Machine Learning

Neural Networks

Introduction

Artificial Neural Network is based on the

biological nervous system as Brain

It is composed of interconnected computing

units called neurons

ANN like human, learn by examples

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Why Artificial Neural Networks?There are two basic reasons why we are interested in building artificial neural networks (ANNs):

Technical viewpoint: Some problems such as character recognition or the prediction of future states of a system require massively parallel and adaptive processing.

Biological viewpoint: ANNs can be used to replicate and simulate components of the human (or animal) brain, thereby giving us insight into natural information processing.

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Science: Model how biological neural systems, like human brain, work?

How do we see? How is information stored in/retrieved

from memory? How do you learn to not to touch fire? How do your eyes adapt to the amount

of light in the environment? Related fields: Neuroscience,

Computational Neuroscience, Psychology, Psychophysiology, Cognitive Science, Medicine, Math, Physics.

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Brief HistoryOld Ages: Association (William James; 1890) McCulloch-Pitts Neuron (1943,1947) Perceptrons (Rosenblatt; 1958,1962) Adaline/LMS (Widrow and Hoff; 1960) Perceptrons book (Minsky and Papert; 1969)

Dark Ages: Self-organization in visual cortex (von der Malsburg; 1973) Backpropagation (Werbos, 1974) Foundations of Adaptive Resonance Theory (Grossberg; 1976) Neural Theory of Association (Amari; 1977)

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History

Modern Ages: Adaptive Resonance Theory (Grossberg; 1980) Hopfield model (Hopfield; 1982, 1984) Self-organizing maps (Kohonen; 1982) Reinforcement learning (Sutton and Barto; 1983) Simulated Annealing (Kirkpatrick et al.; 1983) Boltzmann machines (Ackley, Hinton, Terrence; 1985) Backpropagation (Rumelhart, Hinton, Williams; 1986) ART-networks (Carpenter, Grossberg; 1992) Support Vector Machines

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Hebb’s Learning Law In 1949, Donald Hebb formulated William James’ principle of

association into a mathematical form.

• If the activation of the neurons, y1 and y2 , are both on (+1) then the weight between the two neurons grow. (Off: 0)

• Else the weight between remains the same.

• However, when bipolar activation {-1,+1} scheme is used, then the weights can also decrease when the activation of two neurons does not match.

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Real Neural Learning

Synapses change size and strength with experience.

Hebbian learning: When two connected neurons are firing at the same time, the strength of the synapse between them increases.

“Neurons that fire together, wire together.”

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Biological Neurons Human brain = tens of thousands

of neurons Each neuron is connected to

thousands other neurons A neuron is made of:

– The soma: body of the neuron– Dendrites: filaments that provide

input to the neuron– The axon: sends an output signal– Synapses: connection with other

neurons – releases certain quantities of chemicals called neurotransmitters to other neurons

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Modeling of Brain Functions

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The biological neuron

The pulses generated by the neuron travels along the axon as an electrical wave.

Once these pulses reach the synapses at the end of the axon open up chemical vesicles exciting the other neuron.

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How do NNs and ANNs work? Information is transmitted as a series of

electric impulses, so-called spikes.

The frequency and phase of these spikes encodes the information.

In biological systems, one neuron can be connected to as many as 10,000 other neurons.

Usually, a neuron receives its information from other neurons in a confined area

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Navigation of a car

Done by Pomerlau. The network takes inputs from a 34X36 video image and a 7X36 range finder. Output units represent “drive straight”, “turn left” or “turn right”. After training about 40 times on 1200 road images, the car drove around CMU campus at 5 km/h (using a small workstation on the car). This was almost twice the speed of any other non-NN algorithm at the time.

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Automated driving at 70 mph on a public highway

Camera image

30x32 pixelsas inputs

30 outputsfor steering

30x32 weightsinto one out offour hiddenunit

4 hiddenunits

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Computers vs. Neural Networks

“Standard” Computers Neural Networks

one CPU highly parallelprocessing

fast processing units slow processing units

reliable units unreliable units

static infrastructure dynamic infrastructure

Neural Network

Neural Network Application

•Pattern recognition can be implemented using NN

•The figure can be T or H character, the network should identify each class of T or H.

Simple Neuron

X1

X2

Xn

OutputInputs

b

An Artificial Neuron

x1

x2

xn

…

Wi,1Wi,2

…

Wi,n

n

jjjii txtwt

1, )()()(net

xi

neuron i

net input signal

synapses

output ))(()(x tnetft iii

Neural Network

Input Layer Hidden 1 Hidden 2 Output Layer

Network Layers

The common type of ANN consists of three

layers of neurons: a layer of input neurons

connected to the layer of hidden neuron

which is connected to a layer of output

neurons.

Architecture of ANN

Feed-Forward networks

Allow the signals to travel one way from input to output

Feed-Back Networks

The signals travel as loops in the network, the output is connected to the input of the network

How do NNs and ANNs Learn?

NNs are able to learn by adapting their connectivity patterns so that the organism improves its behavior in terms of reaching certain (evolutionary) goals.

The NN achieves learning by appropriately adapting the states of its synapses.

Learning Rule

The learning rule modifies the weights of

the connections.

The learning process is divided into

Supervised and Unsupervised learning

Supervised Network

Which means there exists an external

teacher. The target is to minimization of the

error between the desired and computed

output

Unsupervised Network

Uses no external teacher and is based upon

only local information.

Perceptron

It is a network of one neuron and hard limit transfer function

Inputs f

X1

X2

Xn

Output

W1

W2

Wn

Perceptron

The perceptron is given first a randomly

weights vectors

Perceptron is given chosen data pairs (input

and desired output)

Preceptron learning rule changes the

weights according to the error in output

Perceptron Learning Rule

W new = W old + (t-a) X

Where W new is the new weight

W old is the old value of weight

X is the input value

t is the desired value of output

a is the actual value of output

Example

Let – X1 = [0 0] and t =0– X2 = [0 1] and t=0– X3 = [1 0] and t=0– X4 = [1 1] and t=1

W = [2 2] and b = -3

AND Network

This example means we construct a network for AND operation. The network draw a line to separate the classes which is called Classification

Perceptron Geometric ViewThe equation below describes a (hyper-)plane in the input space

consisting of real valued m-dimensional vectors. The plane splits the input space into two regions, each of them describing one class.

0 wxw 0

m

1iii

x2

C1

C2x1

decisionboundary

w1x1 + w2x2 + w0 = 0

decisionregion for C1

w1x1 + w2x2 + w0 >= 0

Problems

Four one-dimensional data belonging to two classes are

X = [1 -0.5 3 -2]

T = [1 -1 1 -1]

W = [-2.5 1.75]

Boolean Functions

Take in two inputs (-1 or +1) Produce one output (-1 or +1) In other contexts, use 0 and 1 Example: AND function

– Produces +1 only if both inputs are +1 Example: OR function

– Produces +1 if either inputs are +1 Related to the logical connectives from F.O.L.

The First Neural Neural Networks

AND Function

1

1X1

X2

Y

AND

X1 X2 Y

1 1 1

1 0 0

0 1 0

0 0 0

Threshold(Y) = 2

Simple Networks

t = 0.0

y

x

W = 1.5

W = 1

-1

Exercises

Design a neural network to recognize the problem of

X1=[2 2] , t1=0 X=[1 -2], t2=1 X3=[-2 2], t3=0 X4=[-1 1], t4=1

Start with initial weights w=[0 0] and bias =0

Perceptron: Limitations

The perceptron The perceptron can only model linearly separable classes, linearly separable classes, like (those described by) the following Boolean functions:

ANDAND OROR COMPLEMENTCOMPLEMENT It cannot cannot model the XORXOR.

You can experiment with these functions in the Matlab practical lessons.

Types of decision regions

022110 xwxww

022110 xwxww

x1

1

x2 w2

w1

w0

Convexregion

L1L2

L3L4 -3.5

Networkwith a singlenode

One-hidden layer network that realizes the convex region

1

1

1

1

1

x1

x2

1

Gaussian NeuronsAnother type of neurons overcomes this problem by using a Gaussian activation function:

11

00

11

ffii(net(netii(t))(t))

netnetii(t)(t)-1-1

2

1)(net

))(net(

t

ii

i

etf

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Gaussian NeuronsGaussian neurons are able to realize non-linear functions.

Therefore, networks of Gaussian units are in principle unrestricted with regard to the functions that they can realize.

The drawback of Gaussian neurons is that we have to make sure that their net input does not exceed 1.

This adds some difficulty to the learning in Gaussian networks.

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Sigmoidal NeuronsSigmoidal neurons accept any vectors of real numbers as input, and they output a real number between 0 and 1.

Sigmoidal neurons are the most common type of artificial neuron, especially in learning networks.

A network of sigmoidal units with m input neurons and n output neurons realizes a network function f: Rm (0,1)n

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Sigmoidal Neurons

The parameter controls the slope of the sigmoid function, while the parameter controls the horizontal offset of the function in a way similar to the threshold neurons.

11

00

11

ffii(net(netii(t))(t))

netnetii(t)(t)-1-1

/))(net(1

1))(net(

tii ietf

= = 11

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Sigmoidal NeuronsThis leads to a simplified form of the sigmoid function:

)(1

1)(

netenetS

We do not need a modifiable threshold , because we will use “dummy” inputs as we did for perceptrons.

The choice = 1 works well in most situations and results in a very simple derivative of S(net).

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Sigmoidal Neurons

This result will be very useful when we develop the backpropagation very useful when we develop the backpropagation algorithm.algorithm.

xexS

1

1)(

2)1(

)()('

x

x

e

e

dx

xdSxS

22 )1(

1

1

1

)1(

11xxx

x

eee

e

))(1)(( xSxS

Multi-layers Network

Let the network of 3 layers– Input layer– Hidden layer– Output layer

Each layer has different number of neurons The famous example to need the multi-layer

network is XOR unction

Learning rule

The perceptron learning rule can not be

applied to multi-layer network

We use BackPropagation Algorithm in

learning process

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Feed-forward + Backpropagation

Feed-forward: – input from the features is fed forward in the network from input

layer towards the output layer Backpropagation:

– Method to asses the blame of errors to weights– error rate flows backwards from the output layer to the input

layer (to adjust the weight in order to minimize the output error)

Backprop

Back-propagation training algorithm illustrated:

Backprop adjusts the weights of the NN in order to minimize the network total mean squared error.

Network activationError computationForward Step

Error propagationBackward Step

Correlation LearningHebbian Learning (1949):

“When an axon of cell A is near enough to excite a cell B and repeatedly or persistently takes place in firing it, some growth process or metabolic change takes place in one or both cells such that A’s efficiency, as one of the cells firing B, is increased.”

Weight modification rule:

wi,j = cxixj

Eventually, the connection strength will reflect the correlation between the neurons’ outputs.

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Competitive Learning• Nodes compete for inputs

• Node with highest activation is the winner

• Winner neuron adapts its tuning (pattern of weights) even further towards the current input

• Individual nodes specialize to win competition for a set of similar inputs

• Process leads to most efficient neural representation of input space

• Typical for unsupervised learning

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Backpropagation LearningSimilar to the Adaline, the goal of the Backpropagation learning algorithm is to modify the network’s weights so that its output vector

op = (op,1, op,2, …, op,K)

is as close as possible to the desired output vector

dp = (dp,1, dp,2, …, dp,K)

for K output neurons and input patterns p = 1, …, P.

The set of input-output pairs (exemplars) {(xp, dp) | p = 1, …, P} constitutes the training set.

Bp Algorithm

The weight change rule is

Where is the learning factor <1 Error is the error between actual and trained

value f’ is is the derivative of sigmoid function =

f(1-f)

)('.. ioldij

newij inputferror

Delta Rule

Each observation contributes a variable amount to the output

The scale of the contribution depends on the input Output errors can be blamed on the weights A least mean square (LSM) error function can be

defined (ideally it should be zero)

E = ½ (t – y)2

Example

For the network with one neuron in input layer and one neuron in hidden layer the following values are givenX=1, w1 =1, b1=-2, w2=1, b2 =1, =1 and t=1

Where X is the input valueW1 is the weight connect input to hidden W2 is the weight connect hidden to outputB1 and b2 are biasT is the training value

Exercises

Design a neural network to recognize the problem of

X1=[2 2] , t1=0 X=[1 -2], t2=1 X3=[-2 2], t3=0 X4=[-1 1], t4=1

Start with initial weights w=[0 0] and bias =0

Exercises

Perform one iteration of backprpgation to network of two layers. First layer has one neuron with weight 1 and bias –2. The transfer function in first layer is f=n2

The second layer has only one neuron with weight 1 and bias 1. The f in second layer is 1/n.

The input to the network is x=1 and t=1

Neural NetworkConstruct a neural network to solve the problem

X1 X2 Output

1.0 1.0 1

9.4 6.4 -1

2.5 2.1 1

8.0 7.7 -1

0.5 2.2 1

7.9 8.4 -1

7.0 7.0 -1

2.8 0.8 1

1.2 3.0 1

7.8 6.1 -1

Initialize the weights 0.75 , 0.5, and –0.6

Neural NetworkConstruct a neural network to solve the XOR problem

X1 X2 Output

1 1 0

0 0 0

1 0 1

0 1 1

Initialize the weights –7.0 , -7.0, -5.0 and –4.0

-0.5

-0.5

-2

3

-1

1

1

1-1

1

0.5

The transfer function is linear function.

Consider a transfer function as f(n) = n2. Perform one iteration of BackPropagation with a= 0.9 for neural network of two neurons in input layer and one neuron in output layer. The input values are X=[1 -1] and t = 8, the weight values between input and hidden layer are w11 = 1, w12 = - 2, w21 = 0.2, and w22 = 0.1. The weight between input and output layers are w1 = 2 and w2= -2. The bias in input layers are b1 = -1, and b2= 3.

X1

X2

W11

W22

W12

W1

W21 W2

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Some variations

True gradient descent assumes infinitesmall learning rate (). If is too small then learning is very slow. If large, then the system's learning may never converge.

Some of the possible solutions to this problem are:– Add a momentum term to allow a large learning rate.– Use a different activation function– Use a different error function– Use an adaptive learning rate– Use a good weight initialization procedure.– Use a different minimization procedure

Problems with Local Minima

Backpropagation is gradient descent search– Where the height of the hills is determined by error– But there are many dimensions to the space

• One for each weight in the network

Therefore backpropagation– Can find its ways into local minima

One partial solution:– Random re-start: learn lots of networks

• Starting with different random weight settings

– Can take best network– Or can set up a “committee” of networks to categorise examples

Another partial solution: Momentum

Adding Momentum Imagine rolling a ball down a hill

Without Momentum With Momentum

Gets stuck here

Momentum in Backpropagation For each weight

– Remember what was added in the previous epoch

In the current epoch– Add on a small amount of the previous Δ

The amount is determined by – The momentum parameter, denoted α– α is taken to be between 0 and 1

How Momentum Works If direction of the weight doesn’t change

– Then the movement of search gets bigger

– The amount of additional extra is compounded in each epoch

– May mean that narrow local minima are avoided

– May also mean that the convergence rate speeds up Caution:

– May not have enough momentum to get out of local minima

– Also, too much momentum might carry search

• Back out of the global minimum, into a local minimum

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Momentum

Weight update becomes:

wij (n+1) = (pj opi) + wij(n) The momentum parameter is chosen

between 0 and 1, typically 0.9. This allows one to use higher learning rates. The momentum term filters out high frequency oscillations on the error surface.

What would the learning rate be in a deep valley?

Problems with Overfitting

Plot training example error versus test example error:

Test set error is increasing!– Network is overfitting the data

– Learning idiosyncrasies in data, not general principles

– Big problem in Machine Learning (ANNs in particular)

Avoiding Overfitting Bad idea to use training set accuracy to terminate One alternative: Use a validation set

– Hold back some of the training set during training– Like a miniature test set (not used to train weights at

all)– If the validation set error stops decreasing, but the

training set error continues decreasing• Then it’s likely that overfitting has started to

occur, so stop Another alternative: use a weight decay factor

– Take a small amount off every weight after each epoch

– Networks with smaller weights aren’t as highly fine tuned (overfit)