1. CS621: Artificial Intelligence
Pushpak Bhattacharyya CSE Dept., IIT Bombay
Lecture 44– Multilayer Perceptron Training: Sigmoid Neuron; Backpropagation
9th Nov, 2010

2. The Perceptron Model . Output = y Threshold = θ w1 wn Wn-1 x1 Xn-1
y = 1 for Σwixi >=θ
= 0 otherwise
Threshold = θ w1 wn
Wn-1 x1
Xn-1

3. y 1 θ
Σwixi

4. Perceptron Training Algorithm
Start with a random value of w
ex: <0,0,0…>
Test for wxi > 0
If the test succeeds for i=1,2,…n
then return w
3. Modify w, wnext = wprev + xfail

5. Feedforward Network

6. Limitations of perceptron
Non-linear separability is all pervading
Single perceptron does not have enough computing power
Eg: XOR cannot be computed by perceptron

7. Solutions
Tolerate error (Ex: pocket algorithm used by connectionist expert systems).
Try to get the best possible hyperplane using only perceptrons
Use higher dimension surfaces
Ex: Degree - 2 surfaces like parabola
Use layered network

8. Pocket Algorithm Algorithm evolved in 1985 – essentially uses PTA
Basic Idea:
Always preserve the best weight obtained so far in the “pocket”
Change weights, if found better (i.e. changed weights result in reduced error).

9. XOR using 2 layers Non-LS function expressed as a linearly separable
function of individual linearly separable functions.

10. Example - XOR = 0.5 w1=1 w2=1  Calculation of XOR x1x2 x1x2 x1 x21
Calculation of
x1x2
= 1
w1=-1
w2=1.5 x1 x2

11. Example - XOR
= 0.5
w1=1
w2=1
x1x2 1 1
x1x2
1.5 -1 -1
1.5 x1 x2

12. Some Terminology A multilayer feedforward neural network has
Input layer
Output layer
Hidden layer (assists computation)
Output units and hidden units are called
computation units.

13. 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”

14. Can Linear Neurons Work?h2 h1 x2 x1

15. Claim: A neuron with linear I-O behavior can’t compute X-OR.
Note: The whole structure shown in earlier slide is reducible to a single neuron with given behavior
Claim: A neuron with linear I-O behavior can’t compute X-OR.
Proof: Considering all possible cases:
[assuming 0.1 and 0.9 as the lower and upper thresholds]
For (0,0), Zero class:
For (0,1), One class:

16. A linear neuron can’t compute X-OR.
For (1,0), One class:
For (1,1), Zero class:
These equations are inconsistent. Hence X-OR can’t be computed.
Observations:
A linear neuron can’t compute X-OR.
A multilayer FFN with linear neurons is collapsible to a single linear neuron, hence no a additional power due to hidden layer.
Non-linearity is essential for power.

17. Multilayer Perceptron

18. 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”

19. 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

20. 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 m
wmn n

21. 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.

22. Derivative of Sigmoid function

23. 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)

24. Calculation of ∆wi

25. 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

26. Backpropagation on feedforward network

27. Backpropagation algorithm….
Output layer (m o/p neurons) j
wji …. i
Hidden layers …. ….
Input layer (n i/p neurons)
Fully connected feed forward network
Pure FF network (no jumping of connections over layers)

28. Gradient Descent Equations

29. Backpropagation – for outermost layer

30. Backpropagation for hidden layers….
Output layer (m o/p neurons) k …. j
Hidden layers …. i ….
Input layer (n i/p neurons)
k is propagated backwards to find value of j

31. Backpropagation – for hidden layers

32. General Backpropagation Rule
General weight updating rule:
Where
for outermost layer
for hidden layers

33. How does it work?
Input propagation forward and error propagation backward (e.g. XOR)
w2=1
w1=1
= 0.5
x1x2 -1 x1 x2
1.5 1