Improving Genetic Algorithms Performance via
Deterministic Population Shrinkage
Juan Luis Jimenez Laredo1 Carlos Fernandes1
Juan Julian Merelo1 Christian Gagn´2
e
1 GeNeura Team
Department of Computer Architecture and Technology
University of Granada, Spain
2 Computer Vision and Systems Laboratory (CVSL)
D´partement de g´nie ´lectrique et de g´nie informatique
e e e e
Universit´ Laval, Quebec City (Qu´bec), Canada
e e
GECCO 2009, Montr´al (Qu´bec), Canada
e e
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 1 / 17

Scope
Hypothesis: Different convergence stages of a genetic algorithm may
require different population sizes
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 2 / 17

Scope
Hypothesis: Different convergence stages of a genetic algorithm may
require different population sizes
Model: A Simple Variable Population Sizing (SVPS) scheme where
only population shrinkage is considered
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 2 / 17

Scope
Hypothesis: Different convergence stages of a genetic algorithm may
require different population sizes
Model: A Simple Variable Population Sizing (SVPS) scheme where
only population shrinkage is considered
Aim: Get empirical evidences of performance improvement with
SVPS over a fixed-size scheme
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 2 / 17

Outline
Background on population sizing
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 3 / 17

Outline
Background on population sizing
Methodology
Generalized l-trap function
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 3 / 17

Outline
Background on population sizing
Methodology
Generalized l-trap function
Bisection method for estimating correct population size
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 3 / 17

Outline
Background on population sizing
Methodology
Generalized l-trap function
Bisection method for estimating correct population size
Simple Variable Population Sizing
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 3 / 17

Outline
Background on population sizing
Methodology
Generalized l-trap function
Bisection method for estimating correct population size
Simple Variable Population Sizing
Experimental results
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 3 / 17

Population Sizing
Sizing scheme:
Fixed size: canonical approach
Deterministic methods: function-based adjustment (e.g. Saw-tooth)
Adaptive methods: on-line adjustment (e.g. GAVaPS)
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 4 / 17

Population Sizing
Sizing scheme:
Fixed size: canonical approach
Deterministic methods: function-based adjustment (e.g. Saw-tooth)
Adaptive methods: on-line adjustment (e.g. GAVaPS)
Sizing theory:
Focus is on the correct sizing of population for the fixed-sized scheme
But theory for fixed-size scheme can be helpful for variable-size schemes
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 4 / 17

Generalized l-trap Function
l-trap function (Ackley, 1987):
l: problem size (number of
possible values in range)
a: value of local optimum
b: value of global optimum
z: slope-change location
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 5 / 17

Generalized l-trap Function
l-trap function (Ackley, 1987):
l: problem size (number of
possible values in range)
a: value of local optimum
b: value of global optimum
z: slope-change location
Currently, experiments with
a = l − 1, b = l and z = l − 1
2-trap: not deceptive
3-trap: partially deceptive
4-trap: deceptive
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 5 / 17

Scaling the Problem Difficulty
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 6 / 17

Scaling the Problem Difficulty
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 6 / 17

Working Hypothesis
Minimizing number of solutions evaluated while guaranteeing a
success rate
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 7 / 17

Working Hypothesis
Minimizing number of solutions evaluated while guaranteeing a
success rate
Working hypothesis: larger population required at the beginning
Start with a diverse sampling of the search space
As convergence occurs, smaller population required
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 7 / 17

Working Hypothesis
Minimizing number of solutions evaluated while guaranteeing a
success rate
Working hypothesis: larger population required at the beginning
Start with a diverse sampling of the search space
As convergence occurs, smaller population required
Use a deterministic schedule of the population size
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 7 / 17

Working Hypothesis
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 8 / 17

Simple Variable Population Sizing (SVPS)
Reduce population by a variable ratio at each generation:
τ
g
ng = n0 1 − (1 − ρ)
gmax
n0 : initial population size
ng : population size at generation g
g : current generation number
gmax : last generation number
τ : resizing speed parameter
ρ: resizing severity parameter
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 9 / 17

Simple Variable Population Sizing (SVPS)
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 10 / 17

Estimating the Correct Population Size (SR of 0.98)
1) Rough estimation (ni+1 = 2ni ):
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

Estimating the Correct Population Size (SR of 0.98)
1) Rough estimation (ni+1 = 2ni ):
n1 = 4, SR=0.2
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
n4 = 12, SR=0.99
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
n4 = 12, SR=0.99 n5 = 10, SR=0.982
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
n4 = 12, SR=0.99 n5 = 10, SR=0.982
3) Refinement (ni+1 = 0.99ni ):
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
n4 = 12, SR=0.99 n5 = 10, SR=0.982
3) Refinement (ni+1 = 0.99ni ):
n6 = 9, SR=0.9803
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

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
nimax +nimin nimax −nimin 1
2) Bisection (ni+1 = 2 ), stop when nimin
< 16 :
n4 = 12, SR=0.99 n5 = 10, SR=0.982
3) Refinement (ni+1 = 0.99ni ):
n6 = 9, SR=0.9803
Correct population size is 9 for a success rate of 0.98
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 11 / 17

Population Sizes for a Success Rate of 0.98
m: number of concatenated trap functions
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 12 / 17

Experimental Setting
Selectorecombinative binary Genetic Algorithm:
Population sizes set according to bisection method for a success rate of
0.98
Two parents tournament selection
One-point crossover (probability of 1.0)
No mutation
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 13 / 17

Experimental Setting
Selectorecombinative binary Genetic Algorithm:
Population sizes set according to bisection method for a success rate of
0.98
Two parents tournament selection
One-point crossover (probability of 1.0)
No mutation
Trap problems tested:
Problem sizes, l = {2, 3, 4}
Number of sub-functions, m = {2, 4, 8, 16, 32, 64}
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 13 / 17

Experimental Setting
Selectorecombinative binary Genetic Algorithm:
Population sizes set according to bisection method for a success rate of
0.98
Two parents tournament selection
One-point crossover (probability of 1.0)
No mutation
Trap problems tested:
Problem sizes, l = {2, 3, 4}
Number of sub-functions, m = {2, 4, 8, 16, 32, 64}
SVPS setting:
Speed, τ = 0.125, . . .×1.5 , 32
Severity, ρ = 0.25, . . .+0.05 , 1
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 13 / 17

Speed (τ ) and Severity (ρ)
Size of circles show improvement over fixed-size population
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 14 / 17

Saved Computational Effort
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 15 / 17

Conclusion
SVPS requires a smaller number of evaluations than a fixed
population sizing scheme
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 16 / 17

Conclusion
SVPS requires a smaller number of evaluations than a fixed
population sizing scheme
The improvement is much more noticeable for large population sizes
as the problem instances scale
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 16 / 17

Conclusion
SVPS requires a smaller number of evaluations than a fixed
population sizing scheme
The improvement is much more noticeable for large population sizes
as the problem instances scale
There is not a single but a set of possible strategies for SVPS
(different τ -ρ combinations)
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 16 / 17

Questions
Thanks for your attention!
Laredo et al. (Granada / Laval) Improving GAs via Population Shrinkage GECCO 2009 17 / 17