Building-Block Supply in Genetic Programming

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    Building-Block Supply in Genetic Programming - Presentation Transcript

    1. Building-Block Supply in Genetic Programming K. Sastry1, U.-M. O’Reilly2, D. E. Goldberg1, D. Hill3 1Illinois Genetic Algorithms Laboratory Department of General Engineering University of Illinois at Urbana-Champaign 2 Artificial Intelligence Lab Massachusetts Institute of Technology 3Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign Supported by: AFOSR F49620-00-0163, F49620-03-1-0129, NSF DMI-9908252, and CSE fellowship
    2. Motivation Genetic Programming (GP) Theory Limited attention paid to model population sizing Correct population sizing is critical to GP success Valuable tool for GP-theorists and practitioners Insight on GP mechanisms Practical guide to set population size To obtain good solutions in a robust manner First step towards GP population-sizing Address building-block supply Population size that ensures presence of competing schemas
    3. Outline GA design decomposition Extensions to GP domain Building-block (BB) supply in GP Modeling BB supply Supply of a single BB Supply of all competing schemata Population sizing for BB supply Future Work & Conclusions
    4. GA Design Decomposition Goldberg (1991), Goldberg, Deb, & Clark (1992), Goldberg (2002) 1. Know what GAs are processing—building blocks 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
    5. Building Blocks Minimal, sequential, and superior substructures Holland(1975), Goldberg (2002) Nearly decomposable problems Context, representation, content, … Hierarchical interaction
    6. Problem Difficulty What is a GP hard problem? Koza (1992, 1994); Punch et al (1996); Soule et al (1996); O’Reilly (1998); Daida et al (2001); Daida (2003); GA hardness (Goldberg, 2002) Inter-BB difficulty (deception) Intra-BB difficulty (scaling) Extra-BB difficulty (noise) Key factors Context, Content, and Structure, … Algorithm enhancement and understanding Test problems that thwart GP mechanism Isolate facets of GP difficulty
    7. Building-Block Supply Population size that ensures the presence of competing schema Temporally through genetic operators Spatially (Initial population) Holland, 1975; Goldberg, 1989; Reeves, 1993; Goldberg, Sastry, & Latoza, 2001; Key factors BB Structure Context Temporal supply Goldberg, Sastry & Latoza, GECCO 2001
    8. Building Block Growth Schema theorem Poli & Langdon (1997); Poli (2001); Poli & McPhee (2003) Usually overemphasized Satisfying schema theorem is easy Goldberg & Sastry (2001) BB mixing not guaranteed Goldberg & Sastry, GECCO 2001
    9. Understanding Time Convergence and Takeover time in GAs Goldberg & Deb, 1991; Mühlenbein & Schlierkamp-Voosen, 1993; Báck, 1994; Thierens & Goldberg, 1994; Miller & Goldberg, 1996; Understanding time in GP Goldberg & O’Reilly (1998), O’Reilly & Goldberg (1998) Expression matters Key factors Selection method Varying size Context … Thierens & Goldberg, PPSN 1994; Bäck, ICEC 1994
    10. Building-block Decision Making Stochastic in nature (Holland, 1975) 2-armed bandit problem De Jong (1975); Goldberg (1989); Goldberg & Rudnick (1992); Goldberg, Deb & Clark (1992); Harik, Cantú-Paz, Goldberg, & Miller (1997); Usually dictates population sizing Key GP-specific factors Varying size context, … Harik, Cantu-Paz, Goldberg & Miller, ECJ, 7(3), 1999
    11. Building-block Mixing BB identification & mixing are difficult Goldberg, Thierens & Deb (1993); Thierens & Goldberg (1993); Thierens (1997, 1999); Sastry & Goldberg (1998); Need operators that adapt linkage GA designs that ensure BB mixing Messy GA, fmGA, GemGA, LLGA, PMBGAs, … Key GP-specific factors ADFs, context, structure, … Sastry & Goldberg, GECCO 2003.
    12. GP-specific Design Decomposition Steps GP Workshop: Design of Reuse, Parameterized topologies (Koza, Streeter, & Keane, 2003) Structure and Content time scales (Ryan, 2003; Soule, 2003; Howard, 2003) ???
    13. Building-Block Supply In GP Analyzing BB supply Similar approach as Goldberg, Sastry, & Latoza (2001). Selectorecombinative GP Tree representation Functions (arity = 2), and Terminals Problem definition: Initial supply of BBs Consider only spatial approach At least one copy all competing schemas Given tree sizes (or size distribution), and BB size
    14. Translating GA Schemas to GP Tree fragments (O’Reilly & Oppacher, 1995) No positional anchor Size, k; Functional nodes, Nf; Terminal nodes, Nt F F F F F F T T T T F F F T F Counting tree fragments is not enough How are tree fragments expressed? Goldberg & O’Reilly, 1998; O’Reilly & Goldberg, 1998. Fragments that express partially correct subfunctions
    15. ORDER Expression Mechanism Goldberg & O’Reilly, 1998 Has some GP properties, and easy to analyze Primitive set: Parse program tree inorder Primitive first encountered is expressed. Parsed primitives Expressed primitives
    16. Test Problems: UNITATION & DECEPTION UNITATION: # of expressed primitives, Xi Similar to OneMax in GAs DECEPTION: Expressed primitives
    17. Supply of A Single Building Block Probability that a primitive, Xi, is expressed Xi is present, and appears before its complement Probability that a BB of size, k, is expressed At least one copy of the BB is present
    18. Supply of A Single Building Block l = 32 At least one copy of the Building block: X1 Empirical Runs: UNITATION, k = 1 Population initialized with full trees Average of 1000 independent runs
    19. Supply of A Single Building Block l = 32 l=4 UNITATION At least one copy of the Building block: Empirical Runs: DECEPTION, k = 4 Population initialized with full trees Average of 1000 independent runs
    20. Supply of Competing Schemas Don’t know BBs apriori Ensure supply of all competing schemas At least one copy of competing schemas Individual schema successes are independent Can be modeled more accurately
    21. Supply of All Competing Schemas UNITATION DECEPTION At least one copy of competing schemas UNITATION: DECEPTION: 2k schemas
    22. Population Sizing for BB Supply Population size that ensures At lease one copy of all competing schemas Tolerate a failure probability, ε = 1/m Big enough tree sizes: s À l Same as for GA Goldberg, Sastry, & Latoza, 2001
    23. Supply Models for Tree Fragments Single BB supply Supply of all competing fragments Population sizing
    24. Future Work Handle realistic GP expressions Symbolic regression Tree fragment expresses a correct subfunction Other population-sizing facets Decision making (Goldberg, Deb, & Clark, 1992) Gambler’s ruin (Harik, Cantu-Paz, Goldberg, & Miller, 1997) Mixing (Thierens & Goldberg, 1994; Sastry & Goldberg, 2003) Temporal Supply of BBs Via recombination and mutation Automatically defined functions (ADFs)
    25. Summary & Conclusions Developed facetwise models for BB supply At least one copy of a BB in the initial population At least one copy of all schemas Population sizing dictated by supply ORDER expression mechanism Larger population if good BBs are not expressed Increasing tree size helps Population sizing is identical to GAs Average tree size is greater than the problem size Understanding expression is important Fragments express subfunctions that improve fitness

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