Evolutionary Algorithm for Optimal Connection Weights in Artificial Neural Networks

G.V.R. Sagar, Dr. S. Venkata Chalam & Manoj Kumar Singh
International Journal of Engineering (IJE), Volume (5) : Issue (5) : 2011 333
Evolutionary Algorithm for Optimal Connection Weights
in Artificial Neural Networks
G.V.R. Sagar nusagar@gmail.com
Assoc. Professor,
G.P.R. Engg. College.
Kurnool AP, 518007, India.
Dr. S. Venkata Chalam sv_chalam2003@yahoo.com
Professor,
CVR Engg. College,
Hyderabad AP, India
Manoj Kumar Singh mksingh@manuroresearch.com
Director,
Manuro Tech. Research,
Bangalore, 560097, India
Abstract
A neural network may be considered as an adaptive system that progressively self-organizes in
order to approximate the solution, making the problem solver free from the need to accurately
and unambiguously specify the steps towards the solution. Moreover, Evolutionary computation
can be integrated with artificial Neural Network to increase the performance at various levels; in
result such neural network is called Evolutionary ANN. In this paper very important issue of neural
network namely adjustment of connection weights for learning presented by Genetic algorithm
over feed forward architecture. To see the performance of developed solution comparison has
given with respect to well established method of learning called gradient decent method. A
benchmark problem of classification, XOR, has taken to justify the experiment. Presented method
is not only having very probability to achieve the global minima but also having very fast
convergence.
Keywords: Artificial Neural Network, Evolutionary Algorithm, Gradient Decent Algorithm, Mean
Square Error.
1. INTRODUCTION
An Artificial Neural Network (ANN) is an information-processing paradigm that is inspired by the
way biological nervous systems, such as the brain, process information. The key element of this
paradigm is the novel structure of the information processing system. It is composed of a large
number of highly interconnected processing elements (neurons) working in unison to solve
specific problems. ANNs, like people, learn by example. An ANN is configured for a specific
application, such as pattern recognition or data classification, through a learning process.
Learning in biological systems involves adjustments to the synaptic connections that exist
between the neurons. This is true of ANNs as well. Neural networks, with their remarkable ability
to derive meaning from complicated or imprecise data, can be used to extract patterns and detect
trends that are too complex to be noticed by either humans or other computer techniques. A
trained neural network can be thought of as an "expert" in the category of information it has been
given to analyze. This expert can then be used to provide projections given new situations of
interest and answer "what if “questions. ANN can be viewed as weighted directed graphs in
which artificial neurons are nodes and directed edges (with weights) are connections between
neurons outputs and neuron inputs. Based on the connection pattern (architecture), ANN can be
grouped into two categories: (a) Feed Forward Networks allow signals to travel one-way only,

from input to output. There is no feedback (loops) i.e. the output of any layer does not affect that
same layer as shown in Fig.(1) (b)Recurrent Networks can have signals traveling in both
directions by introducing loops in the network
FIGURE 1: Feedforward architecture
Learning in artificial neural systems may be thought of as a special case of machine learning.
Learning involves changes to the content and organization of a system’s knowledge, enabling it
to improve it’s performance on a particular task or set of tasks. The key feature of neural
networks is that they learn the input/output relationship through training. There are two types of
training/learning used in neural networks, with different types of networks using different types of
training. These are Supervised and Unsupervised training, of which supervised is the most
common for feed forward architecture training modes. Supervised Learning which incorporates an
external teacher, so that each output unit is told what its desired response to input signals ought
to be. During the learning process global information may be required. Paradigms of supervised
learning include error-correction learning. An important issue concerning supervised learning is
the problem of error convergence, i.e. the minimization of error between the desired and
computed unit values. The aim is to determine a set of weights, which minimizes the error. One
well-known method, which is common to many learning paradigms, is the gradient decent based
learning.
The idea behind learning in Neural Network is that, the output depends only in the activation,
which in turn depends on the values of the inputs and their respective weights. The initial weights
are not trained with respect to the inputs, which can result in error. Now, the goal of the training
process is to obtain a desired output when certain inputs are given. Since the error is the
difference between the actual and the desired output, the error depends on the weights, and we
need to adjust the weights in order to minimize the error.
2. GRADIENT DESCENT LEARNING
Gradient descent is a first order optimization algorithm. To find a local minimum of a function
using gradient descent, one takes steps proportional to the negative of the gradient (or of the
approximate gradient) of the function at the current point. If instead one takes steps proportional
to the positive of the gradient, one approaches a local maximum of that function; the procedure is
then known as gradient ascent. Gradient descent is also known as steepest descent, or the
method of steepest descent. A gradient descent based optimization algorithm such as back-
propagation (BP) [6] can then be used to adjust connection weights in the ANN iteratively in order

to minimize the error. The Gradient descent back-propagation algorithm [7] is a gradient descent
method minimizing the mean square error between the actual and target output of multilayer
perceptrons. The Back-propagation [8], [9] networks tend to be slower to train than other types of
networks and sometimes require thousands of epochs. When a reduced number of neurons are
used the Error Back-propagation algorithm cannot converge to the required training error. The
most common mistake is in order to speed up the training process and to reduce the training
errors, the neural networks with larger number of neurons than required. Such networks would
perform very poorly for new patterns nor used for training[10]. Gradient descent is relatively slow
close to the minimum. BP has drawbacks due to its use of gradient descent [11, [12]. It often gets
trapped in a local minimum of the error function and is incapable of finding a global minimum if
the error function is multimodal and/or non differentiable. A detailed review of BP and other
learning algorithms can be found in [13], [14], and [15].
3. EVOLUTIONARY ARTIFICIAL NEURAL NETWORK
Evolutionary artificial neural networks (EANN’s) refer to a special class of artificial neural
networks (ANN’s) in which evolution is another fundamental form of adaptation in addition to
learning [2] – [5]. Evolutionary algorithms (EA’s) are used to perform various tasks, such as
connection weight training, architecture design, learning rule adaptation, input feature selection,
connection weight initialization, rule extraction from ANN’s, etc. One distinct feature of EANN’s is
their adaptability to a dynamic environment. The two forms of adaptation, i.e., evolution and
learning in EANN’s, make their adaptation to a dynamic environment much more effective and
efficient. Evolution has been introduced into ANN’s at roughly three different levels: connection
weights, architectures, and learning rules. The evolution of connection weights introduces an
adaptive and global approach to training, especially in the reinforcement learning and recurrent
network learning paradigm where gradient-based training algorithms often experience great
difficulties. The evolution of architectures enables ANN’s to adapt their topologies to different
tasks without human intervention and thus provides an approach to automatic ANN design as
both ANN connection weights and structures can be evolved.
4. EVOLUTION OF CONNECTION WEIGHTS
Weight training in ANN’s is usually formulated as minimization of an error function, such as the
mean square error between target and actual outputs averaged over all examples, by iteratively
adjusting connection weights. Most training algorithms, such as BP. and conjugate gradient
algorithms are based on gradient descent. There have been some successful applications of BP
in various areas .One way to overcome gradient-descent-based training algorithms’ shortcomings
is to adopt EANN’s, i.e., to formulate the training process as the evolution of connection weights
in the environment determined by the architecture and the learning task. EA’s can then be used
effectively in the evolution to find a near-optimal set of connection weights globally without
computing gradient information. The fitness of an ANN can be defined according to different
needs. Two important factors which often appear in the fitness (or error) function are the error
between target and actual outputs and the complexity of the ANN. Unlike the case in gradient-
descent-based training algorithms, the fitness (or error) function does not have to be differentiable
or even continuous since EA’s do not depend on gradient information. Because EA’s can treat
large, complex, nondifferentiable, and multimodal spaces, which are the typical case in the real
world, considerable research and application has been conducted on the evolution of connection
weights .The aim is to find a near-optimal set of connection weights globally for an ANN with a
fixed architecture using EA’s. Comparisons between the evolutionary approach and conventional
training algorithms, such as BP, will be made over XOR classification problem.
4.1 Evolution of Connection Weights Using GA
% initialization of population
1. sz = total weights in architecture;
2. For i = 1: popsize;
3. pop(i)=sz number of random number;

4. End
% offspring population creation
5. For j=1: popsize/2;
6. pickup two parents randomly through uniform distribution;
7. cp=cross over position defined by randomly pickup any active node position;
8. To create offspring, exchange all incoming weights to selected nodes cp between parents;
9. For each offspring;
10., place of mutation, mp = randomly selected active node;
11. For all incoming weights w to selected node mp;
12. w=w+N (0, 1);
13. End
14. End
15. End
16. Offspring population, off_pop available;
17. npop= [pop; off_pop];
% Define fitness of each solution,
18. For i=1:2*popsize;
19. wt=npop(i);
20. apply wt to ANN architecture to get error value;
21. define fitness as fit(i)=1/error;
22. End
% Tournament selection
23. For r =1:2*popsize;
24. pick P number of Challengers randomly, where P = 10% of popsize;
25. arrange the tournament w.r.t fitness between rth solution and selected P challengers.;
26. define score of tournament for rth solution
27. End
28. Arrange score of all solution in ascending order;
29. sp=pick up the best half score position ;
30. select next generation solution as solution corresponding to position sp;
31. repeat the process from step 5 until terminating criteria does not satisfy
32. final solution=solution with maximum fitness in last generation.
5. EXPERIMENTAL SETUP
A fully interconnected feed forward architecture of size [2-2-1] / [2 3 1] designed .transfer function
in the active node is taken as unimodel sigmoid function. Initial random weights are upgraded by
gradient decent and genetic algorithm respectively. Various learning rates have applied to
capture performance possibilities from gradient decent. To increase the learning and efficiency
‘bias’ in architecture and ‘momentum’ in learning have also included when learning given by
gradient decent. Population size in GA taken as 20 and 10 independent trails have given to get
the generalize behavior. Condition of terminating criteria is taken as fixed iteration and it is equal
to 500 for GA. Because GA works with a population at time where as gradient decent takes only
one solution in each iteration hence to nullify the effect , more number of iterations have given to
gradient decent learning and it is taken as 20*500;
5.1 Performance Shown by Gradient Decent Learning
With the defined size of architecture, bias has applied with +1 input for hidden layer and output
layer. Various learning rate taken from 0.1 to 0.9 with the increment of 0.1 along with momentum
constant as 0.1.in the Fig(2) performance has shown for architecture size [2 2 1] where as in
Fig(3) with size [2 3 1].Mean square error obtained after 10,000 iterations has shown in Table 1
and in Table 2.

FIGURE 2: MSE performance with various learning rate .
FIGURE 3: MSE performance with various learning rate
TABLE 1: performance shown by gradient decent
Learning rate MSE([2 2 1]) MSE([2 3 1])
0.1 1.7352 e-001 3.2613 e-003
0.2 1.2920 e-001 6.3886 e-002
0.3 1.3042 e-001 4.4409 e-004
0.4 1.2546 e-001 3.7232 e-004
0.5 6.2927 e-001 2.6024 e-004
0.6 6.2825 e-001 2.3994 e-004
0.7 6.2915 e-001 2.0970 e-004
0.8 6.2868 e-001 1.3542 e-004
0.9 6.2822 e-001 1.2437 e-004

5.2 Performance Shown by GA Based Learning
FIGURE 4: MSE performance by GA for different trails in [2 2 1]
FIGURE 5: MSE performance by GA for different trails in [2 3 1]
Trail No. MSE([2 2 1]) MSE([2 3 1])
1 8.8709 e-055 3.6112 e-028
2 2.5941 e-022 4.6357 e-031
3 1.2500 e-001 7.5042 e-032
4 1.2500 e-001 4.2375 e-037
5 3.2335 e-044 6.0432 e-049
6 2.5765 e-041 1.1681 e-035
7 9.6010 e-022 9.3357 e-032
8 3.5481 e-047 4.4852 e-030
9 3.9527 e-050 1.1725 e-033
10 6.3708 e-023 2.5171 e-036
TABLE 2: performance shown by GA for different trails
Results shown in Table1 indicate the difficulties associated with gradient decent based learning
rule. Performance is very poor with the architecture size [2 2 1] for all learning rates. In fact
learning failed for this case. This is indication of stuckness in local minima. For architecture [2 3
2] there is an improvement in reduction of mean square error, and with higher value of learning

rate equal to 0.9, best performance has obtained. Convergence characteristics for both cases
have shown in Fig (1) and in Fig (2). convergence characteristics performance of developed form
of GA for weight adjustment shown in Fig(4) and in Fig(5).in both cases it is very clear that very
fast convergence with high reliability can be achieve by GA (except for trail number 3 and 4)as
shown in Table 2.
6. CONCLUSION
Determination of optimal weights in ANN in the phase of learning has obtained by using the
concept of genetic algorithm. Because of direct form realization in defining the solution of weights
there is no extra means required to represent the solution in population. Proposed method of
weights adjustment has compared with the gradient decent based learning and it has shown
proposed method outperform at every level for XOR classification problem. Even with lesser
number of hidden nodes where gradient decent method is completely fail for learning, proposed
method has shown very respectable performance in terms of convergence as well as accuracy
also. Defined solution of learning has generalized characteristics from application point of view
and having simplicity in implementation.
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