Distributed Probabilistic Model-Building Genetic Algorithm

1. Distributed Probabilistic Model-BuildingGenetic Algorithm Tomoyuki Hiroyasu Mitsunori Miki Masaki Sano Hisashi Shimosaka Shigeyoshi Tsutsui Jack Dongarra (Doshisha University) (Doshisha University) (Doshisha University) (Doshisha University) (Hannan University) (University of Tennessee)

2. DPMBGA • Probabilistic Model Building GA • Principle Component Analysis (PCA) • Distributed Population Scheme • Distributed Environment Scheme

3. Genetic Algorithm New search points are generated by Crossover and Mutation. Good characteristics of parents should be inherited to children. Evaluation Selection Crossover Mutation

4. Probabilistic Model-Building GA (1) Select better individuals Estimation of the Distribution Individual (2) Construct a probabilistic model Population Probabilistic Model (3) Generate new individualsand substitute them for old individuals New individuals are generated from estimated probabilistic model instead of crossover and mutation.

5. Classification of PMBGAs (Pelikan, 1999) Type of design variables f (x1, x2) 0 1 0 1 0 1 Bit Strings Real Vectors x1 x2 Correlation among the design variables • The method does not care for the correlation. • The method cares the correlation between the two design variables. • The method cares the correlation among more than three design variables. F(x1, x2, …, xn)=F(x1)+F(x2)+…+F(xn) F(x1, x2, x3)=F(x1,x2)+F(x3)

6. Classification of PMBGA DPMBGA

7. For the effective search • Maintaining the diversity of the solutions • Consideration of the correlation among the design variables Distributed Population Scheme x2 The selected individuals are transferred into another space by PCA. x1 Probabilistic model is constructed. New points are generated. These points are transferred back to the original space.

8. v1 v2 The overflow of the operations x2 (1) Individuals who have better fitness values are selected. Population v1 v2 x1 (4) New individuals are transferred into the original space. (2) Individuals are transferred into the space where there is no correlation among the design variables. (3) new individuals are generated from normal distributed model.

9. Selected Population x2 Some individuals are selected. Population Sample population x1 Best m individuals are chosen from the population.

10. v1 v2 x1 Individuals are transferred into the new space x2 • The Goal • New individuals are generated by consideration of the correlation among the design variables. • The flow of the operations 1. The archive for the PCA is renewed. 2. The individuals in the archive are analyzed by PCA. 3. The Individuals are transferred into the space where there is no correlation among the design variables.

11. Generation one Generation two Generation three Archive for CPA Population Archive • The best individuals in each generation are restored in the archive. • Archive has the size

12. v1 v2 x1 Individuals are transferred into the new space x2 • The Goal • New individuals are generated by consideration of the correlation among the design variables. • The flow of the operations 1. The archive for the CPA is renewed. 2. The individuals in the archive are analyzed by CPA. 3. The Individuals are transferred into the space where there is no correlation among the design variables.

13. x2 v1 v2 Distribution of individuals x1 Principle Component Analysis • PCA analysis for individuals in the archive • Define the Covariance Matrix S in the design field. • Derive the eigen vectors V = (V1, V2, …, VD, ) of S • When one eigen value is bigger than others, the distribution is biased to the direction that is corresponding to the eigen value. • This means that there is strong correlation is existed.

14. x2 v1 v2 x1 Transformation of individuals

15. v1 v2 New individual generation Population New individuals are substituted for some old individuals Moved back to the original space generate new individuals Distribution in the new space Normal distribution

16. Probabilistic Model-Building GA (1) Select better individuals Estimation of the Distribution Individual (2) Construct a probabilistic model Population Probabilistic Model (3) Generate new individualsand substitute them for old individuals Because the model is constructed with the elite individuals, early convergence sometimes happens. The mechanism that keeps the diversity of the solution is needed.

17. Distributed Population Scheme • Distributed GA（DGA）island model • Total population is divided into sub populations. • GA operations are performed in each sub population. • Migration • Parallel Efficiency • Ability to keep the diversity of the solutions. • High searching capability.

18. Overview of DPMBGA

19. Target Problems (1) • Functions that have no correlation between the design variables n=20 n=10

20. Target Problems (2) • Functions that have the correlations n=20 ※ n=20 n=20

21. Parameters

22. Results Optimum value： 1.0E-10 ，Terminal condition ： number of evaluations 3.0E+06

23. Rastrigin, Rosenbrock • History of the number of renewed individuals in the archive

24. Archive is eliminated every 10 generation • Rastrigin ： erase/10 is better • Rosenbrock ： normal is better When the number of renewed individuals becomes small, PCA does not work well.

25. with PCA without PCA Distributed Environment Scheme (DES) • In some sub populations, PCA is performed • In some sub populations, PCA is not performed

26. Results Optimum value： 1.0E-10 ，Terminal condition ： # of evaluations 3.0E+06

27. The Average of number of evaluations to get optimum

28. DPMBGA • Probabilistic Model Building GA • Principle Component Analysis (PCA) • Distributed Population Scheme • Distributed Environment Scheme

29. C1 P2 C2 P1 Parents Children Comparison with UNDX+MGG • Unimodal Normal Distribution Crossover （UNDX）（ Ono et al., 1999） • Typical Real-Coded GA • It has a strong search capability. • Minimal Generation Gap （MGG）（Sato et al., 1997） • Generation Alternate Model • MGG can maintain the diversity of the solutions

30. Parameters

31. Results（Rastrigin, Schwefel）

32. Results（Rosenbrock, Ridge）

33. Result（Griewank）

34. f(x) Extended Region Feasible region x Functions whose optimums locate near the boundary • Problems in Real-Coded GAs • The searching capability may decrease for the problems whose optimum locates near the boundary of the feasible region. • Boundary Extension by Mirroring (BEM) （Tsutsui,1998） • Semi-feasible region is prepared • It is reported that BEM is useful for the problems whose optimum locates near the boundary.

35. Handling the constraints • The operation for the individuals that violate the constraints in DPMBGA • The corresponding individual is pulled back to the edge of the feasible field. • When the optimum point locates the near the boundary, there is a possibility that the probabilistic model cannot be constructed correctly. x2 Out of constraints Feasible field x1

36. x2 x2 Optimum x1 x1 x2 Test Problems • Test problems • The range of design variable is modified. • The optimum locates on the boundary of the feasible region. Example）Rastrigin, Ridge x1

37. Used Parameters

38. History of the search （Rastrigin, Schwefel） • Problems that have not the correlation • The model without BEM derived the better results.

39. History of the search（Rosenbrock, Ridge） • Problems that have the correlation • The model without BEM derived the better results. • The proposed model works for these problems

40. Conclusions • DPMBGA • The diversity of the population is maintained by the distribute population scheme. • The correlation among the design variables are considered by using PCA. • Effectiveness of PCA • Because of the individuals in the archives, sometimes PCA does not work well. • Distributed Environment Scheme is useful. • Comparison with UNDX+MGG • DPMBGA derived the better solutions. • Problems where the solution locates the edge of the design field • BEM or other special mechanism is not necessary.s