Transcript Artificial Neural Networks - University of Northampton

Multilayer Perceptrons 1
ARTIFICIAL INTELLIGENCE
TECHNIQUES
Overview
 Recap of neural network theory
 The multi-layered perceptron
 Back-propagation
 Introduction to training
 Uses
Recap
Linear separability
 When a neuron learns it is positioning a line so
that all points on or above the line give an output
of 1 and all points below the line give an output of
0
 When there are more than 2 inputs, the pattern
space is multi-dimensional, and is divided by a
multi-dimensional surface (or hyperplane) rather
than a line
Pattern space - linearly
separable
X2
X1
Non-linearly separable
problems
 If a problem is not linearly separable, then it is
impossible to divide the pattern space into
two regions
 A network of neurons is needed
Pattern space - non linearly
separable
X2
Decision surface
X1
The multi-layered perceptron
(MLP)
The multi-layered perceptron
(MLP)
Input layer
Hidden layer
Output layer
Complex decision surface
 The MLP has the ability to emulate any
function using one hidden layer with a
sigmoid function, and a linear output layer
 A 3-layered network can therefore produce
any complex decision surface
 However, the number of neurons in the
hidden layer cannot be calculated
Network architecture
 All neurons in one layer are connected to all
neurons in the next layer
 The network is a feedforward network, so all
data flows from the input to the output
 The architecture of the network shown is
described as 3:4:2
 All neurons in the hidden and output layers have
a bias connection
Input layer
 Receives all of the inputs
 Number of neurons equals the number of
inputs
 Does no processing
 Connects to all the neurons in the hidden
layer
Hidden layer
 Could be more than one layer, but theory says
that only one layer is necessary
 The number of neurons is found by experiment
 Processes the inputs
 Connects to all neurons in the output layer
 The output is a sigmoid function
Output layer
 Produces the final outputs
 Processes the outputs from the hidden layer
 The number of neurons equals the number of
outputs
 The output could be linear or sigmoid
Problems with networks
 Originally the neurons had a hard-limiter on
the output
 Although an error could be found between
the desired output and the actual output,
which could be used to adjust the weights in
the output layer, there was no way of
knowing how to adjust the weights in the
hidden layer
The invention of backpropagation
 By introducing a smoothly changing output
function, it was possible to calculate an error
that could be used to adjust the weights in
the hidden layer(s)
Output function
The sigmoid function
1.2
1
0.6
0.4
0.2
net
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
-0
-0.5
-1
-1.5
-2
-2.5
-3
-3.5
-4
-4.5
0
-5
y
0.8
Sigmoid function
 The sigmoid function goes smoothly from 0
to 1 as net increases
 The value of y when net=0 is 0.5
 When net is negative, y is between 0 and 0.5
 When net is positive, y is between 0.5 and 1.0
Back-propagation
 The method of training is called the back-
propagation of errors
 The algorithm is an extension of the delta
rule, called the generalised delta rule
Generalised delta rule
 The equation for the generalised delta rule is
ΔWi = ηXiδ
 δ is the defined according to which layer is
being considered.
 For the output layer, δ is y(1-y)(d-y).
 For the hidden layer δ is a more complex.
Training a network
 Example: The problem could not be
implemented on a single layer - nonlinearly
separable
 A 3 layer MLP was tried with 2 neurons in the
hidden layer - which trained
 With 1 neuron in the hidden layer it failed to
train
The hidden neurons
6
5
4
S e rie s 1
3
S e rie s 2
2
1
0
0
1
2
3
4
5
6
The weights
 The weights for the 2 neurons in the hidden
layer are -9, 3.6 and 0.1 and 6.1, 2.2 and -7.8
 These weights can be shown in the pattern
space as two lines
 The lines divide the space into 4 regions
Training and Testing
 Starting with a data set, the first step is to
divide the data into a training set and a test
set
 Use the training set to adjust the weights
until the error is acceptably low
 Test the network using the test set, and see
how many it gets right
A better approach
 Critics of this standard approach have
pointed out that training to a low error can
sometimes cause “overfitting”, where the
network performs well on the training data
but poorly on the test data
 The alternative is to divide the data into three
sets, the extra one being the validation set
Validation set
 During training, the training data is used to
adjust the weights
 At each iteration, the validation/test data is
also passed through the network and the
error recorded but the weights are not
adjusted
 The training stops when the error for the
validation/test set starts to increase
Stopping criteria
error
Stop here
Validation set
Training set
time
The multi-layered perceptron
(MLP) and Backpropogation
Architecture
Input layer
Hidden layer
Output layer
Back-propagation
 The method of training is called the back-
propagation of errors
 The algorithm is an extension of the delta
rule, called the generalised delta rule
Generalised delta rule
 The equation for the generalised delta rule is
ΔWi = ηXiδ
 δ is the defined according to which layer is
being considered.
 For the output layer, δ is y(1-y)(d-y).
 For the hidden layer δ is a more complex.
Hidden Layer
 We have to deal with the error from the output
layer being feedback backwards to the hidden
layer.
 Lets look at example the weight w2(1,2)
 Which is the weight connecting neuron 1 in the
input layer with neuron 2 in the hidden layer.
 Δw2(1,2)=ηX1(1)δ2(2)
 Where
 X1(1) is the output of the neuron 1 in the hidden
layer.
 δ2(2) is the error on the output of neuron 2 in the
hidden layer.
 δ2(2)=X2(2)[1-X2(2)]w3(2,1) δ3(1)
 δ3(1)
= y(1-y)(d-y)
=x3(1)[1-x3(1)][d-x3(1)]
 So we start with the error at the output and
use this result to ripple backwards altering
the weights.
Example
 Exclusive OR using the network shown
earlier: 2:2:1 network
 Initial weights
 W2(0,1)=0.862518, W2(1,1)=-0.155797, W2(2,1)=0.282885
 W2(0,2)=0.834986, w2(1,2)=-0.505997, w2(2,2)=-0.864449
 W3(0,1)=0.036498, w3(1,1)=-0.430437, w3(2,1)=0.48121
Feedforward – hidden layer
(neuron 1)
 So if
 X1(0)=1 (the bias)
 X1(1)=0
 X1(2)=0
 The output of weighted sum inside neuron 1
in the hidden layer=0.862518
 Then using sigmoid function
 X2(1)=0.7031864
Feedforward – hidden layer
(neuron 2)
 So if
 X1(0)=1 (the bias)
 X1(1)=0
 X1(2)=0
 The output of weighted sum inside neuron 2
in the hidden layer=0.834986
 Then using sigmoid function
 X2(2)=0.6974081
Feedforward – output layer
 So if
 X2(0)=1 (the bias)
 X2(1)=0.7031864
 X2(2)=0.6974081
 The output of weighted sum inside neuron 2 in
the hidden layer=0.0694203
 Then using sigmoid function
 X3(1)=0.5173481
 Desired output=0
 δ3(1)=x3(1)[1-x3(1)][d-x3(1)] =-0.1291812
 δ2(1)=X2(1)[1-X2(1)]w3(1,1) δ3(1)=0.0116054
 δ2(2)=X2(2)[1-X2(2)]w3(2,1) δ3(1)=-0.0131183
 Now we can use the delta rule to calculate the
change in the weights
 ΔWi = ηXiδ
Examples
 If we set η=0.5
 ΔW2(0,1) = ηX1(0)δ2(1)
=0.5 x 1 x 0.0116054
=0.0058027
 ΔW3(2,1) = ηX2(1)δ3(1)
=0.5 x 0.7031864 x –0.1291812
=-0.04545192
 What would be the results of the following?
 ΔW2(2,1) = ηX1(2)δ2(1)
 ΔW2(2,2) = ηX1(2)δ2(2)
 ΔW2(2,1) = ηX1(2)δ2(1)
=0.5x0x0.0116054
=0
 ΔW2(2,2) = ηX1(2)δ2(2)
=0.5 x 0 x –0.131183
=0
 New weights
 W2(0,1)=0.868321
W2(1,1)=-0.155797
W2(2,1)=0.282885
 W2(0,2)=0.828427
w2(1,2)=-0.505997
0.864449
 W3(0,1)=0.028093
w3(1,1)=-0.475856
w3(2,1)=0.436164
w2(2,2)=-