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

2. Diversity Preservation in EAs 2 EAs and Premature Convergence Evolutionary cycleHomogeneous 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

3. Diversity Preservation in EAs 3 GA with Limited Convergence (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

4. Diversity Preservation in EAs 4 GALCO: 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

5. Diversity Preservation in EAs 5 GALCO: replace_with_mask Mask – vector of integer counters; stores a number of 1s for each bit of the representation 50

6. Diversity Preservation in EAs 6 Static Test Problems Multimodal problem Deceptive problem Hierarchical problem Royal Road Problem

7. Diversity Preservation in EAs 7 GALCO: Finding Optimal c multimodal hierarchical royal road deceptive

8. Diversity Preservation in EAs 8 GALCO: Comparison with SGA multimodal hierarchical royal road deceptive

9. Diversity Preservation in EAs 9 GALCO: Multimodal Optimization Initial population SIGA with replace_with_mask without

10. Diversity Preservation in EAs 10 GALCO: Multimodal Optimization (cnd.) Initial populationGALCO SGA

11. Diversity Preservation in EAs 11 GA with 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

12. Diversity Preservation in EAs 12 Pseudo-binary representation – binary gene values coded by real numbers from the interval  0.0, 1.0  Example: ch 1 = [0.92 0.07 0.23 0.62] ch 2 = [0.65 0.19 0.41 0.86] interpretation(ch 1 )=interpretation(ch 2 )=[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

13. Diversity Preservation in EAs 13 GARB: 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 statistic Ex.: 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.

14. Diversity Preservation in EAs 14 GARB: Gene-strength Adaptation cnd. Zero-valued gene: gene’ = gene + c*(1.0-P[i])weakening gene’ = gene – c*P[i]strengthening One-valued gene gene’ = gene + c*(1.0-P[i])strengthening gene’ = 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.

15. Diversity Preservation in EAs 15 GARB: Gene-Strength Adaptation 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

16. Diversity Preservation in EAs 16 GARB: Boosting-up the Exploitation 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.

17. Diversity Preservation in EAs 17 GARB: Algorithm 1begin 2initialize(OldPop) 3repeat 4calculate P[] from OldPop 5repeat 6select Parents from OldPop 7generate Children 8adjust Children genes 9evaluate Children 10if Child is better than Parents 11then rescale Child 12insert Children to NewPop 13until NewPop is completed 14switch OldPop and NewPop 15until termination condition 16end

18. Diversity Preservation in EAs 18 GARB: Results on Static Problems deceptive F101 hierarchical multimodal

19. Diversity Preservation in EAs 19 Single Gene Diversity Monitoring F101 Hierarchical problem

20. Diversity Preservation in EAs 20 GARB: Tracking Moving Optimum Moving optimum Population diversity

21. Diversity Preservation in EAs 21 GARB: Results on Knapsack Problem Oscillating Knapsack Problem  14 objects, w i =2 i, i=0,...,13  f(x)=1/(1+|target - w i x i |)  Target oscillates between two values 12643 and 2837, which differ in 9 bits

22. Diversity Preservation in EAs 22 GARB: Recovering from Homog. State DF3Knapsack problem