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Linkage Learning, Overlapping Building Blocks, and a Systematic Strategy for Scalable Recombination

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This paper aims at an important, but poorly studied area in genetic algorithm (GA) field: How to design the crossover operator for problems with overlapping building blocks (BBs). To investigate this ...

This paper aims at an important, but poorly studied area in genetic algorithm (GA) field: How to design the crossover operator for problems with overlapping building blocks (BBs). To investigate this issue systematically, the relationship between an inaccurate linkage model and the convergence time of GA is studied. Specifically, the effect of the error of so-called false linkage is analogized to a lower exchange probability of uniform crossover. The derived qualitative convergence-time model is used to develop a scalable recombination strategy for problems with overlapping BBs. A set of problems with circularly overlapping BBs exemplify the recombination strategy.

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    Linkage Learning, Overlapping Building Blocks, and a Systematic Strategy for Scalable Recombination Linkage Learning, Overlapping Building Blocks, and a Systematic Strategy for Scalable Recombination Presentation Transcript

    • Linkage Learning, Overlapping Building Blocks, and Systematic Strategy for Scalable Recombination Tian-Li Yu, Kumara Sastry, & David E. Goldberg Presented by Tian-Li Yu tianliyu@illigal.ge.uiuc.edu Illinois Genetic Algorithms Laboratory http://www-illigal.ge.uiuc.edu
    • Roadmap • Motivation • Two linkage errors – Detection failure – False linkage • Effective exchange length • Convergence time elongation • Recombination strategy • Conclusions 2
    • A problem with overlapping building blocks 3 10 12 1 2 3 4 5 6 7 8 9 101112 11 8 Genotype Phenotype 4 5 7 2 6 1 9 • Fitness: 3
    • Gene-wise two-point XO 1 2 3 4 5 6 7 8 9 101112 Genotype Phenotype • {3, 4, 5, 6, 7}, {1, 2, 8, 9, 10, 11, 12} 4
    • BB-wise uniform XO 1 2 3 4 5 6 7 8 9 101112 Genotype Phenotype • {3, 10, 11, 5, 2, 9, 1, 7, 4, 8}, {6, 12} 5
    • Least-disruptive XO 1 2 3 4 5 6 7 8 9 101112 Genotype Phenotype • {12, 3, 10, 11, 5, 2, 6}, {9, 1, 7, 4, 8} 6
    • Different crossover effects Population size: 10 x gambler’s ruin model (Harik et al, 1999) Selection pressure: 2 7
    • Assumptions • Selectorecombinative GA – Crossover probability: 1 – No mutation • Fixed-length χ–ary encoding • Infinite population size – Needed for convergence-time model – 10*n Pop. size from gambler’s ruin model 8
    • Two types of linkage errors • Linkage • False linkage: – Unnecessary correlations are discovered • Detection failure: – Actual linkages are not discovered e.g. [1,2,3], [4,5,6], [7,8,9], [10,11,12] [1,2], [3,4,5,6], [7,8], [9,10,11], [12] 9
    • Effects of detection failure • Increase in run duration (Yu and Goldberg, 2004): • GA fails to converge. 10
    • Effects of false linkage • The false linkage is stationary – BB supply (Goldberg et al, 2001) • Two scenarios – False linkage: concatenate BB1, BB2 together – Uniform XO: exchange BB1, BB2 together Effects are the same, but with different probabilities 11
    • Effective exchange length (EEL) • EEL: effective number of BBs exchanged during crossover • The two scenarios: – vs • EEL of uniform XO, exchange prob = 0.5 12
    • Shortened EEL by false linkages • 4 BBs: probabilities of {0,1,2,3,4} BBs exchanged – 0 false linkage – 1 false linkage – 2 false linkages 13
    • Shortened EEL by false linkages (contd) • For large m – 0 false linkage: EEL~m/2 – (m-2) false linkages: EEL~m/8 m=20 14
    • EEL and uniform XO • Exchange length vs. exchange probability EEL~m/2 – p=0.5 EEL~m/8 – p=0.125 EEL XO: 531462 15
    • What have we been doing? • The relationship between false linkage and convergence time EEL uniform XO with exchange • ef prob • We already have convergence-time model for uniform XO with exchange prob 0.5 • Are we done? 16
    • Convergence time • Convergence-time model (Mühlenbein & Schlierkamp- Voosen, 1993)(Thierens & Goldberg, 1994) – Assumes binomial distribution for calculating variances • Relaxation-time model (Rabani, 1998) – p goes from 1/2 to 1/8, τ increases 2.8 times – Need 3 XOs per individual for binomial distribution to hold • Binomial assumption doesn’t hold well 17
    • Elongation of convergence time • The false linkage elongates • For m=10~50, (m-2) false linkages yields =1.18~1.21 • Assume rf nearly independent of m 18
    • Detect failure vs. false linkage • When detect failure affects GA more than false linkage? 19
    • Recombination strategy (1) Every BB needs to have a nonzero probability to be recombined. (2) The recombination needs to be least disruptive. • Generate a graph G=(V,E) – Nodes BBs – Edges overlaps • Randomly choose two nodes v1 and v2. Partition G into G1=(V1,E1) and G1=(V2,E2), where v1∈V1 v2∈V2 and |E|-|E1|-|E2| is minimal. 20
    • Overlapping-BB problem revisited • Circular overlapping BBs 21
    • Effect of two types of errors • False linkage: constant elongation • Detection failure: inversely proportional to 22
    • Conclusions • Recombination needs to – Consider linkages – Consider linkage group topology • Detection failures affect GA convergence more than false linkages • For problems with overlapping BBs, use least-disruptive recombination 23
    • Future work • Experiments – Random linkage topology – Boundedly overlapping • Weighted overlapping – Probabilistic XO • Real-world problems 24
    • Acknowledgment This work was sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF (F49620-03-1-0129), and by the Technology Research, Education, and Commercialization Center (TRECC), at University of Illinois at Urbana-Champaign, administered by the National Center for Supercomputing Applications (NCSA) and funded by the Office of Naval Research (N00014-01-1-0175). 25
    • Thank you • Questions? 26