UNCOVERING COMMUNICATION DENSITY IN PSO USING...
Transcript of UNCOVERING COMMUNICATION DENSITY IN PSO USING...
UNCOVERING COMMUNICATION DENSITY IN PSO USING COMPLEX
NETWORK
Michal Pluhacek, Roman Senkerik, Adam Viktorin and Tomas Kadavy
Tomas Bata University in Zlin , Faculty of Applied Informatics
Nam T.G. Masaryka 5555, 760 01 Zlin, Czech Republic
{pluhacek, senkerik, aviktorin, kadavy}@fai.utb.cz
KEYWORDS
Swarm Intelligence, Particle Swarm Optimization,
Complex Network, Swarm communication.
ABSTRACT
In this study, we investigate the communication in
particle swarm optimization (PSO) by the means of
network visualization. We measure the communication
density of PSO optimizing four different benchmark
functions. It is presented that the communication density
varies over different fitness landscapes and in different
phases of the optimizing process. We analyze the results
in terms of use for future research.
INTRODUCTION
The Particle Swarm Optimization algorithm (PSO)
(Kennedy, Eberhart 1995, Shi, Eberhart, 1998, Kennedy
1997, Nickabadi et al., 2011) is known as one of the
leading metaheuristic optimizers. Heuristic methods are
widely used for solving industrial problems (Volná,
Kotyrba, 2016). In the past decades the inner dynamic of
the PSO algorithm has been studied in detail and many
modifications were proposed to tackle the known
weaknesses of the method (e.g. premature
convergence).
Recently the interconnection between metaheuristics
and complex networks (CNs) has been (Zelinka 2011a,
2011b, 2013, Senkerik et al., 2016) with interesting
results (Davendra, 2014a, 2014b).
We take inspiration in above mentioned examples of
interconnection of metaheuristics and CNs and use the
network-style visualization to uncover the density of
communication in the PSO. A network structure is
constructed from the inner communication of the swarm
and afterwards analyzed.
The rest of the paper is structured as follows: The PSO
is described in the next section, following is the
description of network construction process. The
experiment design is presented in the next section
followed by the results discussion. The paper concludes
with suggestion for future research.
PARTICLE SWARM OPTIMIZATION
The Particle Swarm Optimization algorithm (PSO) is an
evolutionary optimization algorithm based on the
natural behavior of birds. It was introduced by R.
Eberhart and J. Kennedy in 1995 (Kennedy, Eberhart
1995).
In the PSO algorithm the particles (representing
candidate solutions) fly in the multidimensional space
of possible solutions. The new position of the particle in
the next iteration is obtained as a sum of its actual
position and velocity. The velocity calculation follows
two natural tendencies of the particle: To move to the
best solution found so far by the particular particle
(personal best: pBest). And to move to the overall best
solution found in the swarm (global best: gBest).
In the original PSO the new position of particle is
altered by the velocity given by Eq. 1:
)(
)(
2
1
1
t
ijj
t
ijij
t
ij
t
ij
xgBestRandc
xpBestRandcvwv
(1)
Where: vi
t+1 - New velocity of the ith particle in iteration t+1.
w – Inertia weight value.
vit - Current velocity of the ith particle in iteration t.
c1, c2 - Priority factors.
pBesti – Local (personal) best solution found by the ith
particle.
gBest - Best solution found in a population.
xijt - Current position of the ith particle (component j of
the dimension D) in iteration t.
Rand1j, Rand2j – Pseudo random numbers, interval (0,
1).
The maximum velocity of particles in the PSO is typically limited to 0.2 times the range of the optimization problem and this pattern was followed in this study. The new position of a particle is then given by Eq. 2, where xi
t+1 is the new particle position:
11 t
i
t
i
t
i vxx(2)
Finally the linear decreasing inertia weight (Nickabadi
et al., 2011) is used in this study. Its purpose is to slow
the particles over time and improve the local search
capability in the later phase of the optimization. The
inertia weight has two control parameters wstart and wend.
A new w for each iteration is given by Eq. 3, where t
stands for current iteration number and n stands for the
total number of iterations.
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n
twwww endstartstart
(3)
NETWORK CONSTRUCTION
In this study we use the network structure as a tool to help use represent the communication in the swarm. The nodes in the network represent the particles in different time points (Particle ID with iteration code). This means that the theoretical maximal number of nodes in the network is the number of particles times the number of iterations. However a new node in the network is created only when a particle manages to find a new personal best solution (pBest). When a node is created, two links are also crated. First link is between the newly created node and previous node with the same particle ID (but different iteration code). This represents the information from pBest according to (1). Similarly the information from gBest represented by a link between the newly created node and a node that represents the last update of gBest.
THE EXPERIMENT
The following four well known test functions were used
in this study: Sphere function, Rosenbrock function,
Rastrigin function, Schwefel function with dimension
setting 10 and 100.
In the experiment the PSO was set in the following way:
Iterations: 1000;
Population size: 20;
c1,c2: 2;
wstart: 0.9;
wend: 0.4;
During the run of the algorithm the communication
network was constructed according to the rules
presented in the previous section.
Following is the visualization of the final networks. In the
network visualizations a color coding is used to differentiate
the phases of the run as percentage of the final number of cost
functions evaluations (CFE). (The first 20% of CFE are
represented by red color, magenta represents the 20-40% of
CFE, green is the 40-60% CFE., 60-80% CFE is represented
by yellow color and finally the 80-100% CFE is represented as
cyan).
The network visualizations for Sphere function are
presented in Fig. 1 (dim =10) and Fig. 2 (dim = 100)
alongside the gBest development in Fig. 3 and Fig. 4.
Similarly the network visualizations and gBest history
are presented in Fig. 5 – 8 for Rosenbrock function, Fig.
9 – 12 for Schwefel function in Fig 13 – 16 for Rastrigin
functions.
It is clear from the visualizations that the number of
newly created links in different phases of the algorithm
varies. The numerical representation of newly created
links is presented in Table 1 – 4.
Figure 1: Network visualization with highlighted phases - Sphere function; dim = 10
Figure 2: Network visualization with highlighted phases - Sphere function; dim = 100
Figure 3: gBest history - Sphere function; dim = 10
Figure 4: gBest history - Sphere function; dim = 100
Figure 5: Network visualization with highlighted phases - Rosenbrock function; dim = 10
Figure 6: Network visualization with highlighted phases - Rosenbrock function; dim = 100
Figure 7: gBest history - Rosebrock function; dim = 10
Figure 8: gBest history - Rosebrock function; dim = 100
Figure 9: Network visualization with highlighted phases - Schwefel function; dim = 10
Figure 10: Network visualization with highlighted phases - Schwefel function; dim = 100
Figure 11: gBest history - Sschwefel function; dim = 10
Figure 12: gBest history - Sschwefel function; dim = 100
Figure 13: Network visualization with highlighted phases – Rastrigin function; dim = 10
Figure 14: Network visualization with highlighted phases – Rastrigin function; dim = 100
Figure 15: gBest history - Rastrigin function; dim = 10
Figure 16: gBest history - Rastrigin function; dim = 100
Table 1: Newly created link overview – Sphere function
Newly created links by CFE %
dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
10 448 334 730 1520 2396
100 584 604 550 678 1280
Table 2: Newly created link overview – Rosenbrock
function
Newly created links by CFE %
dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
10 424 298 684 1030 1030
100 440 404 452 816 1156
Table 3: Newly created link overview – Schwefel
function
Newly created links by CFE %
dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
10 330 126 350 1408 1810
100 268 128 466 754 1378
Table 4: Newly created link overview – Rastrigin
function
Newly created links by CFE %
dim 0 - 20 20 - 40 40 - 60 60 - 80 80 - 100
10 248 294 942 1666 22
100 230 222 374 938 1472
RESULTS DISSCUSION
Firstly, according to the results presented in the
previous section it is clear that the number of newly
created nodes in different phases of the algorithm
varies. However when put into context with the history
of gBest, often the smallest part of newly created nodes
represents the most dramatic improvement of gBest
value and vice versa.
Secondly, the shape of the network and number of
newly created nodes seems to be affected by the fitness
landscapes in terms of modality and complexity.
Further, there seems to be a tendency for some particles
to improve after a very long time window without
improvement (possibly escaping local optima), this
trend can be observed namely in Figs. 5, 10 and 13.
In most cases the majority of the newly created nodes is
crated in the last phases of the optimization. This is
most likely due to the decreasing inertia weight and
small velocities of the particles.
CONCLUSION
In this study we have presented the possible use of
network visualization to highlight the trends in
communication density in the particle swarm
optimization. We have concluded that the
communication density varies significantly in different
phases of the optimization and also varies based on the
fitness landscape.
There are two main directions for our future research.
First is employment of more advanced network analysis
for classification of various fitness landscapes and
second is a feedback-loop style control of the swarm
based on the number of newly created links in a
specified time window.
ACKNOWLEDGEMENT
This work was supported by Grant Agency of the Czech
Republic – GACR P103/15/06700S, further by the
Ministry of Education, Youth and Sports of the Czech
Republic within the National Sustainability Programme
Project no. LO1303 (MSMT-7778/2014. Also by the
European Regional Development Fund under the
Project CEBIA-Tech no. CZ.1.05/2.1.00/03.0089 and
by Internal Grant Agency of Tomas Bata University
under the Projects no. IGA/CebiaTech/2017/004
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AUTHOR BIOGRAPHIES
MICHAL PLUHACEK was born in the Czech
Republic, and went to the Faculty of
Applied Informatics at Tomas Bata
University in Zlín, where he studied
Information Technologies and obtained
his MSc degree in 2011 and Ph.D. in 2016
with the dissertation topic: Modern
method of development and modifications of
evolutionary computational techniques. He now works
as a researcher at the same university. His email address
ROMAN SENKERIK was born in the Czech Republic,
and went to the Tomas Bata University in
Zlin, where he studied Technical
Cybernetics and obtained his MSc degree
in 2004, Ph.D. degree in Technical
Cybernetics in 2008 and Assoc. prof. in
2013 (Informatics). He is now an Assoc.
prof. at the same university (research and courses in:
Evolutionary Computation, Applied Informatics,
Cryptology, Artificial Intelligence, Mathematical
Informatics). His email address is: [email protected]
ADAM VIKTORIN was born in the Czech Republic,
and went to the Faculty of Applied
Informatics at Tomas Bata University in
Zlín, where he studied Computer and
Communication Systems and obtained his
MSc degree in 2015. He is studying his
Ph.D. at the same university and the field
of his studies are: Artificial intelligence, data mining
and evolutionary algorithms. His email address is:
TOMAS KADAVY was born in the Czech Republic,
and went to the Faculty of Applied
Informatics at Tomas Bata University in
Zlín, where he studied Information
Technologies and obtained his MSc degree
in 2016. He is studying his Ph.D. at the
same university and the fields of his
studies are: Artificial intelligence and evolutionary
algorithms. His email address is: [email protected]