Statistical Mechanics For G As
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Transcript of Statistical Mechanics For G As
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Statistical Mechanics for GAsA gentle introduction
Presented by :
Yann SEMET
Universite de Technologie de Compiegne
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Overview
Motivations and general idea
Definitions
Selection analysis (pb. independent)
Problem specific analysis : mutation, crossover
Results on OneMax
Beyond simple problems
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2 documents
An analysis of Genetic Algorithms Using Statistical Mechanics
Prugel-Bennett, Shapiro. 1994
Modeling the dynamics of Gas using Statistical Mechanics
Rattray. 1996
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Motivations
Markov chains :Exact model
Gets intractable with size
Statistical mechanicsProbabilistic model
More compact
Macroscopic description
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Macroscopics
Cumulants
Mean correlation
Evaluates the evolution of fitness distribution
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Modelling the dynamics
Each genetic operator :A set of difference equations (on macroscopics)
Iteration
Non trivial terms : Maximum entropy ansatz
Finite population effectsA finite sample from an infinite population
Selection
Infinite population again
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Definitions
Genotype, phenotype and fitness :
Fitness distribution :
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Cumulants
Definition :
First two :
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Cumulants (cont.)
Infinite population :
Finite sample corrections :
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Gram-Charlier expansion
A convenient approximation
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Correlation
A measure of genotype similarity
Mean value :
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Best population member
Our goal after all
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Modelling selection
Problem independent
A general scheme :
2 stages :Random sampling from infinite population
Generating a new infinite one
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Selection (cont.)
Generating the cumulants :
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Selection (cont.)
Expansion :
Finally :
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Correlation after selection
2 terms :
Duplication
Natural change (problem specific)
Final approximation :
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Selection schemes
Particular schemes :Boltzmann
Truncation
Ranking
Tournament
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Tackling problems
Problem specific operators
A convenient class of problems : Functions of an additive genotype
Cumulants :
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Mutation (1)
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Mutation (2)
Cumulants :
Notice non trivial terms
Correlation :
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Crossover
A generalized form of uniform crossover :
Cumulants :
Mean correlation unchanged
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Maximum entropy ansatz
Calculate terms non trivially related to known macroscopics
Assumptions on allele distribution
2 constraintsMean phenotype
Correlation
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Results
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Onemax problem
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2nd class of problems
Fitness=stochastic function of phenotype
Test problem : perceptron with binary weights
Competent model :Size population accurately to remove noise
Size training batch consequently
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A NP hard problem
Storing random pattern in a binary perceptron
Insight gained
Half failure :Technical difficulties
Inconsistencies
Incomplete model
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Extensions of the model
Two tests :1 simple diploid GA
1 temporally varying fitness
Successful description under :Bit-simulated crossover
Extra constraints for MEA
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Summary
Motivation of macroscopic models
Cumulants
Mean correlation
Dynamics modeling
Simple problems
Extensions
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Conclusions
StrengthsCompactness and accuracy
Finite population effects
WeaknessesLimitations of MAE
NP hard problem inaccurately modeled
Technical limitation
Fundamental limitations ?
Punctuated equilibria