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ERROR BACKPROPAGATION ALGORITHM
Why Error Back Propagation Algorithm is required?
Lack of suitable training methods for multilayer perceptrons (MLP)s led to a waning of
interest in NN in 1960s and 1970s. This was changed by the reformulation of the
backPropagation training method for MLPs in the mid-1980s by Rumelhart et al.
Backpropagation was created by generalizing the Widrow-Hoff learning rule to multiplelayer networks and nonlinear differentiable transfer functions. Standard
backpropagation is a gradient descent algorithm, as is the Widrow-Hoff learning rule, in
which the network weights are moved along the negative of the gradient of the
performance function. The term backpropagation refers to the manner in which the
gradient is computed for nonlinear multilayer networks.
Before delta learning rule, an introduction to single-layer continuous perceptron
networks for linearly separable classifications.
Single-layer continuous perceptron networks
TLU in perceptron is replaced by continuous activation function. This helps in gaining a
finer control over the training procedure. It also helps in working with differentiable
characteristics of the threshold element.
The basic procedure of descent is quite simple. Starting from an arbitrary chosen weight
vector w, the gradient of the current error function is computed. The next value of w is
obtained by moving in the direction of the negative gradient along the multidimensional
error surface. The direction of negative gradient is called steepest descent.
The error minimization algorithm requires computation of the gradient of the error.
This is given by
Delta learning rule for multiperceptron layer
As in simple cases of the delta learning rule training studied before, input patterns are
submitted during the back-propagation training sequentially. If a pattern is submitted
and its classification or association is determined to be erroneous, the synaptic weights
as well as the thresholds are adjusted so that the current least mean square classification
error is reduced. The input / output mapping, comparison of target and actual values,
and adjustment, if needed, continue until all mapping examples from the training set are
learned within an acceptable overall error. Usually, mapping error is cumulative and
computed over the full training set.
During the association or classification phase, the trained neural network itself operates
in a feedforward manner. However, the weight adjustments enforced by the learning
rules propagate exactly backward from the output layer through the so-called "hidden
layers" toward the input layer.
The input and output values of the network are denoted yj and ok, respectively. We thus,
denote yj, for j = 1, 2, . . . , J, and ok, for k = 1, 2, . . . , K, as signal values at the j'th
column of nodes, and k'th column of nodes, respectively. As before, the weight wkj
connects the output of the j'th neuron with the input to the k'th neuron.
The activation function netk of layer k is expressed as
Eqn. 1
The error expression generalized to include all squared errors at the outputs k=1,2,3…K
Eqn:2
Where p is a specific pattern and p=1 2……P
Delta learning rule can be formally derived for a multiperceptron layer. Assumptions
made are
1. gradient descent search is performed to reduce the error Ep through adjustments
of weights
2. threshold values are adjustable with other weights and no distinction is made
between threshold and weights during learning
3. Fixed input of value
during both the training and recall phases
Minimization of error requires the weight changes to be in the negative gradient
direction. Individual weight adjustments are computed as follows
Eqn:3
Error E is defined in Eqn:2.
Now for each node in layer k where k=1,2,….K
Eqn:4
And the corresponding neuron output is given by
Eqn:5
Eqn:6
Eqn:7
Since
Eqn:8
Substituting Eqn 8, Eqn 6 in Eqn 7 we get
Eqn:9
The weight adjustment formula of Eqn 3 can accordingly be rewritten as
Eqn: 10
Eqn 10 represents the general formula for delta training/learning weight adjustments for
a single-layer network. It also follows that the adjustments of weight wkj is proportional
to the input activation yj, and to the error signal value at the kth neuron’s output.
The delta value needs to be explicitly computed for specifically chosen activation
functions.
Eqn: 11
Thus we have from equation 6
Eqn: 12
Denoting the second term in the above equation as a derivative of activation function
Eqn: 13
And
Eqn: 14
And rewriting eqn 12 we have
Eqn: 15
Eqn 15 shows that the error signal term depicts the local error (dk-ok) at the output of the
k’th neuron scaled by the multiplicative factor f’k(netk).
The final formula for the weight adjustment of the single-layer network can be obtained
from Eqn 10 as
Eqn: 16
Eqn 16 is identical to the delta training rule. The updated weight values become
Eqn: 17
Delta Training rules for unipolar continuous activation function:
Eqn: 18
Eqn: 19
or
Eqn: 20
Therefore the delta value for unipolar activation function becomes
Eqn: 21
Delta Training rules for bipolar continuous activation function:
The activation function in the case of bipolar continuous activation function is given by
We obtain
An useful identity can be applied here
Verification of identity
Letting o=f(net)
LHS=RHS
The delta value for a bipolar continuous activation function is given by
Summarzing the updated weights are given by
The updated weights under the delta training rule for the single-layer network can be
expressed using the vector notation
where the error signal δo is defined as a column vector consisting of the individual error
signal terms
Generalized Delta Learning Rule
The negative gradient neurons for the hidden neurons is given by
There are two modes of updation of weights
1. Batch mode
2. Incremental mode
When the weights are being changed immediately after a training pattern is presented
then it is called as incremental approach.
When the weights are changed only after all the training patterns are presented then it is
called as batch mode. This mode requires additional local storage for each connection to
maintain the immediate weight changes.
The BP learning algorithm is an example of optimization problem. [Note:- an
optimization problem is the problem of finding the best solution from all feasible
solutions]. The essence of the error back-propagation algorithm is the evaluation of the
contribution of each particular weight to the output error. There are many difficulties
that arise in the implementation of the algorithm. One of the problems is that the error
minimization procedure may produce only a local minimum of the error function.
The learning is successful if it is well below the acceptable Erms value. Erms (Root Mean
Square Normalized Error) and is given by the following formula
Where P=number of training patterns K=number of outputs
But there are 2 such troughs in wl1 and wl2. So if the learning commences at point 2 we
may end up in a local minima instead of a global minima wg. Thus the trained network
will be unable to produce the desired performance in terms of its acceptable terminal
error. To ensure convergence to a satisfactory minimum the starting point should be
changed to 1.
The problem of local minima can however be avoided by inserting some form of
randomness to the training.
The convergence of EBPTA depends on various factors. To name a few we have
1. learning rate
2. Selection of initial weights
3. Momentum
4. Number of training data
5. Number of hidden layer nodes
Selection of Initial weights
The weights of the network to be trained are typically initialized at small random values.
The initialization strongly affects the ultimate solution.
 If all weights start out with equal weight values, and if the solution requires that
unequal weights be developed the network may not train properly.
 Weights can’t be very high because the sigmoidal activation function used may
get saturated from the beginning itself and the system may be stuck at a local
minima or at a very flat plateau at the starting point itself
 One method of choosing the weight wij is choosing it in the range of
3 
 3
 oi
oi  where oi is the number of processing elements j that feed-forward to



processing element i.
Steepness of activation function
λ is the steepness factor in the activation function. It was assumed to be 1 in the
computation of f’(net). f’(net) serves as a multiplying factor in the computation of
error signals. Thus the choice and shape of the activation function would strongly
affect the speed of network learning.
The derivation of activation function can be computed as follows
and it reaches a maximum of 1/2 λ when net=0.
Since the weights are adjusted in proportion to the f’(net), the weights that are
connected to the midrange are changed the most. Since the error signals are
computed with f’(net) as multiplier, the back propagated errors are large for only
those neurons which are in the steep thresholding mode.
The other feature which is apparent from the graph is that for fixed learning constant
all adjustments in weight are in proportion to steepness coefficient. This observation
leads to a conclusion that using activation functions with larger values of λ may yield
results with larger learning constant. So it is advisable to keep λ fixed at 1 and
control only the learning constant, rather than controlling both.
Effect of learning rate
Affects the convergence of BPA. A larger value of α speeds up the convergence but
might result in overshooting, while a smaller value of α results in overshooting and
vice versa. The learning constants should be chosen experimentally for each problem.
The range of learning constants are from 10-3to 10 have been reported throughout the
technical literature as successful for many computational back-propagation
experiments.
Based on the above observations some heuristics for improving the rate of
convergence are proposed.
Momentum Method
This method is used for accelerating the convergence of EBPTA. This method
involves supplementing the current weight adjustments with a fraction of most recent
weight adjustments. This is usually done according to the formula
where t and t-1 represents the current and most recent training step respectively and
a is user-selected positive momentum constant. This second term is called as
momentum term. For N steps using momentum method, the current weight is
expressed as
Typically a is choosen between 0.1 and 0.8.
What is the significance of this momentum term?
From the above figure it is seen that in the case of A’and A”the signs are same. So
combining the gradient component of adjacent step would result in convergence
speed-up. But in the case of B’ and B” the signs are different. This shows that if the
gradient component changes sign in two consecutive iterations, the learning rate
along this axis should be decreased.
This indicates that the momentum term typically helps to speed up convergence and
to achieve an efficient and more reliable learning profile.
Momentum term technique can be recommended for problems where convergence
occur too slowly or for cases when learning is difficult to achieve.
Network architecture versus data representation
Starting from a simple case of single hidden layer the number of input nodes are
determined by the dimension, size of the input vector to be classified, generalized or
associated with a certain output quantity.
The input vector size corresponds to the number of inputs to be classified, generalized
or associated with a certain output quantity.
In planar images, size of input vector is sometimes made equal to the total number of
pixels in the evaluated images.
The conditions for selecting the number of output neurons depends on the type of
neural processing. In the case of auto-associator which associates the distorted input
vector with undistorted class prototype then we have I=K.
In the case of classifier the number of output neurons are equal to the number of
classes.
Necessary number of Hidden neurons
The number of Hidden neurons depends on the dimension n of the input vector and on
the number of separable regions in n-dimensional input space.
Questions
1. Explain Error back propagation training algorithm with the help of a flowchart(3)
2. Prove the following identities
For a unipolar continuous activation function f(net)=o(1-o)
For a bipolar continuous activation function f(net)=1/2(1-o2)
3. Explain EBPTA Training algorithm;
4. Explain Single continuous perceptron training algorithm
5. Multicategory perceptron training algorithm