neural activation
• The McCulloch-Pitts neuron model (Threshold Logic
  Unit) is a crude approximation to real neurons
  that performs a simple summation and thresholding
  function on activation levels
• Appropriate mathematical notation facilitates the
  specification and programming of artificial
  neurons and networks of artificial neurons.

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Networks of McCulloch-Pitts Neurons

• One neuron cant do much on its own. Usually we
  will have many neurons labeled by indices k, i, j
  and activation flows between them via synapses
  with strengths wki, wij

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The Perceptron

• We can connect any number of McCulloch-Pitts
  neurons together in any way we like.
• An arrangement of one input layer of
  McCulloch-Pitts neurons feeding forward to one
  output layer of McCulloch-Pitts neurons is known
  as a Perceptron.

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Logic Gates with MP Neurons

• We can use McCulloch-Pitts neurons to implement
  the basic logic gates.
• All we need to do is find the appropriate
  connection weights and neuron thresholds to
  produce the right outputs for each set of inputs.
• We shall see explicitly how one can construct
  simple networks that perform NOT, AND, and OR.
• It is then a well known result from logic that we
  can construct any logical function from these
  three operations.
• The resulting networks, however, will usually
  have a much more complex architecture than a
  simple Perceptron.
• We generally want to avoid decomposing complex
  problems into simple logic gates, by finding the
  weights and thresholds that work directly in a
  Perceptron architecture.

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Implementation of Logical NOT, AND, and OR

• Logical OR
• x1 x2 y
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 1

x1
?2
2
y
x2
2
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Implementation of Logical NOT, AND, and OR

• Logical AND
• x1 x2 y
• 0 0 0
• 0 1 0
• 1 0 0
• 1 1 1

x1
?2
1
y
x2
1
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Implementation of Logical NOT, AND, and OR

• Logical NOT
• x1 y
• 0 1
• 1 0

x1
?2
-1
y
1
2
bias
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Implementation of Logical NOT, AND, and OR

• Logical AND NOT
• x1 x2 y
• 0 0 0
• 0 1 0
• 1 0 1
• 1 1 0

x1
?2
2
y
x2
-1
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Logical XOR

• Logical XOR
• x1 x2 y
• 0 0 0
• 0 1 1
• 1 0 1
• 1 1 0

x1
?
y
x2
?
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Logical XOR

• How long do we keep looking for a solution? We
  need to be able to calculate appropriate
  parameters rather than looking for solutions by
  trial and error.
• Each training pattern produces a linear
  inequality for the output in terms of the inputs
  and the network parameters. These can be used to
  compute the weights and thresholds.

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Finding the Weights Analytically

• We have two weights w1 and w2 and the threshold
  q, and for each training pattern we need to
  satisfy

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Finding the Weights Analytically

• For the XOR network
• Clearly the second and third inequalities are
  incompatible with the fourth, so there is in fact
  no solution. We need more complex networks, e.g.
  that combine together many simple networks, or
  use different activation/thresholding/transfer
  functions.

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ANN Topologies

• Mathematically, ANNs can be represented as
  weighted directed graphs. For our purposes, we
  can simply think in terms of activation flowing
  between processing units via one-way connections
• Single-Layer Feed-forward NNs One input layer and
  one output layer of processing units. No
  feed-back connections. (For example, a simple
  Perceptron.)
• Multi-Layer Feed-forward NNs One input layer, one
  output layer, and one or more hidden layers of
  processing units. No feed-back connections. The
  hidden layers sit in between the input and output
  layers, and are thus hidden from the outside
  world. (For example, a Multi-Layer Perceptron.)
• Recurrent NNs Any network with at least one
  feed-back connection. It may, or may not, have
  hidden units. (For example, a Simple Recurrent
  Network.)

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ANN Topologies
39
Detecting Hot and Cold

• It is a well-known and interesting psychological
  phenomenon that if a cold stimulus is applied to
  a persons skin for a short period of time, the
  person will perceive heat.
• However, if the same stimulus is applied for a
  longer period of time, the person will perceive
  cold. The use of discrete time steps enables the
  network of MP neurons to model this phenomenon.

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Detecting Hot and Cold

• The desired response of the system is that cold
  is perceived if a cold stimulus is applied for
  two time steps
• y2(t) x2(t-2) AND x2(t-1)
• It is also required that heat be perceived if
  either a hot stimulus is applied or a cold
  stimulus is applied briefly (for one time step)
  and then removed
• y1(t) x1(t-1) OR x2(t-3) AND NOT x2(t-2)

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Detecting Heat and Cold
2
Heat
x1
y1
2
z1
-1
2
1
2
z2
x2
y2
Cold
1
42
Detecting Heat and Cold
Heat
0
Apply Cold
1
Cold
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Detecting Heat and Cold
Heat
0
0
Remove Cold
1
0
Cold
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Detecting Heat and Cold
Heat
0
1
0
0
Cold
45
Detecting Heat and Cold
Heat
1
Perceive Heat
0
Cold
46
Detecting Heat and Cold
Heat
0
Apply Cold
1
Cold
47
Detecting Heat and Cold
Heat
0
0
1
1
Cold
48
Detecting Heat and Cold
Heat
0
0
1
1
Cold
Perceive Cold
49
Example Classification

• Consider the example of classifying airplanes
  given their masses and speeds
• How do we construct a neural network that can
  classify any type of bomber or fighter?

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A General Procedure for Building ANNs

• 1. Understand and specify your problem in terms
  of inputs and required outputs, e.g. for
  classification the outputs are the classes
  usually represented as binary vectors.
• 2. Take the simplest form of network you think
  might be able to solve your problem, e.g. a
  simple Perceptron.
• 3. Try to find appropriate connection weights
  (including neuron thresholds) so that the network
  produces the right outputs for each input in its
  training data.
• 4. Make sure that the network works on its
  training data, and test its generalization by
  checking its performance on new testing data.
• 5. If the network doesnt perform well enough, go
  back to stage 3 and try harder.
• 6. If the network still doesnt perform well
  enough, go back to stage 2 and try harder.
• 7. If the network still doesnt perform well
  enough, go back to stage 1 and try harder.
• 8. Problem solved move on to next problem.

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Building a NN for Our Example

• For our airplane classifier example, our inputs
  can be direct encodings of the masses and speeds
• Generally we would have one output unit for each
  class, with activation 1 for yes and 0 for no
  
• With just two classes here, we can have just one
  output unit, with activation 1 for fighter and
  0 for bomber (or vice versa)
• The simplest network to try first is a simple
  Perceptron
• We can further simplify matters by replacing the
  threshold by using a bias

52
Building a NN for Our Example
53
Building a NN for Our Example
54
Decision Boundaries in Two Dimensions

• For simple logic gate problems, it is easy to
  visualize what the neural network is doing. It
  is forming decision boundaries between classes.
  Remember, the network output is
• The decision boundary (between out 0 and out
  1) is at
• w1in1 w2in2 - ? 0

55
Decision Boundaries in Two Dimensions
In two dimensions the decision boundaries are
always on straight lines
56
Decision Boundaries for AND and OR
57
Decision Boundaries for XOR

• There are two obvious remedies
• either change the transfer function so that it
  has more than one decision boundary
• use a more complex network that is able to
  generate more complex decision boundaries

58
Logical XOR (Again)

• z1 x1 AND NOT x2
• z2 x2 AND NOT x1
• y z1 OR z2

2
x1
z1
2
-1
y
-1
2
x2
z2
2
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Decision Hyperplanes and Linear Separability

• If we have two inputs, then the weights define a
  decision boundary that is a one dimensional
  straight line in the two dimensional input space
  of possible input values
• If we have n inputs, the weights define a
  decision boundary that is an n-1 dimensional
  hyperplane in the n dimensional input space
• w1in1 w2in2 wninn - ? 0

60
Decision Hyperplanes and Linear Separability

• This hyperplane is clearly still linear (i.e.
  straight/flat) and can still only divide the
  space into two regions. We still need more
  complex transfer functions, or more complex
  networks, to deal with XOR type problems
• Problems with input patterns which can be
  classified using a single hyperplane are said to
  be linearly separable. Problems (such as XOR)
  which cannot be classified in this way are said
  to be non-linearly separable.

61
General Decision Boundaries

• Generally, we will want to deal with input
  patterns that are not binary, and expect our
  neural networks to form complex decision
  boundaries
• We may also wish to classify inputs into many
  classes (such as the three shown here)

62
Learning and Generalization

• A network will also produce outputs for input
  patterns that it was not originally set up to
  classify (shown with question marks), though
  those classifications may be incorrect
• There are two important aspects of the networks
  operation to consider
• Learning The network must learn decision surfaces
  from a set of training patterns so that these
  training patterns are classified correctly
• Generalization After training, the network must
  also be able to generalize, i.e. correctly
  classify test patterns it has never seen before
• Usually we want our neural networks to learn
  well, and also to generalize well.

63
Learning and Generalization

• Sometimes, the training data may contain errors
  (e.g. noise in the experimental determination of
  the input values, or incorrect classifications)
• In this case, learning the training data
  perfectly may make the generalization worse
• There is an important tradeoff between learning
  and generalization that arises quite generally

64
Generalization in Classification

• Suppose the task of our network is to learn a
  classification decision boundary
• Our aim is for the network to generalize to
  classify new inputs appropriately. If we know
  that the training data contains noise, we dont
  necessarily want the training data to be
  classified totally accurately, as that is likely
  to reduce the generalization ability.

65
Generalization in Function Approximation

• Suppose we wish to recover a function for which
  we only have noisy data samples
• We can expect the neural network output to give a
  better representation of the underlying function
  if its output curve does not pass through all the
  data points. Again, allowing a larger error on
  the training data is likely to lead to better
  generalization.

66
Training a Neural Network

• Whether our neural network is a simple
  Perceptron, or a much more complicated multilayer
  network with special activation functions, we
  need to develop a systematic procedure for
  determining appropriate connection weights.
• The general procedure is to have the network
  learn the appropriate weights from a
  representative set of training data
• In all but the simplest cases, however, direct
  computation of the weights is intractable

67
Training a Neural Network

• Instead, we usually start off with random initial
  weights and adjust them in small steps until the
  required outputs are produced
• We shall now look at a brute force derivation of
  such an iterative learning algorithm for simple
  Perceptrons.
• Later, we shall see how more powerful and general
  techniques can easily lead to learning algorithms
  which will work for neural networks of any
  specification we could possibly dream up

68
Perceptron Learning

• For simple Perceptrons performing classification,
  we have seen that the decision boundaries are
  hyperplanes, and we can think of learning as the
  process of shifting around the hyperplanes until
  each training pattern is classified correctly
• Somehow, we need to formalize that process of
  shifting around into a systematic algorithm
  that can easily be implemented on a computer
• The shifting around can conveniently be split
  up into a number of small steps.

69
Perceptron Learning

• If the network weights at time t are wij(t), then
  the shifting process corresponds to moving them
  by an amount Dwij(t) so that at time t1 we have
  weights
• wij(t1) wij(t) Dwij(t)
• It is convenient to treat the thresholds as
  weights, as discussed previously, so we dont
  need separate equations for them

70
Formulating the Weight Changes

• Suppose the target output of unit j is targj and
  the actual output is outj sgn(S ini wij), where
  ini are the activations of the previous layer of
  neurons (e.g. the network inputs)
• Then we can just go through all the possibilities
  to work out an appropriate set of small weight
  changes

71
Perceptron Algorithm

• Step 0 Initialize weights and bias
• For simplicity, set weights and bias to zero
• Set learning rate a (0 lt a lt 1) (h)
• Step 1 While stopping condition is false do
  steps 2-6
• Step 2 For each training pair st do steps 3-5
• Step 3 Set activations of input units
• xi si

72
Perceptron Algorithm

• Step 4 Compute response of output unit

73
Perceptron Algorithm

• Step 5 Update weights and bias if an error
  occurred for this pattern
• if y ! t
• wi(new) wi(old) atxi
• b(new) b(old) at
• else
• wi(new) wi(old)
• b(new) b(old)
• Step 6 Test Stopping Condition
• If no weights changed in Step 2, stop, else,
  continue

74
Convergence of Perceptron Learning

• The weight changes Dwij need to be applied
  repeatedly for each weight wij in the network,
  and for each training pattern in the training
  set. One pass through all the weights for the
  whole training set is called one epoch of
  training
• Eventually, usually after many epochs, when all
  the network outputs match the targets for all the
  training patterns, all the Dwij will be zero and
  the process of training will cease. We then say
  that the training process has converged to a
  solution

75
Convergence of Perceptron Learning

• It can be shown that if there does exist a
  possible set of weights for a Perceptron which
  solves the given problem correctly, then the
  Perceptron Learning Rule will find them in a
  finite number of iterations
• Moreover, it can be shown that if a problem is
  linearly separable, then the Perceptron Learning
  Rule will find a set of weights in a finite
  number of iterations that solves the problem
  correctly

76
Overview and Review

• Neural network classifiers learn decision
  boundaries from training data
• Simple Perceptrons can only cope with linearly
  separable problems
• Trained networks are expected to generalize, i.e.
  deal appropriately with input data they were not
  trained on
• One can train networks by iteratively updating
  their weights
• The Perceptron Learning Rule will find weights
  for linearly separable problems in a finite
  number of iterations.

77
Hebbian Learning

• In 1949 neuropsychologist Donald Hebb postulated
  how biological neurons learn
• When an axon of cell A is near enough to excite
  a cell B and repeatedly or persistently takes
  part in firing it, some growth process or
  metabolic change takes place on one or both cells
  such that As efficiency as one of the cells
  firing B, is increased.
• In other words
• 1. If two neurons on either side of a synapse
  (connection) are activated simultaneously (i.e.
  synchronously), then the strength of that synapse
  is selectively increased.
• This rule is often supplemented by
• 2. If two neurons on either side of a synapse are
  activated asynchronously, then that synapse is
  selectively weakened or eliminated.
• so that chance coincidences do not build up
  connection strengths.

78
Hebbian Learning Algorithm

• Step 0 Initialize all weights
• For simplicity, set weights and bias to zero
• Step 1 For each input training vector do steps
  2-4
• Step 2 Set activations of input units
• xi si
• Step 3 Set the activation for the output unit
• y t
• Step 4 Adjust weights and bias
• wi(new) wi(old) yxi
• b(new) b(old) y

79
Hebbian vs Perceptron Learning

• In the notation used for Perceptrons, the Hebbian
  learning weight update rule is
• wij (new) outj . ini
• There is strong physiological evidence that this
  type of learning does take place in the region of
  the brain known as the hippocampus.
• Recall that the Perceptron learning weight update
  rule we derived was
• wij (new) h. targj . ini
• There is some similarity, but it is clear that
  Hebbian learning is not going to get our
  Perceptron to learn a set of training data.

80
Adaline

• Adaline (Adaptive Linear Network) was developed
  by Widrow and Hoff in 1960.
• Uses bipolar activations (-1 and 1) for its input
  signals and target values
• Weight connections are adjustable
• Trained using the delta rule for weight update
• wij(new) wij(old) a(targj-outj)xi

81
Adaline Training Algorithm

• Step 0 Initialize weights and bias
• For simplicity, set weights (small random values)
  Set learning rate a (0 lt a lt 1) (h)
• Step 1 While stopping condition is false do
  steps 2-6
• Step 2 For each training pair st do steps 3-5
• Step 3 Set activations of input units
• xi si

82
Adaline Training Algorithm

• Step 4 Compute net input to output unit
• y_in b S xiwi
• Step 5 Update bias and weights
• wi(new) wi(old) a(t-y_in)xi
• b(new) b(old) a(t-y_in)
• Step 6 Test for stopping condition

83
Autoassociative Net

• The feed forward autoassociative net has the
  following diagram
• Useful for determining is something is a part of
  the test pattern or not
• Weight matrix diagonal is usually zeroimproves
  generalization
• Hebbian learning if mutually orthogonal vectors
  are used

x1
y1
xi
yj
xn
ym
84
BAM Net

• Bidirectional Associative Net