Diversity Preservation in Evolutionary Algorithms

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

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

6. Static Test Problems • Deceptive problem • Multimodal problem • Hierarchical problem • Royal Road Problem

7. deceptive multimodal royal road hierarchical GALCO: Finding Optimal c

8. deceptive multimodal hierarchical royal road GALCO: Comparison with SGA

9. withreplace_with_maskwithout GALCO: Multimodal Optimization Initial population SIGA

10. GALCO: Multimodal Optimization (cnd.) Initial population GALCO SGA

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. GARB: Representation • Pseudo-binaryrepresentation – 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

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. 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. GARB: Gene-Strength Adaptation cnd. • Effect • if some allele begines to prevail in the population, • the corresponding genes are weakened in subsequent generations, • at some point they are moved to the other side of the threshold 0.5, • so that they change their interpretation and the frequency of the allele decreases. • frequency of a given allele is controled bycontradictory pressures • the convergence to optimal solution pressure and • the population diversity preservation pressure

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 • Genessurvive with uchanged interpretation through more generations.

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

18. deceptive hierarchical multimodal GARB: Results on Static Problems F101

19. Single Gene Diversity Monitoring F101 Hierarchical problem

20. Population diversity Moving optimum GARB: Tracking Moving Optimum

21. GARB: Results onKnapsackProblem Oscillating Knapsack Problem • 14 objects, wi=2i, i=0,...,13 • f(x)=1/(1+|target-wixi|) • Target oscillates between two values12643 and2837, which differ in 9 bits

22. GARB: Recovering from Homog. State DF3 Knapsack problem