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
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A problem with overlapping building blocks
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12
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11
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Genotype Phenotype
4
5
7
2
6
1
9
• Fitness:
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Gene-wise two-point XO
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Genotype
Phenotype
• {3, 4, 5, 6, 7}, {1, 2, 8, 9, 10, 11, 12}
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BB-wise uniform XO
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Genotype
Phenotype
• {3, 10, 11, 5, 2, 9, 1, 7, 4, 8}, {6, 12}
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Least-disruptive XO
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Genotype
Phenotype
• {12, 3, 10, 11, 5, 2, 6}, {9, 1, 7, 4, 8}
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Different crossover effects
Population size: 10 x gambler’s ruin model (Harik et al, 1999)
Selection pressure: 2
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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
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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]
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Effects of detection failure
• Increase in run duration (Yu and Goldberg, 2004):
•
GA fails to
converge.
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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
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Effective exchange length (EEL)
• EEL: effective number of BBs exchanged
during crossover
• The two scenarios:
– vs
• EEL of uniform XO, exchange prob = 0.5
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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
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Shortened EEL by false linkages (contd)
• For large m
– 0 false linkage: EEL~m/2
– (m-2) false linkages: EEL~m/8
m=20
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EEL and uniform XO
• Exchange length vs. exchange probability
EEL~m/2
– p=0.5
EEL~m/8
– p=0.125
EEL XO:
531462
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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?
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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
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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
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Detect failure vs. false linkage
• When detect failure affects GA more than
false linkage?
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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.
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Overlapping-BB problem revisited
• Circular overlapping BBs
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Effect of two types of errors
• False linkage: constant elongation
• Detection failure: inversely proportional to
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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
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Future work
• Experiments
– Random linkage topology
– Boundedly overlapping
• Weighted overlapping
– Probabilistic XO
• Real-world problems
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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).
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Thank you
• Questions?
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