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Diversity Preservation in Evolutionary Algorithms Jiří Kubalík Intelligent Data Analysis Group Department of Cybernetics CTU Prague EAs 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 Diversity Preservation in EAs GA with Limited Convergence (GALCO) Motivation Realization ‹#› to maintain a diversity of the evolved population and extend the explorative power of the algorithm 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 Diversity Preservation in EAs GALCO: Algorithm 1. Generate initial population 2. Choose parents 3. 4. Create offspring 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 Diversity Preservation in EAs GALCO: replace_with_mask Mask – vector of integer counters; stores a number of 1s for each bit of the representation 50 ‹#› Diversity Preservation in EAs Static Test Problems ‹#› Multimodal problem Deceptive problem Hierarchical problem Royal Road Problem Diversity Preservation in EAs GALCO: Finding Optimal c ‹#› multimodal deceptive royal road hierarchical Diversity Preservation in EAs GALCO: Comparison with SGA multimodal royal road ‹#› deceptive hierarchical Diversity Preservation in EAs GALCO: Multimodal Optimization Initial population SIGA with ‹#› replace_with_mask without Diversity Preservation in EAs GALCO: Multimodal Optimization (cnd.) Initial population ‹#› GALCO SGA Diversity Preservation in EAs GA with Real-coded Binary Rep. Motivation Realization real coded binary representation Effect ‹#› 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 population can not converge to the homogeneous state so that the premature convergence can not take place Diversity Preservation in EAs GARB: Representation 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 ‹#› Diversity Preservation in EAs 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. ‹#› Diversity Preservation in EAs GARB: Gene-strength Adaptation cnd. Zero-valued gene: gene’ = gene + c*(1.0-P[i]) gene’ = gene – c*P[i] One-valued gene gene’ = gene + c*(1.0-P[i]) gene’ = gene – c*P[i] ‹#› weakening strengthening strengthening 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. Diversity Preservation in EAs GARB: Gene-Strength Adaptation cnd. Effect if some allele begines to prevail in the population, 1. 2. 3. ‹#› 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 by contradictory pressures the convergence to optimal solution pressure and the population diversity preservation pressure Diversity Preservation in EAs 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. Diversity Preservation in EAs GARB: Algorithm 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 ‹#› begin initialize(OldPop) repeat calculate P[] from OldPop repeat select Parents from OldPop generate Children adjust Children genes evaluate Children if Child is better than Parents then rescale Child insert Children to NewPop until NewPop is completed switch OldPop and NewPop until termination condition end Diversity Preservation in EAs GARB: Results on Static Problems 1500 deceptive 2304 1400 GARB SGA 2000 GARB SGA 1350 1300 0 fitness 100 hierarchical 1500 200 300 400 500 f itness ev aluations (x1000) 1000 500 0 -700 100 200 300 400 f itness ev aluations (x1000) -800 fitness fitness 1450 500 GARB SGA multimodal F101 -900 -955 0 ‹#› 100 200 300 400 f itness ev aluations (x1000) Diversity Preservation in EAs 500 Single Gene Diversity Monitoring Hierarchical problem F101 ‹#› Diversity Preservation in EAs GARB: Tracking Moving Optimum Moving optimum ‹#› Population diversity Diversity Preservation in EAs GARB: Results on Knapsack Problem 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 Diversity Preservation in EAs GARB: Recovering from Homog. State DF3 ‹#› Knapsack problem Diversity Preservation in EAs