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On the Supply of Building Blocks

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This study addresses the issue of building-block supply in the initial population. Facetwise models for supply of a single building block as well as for supply of all schemata in a partition have been ...

This study addresses the issue of building-block supply in the initial population. Facetwise models for supply of a single building block as well as for supply of all schemata in a partition have been developed. An estimate for the population size required to ensure the presence of all raw building blocks has been derived using these facetwise models. The facetwise models and the population-sizing estimate are verified with computational results.

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    On the Supply of Building Blocks On the Supply of Building Blocks Presentation Transcript

    • On The Supply of Building Blocks David E. Goldberg Kumara Sastry Thomas Latoza Illinois Genetic Algorithms Laboratory University of Illinois at Urbana-Champaign Urbana, IL 61801 http://www-illigal.ge.uiuc.edu Genetic and Evolutionary Computation Conference (GECCO-2001) July 7-11, 2001 San Francisco, CA
    • 1 On The Supply of Building Blocks Overview • GA design decomposition • Building block supply • Objective • Supply of a single BB • Supply for partition success • Population-sizing for BB supply • Summary Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 2 On The Supply of Building Blocks GA Design Decomposition • Goldberg, Deb, & Clark, 1992 1. Know what GAs are processing—building blocks (BBs) 2. Know thy BB challengers—BB-wise difficult problems. 3. Ensure an adequate supply of raw BBs 4. Ensure increased market share for superior BBs 5. Know BB takeover and convergence time 6. Make decisions well among competing BBs 7. Mix BBs well Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 3 On The Supply of Building Blocks Building Block Supply • Previous work: – Holland, 1975; Goldberg, 1989; Reeves, 1993. • Spatial & temporal approach • Selectorecombinative GAs • BB growth usually supersedes supply • Supply sometimes governs population size • Complex models exist (Harik et al, 1997) • Better models are needed Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 4 On The Supply of Building Blocks Objective • Develop facetwise models – Supply of a single BB – Supply of all schemata in a partition • Population sizing for BB supply • Verify models with empirical results Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 5 On The Supply of Building Blocks Supply Of A Single Building Block • consider a string: – Of length – Containing alphabets of cardinality χ – Has schemata of order k n randomly generated strings • • At least one string has the desired schema Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 6 On The Supply of Building Blocks Single Building Block Supply χk • Total number of schemata: 1 1− • Desired schema is not in a string: χk n 1 n strings: 1 − • It is not in any of χk • At least one string has the desired schema n 1 n pk = 1 − 1 − ≈ 1 − exp − k χk χ n > χk : pk → 1 • Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 7 On The Supply of Building Blocks Results: Supply Of A Single BB 1 1 k=1 0.8 0.8 0.6 0.6 k=8 k=4 pk pk k=8 0.4 0.4 0.2 0.2 χ=2 χ=3 0 0 0 1 2 3 0 1 2 3 10 10 10 10 10 10 10 10 Population size Population size, n 1 1 k=1 k=1 0.8 0.8 0.6 0.6 k=8 pk pk k=8 0.4 0.4 0.2 0.2 χ=5 χ=4 0 0 0 1 2 3 0 2 10 10 10 10 10 10 Population size, n Population size, n Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 8 On The Supply of Building Blocks Supply For Partition Success • Earlier we considered only one schema • At least one copy of all schemata – Don’t know which schema is good – Want a good share of all of them • Exact model is complex • Approximations – n > nk – BB successes are independent Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 9 On The Supply of Building Blocks Exact Model • Consider not having at least one schema • Illustration through a test case – Unit length BB, k = 1 – Tertiary alphabet, χ = 3 • Generalize the result Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 10 On The Supply of Building Blocks Partition Success: k = 1, χ = 3 h1 : **0**, h2 : **1**, h3 : **2** • • Absence of at least one schema – (all h1 ) OR (all h2 ) OR (all h3 ) – (h1 or h2 ) OR (h1 or h3 ) OR (h2 or h3 ) 3 · 2n − 3 • Total ways of absence: χkn = 3n • Total possible ways: • Partition success 2n − 1 ps = 1 − n−1 3 Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 11 On The Supply of Building Blocks Exact Model 1 χ = 2: ps = 1 − • 2n−1 2n −1 χ = 3: ps = 1 − • 3n−1 3n −3·2n−1 +1 χ = 4: ps = 1 − • 4n−1 • Generalize the result k  χ −1 χk 1 n (−1)i−1 χk − 1 ps = 1 − nk  χ i i=1 Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 12 On The Supply of Building Blocks Approximation Of Exact Model • Consider the exact model k    χ −1 χk 1 n  χk − 1 (−1)i−1  ps = 1 − nk  χ i i=1 n > χk : (χk − 1)n > (χk − 2)n > · · · > 1 • Assume • Consider only the first term n 1 = 1 − χk 1 − ps χk n k ≈ 1 − χ exp − k ps χ Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 13 On The Supply of Building Blocks Approximate Model • Schema partition success is independent • Recall, single BB success probability: n k pk = 1 − χ exp − k χ • Using independence assumption χk n ps = 1 − χk exp − k χ (1 − r/n)n ≈ e−r . n > χk , (1 − x)n ≈ 1 − nx • n ps ≈ 1 − χk exp − χk Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 14 On The Supply of Building Blocks Exact Model vs Approximate Model 1 0.9 Probability of one or more copies of each of nk schemata, ps χ=2 0.8 0.7 0.6 0.5 χ=3 χ=4 0.4 0.3 0.2 0.1 0 2 4 6 8 10 12 14 16 18 20 Population size, n Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 15 On The Supply of Building Blocks Results: Partition Success 1 1 χ=3 χ=2 0.8 0.8 0.6 0.6 k=1 ps ps 0.4 0.4 k=4 k=8 k=2 0.2 0.2 0 0 0 2 4 0 2 10 10 10 10 10 Population size, n Population size, n 1 1 χ=4 χ=5 0.8 0.8 k=1 0.6 0.6 k=1 ps ps 0.4 0.4 0.2 0.2 0 0 0 2 0 2 4 10 10 10 10 10 Population size, n Population size, n Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 16 On The Supply of Building Blocks Population Sizing For BB Supply • Require a population size – All schemata are present – Success probability asymptotically reaches 1 – Tolerate small failure probability α n – Partition success, ps = 1 − χk exp − χk – Equate ps = 1 − α n = χk (k log χ − log α) Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 17 On The Supply of Building Blocks Population-Sizing Model m building blocks of length k • • At most one BB need not be present 1 α= • Failure probability, m n = χk (k log χ − log α) χk m • Large problem: n = O χk log m χk m • Complex problem: n = O χk log m Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 18 On The Supply of Building Blocks Results: Population Sizing 3 10 k=8 3 k=8 10 Population size, n Population size, n k=4 2 10 k=4 2 10 k=2 k=2 1 10 k=1 1 χ=3 10 χ=2 2 2 10 10 Number of BBs, m Number of BBs, m 3 10 k=4 3 k=4 10 Population size, n Population size, n k=2 k=2 2 10 2 10 k=1 k=1 χ=4 χ=5 1 1 10 10 2 2 10 10 Number of BBs, m Number of BBs, m Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 19 On The Supply of Building Blocks Summary • Facetwise models – Supply of a single schema – Supply of all raw BBs – Population sizing for BB supply • Two asymptotic cases – Large problem: O χk log m – Complex problem: n = O χk log m Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu
    • 20 On The Supply of Building Blocks Acknowledgments • Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant F49620-00-1-0163. • National Science Foundation under grant DMI-9908252. • U. S. Army Research Laboratory under the Federated Laboratory Program, Cooperative Agreement DAAL01-96-2-0003. Illinois Genetic Algorithms Laboratory Department of General Engineering GECCO, July 7-11, 2001 University of Illinois at Urbana-Champaign Urbana, IL 61801. USA. D.E. Goldberg, K. Sastry, T. Latoza http://www-illigal.ge.uiuc.edu