Robotics 15

8/6/2019 Robotics 15

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EXPERT SYSTEMS AND SOLUTIONS

Email: [email protected]

[email protected]

Cell: 9952749533www.researchprojects.info

PAIYANOOR, OMR, CHENNAI

Call For Research Projects Final

year students of B.E in EEE, ECE, EI,

M.E (Power Systems), M.E (Applied

Electronics), M.E (Power Electronics)

Ph.D Electrical and Electronics.

Students can assemble their hardware in our

Research labs. Experts will be guiding theprojects.


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Learning in Neural Networks

Sibel KAPLAN

ZMR-2004


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Program...

History

Introduction

Models of Processing Elements (neurons) Models of Synaptic Interconnections

Learning Rules for Updating the Weights

Supervised learning

Perceptron Learning

Adaline Learning

Back Propagation (an example)

Reinforcement Learning

Unsupervised Learning

Hebbian Learning Rule


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1943: ANNs were first studied by McCollach and Pitts

1949: Donal Hebb used ANN to store knowledge

1958: Rosenblatt found perceptron and it was started to use for patternrecognition

1960: Widrow-Marcian Hoff found ADALINE

1982: J.J. Hopfield published his study about single-layer feedback

neural network with symmetric weights1990s: It was applied to different kinds of problems, softwares werebegun to be used

History...


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The Real and Artificial Neurons


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ANNs are systems that are constructed to make use of some

organizational principles resembling those of the human brain

ANNs are good at tasks such as; pattern matching and classification

function approximation

optimization

vector quantitization

data clustering


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Input

values

weights

SummingFunction (net input)

Bias

b

ActivationfunctionLocal

Field

vOutput

y

x1

x2

xm

w2

wm

w1

/ /

)(N

The Neuron (Processing Element)


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Models of ANNs are specified by three basic entitites:

Models of the neurons themselves,

Models of synaptic interconnections and structures,

Training or learning rules for updating the connecting

weights


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There are three important parameters about a neuron:

I. An integrated function associated with the input of aneuron to calculate the net input (for M-P neuron):

fi neti =

II. A set of links, describing the neuron inputs, with

weights W1, W2, , Wm

!

m

j

ijij Qtxw

1

)(


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Activation functions [a(f)] output an activation value as a function of

its net input. Some commonly used activation functions are;

Step Function

Hard limiter (thresold function)

Ramp function

Unipolar sigmoid function

Bipolar sigmoid function

III. Activation Function


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Connections

An ANN consists of a set of highly interconnected neurons

such that each neuron output is connected throughweights to other neurons or to itself.


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y1

y2

yn

x1

x2

xm

w11

w21

wnm

w23

Single-Layer Feedforward

Network

Single Node with

Feedback to Itself

x1

x2

xm

y1

y2

yn

Single Layer Recurrent

Network

Basic types of connection geometries (Nelson

and Illingworth, 1991).


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x2

x1

xm

y1

y2

yn

InputLayer

Hidden Layers OutputLayer

Multilayer Feedforward Network

x1

x2

xm

y1

y2

yn

Multilayer Recurrent Network

Basic types of connection geometries (Nelson and

Illingworth, 1991).


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L

earning Rules

Generally, we can classify learning in ANNs in two broad

classes:

Parameter learning which is concerned with updating ofthe connecting weights

Structure learning which focuses on the change in the

network structure, including the number of PEs and theirconnection types.

These two kinds of learning can be performed

simultaneously or separately.


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In weight learning, we have to develop learning rules to

efficiently guide the weight matrix W in approaching a

desired matrix that yields the desired networkperformance. In general, learning rules are classified into

three categories:

Supervised Learning (learning with a teacher)

Reinforcement Learning (learning with a critic)

Unsupervised learning


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In supervised learning;

when input is applied to an ANN, the corresponding desired response of

the system is given. An ANN is supplied with a sequence of examples (x1,d1) , ( x2, d2)...(xk, dk) of desired input-output pairs.

In reinforcement learning;

only less detailed information than supervised learning is available.There is only a single bit of feedback information indicating whether the

output is right or wrong. That is, it just says how good or how bad a

particular output is and provides no hint as to what the right answer

should be.


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In unsupervised learning;

there is no teacher to provide any feedback information. Thenetwork must discover for itself patterns, features,regularities, correlations or categories in the input data andcode for them in the output. While discovering these

features, the network undergoes changes in its parameters;this process is called self-organizing.


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Three Categories of Learning

Supervised Learning Reinforcement Learning

Unsupervised Learning


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Machine Learning

Supervised Unsupervised

Data:

Labeled examples

(input , desired output)Problems:classificationpattern recognitionregression

NN models:perceptronadalineback-propagation

Data:Unlabeled examples

(different realizations of theinput)

Problems:clusteringcontent addressable memory

NN models:self-organizing maps (SOM)HammingnetworksHopfieldnetworkshierarchicalnetworks


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Knowledge about the learning task is given in the form of

examples called training examples.

The aim of the training process is to obtain a NN that

generalizes well, that is, that behaves correctly on new

instances of the learning task.


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A general form of the weight learning rule indicates that the

incremet of the weight vector wi produced by the learning

step at time t is proportional to the product of the learningsignal r and the input x(t);

r = fr (wi, x , di ) (learning signal)Hence, the increse in the weight vector;

)()( trxti L!(

)()()()()()1( ),,( ttitt

ir

t

i

t

i xdxwfww L!


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Perceptron Learning Rule

The perceptron is used for binary classification.

Training is done by error back-propagation algorithm.

For simple perceptrons with linear threshold units (LTUs),

the desired outputs di(k) can take only 1 values. Then;

It is necessary to find a hyperplane that divides inputs that

have positive and negative targets (or outputs).

)()()(

)sgn(k

i

kT

i

k

i dxwy !!


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Linear Separability in Perceptron

?

I1

I2

I1 I1

I2 I2

I1 and I2 I1 or I2I1 xor I2

w1x1 + w2x2 + w0 = 00wxw0

m

1i

ii !!

decision

boundary

C1

C2


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The condition for solvability of a pattern classificationproblem by a simple perceptron depends on whether the

problem is linearly separable or not. If a classification problem is linearly separable by a simpleperceptron then;

(desired output = +1) (desired output = -1)

0HxT

i0R

xw

T

i


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Adaline (Widrow-Hoff) Learning Rule (1962)

When the two classes are not linearly separable, it may be

desirable to obtain a linear separator that minimizes the

mean squared error.

Adaline (Adaptive LinearElement);

uses a linear neuron model

uses the Least-Mean-Square (LMS) learning algorithm

useful for robust linear classification and regression


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To update the weights, we define a cost function E(w)

which measures the systems performance error:

For weight adjustment;

!

!p

k

kk ydwE1

2)()( )(2

1)(

)(wEw w!( L

!

!x

x!(

p

k

k

j

kT

k

j

j xxwdw

Ew

1

)()( )(LL


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Back-Propagation

One of the most important historical development in ANNs.

Applied to multilayer feedforward networks consisting of processing

elements with continuous differentiable activation functions.Process:

The input patterns is propagated from the input layer to the outputlayer and it produces an actual output

The error signals resulting from the difference between dk and yk areback-propagated from the output layer to the previous layers for them toupdate their weights.


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Three-Layer Back-Propagation Network

yi

zq

Xj

wiq

vqj

x1 xj xm

J=1,2,...,m

q=1,2,...,l

i=1,2,...,n

!

!m

j

jqjq xvnet1

)()(1!!!

n

j

jqjqq xvanetaz

))(()()(1 11

! !!

!!!l

q

m

j

jqjiq

l

q

qiqii xvawazwanetay

These indicate the forward propagation of input signals through the

layers of neurons.


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? A

! !!!

n

i

n

i

iiii netadyd1 1

22 )(

2

1)(

2

1)(

iq

iqxx!( L

? A? A? A qoiqiiiiq zznetaydw LHL !!( )(,

Then according to the gradient-descent method, the weights in thehidden to output connections are updated by;

A cost function is defined as in Adaline Learning Rule;


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According to the law;

when an axonal input from Neuron A to neuron B causes

neuron B to immediately emit a pulse (fire) and thissituation happens repeatedly or persistently, then the

chance of that axonal input in terms of its ability to help

neuron B to fire in the future is somehow increased. It is a

rule used also for other learning rules.

Hebbs Learning Law (1949)


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According to Hebbian Hypothesis;

r a (wiTx ) = yi

a(.): activation function of neuron

In this equation, the learning signal r is simply set as the

neurons current output. Then the incremet in the weights:

(i = 1, 2,.........n); (j = 1, 2,..............m)

jijT

iij xyxxwaw LL !!( )(

$


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It is suggested choosing the initial weights between

(Wessels and Barnard, 1992)

Learning constant ( ) is usually chosen experimentally for

each problem. A larger value could speed up the

convergence but might result in overshooting. Valuesranging from 10-3 to 10 have been used succesfully for many

applications.

? Ak

i

kk /3,/3

Tips about important learning factors

L


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I1

I2

w11

w23

w13

G11

G32

Q1

Q2

W11 =0.2 ,W21 =0.1 (i=1,2)W12 =0.4, W22 =0. 2 (j=1,2,3)W13 =0.7, W23 =0.2 (k=1,2)

Gw11 = 0.1,Gw21=0.3,Gw31 =0.4 (j=1,2,3)

Gw12 = 0.2,Gw22 = 0.5,Gw32 = 0.3 ( k=1,2)

L : 0,3

:E 0,7I1=0.6, I2=0.7

11

11 Nete

f

! sigmoidfunction

An example about back-propagation...


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(for j=1)

= (B11=0)

=0.6*0.2+0.7*0.1=0.19

= 0.547

!

!2

1i

ijijiJ BINet

!

!2

1

1111

i

iiwINet 2121111 wIwINet !

19.011

1

!e

A


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!

!3

1

11

j

jjGwNet 313212111GwGwGw =

= 0.547*0.1+0.593*0.3+0.636*0.4

= 0.487

= 0.619 (final output)487.01

1

1

!

eError of the first output: E1= (Q1 desired-Q1 real)

= 0.5-0.619 = -0.119

Total cost (error) =

= 0.1555

)(2

1 22

2

1EE ? A22 )5449.0()119.0(

2

1=

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