Diversity Preservation in Evolutionary Algorithms Jiří Kubalík Intelligent Data Analysis Group Department of Cybernetics CTU Prague

Diversity Preservation in Evolutionary Algorithms

Jiří Kubalík

Intelligent Data Analysis Group

Department of Cybernetics

CTU Prague

Diversity Preservation in EAs2

EAs and Premature ConvergenceEAs and Premature Convergence Evolutionary cycle Homogeneous

population

Premature convergence - as the population gets homogeneous, only a little new can be evolved and EA converges to suboptimal solution.

Causes of premature convergence: improper representation and genetic operators, improper

selection pressure, insufficient population size, deception


GA with Limited ConvergenceGA with Limited Convergence (GALCO)(GALCO)

Motivation to maintain a diversity of the evolved population and extend the

explorative power of the algorithm Realization

Convergence of the population is allowed up to specified extent Convergence at individual positions of the representation is

controlled Convergence rate

specifies a maximal difference in the frequency of ones and zeroes in every column of the population

ranges from 0 to PopSize/2 Principal condition

at any position of the representation neither ones nor zeroes can exceed the frequency constraint

Specific way of modifying the population genotype


GALCOGALCO:: Algorithm Algorithm

1. Generate initial population

2. Choose parents

3. Create offspring

4. if (offspring > parents)

then

replace parents with offspring

else{

find(replacement)

replace_with_mask(child1, replacement)

find(replacement)

replace_with_mask(child2, replacement) }

5. if (not finished) then go to step 2


GALCO: replace_with_mask GALCO: replace_with_mask

Mask – vector of integer counters; stores a number of 1s for each bit of the representation

50


Static Test ProblemsStatic Test Problems

Multimodal problem Deceptive problem

Hierarchical problem Royal Road Problem


GALCO: Finding Optimal cGALCO: Finding Optimal c

multimodal

hierarchicalroyal road

deceptive


GALCO: Comparison with SGAGALCO: Comparison with SGA

multimodal

hierarchicalroyal road

deceptive


GALCO: Multimodal Optimization GALCO: Multimodal Optimization

Initial population SIGA

with replace_with_mask without


GALCO: Multimodal Optimization GALCO: Multimodal Optimization (cnd.)(cnd.)

Initial population GALCO SGA


GA with GA with Real-coded Binary Rep.Real-coded Binary Rep.

Motivation using redundant representation, where many different

genotypes map to the same phenotype would increase the explorative power of the EA and decrease the probability of getting stuck in a local optimum

Realization real coded binary representation

Effect population can not converge to the homogeneous state so that

the premature convergence can not take place


Pseudo-binary representation – binary gene values coded by real numbers from the interval 0.0, 1.0

Example:

ch1 = [0.92 0.07 0.23 0.62]

ch2 = [0.65 0.19 0.41 0.86]

interpretation(ch1)=interpretation(ch2)=[1001]

Gene strength – gene’s stability measure The closer the real value is to 0.5 the weaker the gene is „one-valued genes“: 0.92 > 0.86 > 0.65 > 0.62 „zero-valued genes“: 0.07 > 0.19 > 0.23 > 0.41

GARB: Representation GARB: Representation


GARB: Gene-strength AdaptationGARB: Gene-strength Adaptation

Every offspring gene is adjusted depending on its interpretation the relative frequency of ones at given position in the

population

Vector P[] stores the population statisticEx.: P[0.82 0.17 0.35 0.68]

82% of ones at the first position, 17% of ones at the second position, 35% of ones at the third position, 68% of ones at the fourth position.


GARB: Gene-strength Adaptation GARB: Gene-strength Adaptation cnd.cnd.

Zero-valued gene:gene’ = gene + c*(1.0-P[i]) weakeninggene’ = gene – c*P[i] strengthening

One-valued genegene’ = gene + c*(1.0-P[i]) strengtheninggene’ = gene – c*P[i] weakening

c stands for a maximal gene-adaptation step: c (0.0,0.2

Gene value interpreted with above-average frequency at given position in the chromosome is weakened, the other one is strengthened.


GARB: Gene-Strength Adaptation GARB: Gene-Strength Adaptation cnd.cnd.

Effect if some allele begines to prevail in the population,

1. the corresponding genes are weakened in subsequent generations,

2. at some point they are moved to the other side of the threshold 0.5,

3. so that they change their interpretation and the frequency of the allele decreases.

frequency of a given allele is controled by contradictory pressures

the convergence to optimal solution pressure and the population diversity preservation pressure


GARB: Boosting-up the GARB: Boosting-up the ExploitationExploitation

Genotype of promising solutions should be stabilized for subsequent generations in order to disable rapid changes in their genotype

interpretation Newly generated solutions that are better than their parents

all genes are rescaled (strengthened) - zero-valued genes are set to be close to 0.0 and one-valued genes are set to be close to 1.0

Ex.:

ch = (0.71, 0.45, 0.18, 0.57)

ch’= (0.97, 0.03, 0.02, 0.99)

Effect Genes survive with uchanged interpretation through more

generations.


GARB: AlgorithmGARB: Algorithm

1 begin2 initialize(OldPop)3 repeat4 calculate P[] from OldPop5 repeat6 select Parents from OldPop7 generate Children8 adjust Children genes9 evaluate Children10 if Child is better than Parents11 then rescale Child12 insert Children to NewPop13 until NewPop is completed14 switch OldPop and NewPop15 until termination condition16 end


GARB: Results on Static ProblemsGARB: Results on Static Problems

0 100 200 300 400 5001300

1350

1400

1450

1500

f itness ev aluations (x1000)

fitn

es

s

GARBSGA

deceptive

F101

0 100 200 300 400 500500

1000

1500

2000

2304


fitn

ess

GARBSGA

hierarchical

0 100 200 300 400 500-955

-900

-800

-700


fitne

ss

GARBSGA

multimodal


Single Gene Diversity MonitoringSingle Gene Diversity Monitoring

F101H

iera

rch

ical p

rob

lem


GARB: Tracking Moving OptimumGARB: Tracking Moving Optimum

Moving optimum Population diversity


GARB: Results onGARB: Results on KnapsackKnapsack PProblrobleemm

Oscillating Knapsack Problem 14 objects, wi=2i, i=0,...,13

f(x)=1/(1+|target - wixi|)

Target oscillates between two values 12643 and 2837, which differ in 9 bits


GARB: RecoveringGARB: Recovering from Homog. from Homog. StateState

DF3 Knapsack problem

Documents

Diversity Preservation in Evolutionary Algorithms Jiří Kubalík Intelligent Data Analysis Group Department of Cybernetics CTU Prague