A Compass to Guide Genetic Algorithms

Jorge Maturana and Fr´ed´eric Saubion

{maturana,saubion}@info.univ-angers.fr

LERIA,Universit´e d’Angers

2,Bd Lavoisier 49045 Angers (France)

Abstract.Parameter control is a key issue to enhance performances of

Genetic Algorithms (GA).Although many studies exist on this problem,

it is rarely addressed in a general way.Consequently,in practice,param-

eters are often adjusted manually.Some generic approaches have been

experimented by looking at the recent improvements provided by the op-

erators.In this paper,we extend this approach by including operators’

eﬀect over population diversity and computation time.Our controller,

named Compass,provides an abstraction of GA’s parameters that al-

lows the user to directly adjust the balance between exploration and

exploitation of the search space.The approach is then experimented on

the resolution of a classic combinatorial problem (SAT).

1 Introduction

Genetic Algorithms (GA) are metaheuristics inspired by natural evolution,which

manage a population of individuals that evolve thanks to operators’ applications.

Since their introduction,GAs have been successfully applied to solve various

complex optimization problems.From a general point of view,the performance

of a GA is related to its ability to correctly explore and exploit the interesting

areas of the search space.Several parameters are commonly used to adjust this

exploration/exploitation balance (EEB),and the operator application rates are

probably among the most inﬂuential ones.A suitable control of parameters is

crucial to avoid two well-known problems:premature convergence,that occurs

when the population gets trapped in a local optima,and the loss of computation

time,due to the inability of the GA to detect the most promising areas of the

search space.Most of the eﬀorts on this subject are only applicable to speciﬁc

algorithms,thus,in practice,parameter control is often achieved manually,sup-

ported by empirical observations.More recently,new methods have begun to rise

up,proposing more generic control mechanisms.In this trend,our motivation is

to design a new controller in which parameters could be handled by more general

and abstract concepts,in order to be used by a wide range of GAs.

Techniques for assigning values to parameters can be classiﬁed according to

the taxonomy proposed by Eiben et al.[1].A general class,named Parameter

Setting [2],is divided in Parameter Tuning,where parameters are ﬁxed before

the run,and Parameter Control,where parameters are modiﬁed during the run.

Parameter Control is further divided in Deterministic,where parameters are

2

modiﬁed according to a ﬁxed and predeﬁned scheduling;Adaptive,where the

current state of the search is used to modify parameters by means of rules;

and Self-Adaptive [3],where parameters are encoded in the genotype and evolve

together with the population.

Within adaptive control,the central issue is to design rules able to guide

the search and to make the suitable choices.A straightforward way consists in

performing test runs to extract pertinent information in order to feed the system.

However,this approach involves an extra computational time and does not really

correspond to the idea of an “automatic self-driven” algorithm.

A more sophisticated way to build a control system consists in adding a

learning component,which is able to identify a correct control procedure.This

reduces prior eﬀort and increases adaptation abilities,according to the needs of

diﬀerent algorithms.In this context,two perspectives could be identiﬁed:

The ﬁrst approach consists in modeling the behavior of the GA using dif-

ferent parameters,typically during a learning phase.[4] presents two methods

including a learning phase that tries diﬀerent combinations of parameters and

encodes the results in tables or rules.A similar approach is presented in [5],

where population’s diversity and ﬁtness evaluation are embedded in fuzzy logic

controllers.Later this controllers are used to guide the search according to a

high level strategy.[6] proposes an algorithm divided in periods of learning and

control of parameters,by adjusting central and limit values of them.

A second approach consists in providing a fast control,neglecting the mod-

eling aspect.[7] presents a controller that adjusts operators’ rates according to

recent performances.Similar ideas are presented in [8,9].In [10],this approach

is extended by considering several statistics of individuals ﬁtness and survival

rate to evaluate operator quality.In [11],the population is resized,depend-

ing of several criteria based on the improvement of the best historical ﬁtness.

[12] presents an algorithm that oscillates between exploration and exploitation

phases when diversity thresholds are crossed.[13] modiﬁes parameters according

to best ﬁtness value.Some methods in this class require special features fromthe

GA,such as [14],that maintains several populations with diﬀerent parameter

values,and moves the parameter’s values toward the value that produces the

best results.In [15],a forking scheme is used:a parent population is in charge of

exploration,while several child populations exploit particular areas of the search

space.In [16],a parameterless GA gets rid of popsize parameter by comparing

the performance of multiple populations of diﬀerent size.

In this paper,we investigate a combination of these two general approaches

in order to beneﬁt from their complementary strengths,providing an original

abstract control of GAs’ operators.Our controller measures the variations of

population’s diversity and mean ﬁtness resulting from an operator application,

as well as its execution time.A unique control parameter (Θ) allows us to adjust

the desired level of EEB and determines the application rates assigned to each

operator.We have tested our approach on the resolution of the famous boolean

satisfaction problem (SAT) and compared it to other adaptive control methods.

3

The paper is organized as follows.Sect.2 exposes our approach,Sect.3 de-

scribes the experimental framework we have used,and Sect.4 discusses results.

Finally,main conclusions and future directions are drawn in Sect.5.

2 Method Overview

We consider here a basic steady-state GA:at each step an operator is selected

among several ones,according to a variable probability.Asexual operators are

applied to the best of two randomly chosen individuals of the population,and the

resulting individual replaces the worst one.Sexual operators work on two ran-

domly chosen individuals,modifying them directly.The parameters considered

here are therefore operators’ application rates.

As mentioned in the introduction,adaptive control can be considered from

two diﬀerent points of view.In order to illustrate more precisely these diﬀerences,

we may detail two recent and representative approaches by comparing the work

of Thierens [7] and a method proposed by Wong et al.[6].

In [7],Adaptive Pursuit (AP) aims at adjusting the probabilities of associ-

ated operators,depending on their performances,measured typically by ﬁtness

improvement during previous applications.This method is able to quickly adapt

these probabilities in order to award the most successful operators.AP does not

care about understanding the behavior of the algorithm and focuses immedi-

ately on the best values,in order to increase the performance.At this point,we

may remark that algorithms that are solely based on ﬁtness improvement may

experience premature convergence.

In the APGAIN method [6],the search is divided in epochs,further divided

in two periods.The ﬁrst one is devoted to the measurement of operators’ perfor-

mance by applying themrandomly,and the second one applies operators accord-

ing to a probability which is proportional to the observed performance.Three

values (low,medium,high) are considered for each parameter,and adjusted by

moving them towards the most successful value.Finally,a diversiﬁcation mech-

anism is included in the ﬁtness function.Roughly a quarter of the generations

is dedicated to the ﬁrst (learning) period,what could be harmful if there are

disrupting operators.

Here,we propose a new controller (Compass) based on the idea presented

in [5],that considers both diversity and quality as pertinent criteria to evaluate

algorithms’ performance.Parameters are abstracted,in order to guide the search

by inducing a required level of EEB.The operators are evaluated after each

application and,in addition to diversity and ﬁtness variation,a third measure

–operator’s execution time– is also considered.To get rid of previous drawbacks,

we include some controllers’ features that adapt parameters’ rates during the

search [7–9],namely the speed of response,to update the model.Operators are

applied according to their application rate,which is updated at every generation.

Since we are interested in a controller which could be used by any GA,it must be

independent and placed at a diﬀerent layer.We have then implemented Compass

in a C++ class,included by the GA.

4

2.1 Operator Evaluation and Applications Rates Updating

Given an operator i ∈ [1...k] and a generation number t,let d

it

,q

it

,T

it

be,

respectively,the population’s mean diversity variation,mean quality (ﬁtness)

variation,and mean execution time of i over the last τ applications of this

operator.At the beginning of the run,all operators can be applied with the

same probability.

We deﬁne a vector o

it

= (d

it

,q

it

) to characterize the eﬀects of the operator

over the population in terms of variation of quality and diversity (axis ΔD

and ΔQ of Fig.1).Note that,since both quality and diversity improvements

correspond to somewhat opposite goals,most vectors will lay on quadrants II

(improvement of quality but a decrease in diversity) and IV (increase in diversity

and a reduction of mean ﬁtness),shown in Fig.1a.

Algorithms that just consider the ﬁtness improvement to adjust the operator

probabilities would only use the projection of o

it

over the y-axis (dotted lines

in Fig.1b).On the other hand,if diversity is solely taken in account,measures

would be considered as the projection over the x-axis (Fig.1c).

Our goal is to control these two criteria together by choosing a search direc-

tion which will be expressed by a vector c (deﬁned by its angle Θ ∈ [0,

π

2

]) that

characterizes also its orthogonal plane P (see Fig.1d).

Since measures of diversity and quality usually have diﬀerent magnitudes,

they are normalized as:

d

n

it

=

d

it

max

i

{|d

it

|}

and q

n

it

=

q

it

max

i

{|q

it

|}

We thus have vectors o

n

it

= (d

n

it

,q

n

it

).Rewards are then based on the projection

of vectors o

n

it

over c,i.e.,|o

it

|cos(α

it

),α

it

being the angle between o

it

and c.A

value of Θ close to 0 will encourage exploration,while a value close to

π

2

will

favor exploitation.In this way,the management of application rates is abstracted

by the angle Θ,that guides the direction of the search as the needle of a compass

shows the north.

Projections are turned into positive values by subtracting the smallest one

and dividing themby execution time,in order to award faster operators (Fig.1e).

δ

it

=

|o

n

it

|cos(α

it

) −min

i

{|o

n

it

|cos(α

it

)}

T

it

Application rates are obtained proportionally to values of δ

it

plus a constant ξ

t

,

that ensures that the smallest rate is equal to a minimal rate,P

min

,preventing

the disappearance of the corresponding operator (Fig.1f).

p

it

=

δ

it

+ξ

t

k

i=1

δ

it

+ξ

t

2.2 Operator Application

Operators’ application rates are updated at every generation.An interesting phe-

nomenon,observed during previous experiments,is the displacement of points

5

Fig.1.(a) points (d

it

,q

it

) and corresponding vectors o

it

,(b) quality-based ranking,(c)

diversity-based ranking,(d) proposed approach,(e) values of δ

it

,(f) ﬁnal probabilities

(d

n

it

,q

n

it

) in the graphic during execution.Consider for instance that Θ is set to

π

4

,

so an equal importance is given to ΔQ and ΔD.At the beginning of the search,

just after the population was randomly created,the population diversity is high

and mean ﬁtness is low,thus it is easy for most operators to be situated in the

quadrant II.After some generations,the population starts to converge to some

optimum,so improvement becomes diﬃcult,and points in II corresponding to

exploitative operators move near x-axis.When improvements in this zone are

exhausted,exploitation operators obtain worst rewards than exploration ones,

causing a shift of the search to diversiﬁcation,and escaping from that optimum.

Such a visualization tool could be useful to understand the behavior of operators

as well as for debugging purposes.

3 Experimentation

For our experiments,we focus on the use of GAs for the resolution of combi-

natorial problems.Among the numerous possible classes,we have chosen the

Boolean satisﬁability problem (SAT) [17],which consists in assigning values to

binary variables in order to satisfy a Boolean formula.

The ﬁrst reason is that this is probably the most known combinatorial prob-

lem,since it has been the ﬁrst to be proved NP complete and therefore it has

been used to encode and solve problems frommany application areas.The second

reason is that there exists an impressive library [18] of instances and their dif-

ﬁculty has been deeply studied with several interesting theoretical results (e.g.,

6

phase transition),which allows us to select diﬀerent instances with various search

landscapes’ properties.

More formally,an instance of the SAT problem is deﬁned by a set of Boolean

variables X = {x

1

,...,x

n

} and a Boolean formula F:{0,1}

n

→{0,1}.The for-

mula is said to be satisﬁable if there exists an assignment v:X →{0,1}

n

satis-

fying F and unsatisﬁable otherwise.Instances are classically formulated in con-

junctive normal form (conjunctions of clauses) and therefore one has to satisfy

all these clauses.

To solve this problem,we consider a GA with a binary population that ap-

plies one operator at each generation.The ﬁtness function evaluates the number

of clauses satisﬁed by an individual and the associated problemis thus obviously

a maximization one.The diversity is classically computed as the Hamming dis-

tance entropy (see [19]).

In order to evaluate our control approach,we compare it with Adaptive Pur-

suit (AP) [7] and APGAIN [6].As mentioned in Sect.2,AP is representative

of many controllers that consider ﬁtness improvement as their guiding crite-

rion while APGAIN is representative of methods that try to learn and model

the behavior of the operators.Additionally,we also included a uniform choice

(UC) among operators as the baseline of the comparison.In order to check the

robustness of our method –but restricted by the lack of space–,we present 13

diﬀerent instances fromthe SATLIB repository [18],mixing problems of diﬀerent

sizes and nature,including random-generated instances,graph coloring,logistics

planning and blocks world problems.

3.1 Operators

The goal of this work is to create an abstraction of operators,regardless of their

quality,and to compare controllers,and not to develop an eﬃcient GA for SAT.

The idea is also to use non standard operators,whose eﬀect over diversity and

quality is a priori unknown.Therefore,we propose six operators with diﬀerent

features,more or less specialized with regards to the SAT problem.

One-point crossover chooses randomly two individuals and crosses them at

a random position.In this operator exclusively,the best child replaces the

worst parent.

Contagion chooses randomly two individuals,and the variables in false clauses

of the worst one are replaced with corresponding values of the best individual.

Hill climbing checks all neighbors by swapping one variable,moves to the bet-

ter one and repeats while improvement is possible.

Tunneling swaps variables without decreasing the number of true clauses ac-

cording to a tabu list of length equal to

1

4

of the number of variables.

Badswap swaps all variables that appear in false clauses.

Wave chooses the variable that appears in the highest number of false clauses

and in the minimum number of clauses only supported by it,and swaps it.

It repeats the same process at most

1

2

times the number of variables.

7

In order to observe the eﬀect of population size over the performance of

controllers,we performed experiments with populations of 3,5,10 and 20 in-

dividuals.10.000 generations were processed,in order to observe the long-term

behavior of controllers.

3.2 Control strategy

Previous experiments have shown that values of Θ around 0.25π produced good

results.To observe the sensitivity of this value,we ran experiments with values of

0.20π,0.25π and 0.30π.Note that,even when the value of Θ remains ﬁxed along

the run,it does not mean that Compass falls in the category of parameter tuning.

It is necessary to distinguish the parameters of the GA (operator’s application

rates) from the parameter(s) of the control strategy (θ in this case).Controller

parameters provide an abstraction of GA’s parameters.It is pertinent to wonder

whether it is worth replacing GA parameters by controller parameters.We think

that this substitution is beneﬁcial in two cases:

– When the eﬀect of controller parameters is less sensitive than GA’s parame-

ters.Consider,for instance,the case of mutation rate:small changes in this

parameter have a drastic eﬀect over GA performances;so it is interesting to

use a controller which is able to wrap these parameters,providing a more

stable operation,even by including additional control parameters.

– When the controller provides a more comprehensible abstraction of GA pa-

rameters.This is the case in our approach:it is easier for a human to think in

terms of raising and lowering EEB instead of modifying multiple operators’

parameters,specially when their behavior is ill-known.

The parameter τ is set to 100,and P

min

to

1

3k

(see Sect.2.1).Each run,

consisting of a speciﬁc problem instance,population size,controller and Θ (just

for Compass),was replicated 30 times for signiﬁcant statistical comparisons.AP

and APGAIN parameters were set to published values,or tuned to obtain good

performance.According to the notations used in [7,6],for AP:α = 0.8,β = 0.8,

P

min

=

1

2k

.For APGAIN:v

L

= 0,v

U

= 1,δ = 0.05,σ = 700,ρ =

σ

4

,ξ = 10,

φ = 0.045 (about 10% of re-evaluations).

4 Results and discussion

The average number of false clauses obtained over 30 runs is shown in table 1.

Comparisons were done using a student-t test with a signiﬁcance level of 5%.

Values are boldfaced when Compass outperforms UC,and italicized when UC

is better than Compass.No font modiﬁcation means that results are statistically

indistinguishable.Cells are

grey when Compass outperforms AP,and

black

when AP outperforms Compass.White cells means indistinguishability.Finally,

Compass outperformed APGAIN in all cases,except in those indicated with

underlined

values,where results are indistinguishable.Average execution times

of AP,APGAIN and Compass,relative to those of UC,are shown at the rightest

8

column of the table.Total number of clauses of each problem appear in the

bottom of the table.From now on we will refer as C.2,C.25,C.3 to Compass

with Θ values of 0.20π,0.25π and 0.30π,respectively.

Table 1.Average false clauses and comparative execution times

popsize

control

4-blocks

aim

f1000

CBS

ﬂat200

logistics

medium

par16

sw100-0

sw100-1

uf250

uuf250

time

UC 12.9 3.2 52.6 5.4 38.4 16.7 7.7 124.2 23.2 9.4 8.3 11.7 1.00

AP 7.5 2.1 37.7 3.4 19.6 8.9 3.5 71.3 16.2 3.3 5.6 8.3 0.86

APGAIN 11.8 3.3 51.6 5.0 27.5 14.0 5.4 109.2 20.6 6.0 8.8 10.8 0.97

3 C.2

5.8 2.1

25.5

2.1

13.2

8.1

2.0

41.6

13.7

1.3

3.3

5.5 0.88

C.25

6.4

1.6

26.7

2.3

11.7

8.1

2.0

38.4

13.4

1.8

3.6

5.9 0.89

C.3

6.1

1.6

26.8 2.8

15.9

8.1 3.0

47.2 15.5

2.2

4.3

6.4 0.73

UC 13.8 3.3 61.4 7.6 34.6 16.5 7.9 126.2 24.6 8.3 11.2 14.0 1.00

AP 8.9 3.0 47.4 4.8 23.3 11.3 4.9 88.2 18.4 4.5 8.3 10.2 0.88

APGAIN 11.2 4.9 60.1 6.6 31.1 14.0 6.4 118.7 20.1 6.0 9.8 13.6 1.09

5 C.2

6.2

2.2

27.5

3.0

15.5

9.1

2.6

45.0

15.0

2.8

4.6

6.7 0.89

C.25

6.3

1.9

27.2

2.7

16.2

8.8

2.6

43.4

14.8

2.9

4.4

7.1 0.91

C.3

7.8 2.5

36.5 4.2

20.0

9.0

3.5

66.7

16.8

3.4

5.9 10.3 0.80

UC 13.8 3.3 54.2 6.3 28.8 15.5 7.0 110.0 19.4 6.1 10.1 12.2 1.00

AP 9.9 3.9 55.2 5.0 26.3 12.9 5.7 98.6 18.1 5.9 9.3 12.2 1.00

APGAIN 11.7 4.8 66.0 5.8 30.4 16.3 6.2 120.4 18.8 6.8 11.3 13.9 0.62

10 C.2

8.3 3.5

44.9

3.9

21.9

11.3

4.5

72.5 17.9

4.3

7.5

10.4 0.92

C.25

8.0

2.9

42.7 4.4

23.1

11.2

4.2

72.6 17.8

4.6

7.2

9.5 0.83

C.3

9.1 3.7

49.3 5.7

24.7

11.5 5.1 96.8 19.1

5.9

9.1 12.6

0.78

UC 13.8 2.8 42.0 4.5 23.1 13.8 5.2 88.5 16.5 3.4 6.4 10.3 1.00

AP 9.1 2.4 54.1 4.9 26.0 14.6 5.3 102.6 17.2 5.7 8.4 11.3 1.28

APGAIN 11.3 4.3 68.0 5.5 30.0 17.8 6.2 120.1 18.7 6.3 11.1 13.7 2.38

20 C.2 9.1

3.5

55.2 4.8

26.0

13.3 5.2

90.4

18.9

6.2

8.1 12.5

0.92

C.25 9.2

3.3

53.2 4.8

25.2

13.3 4.8

90.6 17.8

6.2

8.7

13.2

0.93

C.3 9.4

3.9

58.6

4.9

27.5 13.4 6.2

99.2

18.7

6.1

10.4

12.4

0.88

Total clauses 47820 320 4250 449 2237 6718 953 3310 3100 3100 1065 1065

The mean number of generations required to reach the best values varies

between 1000 and 7000,therefore,the 10000 allowed generations seem suﬃcient

for all controllers to insure a fair comparison.Of course,the results are not

competitive with speciﬁc SAT evolutionary solvers,since we do not use the best

dedicated operators and neither try to optimize ours.Our purpose here is rather

to highlight the diﬀerences between controllers.Better results for SAT using GAs

were obtained by a hierarchical memetic algorithm[19].However,given the early

stage of this research,we preferred a simpler GA that applies one operator each

time,in order to facilitate understanding.Further research will consider more

complex operator architectures.

The predominance of Compass,and specially C.25 over UC,AP and AP-

GAIN is noticeable,particularly for small populations.Something similar hap-

pens with C.2,and,to some extent,with C.3.As mentioned previously,a value

Θ = 0.25π works well with all kind of problems.

9

Small populations lose diversity easily,so controlling diversity is a critical

issue.APGAIN does it by penalizing common individuals.However,when all in-

dividuals are the same,this penalization is not eﬀective.It seems that in practice,

diversity is mostly induced by the ﬁrst period in APGAIN (operator evaluation).

AP controls diversity by deﬁning a minimum application rate equal to

1

2k

.This

value could be excessive if operators are mostly exploitative.A smaller value of

1

3k

,used in Compass,grants the controller a greater range to balance EEB.

Small populations provide better results than larger ones.This is probably

due to the operators,that were inspired by local search heuristics:they are

applied more repetitively over the same individual in smaller populations than

in large ones,thus producing better results.Surprisingly,UC is quite competitive

as population size increases.It seems that applying operators of both low d

it

and

q

it

produce “bad” individuals that are able,however,to escape fromlocal optima.

Nevertheless,this practice is beneﬁcial only if the population is big enough to

keep their good elements at the same time.

Execution times of AP and APGAIN are shorter than those of UC for the

smallest populations.Compass has stable short execution times for diﬀerent

population sizes.This is interesting because it means that the eﬀort spent in

performing control induces savings in total execution time.

From an implementation point of view,we found that Compass and AP were

more independent from the logic of the GA than APGAIN,which introduces its

diversity control mechanism in the GA ﬁtness function.Both AP and Compass

provide a separate layer of control.Parameterization of Compass is quite intu-

itive.We have already discussed the eﬀect of Θ and P

min

.The last parameter,

τ,is quite stable,we have replicated experiments with several values for this

parameter without detecting a considerable inﬂuence over the performances.

5 Conclusions

In this paper we have presented Compass,a GA controller that provides an

abstraction of parameters and simpliﬁes control by adjusting the level of explo-

ration/exploitation along the search.This controller measures operators’ eﬀects

over population’s mean ﬁtness,diversity and execution time.Compass is inde-

pendent from the GA,in order to provide an additional control layer that could

be used by other of population-based algorithms.Experiments were performed

using a 6-operators GA to solve instances of the SAT problem.Results were fa-

vorably compared against a basic uniformchoice and state-of-the-art controllers.

The twofold evaluation of operators (quality and diversity variation) is co-

herent with the guiding principles of population-based search algorithms,i.e.

maximizing quality of solutions while avoiding the concentration of the popula-

tion,in order to beneﬁt fromtheir parallel nature.By considering both measures,

we observed a natural mechanism to escape from local optima.

The search direction is easily apprehensible by observing a dynamic vectorial

representation,thus Compass could also be used as a tool for understanding the

role of operators.

10

The management of nonstandard unknown operators also opens the perspec-

tive of using Compass to evaluate operators generated automatically,for example

by means of Genetic Programming.

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