1. CS621: Artificial Intelligence Lecture 22-23: Sigmoid neuron, Backpropagation (Lecture 20 and 21 taken by Anup on Graphical Models)
Pushpak Bhattacharyya
Computer Science and Engineering Department
IIT Bombay

2. 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”

3. 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

4. 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

5. 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.

6. Derivative of Sigmoid function

7. 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)

8. Calculation of ∆wi

9. 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

10. Backpropagation on feedforward network

11. 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)

12. Gradient Descent Equations

13. Backpropagation – for outermost layer

14. 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

15. Backpropagation – for hidden layers

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

17. 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