This document presents research on improving genetic algorithms via deterministic population shrinkage. The researchers hypothesize that genetic algorithms require different population sizes at different convergence stages, with larger populations needed early on. They propose a Simple Variable Population Sizing (SVPS) scheme where the population shrinks deterministically over generations. Experiments on generalized l-trap functions of varying difficulty show that SVPS can improve performance over a fixed population size by minimizing the number of solutions evaluated while maintaining solution quality.

1. 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

2. 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

3. 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

4. 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

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

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

7. 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

8. 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

9. 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

10. 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

11. 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

12. 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

13. 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

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

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

16. 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

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

18. 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

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

20. 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

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

22. 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

23. 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

24. 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

25. 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

26. 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

27. 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

28. 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

29. 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

30. 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

31. 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

32. 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

33. 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

34. 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

35. 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

36. 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

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

38. 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

39. 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

40. 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

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