Post on 15-Jan-2016
Auto-Tuning Fuzzy Granulation for Evolutionary Optimization
Auto-Tuning Fuzzy Granulation for Evolutionary Optimization
Mohsen Davarynejad, Ferdowsi University of Mashhad, Mohsen Davarynejad, Ferdowsi University of Mashhad, davarynejad@kiaeee.org
M.-R.Akbarzadeh-T., Ferdowsi University of Mashhad. M.-R.Akbarzadeh-T., Ferdowsi University of Mashhad. akbarzadeh@ieee.org
Carlos A. Coello Coello, CINVESTAV-IPN, Carlos A. Coello Coello, CINVESTAV-IPN, ccoello@cs.cinvestav.mx
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Brief Introduction to Evolutionary Algorithms
• Generic structure evolutionary algorithms– Problem representation (encoding) – Selection, recombination and mutation– Fitness evaluations
• Evolutionary algorithms– Genetic algorithms – Evolution strategies – Genetic programming …
• Pros and cons– Stochastic, global search – No requirement for derivative information– Need large number of fitness evaluations– Not well-suitable for on-line optimization
Motivations(When fitness approximation is necessary?)
• No explicit fitness function exists: to define fitness quantitatively
• Fitness evaluation is highly time-consuming: to reduce computation time
• Fitness is noisy: to cancel out noise• Search for robust solutions: to avoid additional
expensive fitness evaluations
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Fitness Approximation Methods
• Problem approximation– To replace experiments with simulations– To replace full simulations /models with reduced simulations / models
• Ad hoc methods– Fitness inheritance (from parents)– Fitness imitation (from brothers and sisters)
• Data-driven functional approximation (Meta-Models)– Polynomials (Response surface methodology)– Neural networks, e.g., multilayer perceptrons (MLPs), RBFN– Support vector machines (SVM)
Fitness Approximation Methods
• Problem approximation– To replace experiments with simulations– To replace full simulations /models with reduced simulations / models
• Ad hoc methods– Fitness inheritance (from parents)– Fitness imitation (from brothers and sisters)
• Data-driven functional approximation (Meta-Models)– Polynomials (Response surface methodology)– Neural networks, e.g., multilayer perceptrons (MLPs), RBFN– Support vector machines (SVM)
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Meta-Model in Fitness Evaluations*
• Use Meta-Models Only: Risk of False Convergence
• So a combination of meta-model with the original fitness function is necessary– Generation-Based Evolution Control – Individual-Based Evolution Control (random strategy
or a best strategy)
*Yaochu Jin. “A comprehensive survey of fitness approximation inevolutionary computation”. Soft Computing, 9(1), 3-12, 2005
Generation-Based Evolution Control
Generation t Generation t+1 Generation t+2 Generation t+3 Generation t+4
Individual-Based Evolution Control
Generation t Generation t+1 Generation t+2 Generation t+3 Generation t+4
Fitness Approximation Methods
Problem approximation Meta-Model Ad hoc methods
Generation-Based Individual-Based
Fitness Approximation Methods
Problem approximation Meta-Model Ad hoc methods
Generation-Based Individual-Based
ADAPTIVE FUZZY FITNESS GRANULATION is Individual-Based Meta-Model Fitness Approximation
Method
GRADUATION AND GRANULATION
• The basic concepts of graduation and granulation form the core of FL and are the principal distinguishing features of fuzzy logic.
• More specifically, in fuzzy logic everything is or is allowed to be graduated, or equivalently, fuzzy.
• Furthermore, in fuzzy logic everything is or is allowed to be granulated, with a granule being a clump of attribute-values drawn together by indistinguishability, similarity, proximity or functionality.
• Graduated granulation, or equivalently fuzzy granulation, is a unique feature of fuzzy logic.
• Graduated granulation is inspired by the way in which humans deal with complexity and imprecision.*
*L. A. Zadeh, www.fuzzieee2007.org/ZadehFUZZ-IEEE2007London.pdf
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Fuzzy Similarity Analysis based on Granulation Pool
Yes FF Evaluation
ADAPTIVE FUZZY FITNESS
GRANULATION (AFFG)
FF Association No
Phenospace
Fitness of Individual
Update Granulation Table
Create initial random population
Evaluate Fitness of solution exactly
Termination Criterion Satisfied?
Designed Results
Is the solution similar to an existing
granule?
Gen:=0
Gen:=Gen+1
Yes
No
Yes
No
Look up fitness from the queue of
granules
End
Select a solution
Update table life function and granule radius
Yes
Add to the queue as a new granule
No
AFFG Part
Finished Fitness Evaluation of population?
Perform Reproduction
Perform Crossover and Mutation
Flowchart of the Purposed
AFFG Algorithm
AFFG (III)
• Random parent population is initially created.
• Here,
• G is a set of fuzzy granules,
1112
110 , ... , , ... , , tj XXXXP
1112
110 , ... , , ... , , tj XXXXP
imj
irj
ij
ij
ij xxxxX ,,2,1, , ... , , ... , ,
},...,1,,,|,,{ lkLCLCG kkm
kkkk
AFFG (IV)
• Calculate average similarity of a new solution to each granule is:kG
m
r
irjrk
kj m
x
1
,,,
)(
otherwiseXf
CfXf
ij
ikj
lkKi
j
function fitnessby computed
max if , ..., 2, 1,
AFFG (V)
1
112
11 )(...),(),(
.
i
it
iii
f
XfXfXfMax
)(
1kCf
ke
i
larger and Since members of the initial populations generally have less fitness, larger and is smaller, fitness is assumed more often initially by estimation/association to the granules.
ki
,/exp 2,
2
,,,, rkrkirj
irjrk Cxx
AFFG (V)
1
112
11 )(...),(),(
.
i
it
iii
f
XfXfXfMax
)(
1kCf
ke
i
larger and Since members of the initial populations generally have less fitness, larger and is smaller, fitness is assumed more often initially by estimation/association to the granules.
ki
,/exp 2,
2
,,,, rkrkirj
irjrk Cxx
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
AFFG - FS
• In order to avoid tuning parameters, a fuzzy supervisor as auto- tuning algorithm is employed with three inputs.
• Number of Design Variables (NDV)
• Maximum Range of Design Variables (MRDV)
• Percentage of Completed Generations (PCG)
• The knowledge base of the above architecture has large number of rules and the extraction of these rules is very difficult. Consequently the new architecture is proposed in which the controller is separated in two controllers to diminish the complexity of rules.
AFFG – FS • Granules compete for their survival through a life index.
• is initially set at N and subsequently updated as below:
• Where M is the life reward of the granule and K is the index of the winning granule for each individual in generation i. Here we set M = 5.
• At each table update, only granules with highest index are kept, and others are discarded.
kL
OtherwiseL
KkifMLL
k
kk
GN kL
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
n
i
in
i
i nii
xx
1 1
2
;:1,)cos(4000
1
;600600 ix
n
iii xxnxf
1
2 ))2cos(10(10)(
;12.512.5;:1 ixni
n
ii
n
ii x
nx
neeexf 11
2 2cos11
2.0
2020)(
;768.32768.32;:1 ixni
List of Test benchmark Functions
Function Formula
Griewangk
Rastrigin
Akley
Parameters used for AFFG
Function
Griewangk 0.00012 190
Rastrigin 0.004 0.15
Akley 0.02 0.25
is considered as a constant parameter and is equal to 0.9 for all simulation results
0 100 200 300 400 500
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Griewangk, (5-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 200 400 600 800 10000
1
2
3
4
5
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Griewangk, (10-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 20000
1
2
3
4
5
6
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Griewangk, (20-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 2000 2500 3000
1
2
3
4
5
6
7
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Griewangk, (30-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 100 200 300 400 500
2
2.5
3
3.5
4
4.5
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Rastrigin, (5-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 200 400 600 800 1000
3.2
3.4
3.6
3.8
4
4.2
4.4
4.6
4.8
5
5.2
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Rastrigin, (10-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 20004
4.2
4.4
4.6
4.8
5
5.2
5.4
5.6
5.8
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Rastrigin, (20-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 2000 2500 30004.4
4.6
4.8
5
5.2
5.4
5.6
5.8
6
6.2
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Rastrigin, (30-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 100 200 300 400 500
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Akley, (5-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 200 400 600 800 1000
2.4
2.5
2.6
2.7
2.8
2.9
3
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Akley, (10-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 2000
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Akley, (20-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
0 500 1000 1500 2000 2500 3000
2.6
2.7
2.8
2.9
3
3.1
3.2
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Akley, (30-D)
GAAFFGAFFG-FSFES-.5FES-0.7FES-0.9
Effect of on the convergence
behavior
• only granules with highest index are kept, and others are discarded.
• The effect of varying number of granules on the convergence behavior of AFFG and AFFG-FS is studied.
GN kL
GN
0 200 400 600 800 1000
0
0.5
1
1.5
2
2.5
3
3.5
4
4.5
5
5.5
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Griewangk, (10-D)
AFFG, 20AFFG, 50AFFG, 100AFFG-FS, 20AFFG-FS, 50AFFG-FS, 100
0 200 400 600 800 10003
3.5
4
4.5
5
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Rastrigin, (10-D)
AFFG, 20AFFG, 50AFFG, 100AFFG-FS, 20AFFG-FS, 50AFFG-FS, 100
0 200 400 600 800 1000
2.2
2.3
2.4
2.5
2.6
2.7
2.8
2.9
3
3.1
Exact Function Evaluation Count
log
(Fitn
ess
Va
lue
)
Akley, (10-D)
AFFG, 20AFFG, 50AFFG, 100AFFG-FS, 20AFFG-FS, 50AFFG-FS, 100
Outline of the presentation
• Brief introduction to evolutionary algorithms• Fitness Approximation Methods• Meta-Models in Fitness Evaluations• Adaptive Fuzzy Fitness Granulation (AFFG)• AFFG with fuzzy supervisory• Numerical Results• Summery and Contributions
Summery
• Fitness approximation Replaces an accurate, but costly function evaluation with approximate, but cheap function evaluations.
• Meta-modeling and other fitness approximation techniques have found a wide range of applications.
• Proper control of meta-models plays a critical role in the success of using meta-models.
• Small number of individuals in granule pool can still make good results.
Contribution• Why AFFG-FS?
– Avoids initial training.– Uses guided association to speed up search process. – Gradually sets up an independent model of initial training data to compensate
the lack of sufficient training data and to reach a model with sufficient approximation accuracy.
– Avoids the use of model in unrepresented design variable regions in the training set.
– In order to avoid tuning of parameters, A fuzzy supervisor as auto-tuning algorithm is being employed.
• Features of AFFG-FS: – Exploits design variable space by means of Fuzzy Similarity Analysis (FSA) to
avoid premature convergence.– Dynamically updates the association model with minimal overhead cost.