gecco-09-ga-improvement-with-svps

Improving Genetic Algorithms Performance viaDeterministic Population Shrinkage

Juan Luis Jimenez Laredo1 Carlos Fernandes1

Juan Julian Merelo1 Christian Gagne2

1GeNeura TeamDepartment of Computer Architecture and Technology

University of Granada, Spain

2Computer Vision and Systems Laboratory (CVSL)Departement de genie electrique et de genie informatique

Universite Laval, Quebec City (Quebec), Canada

GECCO 2009, Montreal (Quebec), Canada

Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 1 / 17

Hypothesis: Different convergence stages of a genetic algorithm mayrequire different population sizes

Model: A Simple Variable Population Sizing (SVPS) scheme whereonly population shrinkage is considered

Aim: Get empirical evidences of performance improvement withSVPS over a fixed-size scheme

Outline

Background on population sizing

Methodology

I Generalized l-trap functionI Bisection method for estimating correct population sizeI Simple Variable Population Sizing

Experimental results

Outline

MethodologyI Generalized l-trap function

I Bisection method for estimating correct population sizeI Simple Variable Population Sizing

Outline

MethodologyI Generalized l-trap functionI Bisection method for estimating correct population size

I Simple Variable Population Sizing

Outline

MethodologyI Generalized l-trap functionI Bisection method for estimating correct population sizeI Simple Variable Population Sizing

Outline

MethodologyI Generalized l-trap functionI Bisection method for estimating correct population sizeI Simple Variable Population Sizing

Population Sizing

Sizing scheme:I Fixed size: canonical approachI Deterministic methods: function-based adjustment (e.g. Saw-tooth)I Adaptive methods: on-line adjustment (e.g. GAVaPS)

Sizing theory:

I Focus is on the correct sizing of population for the fixed-sized schemeI But theory for fixed-size scheme can be helpful for variable-size schemes

Population Sizing

Sizing scheme:I Fixed size: canonical approachI Deterministic methods: function-based adjustment (e.g. Saw-tooth)I Adaptive methods: on-line adjustment (e.g. GAVaPS)

Sizing theory:I Focus is on the correct sizing of population for the fixed-sized schemeI But theory for fixed-size scheme can be helpful for variable-size schemes

Generalized l-trap Function

l-trap function (Ackley, 1987):I l : problem size (number of

possible values in range)I a: value of local optimumI b: value of global optimumI z : slope-change location

Currently, experiments witha = l − 1, b = l and z = l − 1

I 2-trap: not deceptiveI 3-trap: partially deceptiveI 4-trap: deceptive

Generalized l-trap Function

l-trap function (Ackley, 1987):I l : problem size (number of

possible values in range)I a: value of local optimumI b: value of global optimumI z : slope-change location

Currently, experiments witha = l − 1, b = l and z = l − 1

I 2-trap: not deceptiveI 3-trap: partially deceptiveI 4-trap: deceptive

Scaling the Problem Difficulty

Working Hypothesis

Minimizing number of solutions evaluated while guaranteeing asuccess rate

Working hypothesis: larger population required at the beginning

I Start with a diverse sampling of the search spaceI As convergence occurs, smaller population required

Use a deterministic schedule of the population size

Working Hypothesis

Working hypothesis: larger population required at the beginningI Start with a diverse sampling of the search spaceI As convergence occurs, smaller population required

Working Hypothesis

Working hypothesis: larger population required at the beginningI Start with a diverse sampling of the search spaceI As convergence occurs, smaller population required

Working Hypothesis

Simple Variable Population Sizing (SVPS)

Reduce population by a variable ratio at each generation:

ng = n0

(1− (1− ρ)

)τ)I n0: initial population sizeI ng : population size at generation gI g : current generation numberI gmax : last generation numberI τ : resizing speed parameterI ρ: resizing severity parameter

Simple Variable Population Sizing (SVPS)

Estimating the Correct Population Size (SR of 0.98)

1) Rough estimation (ni+1 = 2ni ):

n1 = 4, SR=0.2 n2 = 8, SR=0.95 n3 = 16, SR=0.995

2) Bisection (ni+1 =nmax

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

3) Refinement (ni+1 = b0.99nic):n6 = 9, SR=0.9803

Correct population size is 9 for a success rate of 0.98

1) Rough estimation (ni+1 = 2ni ):n1 = 4, SR=0.2

n2 = 8, SR=0.95 n3 = 16, SR=0.995

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

1) Rough estimation (ni+1 = 2ni ):n1 = 4, SR=0.2 n2 = 8, SR=0.95

n3 = 16, SR=0.995

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

1) Rough estimation (ni+1 = 2ni ):n1 = 4, SR=0.2 n2 = 8, SR=0.95 n3 = 16, SR=0.995

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99

n5 = 10, SR=0.982

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

3) Refinement (ni+1 = b0.99nic):

n6 = 9, SR=0.9803

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

i +nmini

2 ), stop whennmax

i −nmini

< 116 :

n4 = 12, SR=0.99 n5 = 10, SR=0.982

Population Sizes for a Success Rate of 0.98

m: number of concatenated trap functions

Experimental Setting

Selectorecombinative binary Genetic Algorithm:I Population sizes set according to bisection method for a success rate of

0.98I Two parents tournament selectionI One-point crossover (probability of 1.0)I No mutation

Trap problems tested:

I Problem sizes, l = {2, 3, 4}I Number of sub-functions, m = {2, 4, 8, 16, 32, 64}

SVPS setting:

I Speed, τ = 0.125, . . .×1.5 , 32I Severity, ρ = 0.25, . . .+0.05 , 1

Trap problems tested:I Problem sizes, l = {2, 3, 4}I Number of sub-functions, m = {2, 4, 8, 16, 32, 64}

SVPS setting:

I Speed, τ = 0.125, . . .×1.5 , 32I Severity, ρ = 0.25, . . .+0.05 , 1

Trap problems tested:I Problem sizes, l = {2, 3, 4}I Number of sub-functions, m = {2, 4, 8, 16, 32, 64}

SVPS setting:I Speed, τ = 0.125, . . .×1.5 , 32I Severity, ρ = 0.25, . . .+0.05 , 1

Speed (τ) and Severity (ρ)

Size of circles show improvement over fixed-size population

Saved Computational Effort

Conclusion

SVPS requires a smaller number of evaluations than a fixedpopulation sizing scheme

The improvement is much more noticeable for large population sizesas the problem instances scale

There is not a single but a set of possible strategies for SVPS(different τ -ρ combinations)

Conclusion

Questions

Thanks for your attention!

gecco-09-ga-improvement-with-svps

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