cs621 : artificial intelligence

CS621 : Artificial Intelligence

Pushpak BhattacharyyaCSE Dept., IIT Bombay

Lecture 24

Sigmoid neuron and backpropagation

Feedforward n/w

• A multilayer feedforward neural network has – Input layer– Output layer– Hidden layer (assists computation)

Output units and hidden units are called

computation units.

Architecture of the n/w

• Fully connected feed forward network• Pure FF network (no jumping of connections

over layers)

Hidden layers

Input layer (n i/p neurons)

Output layer (m o/p neurons)

j

i

wji

….

….

….

….

Training of the MLP

• Multilayer Perceptron (MLP)

• Question:- How to find weights for the hidden layers when no target output is available?

• Credit assignment problem – to be solved by “Gradient Descent”

Gradient Descent Technique

• Let E be the error at the output layer

• ti = target output; oi = observed output

• i is the index going over n neurons in the outermost layer

• j is the index going over the p patterns (1 to p)• Ex: XOR:– p=4 and n=1

p

j

n

ijii otE

1 1

2)(2

1

Weights in a ff NN

• wmn is the weight of the connection from the nth neuron to the mth neuron

• E vs surface is a complex surface in the space defined by the weights wij

• gives the direction in which a movement of the operating point in the wmn co-ordinate space will result in maximum decrease in error

W

m

n

wmn

mnw

E

mnmn w

Ew

Sigmoid neurons• Gradient Descent needs a derivative computation

- not possible in perceptron due to the discontinuous step function used!

Sigmoid neurons with easy-to-compute derivatives used!

• Computing power comes from non-linearity of sigmoid function.

xy

xy

as 0

as 1

Derivative of Sigmoid function

)1(1

11

1

1

)1()(

)1(

1

1

1

22

yyee

e

ee

edx

dy

ey

xx

x

xx

x

x

Training algorithm

• Initialize weights to random values.

• For input x = <xn,xn-1,…,x0>, modify weights as follows

Target output = t, Observed output = o

• Iterate until E < (threshold)

ii w

Ew

2)(2

1otE

Calculation of ∆wi

ii

ii

i

i

n

iii

ii

xoootw

w

Ew

xooot

W

net

net

o

o

E

xwnetwhereW

net

net

E

W

E

)1()(

)10 constant, learning(

)1()(

:1

0

Observations

Does the training technique support our intuition?

• The larger the xi, larger is ∆wi

– Error burden is borne by the weight values corresponding to large input values

Observations contd.

• ∆wi is proportional to the departure from target

• Saturation behaviour when o is 0 or 1

• If o < t, ∆wi > 0 and if o > t, ∆wi < 0 which is consistent with the Hebb’s law

Hebb’s law

• If nj and ni are both in excitatory state (+1)– Then the change in weight must be such that it enhances

the excitation– The change is proportional to both the levels of excitation

∆wji is prop. to e(nj) e(ni)

• If ni and nj are in a mutual state of inhibition ( one is +1 and the other is -1),– Then the change in weight is such that the inhibition is

enhanced (change in weight is negative)

nj

ni

wji

Saturation behavior

• The algorithm is iterative and incremental

• If the weight values or number of input values is very large, the output will be large, then the output will be in saturation region.

• The weight values hardly change in the saturation region

Backpropagation algorithm

• Fully connected feed forward network• Pure FF network (no jumping of connections

over layers)

Hidden layers



j

i

wji

….

….

….

….

Gradient Descent Equations

ji

jji

j

thj

ji

j

jji

jiji

w

netjw

jnet

E

netw

net

net

E

w

E

w

Ew

)layer j at theinput (

)10 rate, learning(

Example - Character Recognition

• Output layer – 26 neurons (all capital)

• First output neuron has the responsibility of detecting all forms of ‘A’

• Centralized representation of outputs

• In distributed representations, all output neurons participate in output

Backpropagation – for outermost layer

ijjjjji

jjjj

m

ppp

thj

j

j

jj

ooootw

oootj

otE

netnet

o

o

E

net

Ej

)1()(

))1()(( Hence,

)(2

1

)layer j at theinput (

1

2

Backpropagation for hidden layers

Hidden layers



j

i

….

….

….

….

k

k is propagated backwards to find value of j

Backpropagation – for hidden layers

ijjk

kkj

jjk

kjkj

jjk jk

jjj

j

j

jj

iji

ooow

oow

ooo

netk

net

E

ooo

E

net

o

o

E

net

Ej

jow

)1()(

)1()( Hence,

)1()(

)1(

layernext

layernext

layernext

General Backpropagation Rule

ijjk

kkj ooow )1()(layernext

)1()( jjjjj ooot

iji jow • General weight updating rule:

• Where for outermost layer

for hidden layers

Issues in the training algorithm

• Algorithm is greedy. It always changes weight such that E reduces.

• The algorithm may get stuck up in a local minimum.

• If we observe that E is not getting reduced anymore, the following may be the reasons:

Issues in the training algorithm contd.

1. Stuck in local minimum.

2. Network paralysis. (High –ve or +ve i/p makes neurons to saturate.)

3. (learning rate) is too small.

Diagnostics in action(1)

1) If stuck in local minimum, try the following:

• Re-initializing the weight vector.

• Increase the learning rate.

• Introduce more neurons in the hidden layer.

Diagnostics in action (1) contd.

2) If it is network paralysis, then increase the number of neurons in the hidden layer.

Problem: How to configure the hidden layer ?

Known: One hidden layer seems to be sufficient. [Kolmogorov (1960’s)]


Kolgomorov statement:

A feedforward network with three layers (input, output and hidden) with appropriate I/O relation that can vary from neuron to neuron is sufficient to compute any function.

More hidden layers reduce the size of individual layers.


3) Observe the outputs: If they are close to 0 or 1, try the following:

1. Scale the inputs or divide by a normalizing factor.

2. Change the shape and size of the sigmoid.

Diagnostics in action(3)1. Introduce momentum factor.

Accelerates the movement out of the trough. Dampens oscillation inside the trough. Choosing : If is large, we may jump over

the minimum.

iterationthnjiijiterationnthji wOw )1()()(

An application in Medical Domain

Expert System for Skin Diseases Diagnosis

• Bumpiness and scaliness of skin

• Mostly for symptom gathering and for developing diagnosis skills

• Not replacing doctor’s diagnosis

Architecture of the FF NN

• 96-20-10• 96 input neurons, 20 hidden layer neurons, 10

output neurons• Inputs: skin disease symptoms and their

parameters– Location, distribution, shape, arrangement, pattern,

number of lesions, presence of an active norder, amount of scale, elevation of papuls, color, altered pigmentation, itching, pustules, lymphadenopathy, palmer thickening, results of microscopic examination, presence of herald pathc, result of dermatology test called KOH

Output

• 10 neurons indicative of the diseases:– psoriasis, pityriasis rubra pilaris, lichen

planus, pityriasis rosea, tinea versicolor, dermatophytosis, cutaneous T-cell lymphoma, secondery syphilis, chronic contact dermatitis, soberrheic dermatitis

Training data

• Input specs of 10 model diseases from 250 patients

• 0.5 is some specific symptom value is not knoiwn

• Trained using standard error backpropagation algorithm

Testing

• Previously unused symptom and disease data of 99 patients

• Result:• Correct diagnosis achieved for 70% of papulosquamous

group skin diseases• Success rate above 80% for the remaining diseases

except for psoriasis• psoriasis diagnosed correctly only in 30% of the cases• Psoriasis resembles other diseases within the

papulosquamous group of diseases, and is somewhat difficult even for specialists to recognise.

Explanation capability

• Rule based systems reveal the explicit path of reasoning through the textual statements

• Connectionist expert systems reach conclusions through complex, non linear and simultaneous interaction of many units

• Analysing the effect of a single input or a single group of inputs would be difficult and would yield incor6rect results

Explanation contd.

• The hidden layer re-represents the data

• Outputs of hidden neurons are neither symtoms nor decisions

Figure : Explanation of dermatophytosis diagnosis using the DESKNET expert system.

5 (Dermatophytosis node)

0( Psoriasis node )

Disease diagnosis

-2.71

-2.48

-3.46

-2.68

19

14

13

0

1.621.43

2.13

1.68

1.58

1.22

Symptoms & parametersDuration

of lesions : weeks 0

1

6

10

36

171

95

96

Duration of lesions : weeks

Minimal itching

Positive KOH test

Lesions locatedon feet

Minimalincrease

in pigmentation

Positive test forpseudohyphae

And spores

Bias

Internalrepresentation

1.46

20 Bias

-2.86

-3.31

9 (Seborrheic dermatitis node)

Discussion

• Symptoms and parameters contributing to the diagnosis found from the n/w

• Standard deviation, mean and other tests of significance used to arrive at the importance of contributing parameters

• The n/w acts as apprentice to the expert

cs621 : artificial intelligence

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