(Paper) Self adaptive island GA

Self Adaptive Island GA
Eiichi Takashima, Yoshihiro Murata, Naoki Shibata and Minoru Ito
Graduate School of Information Science
Nara Institute of Science and Technology
8916-5, Takayama, Ikoma, Nara 630-0192, Japan

e-mail: eiichi-t, yosihi-m, n-sibata, ito @is.aist-nara.ac.jp
¡

Abstract- Exploration efficiency of GAs largely depends ciently. But, if there is only one problem to solve, A-SAGA
on parameter values. But, it is hard to manually adjust is not always efficient since it requires extra cost of training.
these values. To cope with this problem, several adaptive In this paper, we propose Self Adaptive Island
GAs which automatically adjust parameters have been GA(SAIGA) which is based on A-SAGA and works effi-
proposed. However, most of the existing adaptive GAs ciently even if there is only one problem to solve. SAIGA
can adapt only a few parameters at the same time. Al- adapts parameter values using a similar mechanism to meta-
though several adaptive GAs can adapt multiple param- GA, but it requires no training. Throughout our evaluation
eters simultaneously, these algorithms require extremely experiments, we confirmed that our algorithm outperforms a
large computation costs. In this paper, we propose Self simple GA using De Jong’s rational parameters[8], and has
Adaptive Island GA(SAIGA) which adapts four param- performance close to a simple GA using manually tuned pa-
eter values simultaneously while finding a solution to a rameter values. We describe related works in section 2, pro-
problem. SAIGA is a kind of island GA, and it adapts posed algorithm in section 3, experimental result and con-
parameter values using a similar mechanism to meta- sideration in section 4, and conclusion in section 5.
GA. Throughout our evaluation experiments, we con-
firmed that our algorithm outperforms a simple GA us- 2 Related works
ing De Jong’s rational parameters, and has performance
close to a simple GA using manually tuned parameter F. G. Lobo et al. have proposed an adaptive GA which ef-
values. fectively works when optimal number of individuals is not
known[9]. In this method, parallel searches are performed
1 Introduction using different numbers of individuals such as 16, 32, 64,
and so on, expecting one or more of them with appropri-
Genetic algorithm(GA) is an approximation algorithm for ate number of individuals would yield a good result. But,
combinatorial optimization problems inspired by evolu- it is not realistic to perform a large number of searches in
tion mechanisms in nature. Exploration efficiency of GAs parallel. Accordingly, each GA is made to use a different
largely depends on parameter values such as crossover rate number of evaluations per unit time so that a GA with a
and mutation rate. But, adjusting these parameter values small number of individuals uses a larger number of evalu-
takes a great deal of time, since the optimal parameter val- ations per unit time. In the case where a GA with a small
ues depend on the problem to solve. Besides it, the optimal number of individuals can find a near optimal solution, not
value of each parameter depends on crossover method and so much number of evaluations are wasted since GAs with
mutation method. To cope with this problem, several adap- a large number of individuals are only assigned a relatively
tive GAs which automatically adjust parameters have been small number of evaluations. In the case where only a GA
proposed[1, 2, 3, 4]. However, most of the existing adaptive with a large number of individuals can find a near optimal
GAs can adapt only a few parameters at the same time. Al- solution, GAs with a small number of individuals are dis-
though several adaptive GAs can adapt multiple parameters continued when a GA with a larger number of individual
simultaneously, these algorithms require extremely large finds a better solution.
computation costs. We have already proposed an algorithm Hinterding et al. have proposed an adaptive GA which
called A-SAGA[5] which is a combination of meta-GA[6] runs three GAs with the different numbers of individuals in
and GA with distributed environment scheme[7], where GA parallel[10]. The process of search is divided into epochs.
with distributed environment scheme is a kind of island GA. At each epoch, fitness values of elite individuals are com-
A-SAGA can adapt any combinations of parameters which pared, and the number of individuals are changed according
are assigned to each island, although it only requires def- to the result. For example, if the GA with the largest num-
initions of a fitness evaluation function and GA operators. ber of individuals yielded the best result, all GAs will use a
A-SAGA requires training before solving a problem. In the larger number of individuals in the next epoch.
training phase, many problems are solved. By using the B¨ ck has proposed an adaptive GA[1] whose individual
a
result of training, A-SAGA solves similar problems effi- has its own mutation rate encoded in its gene. Individuals

with good mutation rates are expected to survive. However, Tongchim et al. have proposed an adaptive GA which
since individuals with high mutation rates die in high prob- adapts mutation rate and crossover rate[16]. This GA is
ability, only individuals with low mutation rates tend to sur- based on IGA. As the search progresses, increases in aver-
vive in the last phase of search. Actually, literature [11] age fitness of each island are compared to those of neighbor
points out that this algorithm shows only low performance islands. If an increase of a neighbor island is larger, param-
in the last phase of search. eter values are changed according to that of the neighbor
Espinoza, et al. have proposed another adaptive GA [2] island.
whose individuals can independently search solutions us- Miki et al. have proposed a GA with distributed envi-
ing local search. In this algorithm, search efficiency per ronment scheme[7]. In this GA, different parameter vectors
unit number of evaluations has been improved by adapting a are given to islands of IGA expecting good solutions to be
ratio between computation costs of crossover/mutation and found in islands with good parameter vectors. This is not an
local search. adaptive GA, since parameters have to be given manually.
An adaptive GA proposed by Krink, et al.[3] determines
crossover rate and mutation rate of each individual by its 3 Proposed algorithm
location in 2-dimensional lattice space. Individuals are ex-
pected to move towards a location with better parameters. 3.1 Overview
The algorithm keeps diversity of these parameter values by
SAIGA uses two layers of GAs. The lower layer is an is-
limiting the number of individuals in each lattice.
land GA called low level GA which consists of two or more
Goldberg, et al. has proposed a theoretical way to com-
islands. The set of islands in the low level GA is denoted by
pute effective value ranges of parameters when applying a " "
. Each island in searches for the solution to a given prob-
GA to the one max problem[12, 13]. The derived ranges
lem. We place these islands in ring topology. The upper
are depicted in a graph called control map. In literature
layer is a simple GA called high level GA which searches
[14], properties of several selection techniques have been
for a suitable parameter vector for the low level GA.
analyzed. However, in order to derive detailed parameter "
The parameter vector corresponding to an island in is #
values using such an analytic approach, properties of target 6% ! % 2 % 2 % 0( $
5 4 3 1 % % ) % 1 % 3 % 4
)' , where , , and denote pop-
problems must be formulated from scratch. This formula-
ulation size, tournament size, crossover rate and mutation
tion may be difficult depending on the problem to solve.
rate respectively.
Meta-GA is a general method which uses a GA to de-
Our algorithm executes one generation of the high level
rive a good set of parameter values (called parameter vec-
GA at every predefined number of evaluations of the low
tor, hereafter) used in other GAs for searching solutions.
level GA. We call these predefined number of evaluations
In meta-GA, since the number of evaluations tends to be
era. Islands independently search for a better solution using
large, the whole computation costs must also be high. For
predefined number of evaluations at each era. For example,
example, an ordinary GA with 100 individuals and 100 gen- %
if 100 evaluations are assigned to each island, and ) ©7
¤ §
erations requires 100 ¢100 evaluations. To find a good
is assigned to one of the islands, 9 9 generations of
2@A 8 B C
parameter vector for this GA using a meta-GA with 10 indi- 9
search is performed during that era.
viduals and 20 generations, the number of evaluations will
After each era, fitness values are determined for all pa-
be ! ¥©¥©¨¦¥£ .
¤ ¤ ¤ ¤ ¤ ¤ § ¤ ¤ £ ¢ ¤ ¤ £ ¢ ¤ § ¢ ¤
rameter vectors. In meta-GA, parameter vectors are evalu-
Kee, et al. has improved meta-GA methods [4]. In this
ated by individual fitness of the elite individual. But, this
method, a preliminary search is carried out for training be-
evaluation method does not work well with our algorithm,
fore applying the algorithm to the actual problem. In the
since when one of the immigrants has better fitness value
training phase, states of individuals are classified into sev-
than the original elite individual, the fitness value of the
eral groups depending on given indices. Then, for each
elite individual increases. Accordingly, parameter fitness
group, search efficiency is investigated for several tens of
of our algorithm is given by cumulative increases of indi-
parameter vectors and the parameter vector with the highest
vidual fitness of elite individual, except increase caused by
search efficiency is determined. When searching a solution
immigration.
to the actual problem, it observes the state of individuals and
Strictly speaking, if populations of two islands are dif-
uses the parameter vector with the highest efficiency for that
ferent, the best parameter vectors for these islands might
state.
also be different. But, since the islands share information
Island GA (IGA)[15] is a kind of parallel GAs. In IGA,
by immigration, we regard that the best parameter vectors
each GA is regarded as an island, and all islands are exe-
for these islands are same.
cuted in parallel. Some individuals immigrate to another is-
Using our algorithm, we need not adjust the parameter
lands as the search progresses and in this way islands share
vector for the low level GA, but still need to adjust the pa-
some information and search the solution in cooperation.

rameter values for the high level GA. But, there should be max era count 2500
a parameter vector suitable for any kind of problems, since evaluation count per era 256
the high level GA always solves a problem to find the best high level crossover rate 0.8
parameter vector for the low level GA, and there is not so high level mutation rate
ƒ ƒ 0.6
much difference between these problems unless the set of (the number of islands) 10
parameters are changed, even if the tasks for the low level
GA are different. We use the following notation in out algorithm.
… †„ ˆ b…
3.2 Detailed explanation … „
is the set of individuals in the high level GA. ‡ ‰ 0#
( ) is a high level individual for island . Each
#
3.2.1 Encoding of parameter vector for low level GA island in has
!iR ¥bU d™fG˜ f`©’ ”©©SR
’ e e R • — “ Y — Y– • “X ’ ‘ individuals.
#
ˆ ˆ ˆ ˆ
and
%G! %F! %E)
D 3 D 1 D denote genotypes for population size,
D %H4 u!¥R ¥€U j™fG˜ fi!©’ ”©SR c# bh `è hg£ 0# f ‡
‰ ’ e e R • — “ Y — Y– • “X ’ ‘ ‰ f ‡ ee ‰ ‰
are global vari-
k kˆ ˆ „
tournament size, crossover rate and mutation rate, respec- ables, where is the -th individual in island .
‰ 0# ™‡
‰ f is a # f%
%
tively(figure 1). Each of the genotypes is represented by a set of individuals in the island .
l # is a buffer for receiving
% %
16bit fixed point number whose range is between 0 and 1. immigrants to island .
m is global variable.
l # will retain
The genotypes are converted to corresponding phenotypes elite fitness value of island .
3 in # retains the number of era
%
% %
! ! )
3 1 and %
using the following formulas.
4
% count. retains fitness value of the high level individual o
corresponding to island . d¥n #
3 p retains the number of evalu-
ation performed in this subroutine.
Sr um
s q retains increase of
)
%
fc!5 cb`SP %D SW0ESSQ0I
e d5 §' a YX ) P V' U T R P § (1) the elite fitness. retains the elite fitness in the next
v6hhS6s m
w r vuu t
q%D ¥P % )
p 1 d % iP % hI g
if generation. is the elite individual. {6¥ux
wz v y
% § D 1 )
1 e § q% ¥P % ) (2)
r D 1 if §
% e %D 3 Pseudo code of the proposed algorithm is as follows.
3 (3)
% 4 v5 £ ¤ e Fub`GP %D 0sU G¥Qf©¤ e ¤
e 5
¤ ¤ ¤ t £ ' a YX 4' T R P B ¤ ¤ ¤ (4) Algorithm SAIGA
1 begin
2 for each do
ni ’ si ’ ci ’ mi ’ 3 randomly generate
¦|
~ }
; h| d
‚ €
4 randomly generate –v‹ • ©“fi’2” “iŠ‘6SjiŽSiŒv‹ 0‚h| hƒd{ŠˆŠ‰ˆv†‡‚2…0h| hd
’ † ˆ „ ‚ ƒ
Figure 1: Encoding of parameter vector for low level GA ; S@v‹
‚ –
5 := ; ˜ ™— š
6 := 0; ˜ ›
3.2.2 Crossover and mutation methods for the high level 7 next
8 for := 0 to max era count do
gœ
GA
9 // Each individual for high level GA corresponds to an is-
We use uniform crossover as the crossover method for the land.
high level GA, provided that both of the target individu- 10 for each do ¦ž|
~ }
als have population size more than 2. Otherwise, and %D ) 11 := Executeisland( ,
0˜ † ˜ h† ƒ ˜ hŸ
¢ › ¡ , , );/* Exe- | ˜ › h| € ƒ ˜
‚
cute one era of island. */
are handled as one gene and not separated. This is be-
%D 1
% % 12 next
cause when is 2 or less, is 2 regardless of and thus
) 1 %D 1
13 Perform roulette selection of individuals for high level GA
D is not evaluated properly.
%i1
using fitness value . The elite individual is conserved.
˜ ¡
The mutation operator of the high level GA is as follows. 14 := HighLevelCrossover( );
€ £ € ¤
Mutation operator are applied to every gene. For exam- 15 := HighLevelMutation( );
€ €
ple, mutation operator applied to population size is defined 16 next
as 17 end

w x%D ) ‚€%D )
y 5 £ e ! c'
¤ ¤ Algorithm Executeisland( ‡ f%
„
,
ˆ €…
,
# hz ¦ Fm c#
w ¥ ‰ ,)
5 £ e ! c' 1 begin
where ¤ ¤ is a gauss random number with mean 0 and 2 := 0;
Fgœ
§
variance 0.1. 3 := 0;
ª2© G›
¨
4 Let be phenotypes of
¢ 6¥6iE{Ÿ
† † ¬ † « . h| d€
‚
3.2.3 Pseudo code 5 while evaluation count per era do
¯ugœ
® §
6 Perform crossover to with probability per individual.
ƒ˜
We use the following constants in our algorithm. The elite individual is preserved.

7 Perform mutation to with probability per individual.
£
ƒ˜ 1 Begin
The elite individual is preserved. 2 for each £d{| do
€ } ‚ €
8 Substitute a sufficiently small number for . 2‡””20G›
³ ©²±± °ª 3 Let ¢ Î¦6hvh‡v‡Î {Ÿbe genotypes of
†Î †Î ¬ † « ‚ .
{| €
9 for each newly generated individual do £µ´
ƒ˜ } 4 Generate a uniform random number r, where .
ÍÓQÆ
„ Ê Ë Ê
10 Calculate fitness value of and substitute this value ´ 5 if ® QËhigh level mutation rate then
for ¶ G› 6 Î « := ;
¢ Š@hHØÎ «
„ˆ Æ † ÆŸ × Ö
11 := + 1; ugœ
§ ugœ
§ /*Function ¢ Š@hÙ×
„ˆ Æ † ÆŸ returns a gauss random number
12 if then
¶G›®ž!0”`v‡G›
³ ©²±± °ª with mean 0 and variance 0.1*/
13 := ;
¶g› !0”`v‡S›
³ ©²±± °ª 7 if then := 0; endif
™b«
Æ ® Î Î ™«
14 := ; ´ `0² i´
³¸ · 8 if Ô™b«
„ Ä Îthen := 1; endif
Î™«
15 endif 9 endif
16 next 10 Generate a uniform random number r, where .
ÍµØµÆ
„ Ê Ë Ê
17 Perform tournament selection with size to based on ƒ˜ ¬ 11 if high level mutation rate then
® ÓË
each fitness value , and let ¶ › be re- {‚ « ch | ƒ {Š‰Šv‡2„ ch | ƒ 6¹
º ‚ †ˆˆˆ †‚ ‚ 12 :=
¢ ‰{HØ2¬
„ˆ Æ † ÆŸ × Ö Î ; Î g¬
sult of the selection. The elite individual is preserved. 13 if Î ¬ then := 0; endif
Æ ® Î ¬
18 :=ª +
2© ¨ › ;
`¸ ½ S› €0`”v‡ª › 2© ¨ ›
³ ¼ » ³ ©²±± ° ª 14 if Î ¬ then := 1; endif
„ ÍÄ Î ¬
19 :=
”¸ ½ S›
³ ¼ ; !0””20G›
³ ©²±± °ª 15 := ;
ŠvG”Ú
ÛÎ ¬ È « ¬
20 end while 16 Ò Ã ¤¬
if Ü then ; endif
Å¤¬
Ò ®
/*Process emigration*/ 17 endif
21 ¾ := emigration destination( ); /* Function emigra- | 18 Generate a uniform random number r, where .
ÍµØµÆ
„ Ê Ë Ê
tion destination( ) returns emigration destination from island
| 19 if ® ÓËhigh level mutation rate then
. islands are placed in a ring topology.*/
| 20 Î2 := ¢ ‰{HØ2 ;
„ˆ Æ † ÆŸ × Ö Î
22 ¿:=
À— @`0² · GqÀ—
º ³¸ ´ ¹ Á ¿ 21 if then := 0; endif
Î Æ ® Î
/*Receive immigrants*/ 22 if then := 1; endif
Î „ ÔÄ Î
23 if ÂÃ
then
š ¯˜ — 23 endif
24 Substitute for the individual with highest fitness in .
œ ˜ — 24 Generate a uniform random number r, where .
ÍµØµÆ
„ Ê Ë Ê
25 if the fitness value of then `¸ ½ ¼ Å£œ
³ › Ä 25 if high level mutation rate then
® ÓË
26 := the fitness value of ;
”¸ ½ S›
³ ¼ œ 26 „ Æ † :=
¢ Šˆ6ÆhŸÙ×ÌÖ©¦
Î Î¦ ;
27 Choose an individual from to except 2„ ch | ƒ
‚ ‚ ‚ « ch | ƒ
‚ 27 if Î¦ then
Æ := 0; endif
Ì®€Î
the elite individual, and substitute it for . œ 28 if Î then
„‚Ä Î := 1; endif
28 endif 29 endif
29 endif 30 Let ‚ h| € be a chromosome with genotypes
30 return ;
¢ `¸ ½ G† v© ¨ † ƒ ˜ hŸ
³ ¼ › ª › .
¢ Î 6† Î 6† Î † Î ”Ÿ
¬ «
31 end 31 next
32 end
… †„
Algorithm HighLevelCrossover( )
1 begin 4 Evaluation
2 for i := 1 to high level crossover rate do
…É ~ u™Ç Æ
È É È ˆ
3 Generate a uniform random number r, where ÍQÌµÆ
„ Ê Ë Ê . 4.1 Overview
4 if 0.5 then
® µË
5 swap( ); €`˜ Ï™@† ˜ ÏÎ «
Ñ Ð Î « For evaluation, we apply our algorithm to Traveling Sales-
6 swap( ); b`˜ ÏÎ !† ˜ ÏÎ ¬
Ñ Ð ¬ man Problem (TSP), deceptive problem, minimization
7 endif problems of Rastrigin function and Griewank function. We
8 if Òand
Ä ˜ Ï « then Ò Ä b`˜ Ï «
Ñ Ð compare our algorithm to a simple GA using De Jong’s ra-
9 Generate a uniform random number r, where ÓÌµÆ
Ê Ë Ê tional parameters [8], where mutation rate is 0.001, popu-
. „ lation size is 50, crossover rate is 0.6 and tournament size
10 if 0.5 then swap(
® ÓË ); endif €`˜ ÏÎ @† ˜ ÏÎ ¬
Ñ Ð ¬ is 2. We also compare our algorithm to a simple GA using
11 endif manually tuned parameter values.
12 Generate a uniform random number r, where ÔQQÆ
„ Ê Ë Ê .
® ÓË ); endif €i˜ Ï2h† ˜ Ï!
Ñ Ð Î Î
14 Generate a uniform random number r, where ÔQQÆ
„ Ê Ë Ê . 4.2 Problems
® Ë ); b`˜ Ï6† ˜ ÏÕ
Ñ Ð Î Î endif 4.2.1 Minimization problem of Rastrigin and Griewank
.
¢ €`˜ Ï@† €`˜ Ï26† b`˜ Ï2!† €`˜ Ï™{Ÿ
Ñ Ð Î Ñ Ð Î Ñ Ð Î ¬ Ñ Ð Î « function
16 next
17 end We use Rastrigin(equation 5) and Griewank function (equa-
tion 6) with 30 variables where each variable is encoded by
… W„
Algorithm HighLevelMutation( ) Gray code. We use one point crossover as the low level

4e-06
SAIGA 0.2
3.5e-06 rational SAIGA
manually tuned rational
3e-06 manually tuned
0.15
2.5e-06
fitness

2e-06

fitness
0.1
1.5e-06

1e-06
0.05
5e-07

0
0 0.1 0.2 0.3 0.4 0.5 0.6
0
number of evaluations (*1.0e+06) 0 1 2 3 4 5 6
number of evaluations (*1.0e+06)
Figure 2: Search efficiency on minimization problem of
Rastrigin function Figure 3: Search efficiency on minimization problem of
Griewank function
5
crossover operator and bit reverse as the low level mutation SAIGA
rational
operator. manually tuned
4

ã A 3
âà µ¤ e ß 5 0m ê % Ý é §' è Y ˜ P
á y ç æ% Ý 65 suhSfGSW¤ e Þ (5)

fitness
P Þ Ý'
ä ”%
2
r f©¤ e § ç æ8
V ë
å
Ý
ì q% e fu¤ e §
V ë
1

î A î î %
ã % Ý ð Ý
á íy 7 5 0m á Cç 0
£ Ý' fG˜
è Y ñ (6) 0 1 2 3 4 5 6
ä ”% ©fë
ï ¤ ¤ ¤ ä”% ï æï # number of evaluations (*1.0e+06)
8 r i8B ç e idB ò% Ý ì
§ £ § £
Figure 4: Search efficiency on deceptive problem

Griewank function are shown in figure 2 , 3 , 4 and 5 re-
4.2.2 Deceptive problem
spectively.
We use the problem which is a concatenation of 4 deceptive The vertical axis represents fitness value of the elite in-
functions with an 8bit variable each, shown as expression dividual, and the horizontal axis represents the number of
7 where 5 0€o
is the number of 1’s in expressed in binary
ó' ó evaluations. Each of the results is the average of 300 trials.
form. Each variable is encoded in binary. At the beginning of the search, our algorithm tends to
g be outperformed by a simple GA using the rational param-
¤ if ß 5 0€o
¤ ó' eters, but eventually our algorithm outperforms the simple
!5 0€‡žm
5 ó' o' (7)
ç ô 5 0€o
ó' if e æ5 0€o
¤ õ ó ' GA. This is because our algorithm uses randomly initial-
ized parameter values at first and thus its search efficiency
We use one point crossover as the low level crossover is low, but search efficiency becomes high as the param-
operator and bit reverse as the low level mutation operator. eter vector converges towards the optimal value. Also, our
algorithm has performance close to a simple GA using man-
4.2.3 Traveling salesperson problem ually tuned parameter values. Similarly, the performance of
our algorithm is close to that of a simple GA using manu-
We apply our algorithm to lin 105 problem[17] which has ally tuned parameters on TSP, the minimization problems of
105 cities. The optimal solution to this problem is 14379. Rastrigin function and Griewank function.
We use 2opt method as a mutation operator. Whether the On the deceptive problem, both of the simple GAs stick
mutation operator is applied or not is determined for each after finding a local optima whose fitness value is 4. We can
city. We use EXX[18] as a crossover operator. see that the average of the fitness values after convergence is
a little less than 4. This is because we use the problem which
4.3 Search efficiency is a concatenation of 4 independent deceptive problems, and
The results for search efficiency of TSP, deceptive prob- rarely optimal solutions are found for some of these prob-
lem, the minimization problems of Rastrigin function and lems. Our algorithm escapes the local optima, since it has

35000 80
average
SAIGA standard deviation
rational 70
median
manually tuned
30000 60

population size
50
fitness

25000 40

30
20000
20

10
15000
0
0 1 2 3 4 5 6 50000 45000 40000 35000 30000 25000 20000
number of evaluations (*1.0e+06) fitness

Figure 5: Search efficiency on TSP Figure 7: Transition of population size ) on TSP
80 1
average average
70 standard deviation standard deviation
median median
0.8
60
population size

50
0.6

rate
40

30 0.4

20
0.2
10

0 0
100 1 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 100 1 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14
fitness fitness

Figure 6: Transition of population size ) on Rastrigin func- Figure 8: Transition of crossover rate 3 on Rastrigin func-
tion tion

the characteristic of IGA. comes high.
As shown in figure 2, the performance of our algo- As of TSP, the average population size decreases to 15
rithm when applied to the minimization problem of Rast- momentary, but the average elite population size when fit-
rigin function is similar to a simple GA using the rational ness value is around 20000 is larger than 30, and this seems
parameters. to be because our algorithm is trying to increasing diversity.
As of the minimization problem of Rastrigin function,
4.4 Transition of adapted parameter values population size decreases on the early stages of the search.
This seems to be because local search is effective on the
Processes of parameter adaptations for each problem are
early stages of the search. This is also seen in the transition
contrasted as follows. As of the problem of Rastrigin func-
of parameters on the problem of Griewank function.
tion, population size, mutation rate, crossover rate and tour-
As of the problem of Rastrigin function, mutation rate
nament size are shown in figure 6, 8, 10 and 12 respectively.
decreases as the search proceeds. This seems to be be-
As of TSP, population size, mutation rate, crossover rate and
cause it finds a better solution as the search proceeds, and
tournament size are shown in figure 7, 9, 11 and 13 respec-
local search becomes effective because of the property of
tively. The vertical axis represents each parameter value,
the problem. In contrast, as of TSP, mutation rate does not
and the horizontal axis represents fitness value. In these
change from 0.02. With this rate, two 2opt operations are
figures, average, median and standard deviation of elite pa-
expected to be applied for one individual in one low level
rameter value of 300 trials are shown.
generation.
We can see that population size converges to different
As of TSP, crossover rate does not change at the begin-
values for each problem. Thus, we consider that SAIGA
ning of search. But, it gradually increases as the search
adapts the population size to each problem suitably. This
proceeds. This seems to be because other parameter val-
seems to be because there is difference between degrees of
ues have larger influences on search efficiency at the be-
diversity required for each problem. As diversity in the pop-
ginning, but influences of crossover rate becomes high at a
ulation becomes little, the population is likely to stick in
later stage. We consider that SAIGA successfully adapts
a local optima, although the efficiency of local search be-
crossover rate. As of the problem of Rastrigin function,

1 1
average average
standard deviation standard deviation
median median
0.8
0.1

0.6

rate
rate

0.01

0.4

0.001
0.2

0 0.0001
50000 45000 40000 35000 30000 25000 20000 50000 45000 40000 35000 30000 25000 20000
fitness fitness

Figure 9: Transition of crossover rate on TSP 3 Figure 11: Transition of mutation rate 4 on TSP
1 1
average average
standard deviation standard deviation
median median
0.8 0.5
0.1

0.6

rate
rate

0.01
0.4

0.001
0.2

0.0001 0
100 1 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14 100 1 0.01 0.0001 1e-06 1e-08 1e-10 1e-12 1e-14
fitness fitness

Figure 10: Transition of mutation rate 4 on Rastrigin func- Figure 12: Transition of rate of tournament size D 1 on Rast-
tion rigin function

crossover rate is always around 0.6. But, we consider that 5 Conclusions
SAIGA successfully adapts crossover rate since if it fails
and crossover rate changes randomly, the average should be We proposed a self adaptive Island GA which adapts 4 pa-
0.5. rameter values simultaneously by giving different parame-
As of TSP and the problem of Rastrigin function, tour- ter values to each island and observing search efficiencies.
nament size ratio does not change from around 0.5. This
%D 1 Throughout our evaluation experiments, we confirmed that
value is close to the average value where randomly de- %D 1 our algorithm outperforms a simple GA using De Jong’s ra-
cided, and this might suggest that our algorithm would fail tional parameters, and has performance close to a simple
to adapt the tournament size. However, we observed transi- GA using manually tuned parameter values. Also, we con-
tion of when is initialized to 1.0e-5, and converged
%D 1 %D 1 %D 1 firmed that SAIGA successfully adapts each parameter val-
to 0.5 after fitness value becomes 21500 or lower. Thus, we ues.
consider that SAIGA successfully adapts tournament size. In the future work, we would like to improve handling of
On figure 14, distributions of adapted values of are %D 1 parameter values for the low level GA when the low level
given when fitness values are around 40000 and 19500. We GA is stuck in a local optima.
can see that when fitness value is around 40000, the value
of is mostly 0.7 to 0.8, and when fitness value is around
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