=Paper= {{Paper |id=Vol-2589/Paper5 |storemode=property |title=Training Feedforward Neural Networks Using Hybrid Particle Swarm Optimization, Multi-Verse Optimization |pdfUrl=https://ceur-ws.org/Vol-2589/Paper5.pdf |volume=Vol-2589 |authors=Rabab Bousmaha,Reda Mohamed Hamou,Amine Abdelmalek |dblpUrl=https://dblp.org/rec/conf/citsc/BousmahaHA19 }} ==Training Feedforward Neural Networks Using Hybrid Particle Swarm Optimization, Multi-Verse Optimization== https://ceur-ws.org/Vol-2589/Paper5.pdf

Training feedforward neural networks using hybrid
particle swarm optimization, Multi-Verse
Optimization
*

1st Rabab Bousmaha 2nd Reda Mohamed Hamou 3rd Amine Abdelmalek
GeCoDe Laboratory GeCoDe Laboratory GeCoDe Laboratory
Department of Computer Science Department of Computer Science Department of Computer Science
University of Saida University of Saida University of Saida
Saida, Algeria Saida, Algeria Saida, Algeria
Rabab.bousmaha@gmail.com hamoureda@yahoo.fr amineabd1@yahoo.fr

Abstract—The learning process of artificial neural networks value of weights and biases and the tendency to be trapped
is an important and complex task in the supervised learning in local minima [1].To address these problems, stochastic
field. The main difficulty of training a neural network is the search methods, such as metaheuristics have been proposed as
process of fine-tuning the best set of control parameters in terms
of weight and bias. This paper presents a new training method alternative methods for training feedforward neural network.
based on hybrid particle swarm optimization with Multi-Verse Metaheuristics have many advantages: they apply to any type
Optimization (PMVO) to train the feedforward neural networks. of ANN with any activation function [2], are particularly useful
The hybrid algorithm is utilized to search better in solution for dealing with large complex problems that generate many
space which proves its efficiency in reducing the problems of local optima [3] [4]. Genetic algorithm (GA) and Particle
trapping in local minima. The performance of the proposed
approach was compared with five evolutionary techniques and Swarm Optimization (PSO) considered as the most well-
the standard momentum backpropagation and adaptive learning known nature inspired MLP trainers. Montana and Davis
rate. The comparison was benchmarked and evaluated using proposed one of the earliest works on training the feedforward
six bio-medical datasets. The results of the comparative study neural network (FFNN) with GA [22]. They showed that GA
show that PMVO outperformed other training methods in most outperform BP when solving real problems.Slowik and Bialko
datasets and can be an alternative to other training methods.
Index Terms—Particle swarm optimization, Multi-Verse Op- [23] employed Differential Evolution (DE) for training MLP
timization, Training feedforward neural networks, Real world and showed that it has promising performance compared to
Datasets BP and Levenberg-Marquardt methods.
Others metaheuristics algorithms have been applied in train-
I. I NTRODUCTION ing feedforward MLP, such as the modified BAT [5], Multi-
Artificial neural network (ANN) is one of the most impor- Verse Optimization MVO [6], Whale Optimization Algorithm
tant data mining techniques. It has been successfully applied (WOA) [7], Grey Wolf Optimizer (GWO) [8] [9], Biogeogra-
to many fields. The feedforward multilayer perceptron (MLP) phy Based on Optimizer (BBO) [10], Moth-Flame Optimiza-
is one of the best-known neural networks. The multilayer per- tion (MFO) [11] and Improved Monarch Butterfly Optimiza-
ceptron (MLP) consists of three layers composed of neurons tion (IMBO) [12]. Furthermore, several hybrid algorithms have
organized into input, output and hidden layers. The success been proposed to train a neural network. Tarkhaneh and Shen
of an MLP generally depends on the training process that is [13] suggested a hybrid approach to neural network training
determined by training algorithms. The objective of the train- by combining PSO, Mantegna Levy flight and neighbor search
ing algorithms is to find the best connection between weights (LPSONS). The comparison experiments showed that the
and biases that minimize the classification error. Training proposed algorithm can find optimal results. Khan et al [14]
algorithms can be classified into two classes: gradient-based introduced a new method based on two algorithms, accelerated
and stochastic search methods. Backpropagation (BP) and its particle swarm optimization (APSO) and cuckoo search (CS),
variants are gradient-based methods and considered as one of named HACPSO. The comparison results demonstrated that
the most popular techniques used to train the MLP neural the proposed algorithm outperforms other algorithms in term
network. Gradient-based methods have many drawbacks, such of accuracy, MSE and standard deviation. This paper presents
as the slow convergence, the high dependency on the initial a new training approach based on hybrid particle swarm opti-
mization (PSO) with Multi-Verse Optimization (MVO), called
PMVO, to train the feedforward neural network (FFNN). Six
max min
−w
datasets were solved by the proposed trainer. Moreover, the wmax −( (w maxiteration
)∗iteration
),C1 , C2 two positive constants
t+1
application of the trainer was investigated in bio-medical. The , R1 ,R2 are random numbers in the interval of [0, 1], Vi,j is
performance of PMVO was compared with five well-known the velocity of jth member of ith particle at iteration number
trainer metaheuristics algorithms in the literature: PSO [15], (t) and (t+1). The new position values X t+1 are obtained by
MFO [11], MVO [6], WOA [7], HACPSO [14]. adding the velocity updates determined by the formula given
in Equation (5).
II. ARTIFICIAL NEURAL NETWORKS (ANN S )
An artificial neural network (ANN) is a computational IV. M ULTI - VERSE OPTIMIZATION (MVO)
model based on the structure and functions of the biological Multiverse optimization proposed by Syed Ali mirjalili in
brain and nervous system. The feedforward neural network 2015 [17] As Inspired by the concepts of white holes, black
(FFNN) is one of the most popular types of artificial neural holes, and wormholes in the multi-verse theory and big bang
network [6].FFNN has three interconnected layers. The first theory. In this algorithm, the models of these three concepts are
layer consists of input neurons. These neurons send the data developed to perform exploration and exploitation and local
to the second layer, called the hidden layer, which sends the search. The fitness function for each search agent is indicated
output neurons to the third layer. In FFNN, the information by the inflation rate, and each object and each universe in the
travels in one direction, from the input layer to the output search agent represent a candidate solution and a variable in
layer. The node or the artificial neuron multiplies each of these the candidate solution.
inputs by weight, as shown in (1): In this algorithm, the larger universes tend to send objectives
n
X to smaller universes. A large universe is defined based on
Sj = wi,j Ii + βj (1) inflation rate in the multi-verse theory. The following rules
i=1 are applied to the universes of the MVO:
where, n is the total number of neuron inputs, W ij is the • If the inflation rate rate is higher, the probability of having
connection weight connecting Ij to neuron j and βj is a bias a white hole is higher.
weight [6]. Then, the node or the artificial neuron adds the • If the inflation rate rate is higher, the probability of having
multiplications and sends the sum to a transfer function, for black holes is lower.
example, Sigmoid function presented in (2): • Universes having higher inflation rate rate send the ob-

1 jects through white holes.
f (x) = (2) • Universes having lower inflation rate rate tend to receive
1 + e−x
more objects through black holes.
The output of the neuron j can be described as follows (3):
• The objects of all universes may be replaced by the
n
X objects of the universe with the greater inflation rate.
yj = fj ( wi,j Ii + βj ) (3) The mathematical model of this algorithm is as follows:
i=1

After building the neural network, the set of network weights

are adjusted to approximate the desired results. This process  xj + T DR + ((ubj − lbj ) ∗ r4 + lbj ),if r3 < 0.5
,if r2 < W EP
is carried out by applying a training algorithm to adapt the xji = xj − T DR + ((ubj − lbj ) ∗ r4 + lbj ),if r3 ≥ 0.5
j
weights until error criteria are met [6]. xi , if r2 ≥ W EP

(6)
III. PARTICLE S WARM O PTIMIZATION (PSO)
in 1995 Russell Eberhart and James Kennedy have invented
the particle swarm optimization which is a population-based Where xj indicates the jth variable in the bests universe,
stochastic optimization technique inspired by birds flocking lbi indicates the lower bound in jth variable, ubi shows the
around food sources. like each other evolutionary computa- upper bound in jth variable, r2 ,r3 , r4 are random numbers
tional algorithms. In PSO, each individual is a bird in the in the interval of [0, 1], T DP/W EP are coefficients, and xji
search space. We call it a particle. All of the particles have indicates the jth parameter in ith universe.
fitness values which are evaluated by the fitness function to
be optimized and flies in the space with a velocity which is V. H YBRID PSO-MVO:
dynamically adjusted according to its own flying experience Hybrid PSO-MVO is sequential combination of PSO and
[16]. MVO. The algorithm merges the best strength of both PSO
in exploitation and MVO in exploration towards the optimum
t+1 t solution when the universe value of MVO replace the Pbest
Vi,j = Vi,j W + C1 R1 (P bestt − X t ) + C2 R2 (Gbestt − X t ) value of PSO [20] [21]. In this paper we propose a novel
(4) training algorithm based on this algorithm for the first time in
X t+1 = X t + V t+1 − i = 1, 2...N P )And(j = 1, 2...N G) the following section.The equation can be written as follows:
(5)
t+1 t
where P bestt and Gbestt denote the best particle position Vi,j = Vi,j W + C1 R1 (U niversest − X t ) + C2 R2 (Gbestt − X t )
and best group position and w is inertia weight w = (7)
Step 1: Initialize the PSO values of PMVO was carried out with five approaches used to train
Step 2: Evaluate the fitness function of each particle feedforward neural network in the literature: PSO [15], MFO
Step 3: Determine Gbest from the Pbest value [11], MVO [6], WOA [7], HACPSO [14] In addition, the
Step 4: updated velocity and position values of each particule proposed algorithm was compared with standard momentum
Step 5: verify the solution whether it is feasible or not Back-Propagation and adaptive learning rate and (BP), which
Step 6: steps 2 to 5 were repeated until the maximum number are gradient-based algorithms.
of iterations was reached.
Step 9: Use the optimal solutions of PSO as boundary to VIII. E XPERIMENTAL SETUP
MVO algorithm The proposed trainer and other algorithms were imple-
Step 10: Initialize the MVO values mented with Python language and a personal computer with
Step 11: Evaluate the inflation rate of the universe (fitness Intel(R) Core(TM) CPU 1.60 GHz 2.30 GHz, 64 Bits Windows
function) 7 operating system and 4 GB (RAM).
Step 12: Update the position of the universes The metaheuristics are sensitive to the value of their pa-
Step 13: if the convergence criterion is reached; get the rameters, which requires a careful initialization. Therefore, the
results control parameters recommended in the literature were used
Step 14: if the convergence criterion is not reached; continue [15] [6] [7] and summarized in Table II. All datasets were
the process from step 11-14 divided into 66% for training and 34% for testing. Moreover,
all features were mapped to the interval [0, 1] to eliminate
the effect of features that have different scales. Min-max
VI. PMVO FOR TRAINING MLP
normalization is applied to perform a linear transformation
This section presents the proposed approach based on the on the original data, were v 0 is the normalized value of v in
PMVO to train the MLP network named PMVO.Two impor- the range [minA , maxA ] as given in (10).
tant points are taken into consideration: the fitness function
vi − minA
and the representation of the PMVO solutions. In this work, v0 = (10)
the PMVO algorithm was applied for training MLP network maxA − minA
with a single hidden layer and each PMVO solution (weights In the literature, there is no standard method for selecting the
and biases) was formed by three parts: the connection weights number of hidden neurons. In this work, the method proposed
between the input layer and the hidden layer, the weights in [18] [19] [6] was used; the number of neurons in the hidden
between the hidden layer and the output layer, and the bias layer equals to 2N + 1 , where N , is a number of features in
weights.The length of each solution vector is given by equation the dataset.
(8), where n is the number of input features and m is the
number of neurons in the hidden layer [6]. IX. R ESULTS
All algorithms were tested ten times for every dataset and
IndividualLength = (n × m) + (2 × m) + 1 (8) the population size and the maximum number of generations
were set to 50 and 200, respectively.
PMVO solutions are implemented as real number vectors Table IV shows the statistical results: average, best, worst
when each vector belongs to the interval [-1, 1]. The mean and standard deviation of classification accuracy. The results
square error (MSE) was used to measure the fitness value of of PMVO outperformed other approaches in breast cancer,
PMVO solutions.MSE was calculated based on the difference blood, liver, vertebral with an average accuracy of 0.962,
between the estimated and actual values of the neural network 0.766, 0.752, 0.839. In addition, PMVO was ranked second
using the training datasets, as shown in equation (9), where n in diabetes and Parkinson datasets with an average accuracy
is the number of samples in the training dataset y and ŷ are of 0.783, 0.842 respectively. Moreover, it can also be seen that
respectively the actual and predicted values: the PMVO has a smaller Std which indicates that PMVO is
n stable. Table V shows the average, best and worst MSE with
1X 2
M SE = (y − ŷ) (9) standard deviation, obtained for each algorithm. As a result,
n i=1 it can be noted that PMVO outperforms other techniques in
VII. EXPERIMENTS AND RESULTS four datasets: breast cancer, blood, liver and vertebral with an
average MSE of 0.032, 0.168, 0.176, 0.131, respectively. In
This section presents the evaluation of the proposed PMVO
addition, it can be also noticed that PMVO has small standard
for training MLP networks on six well-known datasets, which
deviation value for all datasets which proves the efficiency and
were selected from University Of California Irvine machine
robustness of this algorithm.
learning (UCI)1 and Kaggle2 dataset repositories. Table I
Figures 1 2 3 4 5 and 6 show the convergence curves of
shows the classification of these datasets in terms of features
all metaheuristics training algorithms based on the average
number, classes, training and testing samples. The comparison
values of MSE.The convergence curves show that PMVO has
1 http://archive.ics.uci.edu/ml/ the lowest value of MSE for four datasets: breast cancer,
2 https://www.kaggle.com/datasets blood, liver and vertebral. Moreover, PMVO has the fastest
convergence speed in liver, vertebral, blood and European
datasets. For diabetes dataset, PMVO provides a very close
performance compared to MVO algorithm. These results show
that PMVO has a faster convergence and a better optimization
than other metaheuristic algorithms.
TableIII shows the average ranks obtained by each optimiza-
tion technique in the Friedman test. The comparative shows
that the proposed algorithm outperforms other algorithms.

TABLE I
S UMMARY OF THE CLASSIFICATION DATASETS

DataSets Features Training samples Testing samples
Blood 4 493 255
Breast cancer 8 461 238
Diabetes 8 506 262
Vertebral 6 204 106
Liver 6 79 41
Parkinson 22 128 67 Fig. 1. MSE convergence curve of Breast cancer

TABLE II
I NITIAL PARAMETERS OF THE OPTIMIZATION ALGORITHMS

Parameter Definition Value
Acceleration constants [2.1,2.1]
PSO Intertia weights [0.9,0.6]
Number of particles 50
Minimum wormhole 0.2
MVO
Maximum wormhole 1
Population size 50
Numer of generations 200
Number of search agents 50
MFO b 1
t [-1,1]
Population size 200
Numer of generations 200
r [0,1]
WOA
Population size 50
B 0.5
HACPSO
Population size 50
Numer of generations 200 Fig. 2. MSE convergence curve of Blood

TABLE III
AVERAGE R ANKING OF THE TECHNIQUES (F RIEDMAN )

Algorithms Ranking
PMVO 1.33
PSO 3.25
MFO 4.08
MVO 3
WOA 4.92
HACPSO 4.91
BP 6.5

X. C ONCLUSION
In this paper, we have proposed a new training approach
based on Particle swarm optimization, Multi-Verse Optimiza-
tion to train the feedforward neural network. The training
method took into account the capabilities of the PMVO in
terms of high exploration and exploitation to locate the optimal Fig. 3. MSE convergence curve of Diabetes
values for weights and biases of FFNN. The approach is
TABLE IV
ACCURACY RESULTS

PMVO PSO MFO MVO WOA HACPSO BP
B.cancer Avg 0.962 0.959 0.958 0.958 0.956 0.959 0.744
STd 0.002 0.003 0.002 0.002 0.006 0.004 0.254
Best 0.964 0.960 0.963 0.960 0.963 0.965 0.945
Worst 0.957 0.953 0.954 0.954 0.947 0.952 0.680
Blood Avg 0.766 0.762 0.763 0.764 0.762 0.760 0.744
STd 0.003 0.002 0.005 0.009 0.004 0.003 0.254
Best 0.768 0.765 0.765 0.784 0.774 0.760 0.945
Worst 0.762 0.760 0.760 0.760 0.758 0.758 0.680
Diabetes Avg 0.783 0.780 0.724 0.792 0.720 0.768 0.619
STd 0.018 0.011 0.008 0.006 0.028 0.006 0.08
Best 0.792 0.796 0.735 0.802 0.750 0.774 0.690
Worst 0.732 0.758 0.709 0.782 0.657 0.764 0.601
Liver Avg 0.752 0.722 0.722 0.737 0.665 0.679 0.519
STd 0.003 0.017 0.010 0.006 0.030 0.020 0.055
Best 0.785 0.748 0.744 0.744 0.700 0.709 0.586
Worst 0.721 0.700 0.709 0.726 0.603 0.669 0.495
Vertebral Avg 0.839 0.836 0.836 0.836 0.774 0.717 0.651
STd 0.006 0.012 0.007 0.005 0.052 0.015 0.170
Best 0.845 0.848 0.843 0.838 0.862 0.833 0.866
Worst 0.828 0.808 0.82 0.828 0.676 0.794 0.627
Parkinson Avg 0.842 0.841 0.802 0.802 0.824 0.875 0.750
STd 0.029 0.040 0.022 0.046 0.016 0.013 0.199
Best 0.882 0.898 0.0.828 0.867 0.852 0.905 0.849
Worst 0.788 0.773 0.773 0.695 0.800 0.863 0.623

TABLE V
MSE R ESULTS

PMVO PSO MFO MVO WOA HACPSO BP
B.cancer Avg 0.032 0.032 0.033 0.032 0.047 0.040 0.049
Std 0.001 0.001 0.001 0.002 0.004 0.001 0.015
Best 0.030 0.030 0.310 0.030 0.043 0.038 0.030
Worst 0.0336 0.326 0.036 0.032 0.058 0.041 0.050
Blood Avg 0.168 0.178 0.176 0.170 0.180 0.169 0.174
Std 0.005 0.003 0.005 0.008 0.004 0.003 0.009
Best 0.160 0.182 0.174 0.155 0.174 0.162 0.172
Worst 0.174 0.182 0.177 0.181 0.187 0.175 0.175
Diabetes Avg 0.151 0.155 0.171 0.147 0.186 0.163 0.179
Std 0.002 0.004 0.005 0.001 0.013 0.002 0.066
Best 0.1491 0.151 0.165 0.145 0.168 0.160 0.168
Worst 0.153 0.157 0.175 0.149 0.213 0.166 0.180
Liver Avg 0.176 0.191 0.193 0.186 0.220 0.212 0.210
Std 0.004 0.003 0.002 0.004 0.007 0.002 0.003
Best 0.170 0.185 0.189 0.179 0.210 0.208 0.190
Worst 0.180 0.195 0.194 0.194 0.233 0.215 0.220
Vertebral Avg 0.131 0.134 0.136 0.133 0.163 0.146 0.168
Std 0.006 0.002 0.002 0.002 0.018 0.002 0.015
Best 0.120 0.132 0.135 0.131 0.137 0.142 0.160
Worst 0.138 0.135 0.137 0.134 0.202 0.147 0.172
Parkinson Avg 0.134 0.141 0.158 0.147 0.165 0.119 0.158
STd 0.014 0.023 0.019 0.020 0.030 0.005 0.018
Best 0.119 0.099 0.135 0.125 0.127 0.112 0.137
Worst 0.157 0.182 0.203 0.197 0.228 0.130 0.203
proposed to minimize the training error and to increase the
accuracy. The approach is benchmarked and evaluated using
six standard bio-medical datasets.
The comparison between the proposed algorithm and PSO,
MFO, MVO, WOA, HACPSO and standard BP with momen-
tum term and adaptive learning rate shows the superiority of
the PMVO algorithm with high accuracy and small MSE in
most of the datasets compared with other training algorithms.
Moreover, the small value of standard deviation shows that
our trainer is robust and stable. Finally, from the experiment,
we can conclude that PMVO can give good results and can
be an alternative to other training methods.
In future works, we focus on how to extend this work to
solve a more real world problem and we test the performance
of PMVO to train other types of neural networks.

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