Distributed Probabilistic ModelBuilding Genetic Algorithm

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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)
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DPMBGA

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

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Genetic Algorithm
New search points are generated by Crossover and
Mutation. Good characteristics of parents should
be inherited to children.
Evaluation
Selection
Crossover
Mutation
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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.
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Classification of PMBGAs (Pelikan, 1999)
Type of design variables
f (x1, x2)
0
1
0
0
1
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)
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Classification of PMBGA
DPMBGA
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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.
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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.
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Selected Population
x2
Some individuals are selected.
Population
Sample population
x1
Best m individuals are chosen from the population.
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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.

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Archive for CPA
Population
Archive

• The best individuals in each generation are
  restored in the archive.
• Archive has the size

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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.

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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.

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Transformation of individuals
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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
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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.
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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.

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Overview of DPMBGA
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Target Problems (1)

• Functions that have no correlation between the
  design variables

n20
n10
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Target Problems (2)

• Functions that have the correlations

n20
?
n20
n20
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Parameters
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Results
Optimum value 1.0E-10 ,Terminal condition
number of evaluations 3.0E06
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Rastrigin, Rosenbrock

• History of the number of renewed individuals in
  the archive

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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.
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Distributed Environment Scheme (DES)

• In some sub populations, PCA is performed
• In some sub populations, PCA is not performed

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Results
Optimum value 1.0E-10 ,Terminal condition of
evaluations 3.0E06
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The Average of number of evaluations to get
optimum
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DPMBGA

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

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Comparison with UNDXMGG

• 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

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Parameters
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Results(Rastrigin, Schwefel)
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Results(Rosenbrock, Ridge)
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Result(Griewank)
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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.

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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
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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
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Used Parameters
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History of the search (Rastrigin, Schwefel)

• Problems that have not the correlation

• The model without BEM derived the better results.

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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

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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 UNDXMGG
• 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

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