november 11, 2004ai: chapter 20.5: neural networks 1 artificial intelligence chapter 20.5: neural...

November 11, 2004 AI: Chapter 20.5: Neural Networks

1

Artificial IntelligenceChapter 20.5: Neural

Networks

Michael SchergerDepartment of Computer

ScienceKent 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.


6

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.


8

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


9

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.


10

Learning in Neural Networks

• There are three broad types of learning:

1. Supervised Learning (i.e. learning with a teacher)

2. Reinforcement learning (i.e. learning with limited feedback)

3. 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 Papert’s 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.


12

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:1. Molecules and Ions2. Synapses3. Neuronal microcircuits4. Dendritic trees5. Neurons6. Local circuits7. Inter-regional circuits8. Central nervous system

• The ANNs we study in this module are crude approximations to levels 5 and 6.


15

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.


16

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 neuron’s 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)

x if 0

x if 1)(xf


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Some Useful Functions

• Binary sigmoid

• Bipolar sigmoid

xexf

1

1)(

11

21)(2)(

xe

xfxg


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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 :


25

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.


26

Networks of McCulloch-Pitts Neurons

• One neuron can’t 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 y0 0 00 1 11 0 11 1 1

x1

x2

y

2

2

θ=2


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• Logical AND

x1 x2 y0 0 00 1 01 0 01 1 1

x1

x2

y

1

1

θ=2


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• Logical NOT

x1 y0 11 0

x1

y

-1θ=2

2

bias

1


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• Logical AND NOT

x1 x2 y0 0 00 1 01 0 11 1 0

x1

x2

y

2

-1

θ=2


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Logical XOR

• Logical XOR

x1 x2 y0 0 00 1 11 0 11 1 0

x1

x2

y

?

?


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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.


35

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.


37

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.)


38

ANN Topologies


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Detecting Hot and Cold

• It is a well-known and interesting psychological phenomenon that if a cold stimulus is applied to a person’s 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

x1

x2

z1

z2 y2

y12

1

1

2

2

-1

2

Heat

Cold


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0

1

Heat

Cold

Apply Cold


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0

0

0

1

Heat

Cold

Remove Cold


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1

0 0

0Heat

Cold


45


0

1Heat

Cold

Perceive Heat


46


0

1

Heat

Cold

Apply Cold


47


0

1

0

1

Heat

Cold


48


0

1 1

0Heat

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 doesn’t perform well enough, go back to stage 3 and try harder.

• 6. If the network still doesn’t perform well enough, go back to stage 2 and try harder.

• 7. If the network still doesn’t 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


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53



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 - θ= 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


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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

x1

x2

z1

z2

y

2

2

-1

2

2

-1


59

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 - θ= 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 network’s 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 don’t 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 wij(t) so that at time t+1 we have weights

wij(t+1) = wij(t) + wij(t)

• It is convenient to treat the thresholds as weights, as discussed previously, so we don’t 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( 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 (0 <= <= 1) ()

• Step 1: While stopping condition is false do steps 2-6

• Step 2: For each training pair s:t do steps 3-5

• Step 3: Set activations of input unitsxi = si


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• Step 4: Compute response of output unit:

- y_in if

y_in - if

y_in if

1

0

1

_

y

wxbinyi

ii


73


• Step 5: Update weights and bias if an error occurred for this patternif y != t

wi(new) = wi(old) + txi

b(new) = b(old) + t

elsewi(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 wij 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 wij 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 A’s 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 unitsxi = si

• Step 3: Set the activation for the output unity = t

• Step 4: Adjust weights and biaswi(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)= . 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) + (targj-outj)xi


81

Adaline Training Algorithm

• Step 0: Initialize weights and bias– For simplicity, set weights (small random values)

Set learning rate (0 <= <= 1) ()

• Step 1: While stopping condition is false do steps 2-6

• Step 2: For each training pair s:t 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 + xiwi

• Step 5: Update bias and weightswi(new) = wi(old) + (t-y_in)xi

b(new) = b(old) + (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 zero…improves generalization

• Hebbian learning if mutually orthogonal vectors are used

x1

xi

xn

y1

yj

ym


84

BAM Net

• Bidirectional Associative Net

november 11, 2004ai: chapter 20.5: neural networks 1 artificial intelligence chapter 20.5: neural...

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