Transcript ppt - IIT Bombay

CS621 : Artificial Intelligence
Pushpak Bhattacharyya
CSE 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
j
wji
i
….
….
….
….
Output layer
(m o/p neurons)
Hidden layers
Input layer
(n i/p neurons)
• Fully connected feed forward network
• Pure FF network (no jumping of
connections over layers)
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
1 p n
E   (ti  oi ) 2j
2 j 1 i 1
• 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
Weights in a ff NN
• wmn is the weight of the
connection from the nth neuron
to the mth neuron
• E vs W surface is a complex
surface in the space defined by
the weights wij
E

• w gives the direction in which
a movement of the operating
point in the wmn co-ordinate
space will result in maximum
decrease in error
mn
m
wmn
n
wmn
E

wmn
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!
y  1 as x  
y  0 as x  
• Computing power comes from non-linearity of
sigmoid function.
Derivative of Sigmoid function
1
y
x
1 e
x
dy
1
e
x

( e ) 
x 2
x 2
dx
(1  e )
(1  e )
1

x
1 e
1 

 y (1  y )
1 
x 
 1 e 
Training algorithm
• Initialize weights to random values.
• For input x = <xn,xn-1,…,x0>, modify weights as
follows
Target output = t, Observed output = o
E
wi  
wi
E
1
(t  o) 2
2
• Iterate until E < 
(threshold)
Calculation of ∆wi
n 1
E
E net 



 where : net   wi xi 
Wi net Wi 
i 0

E o net



o net Wi
 (t  o)o(1  o) xi
E
wi  
(  learning constant, 0    1)
wi
wi   (t  o)o(1  o) xi
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
n
j
wji
ni
• 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)
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
j
wji
i
….
….
….
….
Output layer
(m o/p neurons)
Hidden layers
Input layer
(n i/p neurons)
• Fully connected feed forward network
• Pure FF network (no jumping of
connections over layers)
Gradient Descent Equations
E
w ji  
(  learning rate, 0    1)
w ji
net j
E
E


( net j  input at the jth layer )
w ji net j w ji
E
net j
 j
net j
w ji  j
w ji
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
E
 E o j
th
j  


(net j  input at the j layer )
net j
o j net j
1 m
E   (t p  o p ) 2
2 p 1
Hence, j  ((t j  o j )o j (1  o j ))
w ji   (t j  o j )o j (1  o j )oi
Backpropagation for hidden
layers
k
j
i
….
….
….
….
Output layer
(m o/p neurons)
Hidden layers
Input layer
(n i/p neurons)
k is propagated backwards to find value of j
Backpropagation – for hidden
layers
w ji  joi
o j
E
j  


net j
o j net j
E

E
 o j (1  o j )
o j
E
netk
  (

)  o j (1  o j )
o j
knext layer net k
Hence,  j  

 (w
knext layer
kj
 (
knext layer
k
 wkj )  o j (1  o j )
 k )o j (1  o j )oi
General Backpropagation Rule
• General weight updating rule:
w ji  joi
• Where
 j  (t j  o j )o j (1  o j )

 (w 
knext layer
kj
k
for outermost layer
)o j (1  o j )oi 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)]
Diagnostics in action(2)
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.
Diagnostics in action(3)
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.
(w ji ) nth  iteration  jOi   (wji)( n  1)th  iteration
 Accelerates the movement out of the trough.
 Dampens oscillation inside the trough.
 Choosing  : If  is large, we may jump over
the minimum.
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
Duration
of lesions : weeks
Duration
of lesions : weeks
Symptoms & parameters
0
Internal
representation
Disease
diagnosis
0
1
0
( Psoriasis node )
Minimal itching
6
Positive
KOH test
Lesions located
on feet
1.68
10
13
5
(Dermatophytosis node)
1.62
36
14
Minimal
increase
in pigmentation 71
1
Positive test for
pseudohyphae
95
And spores
19
Bias
Bias
96
9
(Seborrheic dermatitis node)
20
Figure : Explanation of dermatophytosis diagnosis using the DESKNET expert system.
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