Post on 17-Dec-2015
Introduction to Artificial Intelligence (G51IAI)
Dr Matthew HydeNeural Networks
More precisely: “Artificial Neural Networks”Simulating, on a computer, what we understand about neural networks in the brain
G51IAI – Introduction to AI
Lecture Outline
Recap on perceptrons Linear Separability Learning / Training The Neuron’s Activation Function
G51IAI – Introduction to AI
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
6 Z
θ = 6
0.5
-33
2 0 or 1
Truth Tables and Linear
SeparabilityG51IAI – Introduction to AI
G51IAI – Introduction to AI
AND function, and OR function
X1 X2 Z
1 1 11 0 00 1 00 0 0
AND XORX1 X2 Z
1 1 01 0 10 1 10 0 0
These are called “truth tables”
G51IAI – Introduction to AI
AND function, and OR function
X1 X2 Z
T T TT F FF T FF F F
AND XORX1 X2 Z
T T FT F TF T TF F F
These are called “truth tables”
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
G51IAI – Introduction to AI
0,0,1 1,0,1
0,1,11,1,1
0,0,01,0,0
1,1,00,1,0
X axisY axis
Z axis
X Y Z Output
0 0 0 10 0 1 10 1 0 00 1 1 01 0 0 11 0 1 01 1 0 11 1 1 0
3 Inputs means 3-dimensions
G51IAI – Introduction to AI
Linear Separability in 3-dimensions
Instead of a line, the dots are separated by a plane
G51IAI – Introduction to AI
AND
• Functions which can be separated in this way are called Linearly Separable
• Only linearly Separable functions can be represented by a Perceptron
Minsky & Papert
0,0 XOR 1,0
0,11,1
0,0 1,0
0,11,1
AND XORX1 X2 Z
1 1 0
1 0 1
0 1 1
0 0 0
X1 X2 Z
1 1 1
1 0 0
0 1 0
0 0 0
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
How to Train your Perceptron
G51IAI – Introduction to AI
Simple Networks
AND
X
Y
θ=1.5
1
1
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
X
Y
θ=01
1
-1 1.5Both of these
represent the AND function.
It is sometimes convenient to set the threshold to zero, and add a constant negative input
G51IAI – Introduction to AI
Training a NN
AND0,0 1,0
0,11,1
AND
X1 X2 Z
1 1 1
1 0 0
0 1 0
0 0 0
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]
Randomly Initialise the Network
X
Y
θ=00.5
-0.4
-1 0.3
G51IAI – Introduction to AI
Learning
While epoch produces an errorPresent network with next inputs (pattern) from epoch Err = T – OIf Err <> 0 then
Wj = Wj + LR * Ij * Err
End If
End While Get used to this notation!!Make sure that you can reproduce this pseudocode AND understand what all of the terms mean
Epoch
The ‘epoch’ is the entire training set
The training set is the set of four input and output pairs
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
INPUT DESIRED OUTPUT
The learning algorithm
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
INPUT DESIRED OUTPUT
Input the first inputs from the training set into the Neural NetworkWhat 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
First training step
Input 1, 1 Desired output is 1 Actual output is 0
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
1
1
θ=00.5
-0.4
-1 0.3
-0.3 + 0.5 + -0.4= -0.2= Output of 0
First training step
We wanted 1 We got 0 Error = 1 – 0 = 1
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
While epoch produces an errorPresent network with next inputs (pattern) from epoch Err = T – OIf 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
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 = 0.3 + (0.1 * -1 * 1)= 0.2
1 θ=00.5
-1 0.3 0.2
If there is an error, change ALL the weights
Wj = Wj + ( LR * Ij * Err ) New Weight = 0.5 + (0.1 * 1 * 1)
= 0.6
1 θ=00.5
-1 0.2
1 -0.4
0.6
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
X θ=0
-1 0.2
Y -0.3
0.6X
Y
θ=00.5
-0.4
-1 0.3
Epoch not finished yet
The ‘epoch’ is the entire training set
We do the same for the other 3 input-output pairs
X Y Z
1 1 1
1 0 0
0 1 0
0 0 0
INPUT DESIRED OUTPUT
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
The epoch is now finished
If there were no errors, then we have the network that we want
It has been trainedWhile epoch produces an errorPresent network with next inputs (pattern) from epoch Err = T – OIf Err <> 0 then
Wj = Wj + LR * Ij * Err
End If
End While
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 )
Effect of the learning rate
Set too high It may ‘bounce around’ the correct
weights
AND0,0 1,0
0,1 1,1
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
The Neuron’s Activation Function
G51IAI – Introduction to AI
G51IAI – Introduction to AI
Expanding the Model of the Neuron: Outputs other than ‘1’
X1
X2
Y1
2
-5
θ = 5
X3 2
θ = 0
Y2
-2
1
θ = 9
Z
θ = 21
5
1
20
-10
-4
3
6
10
1
1
Output is 1 or 0It doesn’t matter about how far over the threshold we are
Example from last lecture
G51IAI – Introduction to AI
...
...
Left wheel speed
Right wheel speedThe speed of
the wheels is not just 0 or 1
G51IAI – Introduction to AI
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
Modelling a Neuron
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
j
jiji aWin ,
G51IAI – Introduction to AI
Activation Functions
Stept(x) = 1 if x >= t, else 0 Sign(x) = +1 if x >= 0, else –1 Sigmoid(x) = 1/(1+e-x)
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
Summary
Linear Separability Learning Algorithm Pseudocode Activation function (threshold, sigmoid,
etc)