1. Introduction to Artificial Intelligence (G51IAI)
Dr Matthew Hyde
Neural Networks
More precisely:
“Artificial Neural Networks”
Simulating, on a computer, what we understand about neural networks in the brain
Machines can think/act as human being.

2. Lecture Outline Recap on perceptrons Linear Separability
Learning / Training
The Neuron’s Activation Function
G51IAI – Introduction to AI

3. Recap from last lecture
A ‘Perceptron’
Single layer NN (one neuron)
Inputs can be any number
Weights on the edges
Output can only be 0 or 1 5
0.5
θ = 6 6 2 Z
0 or 1 3 -3
G51IAI – Introduction to AI

4. Truth Tables and Linear Separability
G51IAI – Introduction to AI

5. AND function, and OR function
XOR X1 X2 Z 1 X1 X2 Z 1
These are called “truth tables”
G51IAI – Introduction to AI

6. AND function, and OR function
XOR X1 X2 Z T F X1 X2 Z T F
These are called “truth tables”
G51IAI – Introduction to AI

7. Important!!!
You can represent any truth table graphically, as a diagram
The diagram is 2-dimensional if there are two inputs
3-dimensional if there are three inputs
Examples on the board in the lecture, and in the handouts
G51IAI – Introduction to AI

8. 3 Inputs means 3-dimensionsX Y Z
Output 1
0,1,0
1,1,0
0,1,1
1,1,1
Y axis
X axis
Z axis
0,0,0
1,0,0
0,0,1
1,0,1
G51IAI – Introduction to AI

9. Linear Separability in 3-dimensions
Instead of a line, the dots are separated by a plane
G51IAI – Introduction to AI

10. Only linearly Separable functions can be represented by a PerceptronX1 X2 Z 1 X1 X2 Z 1
AND
XOR
0,0
1,0
0,1
1,1
0,0
XOR
1,0
0,1
1,1
Minsky & Papert
Graphically represent
Minsky & Papert (1969)
AND
Functions which can be separated in this way are called Linearly Separable
Only linearly Separable functions can be represented by a Perceptron
G51IAI – Introduction to AI

11. Examples – Handout 3 Linear Separability
Fill in the diagrams with the correct dots
black or white, for an output of 1 or 0
G51IAI – Introduction to AI

12. How to Train your Perceptron
G51IAI – Introduction to AI

13. Simple Networks -1 X Y Z 1 ANDX 1
θ=1.5 Y 1 -1
Both of these represent the AND function.
It is sometimes convenient to set the threshold to zero, and add a constant negative input
1.5
Relation between threshold and weights, how about different number of input?
Or if we want to set threshold to 0, is an extra input helpful? X 1
θ=0 1 Y

14. Training a NN AND X1 X2 Z 1 AND G51IAI – Introduction to AI 0,1 1,1
0,0
1,0
0,1
1,1 X1 X2 Z 1
G51IAI – Introduction to AI

15. Randomly Initialise the Network
We set the weights randomly, because we do not know what we want it to learn.
The weights can change to whatever value is necessary
It is normal to initialise them in the range [-1,1]

16. Randomly Initialise the Network-1
0.3
0.5 X
θ=0 Y
-0.4

17. Learning While epoch produces an error End While
Present network with next inputs (pattern) from epoch
Err = T – O
If Err <> 0 then
Wj = Wj + LR * Ij * Err
End If
End While
Final important thing to learn
Get used to this notation!!
Make sure that you can reproduce this pseudocode AND understand what all of the terms mean
G51IAI – Introduction to AI

18. Epoch The ‘epoch’ is the entire training set
The training set is the set of four input and output pairs
INPUT
DESIRED OUTPUT X Y Z 1

19. The learning algorithm
INPUT
DESIRED OUTPUT X Y Z 1
Input the first inputs from the training set into the Neural Network
What does the neural network output?
Is it what we want it to output?
If not then we work out the error and change some weights

20. First training step 0.3 0.5 -0.3 + 0.5 + -0.4 = -0.2 -0.4X Y Z 1
Input 1, 1
Desired output is 1
Actual output is 0 -1
0.3 1
0.5
θ=0
= -0.2 1
-0.4
= Output of 0

21. First training step We wanted 1 We got 0 Error = 1 – 0 = 1X Y Z 1
We wanted 1
We got 0
Error = 1 – 0 = 1
While epoch produces an error
Present network with next inputs (pattern) from epoch
Err = T – O
If Err <> 0 then
Wj = Wj + LR * Ij * Err
End If
End While
If there IS an error, then we change ALL the weights in the network

22. If there is an error, change ALL the weights
Wj = Wj + ( LR * Ij * Err )
New Weight = Old Weight +
(Learning Rate * Input Value * Error)
New Weight = (0.1 * -1 * 1)
= 0.2 -1
0.3
0.2 1
0.5
θ=0

23. If there is an error, change ALL the weights
Wj = Wj + ( LR * Ij * Err )
New Weight = (0.1 * 1 * 1)
= 0.6 -1
0.2 1
0.5
0.6
θ=0
-0.4 1

24. Effects of the first change
The output was too low (it was 0, but we wanted 1)
Weights that contributed negatively have reduced
Weights that contributed positively have increased
It is trying to ‘correct’ the output gradually -1 -1
0.3
0.2 X
0.5 X
0.6
θ=0
θ=0 Y
-0.4 Y
-0.3

25. Epoch not finished yet The ‘epoch’ is the entire training set
We do the same for the other 3
input-output pairs
INPUT
DESIRED OUTPUT X Y Z 1

26. The epoch is now finished
Was there an error for any of the inputs?
If yes, then the network is not trained yet
We do the same for another epoch, from the first inputs again

27. The epoch is now finished
If there were no errors, then we have the network that we want
It has been trained
While epoch produces an error
Present network with next inputs (pattern) from epoch
Err = T – O
If Err <> 0 then
Wj = Wj + LR * Ij * Err
End If
End While

28. Effect of the learning rate
Set too high
The network quickly gets near to what you want
But, right at the end, it may ‘bounce around’ the correct weights
It may go too far one way, and then when it tries to compensate it will go too far back the other way
Wj = Wj + ( LR * Ij * Err )

29. Effect of the learning rate
Set too high
It may ‘bounce around’ the correct weights
AND
0,0
1,0
0,1
1,1

30. Effect of the learning rate
Set too low
The network slowly gets near to what you want
It will eventually converge (for a linearly separable function)
but that could take a long time
When setting the learning rule, you have to strike a balance between speed and effectiveness
Wj = Wj + LR * Ij * Err

31. The Neuron’s Activation Function
G51IAI – Introduction to AI

32. Expanding the Model of the Neuron: Outputs other than ‘1’
Output is 1 or 0
It doesn’t matter about how far over the threshold we are X1
θ = 5 2 20 Y1 -5
-10 1 X2
θ = 2 -2
θ = 9 1 -4 Z Y2 5 1 1 X3 3 1 2 1
G51IAI – Introduction to AI 6
θ = 0

33. Example from last lecture
Left wheel speed
...
Right wheel speed
The speed of the wheels is not just 0 or 1
...
G51IAI – Introduction to AI

34. Expanding the Model of the Neuron: Outputs other than ‘1’
So far, the neurons have only output a value of 1 when they fire.
If the input sum is greater than the threshold the neuron outputs 1.
In fact, the neurons can output any value that you want.
G51IAI – Introduction to AI

35. Modelling a Neuron aj : Input value (output from unit j)
model in more details
Threshold function seen so far: one of activation function
aj : Input value (output from unit j)
wj,i : Weight on the link from unit j to unit i
ini : Weighted sum of inputs to unit i
ai : Activation value of unit i
g : Activation function
G51IAI – Introduction to AI

36. Activation Functions Stept(x) = 1 if x >= t, else 0
Sign(x) = +1 if x >= 0, else –1
Sigmoid(x) = 1/(1+e-x)
Sigmoid: output as input for other neuron
aj : Input value (output from unit j)
ini : Weighted sum of inputs to unit i
ai : Activation value of unit i
g : Activation function
G51IAI – Introduction to AI

37. Summary Linear Separability Learning Algorithm Pseudocode
Activation function (threshold, sigmoid, etc)
G51IAI – Introduction to AI