Neural Networks: Multilayer Perceptron

CHAPTER 04
MULTILAYER PERCEPTRONS
CSC445: Neural Networks
Prof. Dr. Mostafa Gadal-Haqq M. Mostafa
Computer Science Department
Faculty of Computer & Information Sciences
AIN SHAMS UNIVERSITY
(most of figures in this presentation are copyrighted to Pearson Education, Inc.)

ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq
 Introduction
 Limitation of Rosenblatt’s Perceptron
 Batch Learning and On-line Learning
 The Back-propagation Algorithm
 Heuristics for Making the BP Alg. Perform Better
 Computer Experiment
2
Multilayer Perceptron

Introduction
 AND operation:
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Introduction
 OR Operation:
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Its easy to find a set of weight that satisfy the above inequalities.
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Introduction
 XOR Operation:
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Clearly the second and third inequalities are incompatible with the fourth, so
there is no solution for the XOR problem. We need more complex networks!
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The XOR Problem
 A two-layer Network to solve the XOR Problem
Figure 4.8 (a) Architectural graph of network for solving the XOR problem. (b)
Signal-flow graph of the network.
6
b
ww
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3
1
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b
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The XOR Problem
 A two-layer Network to solve the XOR Problem
Figure 4.9 (a) Decision boundary constructed by hidden neuron 1 of the network in
Fig. 4.8. (b) Decision boundary constructed by hidden neuron 2 of the network. (c)
Decision boundaries constructed by the complete network.
7

ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq 8
MLP: Some Preliminaries
 The multilayer perceptron (MLP) is
proposed to overcome the limitations of the
perceptron
 That is, building a network that can solve
nonlinear problems.
 The basic features of the multilayer perceptrons:
 Each neuron in the network includes a nonlinear activation
function that is differentiable.
 The network contains one or more layers that are hidden from
both the input and output nodes.
 The network exhibits a high degree of connectivity.

 Architecture of a multilayer perceptron
Figure 4.1 Architectural graph of a multilayer perceptron with two hidden layers.
9

 Weight Dimensions
10
If network has n units in layer i , m units in layer i +1 , then the weight
matrix Wij will be of dimension m x (n+1) .
Wij

 Number of neuron in the output layer
11
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ASU-CSC445: Neural Networks Prof. Dr. Mostafa Gadal-Haqq 12
 Training of the multilayer perceptron proceeds in
two phases:
 In the forward phase, the weights of the network are fixed and
the input signal is propagated through the network, layer by
layer, until it reaches the output.
 In the backward phase, the error signal, which is produced by
comparing the output of the network and the desired response,
is propagated through the network, again layer by layer, but in
the backward direction.

 Function Signal:
 is the input signal that comes in
at the input end of the network,
propagates forward (neuron
by neuron) through the network,
and emerges at the output of the
network as an output signal.
 Error Signal:
 originate at the output neuron of
the network and propagates
backward (layer by layer)
through the network.
 Each hidden or output
neuron computes these two
signals.
Figure 4.2 Illustration of the
directions of two basic signal flows
in a multilayer perceptron: forward
propagation of function signals
and back propagation of error
signals.
13

 Function of the Hidden neurons
 The hidden neurons play a critical role in the operation of a
multilayer perceptron; they act as feature detectors.
 The nonlinearity transform the input data into a feature
space in which data may be separated easily.
 Credit Assignment Problem
 Is the problem of assigning a credit or a blame for overall
outcomes to the internal decisions made by the computational
units of the distributed learning system.
 The error-correction learning algorithm is easy to use for
training single layer perceptrons. But its not easy to use it for a
multilayer perceptrons,
 the backpropagation algorithm solves this problem.
14

The Back-propagation Algorithm
 An on-line learning algorithm.
Figure 4.3 Signal-flow graph highlighting the
details of output neuron j.
15



m
i
ijij nynwnv
0
)()()(
))(()( nvny ij 
)()()( nyndne jjj 

 The weights are updated in a manner similar to the LMS and
the gradient descent method. That is, the instantaneous error
and the weight corrections are:
and
 Using the chain rule of calculus, we get:
 We have:
16
(n)e
2
1
(n) 2
jj E (n)w
(n)
ηw
ji
j
ji



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Δ
)(and))((1 ny
(n)w
(n)v
,nv
(n)v
(n)y
,
(n)y
(n)e
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(n)
i
ji
j
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ji
j
j
j
j
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j

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
E
(n)w
(n)v
(n)v
(n)y
(n)y
(n)e
(n)e
(n)
(n)w
(n)
ji
j
j
j
j
j
j
j
ji
j










 EE

 which yields:
 Then the weight correction is given by:
 where the local gradient j (n) is defined by:
17
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)())(( nynv(n)e
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ijjj
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E
)()()(Δ nynηnw ijji 
))(()()( nvnen jjjj  

 That is, the local gradient of neuron j is equal to the product
of the corresponding error signal of that neuron and the
derivative of the associated of the activation function. Then,
we have two distinct cases:
 Case 1: Neuron j is an output node:
 In this case, it is easy to use the credit assignment rule to compute
the error signal ej(n), because we have the desired signal visible to
the output neuron. That is, ej(n)=dj(n) - yj(n).
 Case 2: Neuron j is an hidden node:
 In this case, the desired signal is not visible to the hidden neuron.
Accordingly, the error signal for the hidden neuron would have to be
determined recursively and working backwards in terms of the
error signals of all the neurons to which that hidden neuron is
directly connected.
18

 Case 2: Neuron j is hidden node.
Figure 4.4 Signal-flow graph highlighting the details of output neuron k connected
to hidden neuron j.
19

 We redefine the local gradient for a hidden neuron j as:
 Where the total instantaneous error of the output neuron k:
 Differentiating w. r. t. yj (n) yields:
 But
 Hence
20
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)()(
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))(()()()( nvndnynd(n)e kkkkkk 
))((
)(
nv
nv
e
kk
k
k 



 Also, we have
 Differentiating, yields:
 Then, we get
 Finally, the backpropagation for the local gradient of (hidden)
neuron j, (neuron k is output neuron), is given by:
21

k
kjkjjj nwnnvn )()())(()( 
 


k
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k
kjkkk
j
wnwnvne
ny
n
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
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k 
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
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m
j
jkjk nynwnv
0
)()()(

Figure 4.5 Signal-flow graph of a part of the adjoint system pertaining to back-
propagation of error signals.
22

 We summarize the relations for the back-propagation algorithm:
 First: the correction wji(n) applied to the weight connecting
neuron i to neuron j is defined by the delta rule:
 Second: local gradient j (n) depends on neuron j :
 Neuron j is an output node:
 Neuron j is an hidden node (neuron k is output or hidden):
23









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
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 )(
jneuronof
signalinput
)(
gradient
local
parameter
ratelearning
)(
correction
weight
nynnw ijji 
)()()(;))(()()( nyndnenvnen jjjjjjj  

k
kjkjjj nwnnvn )()())(()( 

The Activation Function
 Differentiability is the only requirement that an activation
function has to satisfy in the BP Algoruthm.
 This is required to compute the  for each neuron.
 Sigmoidal functions are commonly used, since they satisfy
such a condition:
 Logistic Function
 Hyperbolic Tangent Function
24
0a,
)exp(1
1
)( 


av
v )](1)[(
)exp(1
)exp(
)(' vva
av
ava
v  
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
0ba,,)tanh()(  bvav )]()][([)(' vava
a
b
v  

The Rate of Learning
 A simple method of increasing the rate of learning
and avoiding instability (for large learning rate ) is
to modify the delta rule by including a momentum
term as:
Figure 4.6 Signal-flow graph
illustrating the effect of
momentum constant α, which lies
inside the feedback loop.
25
 where  is usually a positive
number called the momentum
constant.
 To ensure convergence, the
momentum constant must be
restricted to
)()()1()(Δ nynηnwnw ijjiji  
10  

Summary of the Back-propagation Algorithm
1. Initialization
2. Presentation of
training example
3. Forward
computation
4. Backward
computation
5. Iteration
Figure 4.7 Signal-flow graphical summary of back-propagation learning. Top part of
the graph: forward pass. Bottom part of the graph: backward pass.
26

Heuristics for making the BP Better
1. Stochastic vs. Batch update
 Stochastic (sequential) mode is computationally faster than the
batch mode.
2. Maximizing information content
 Use an example that results in large training error
 Use an example that is radically different from the others.
3. Activation function
 Use an odd function
 Hyperbolic not logistic function
27
)tanh()( bvav 

4. Target values
 Its very important to choose the
values of the desired response
to be within the range of the
sigmoid function.
5. Normalizing the input
 Each input should be
preprocessed so that its mean
value, averaged over the entire
training sample, is close to zero,
or else it will be small
compared to its standard
deviation.
28
Figure 4.11 Illustrating the operation of mean
removal, decorrelation, and covariance
equalization for a two-dimensional input space.

6. Initialization
 A good choice will be of tremendous help.
 Initialize the weights so that the standard deviation of the
induced local field v of a neuron lies in the transition area
between the linear and saturated parts or its sigmoid function.
7. Learning from hints
 Is achieved by allowing prior information that we may have
about the mapping function, e.g., symmetry, invariances, etc.
8. Learning rate
 All neurons in the multilayer should learn at the same rate,
except for that at the last layer, the learning rate should be
assigned smaller value than that of the front layers.
29

Batch Learning and On-line Learning
 Consider the training sample used to train the network in supervised
manner:
T = {x(n), d(n); n =1, 2, …, N}
 If yj(n) is the functional signal produced at the output neuron j. the
error signal produced at the same neuron is:
ej (n) = dj(n) – yj (n)
 the instantaneous error produced at the output neuron j is:
 the total instantaneous error of the whole network is:
 the total instantaneous error averaged over the training sample:
30



Cj
2
j
Cj
j (n)e
2
1
(n)(n) EE
 
 

N
1n Cj
2
j
N
1n
av (n)e
2N
1
(n)
N
1
(n) EE
(n)e
2
1
(n) 2
jj E

Batch Learning:
 Adjustment of the weights of the MLP is performed after the
presentation of all the N training examples T.
 this is called an epoch of training.
 Thus, weight adjustment is made on epoch-by-epoch basis.
 After each epoch, the examples in the training samples T are randomly
shuffled.
 Advantages:
 Accurate estimation of the gradient vector (the derivates of the cost
function Eav w.r.t. the weight vector w), which therefore guarantee the
convergence of the method of steepest descent to a local minimum.
 Parallelization of the learning process.
 Disadvantages: it is demanding in terms of storage requirements.
31

On-line Learning:
 Adjustment of the weights of the MLP are performed on an example-
by-example basis.
 The cost function to be minimized is therefore the total instantaneous
error E (n).
 An epoch of training is the presentation all the N samples to the
network. Also, in each epoch the examples are randomly shuffled.
 Advantages:
 Its stochastic learning nature, make it less likely to be trapped in
local minimum.
 it is much less demanding in terms of storage requirements.
 Disadvantages:
 We can not Parallelize the learning process.
32

 Batch learning vs. On-line Learning:
33
On-line LearningBatch learning
The learning process is performed
in stochastic manner.
The learning process is performed
by ensemble averaging, which in
statistical context my be viewed as
a form of statistical inference.
It is less likely to be trapper in a
local minimum.
Guarantee for convergence to local
minimum.
Can not be parallelizedCan be parallelized
Require much less storageRequire large storage
Well suited for pattern
classification problems.
Well suited for nonlinear
regression problems.

Generalization
 A network is said to generalize well when
the network input-output mapping is
correct (or nearly so) for the test data.
 If we viewed the learning process as “curve-
fitting”.
 When the network is trained with too many
sample, it may become overfitted, or
overtrained, which lead to wrong
generalization.
 Sufficient training-Sample Size
 Generalization is influenced by three factors:
 The size of the training sample
 The network architecture
 The physical complexity of the problem at hand
 In practice, good generalization is achieved if
we the training sample size, N, satisfies:
 W is number of free parameters in the
network, and  is the fraction of classification
error permitted on test data.
Figure 4.16 (a) Properly fitted nonlinear
mapping with good generalization. (b) Overfitted
nonlinear mapping with poor generalization.
34
)/( WON 

Cross-Validation Method
 Cross-Validation is a standard tool in statistics that
provide appealing guiding principle:
 First: the available data set is randomly partitioned into a
training set and a test set.
 Second: the training set is further partitioned into two disjoint
subsets:
 An estimation subset, used to select the model (estimate the
parameters).
 A validation subset, used to test or validate the model
 The training set is used to assess various models and choose the
“best” one.
 However, this best model may be overfitting the validation data.
 Then, to guard against this possibility, the generalization
performance is measured on the test set, which is different from
the validation subset.
35

 Early-stopping Method
 (Holdout method)
 The training is stopped
periodically, i.e., after so many
epochs, and the network is
assessed using the validation
subset.
 When the validation phase is
complete, the estimation
(training) is resumed for another
period, and the process is
repeated.
 The best model (parameters) is
that at the minimum validation
error.
Figure 4.17 Illustration of the early-
stopping rule based on cross-
validation.
36

 Variant of Cross-Validation
 (Multifold Method)
 Divide the data set of N samples
into K subsets, where K>1.
 The network is validated in each
trial using a different subset.
After training the network using
the other subsets.
 The performance of the model is
assessed by averaging the
squared error under validation
over all trials.
Figure 4.18 Illustration of the multifold
method of cross-validation. For a given trial,
the subset of data shaded in red is used to
validate the model trained on the remaining
data.
37

Computer Experiment
 d= -4
Figure 4.12 Results of the computer experiment on the back-propagation
algorithm applied to the MLP with distance d = –4. MSE stands for mean-square
error.
38

Computer Experiment
 d = -5
Figure 4.13 Results of the computer experiment on the back-propagation
algorithm applied to the MLP with distance d = –5.
39

Real Experiment
 Handwritten Digit Recognition*
*Courtesy of Yann LeCun.
40

•Problems:
•4.1, 4.3
•Computer Experiment
•4.15
Homework 4
41

Kernel Methods and
RBF Networks
Next Time
42

Neural Networks: Multilayer Perceptron

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