Discussion of Search Phases of Probabilistic Model-Building Genetic Algorithms

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Discussion of Search Phases of Probabilistic
Model-Building Genetic Algorithms
Masaki Sano (Doshisha
University) Tomoyuki Hiroyasu (Doshisha
University) Mitsunori Miki (Doshisha
University)
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Genetic Algorithms

• Genetic Algorithms (GAs)
• Optimization method based on the mechanism of
  natural selection and natural genetics.

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Individual
chromosome

• Individuals search points
• Each individual has design variables encoded to
  chromosome and fitness.

population
selection

• Genetic Operators
• Selection reproduces individuals with high
  fitness.
• Crossover produces new individuals by
  recombining parents chromosomes.
• Mutation modifies a chromosome.

crossover
mutation
genetic operators
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Background

• A disadvantage of GAs
• Parameter settings of GAs are difficult because
  there are a lot of parameters which must be set.

• Related studies
• Suitable parameter setting (dejong1975,
  back1993, ochoa1999).
• Algorithm with fixed parameters, e.g.
  Parameter-free GA(sawai1999), Parameter-less
  GA(harik1999).
• Adaptive adjustment of parameters (toshine2001,
  kee2001).

In this study, a guide to a strategy of adaptive
parameter setting of a canonical GA is given by
the transition of the variance of the function
evaluation values
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Test Functions
n30
n30
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Parameters of GA
GA
Population Size 512
Number of Elites 1
Coding Gray Coding
Chromosome Length (L) 300 (30 dim. 10bits)
Selection Roulette Selection
Crossover 1-Point Crossover
Crossover Rate 0.8
Mutation Rate 0.0033333 (1/L)
Trials 20 (average)
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History of Solution of GA

• Best and variance of function evaluation values

• Variance of evaluation values keeps decreasing
  in the early stage
• Variance starts increasing in a certain
  generation
• 160 generation for Rastrigin
• 80 generation for Griewank

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Classification of Search Phase of GA

• Search Process of GA is classified into 3 phases

phase 1

• Solutions converge rapidly
• Variance decreases rapidly

phase 2

• Solutions improve slowly
• Variance increases

phase 3

• The search is converged or the optimum is
  found
• Variance stays

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Distribution of Function Value(Griewank)
phase 1
phase 2

• phase 1 The whole population shifts toward the
  optimum value.
• phase 2 Only the individuals which has
  relatively high fitness progress.

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Effectiveness of Crossover and Mutation

• 3 types of crossover and mutation in each phase
  are discussed.
• In GAs, similar chromosomes spread in population
  as search progress.
• Mutation plays more important role than
  crossover when population loses diversity
  (spears1993).
• Mutation may be more effective than crossover in
  the phase2.

Crossover Rate Crossover Rate
phase 1 phase 2, 3
crossover A 0.8 0.8
crossover B 0.8 0.0
crossover C 0.0 0.0
Mutation Rate Mutation Rate
phase 1 phase 2, 3
mutation A 1/L 1/L
mutation B 1/L 0.0
mutation C 0.0 1/L
L chromosome length 1/L 0.00333
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History of Solution (Rastrigin)

• The search performance of the crossover B (only
  phase 1) is nearly equal to the one of the
  crossover A (with crossover).

• The search performance of the mutation C (only
  phase 2) is nearly equal to the one of the
  mutation A (with mutation).

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Probabilistic Model-Building GAs (PMBGAs)

• Recently proposed models of GAs
• PMBGAs use a probabilistic model of promising
  solutions to guide further exploration of the
  search space

Estimation of Probability Distribution
1. Select better individuals
2. Construct probabilistic model from the
estimated distribution
population
3. Generate new individuals according to the
probabilistic model
Probabilistic Model
PMBGAs use a probabilistic model of promising
individuals instead of crossover
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PMBGA with Principal Component Analysis
x2
1. Select good individuals
population
x1
5. Substitute for old individuals
2. Reduce correlation of design variables by
Principal Component Analysis (PCA)
x2
4. Restore correlation
v1
v2
x1
x2
x1
3. Generate new individuals according to normal
distribution
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Parameters of the PMBGA
PMBGA with PCA
Population 512
Chromosome Length (L) 30 (same as dimension)
Number of Elites 1
Selection Select best individuals
Sampling Rate 0.1
Amplification of Variance 1.2
Mutation Rate 0.0033334 (1 / 10L)
Trials 20
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Experimental Results of the PMBGA

• Best and variance of evaluation values

• Unlike the canonical GA, phase 2 does not exist.
• advantage Whole population explore solutions
  effectively.
• disadvantage There is possibility that all of
  individuals
• converged to the local minima.

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Conclusion

• Discussion of the search process of a canonical
  Genetic Algorithm(GA) and a Probabilistic
  Model-Building GA (PMBGA).
• The search process of the GA is classified into
  3 phases according to the transition of the
  variance of evaluation value.

A strategy which gives greater importance to
crossover in the phase 1, and gives greater
importance to mutation in the phase 2 is
effective.

• The PMBGA find good solutions because whole
  population explore solutions effectively, but
  there is possibility that all of individuals are
  converged on the local minimum.

Maintenance of the diversity of the population is
necessary in the PMBGA.
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Appendix
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Outline of This Presentation

• Comparison of a PMBGA with a canonical GA for
  the search process
• The search process of canonical GA is classified
  by the transition of the variance of function
  evaluation values.
• Effectiveness of the Genetic Operators in each
  search phase is discussed.
• crossover, mutation
• The difference of the search process between a
  PMBGA and GA is discussed.

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Reason for increasing of variance (phase 2)

• In canonical GAs, design variables are encoded
  from real value to bit string (chromosome). A
  slight change of chromosome may cause a big
  change of design variables.

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x2
small change
big change
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x1
x2
x1
chromosome
search space

• As the search progress, good chromosomes spread
  in population.

Crossover and mutation change fitness for the
worse.
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Genetic Algorithms (1)

• Genetic Algorithms (GAs)
• Optimization method based on the mechanism of
  natural selection and natural genetics.
• probabilistic transition rules
• multi-point search

f (x1, x2)
search point
encode

• Individuals search points
• Each individual has design variables and
  fitness.
• Design variables are encoded from real value to
  bit string (chromosome)

population
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1
0
0
1
1
Individual
x1
x2
chromosome
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Genetic Algorithms (2)

• Genetic Operators
• selection reproduce individuals with high
  fitness.
• crossover produce new individuals by
  recombining parents chromosomes.
• mutation modify chromosome.

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selection
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crossover
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0
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mutation
After repeatedly application of genetic
operators, GAs find good solutions
genetic operators
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Test Functions (1)

• Independence between design variables

n20
n10
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Test Functions (2)

• Dependence between design variables

n20
?
n20
n20
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Distoribution of Function Value(Rastrigin)
phase 1
phase 2

• phase 1 whole population go toward the optimum
  value.
• phase 2 only individuals which has relatively
  high fitness progress.

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Effectiveness of Elitism

• Comparison of 4 types of elitism
• It is discussed that individuals with how high
  fitness are play an important role in search in
  each phase.

Number of Elites Number of Elites
phase 1 phase 2, 3
elitism A 1 64
elitism B 64 1
elitism C 1 1
elitism C 64 64
elites
genetic operators
elitism
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Function Value at End of Search

• Average of Best evaluation values at the end of
  searches

• the GA with many elites in the phase 2 finds the
  better solution than with many elites in the
  phase 1.

Effectiveness of elites in phase 2 is higher than
in phase 1
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History of Function Value (Elitism)

• the GA with many elites in the phase 2 finds the
  better solution than with many elites in the
  phase 1.

Effectiveness of elites in phase 2 is higher than
in phase 1
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History of Solution (Crossover Rate)

• the crossover B (rate 0.8 ? 0.0) shows the
  nearly equal to the search performance of the
  crossover A (rate 0.8 in all phases).

Crossover is more important in the phase 1 than
in the phase 2.
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History of Solution (Mutation Rate)

• Accuracy of the solution obtained in the
  mutation A (rate 1/L in all phases) is nearly
  equal to the one in the mutation C (rate 0.0 ?
  1/L).

Mutation is more important in the phase 2 than in
the phase 1.
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Distributed PMBGA

• Distributed PMBGA (DPMBGA)
• The population is divided into subpopulations
• Each Individual has real value chromosome
• Migration of individualsbetween subpopulations
• Maintenance of diversity
• Generate offsprings considering dependence
  between design variables by Principal Component
  Analysis(PCA)

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Distributed Environment Scheme

• Distributed Environment Scheme(DES) (Miki, 1998)
• whole population is divided into subpopulations
• parameters and operators in each subpopulation
  are different from each other.
• crossover rate, mutation rate, selection
  operator , etc.
• DES considering dependence between design
  variables
• use of Principal Component Anarysis(PCA)differs
  .

with PCA
without PCA
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Distributed Genetic Algorithm

• Distributed Genetic Algorithm(DGA)
• island model
• apply genetic operator in each island
• migration
• maintenance of diversity
• high search ability (Tanese, Belding)