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- 1. Empirical Analysis of Ideal Crossover on Random Additively Decomposable Problems Kumara Sastry1, Martin Pelikan2, David E. Goldberg1 1Illinois Genetic Algorithms Laboratory (IlliGAL) University of Illinois at Urbana-Champaign, Urbana, IL 61801 2MissouriEstimation of Distribution Algorithm Lab (MEDAL) University of Missouri at St. Louis, St. Louis, MO ksastry@uiuc.edu, pelikan@cs.umsl.edu, deg@uiuc.edu http://www.illigal.uiuc.edu, http://medal.cs.umsl.edu Supported by AFOSR FA9550-06-1-0096, NSF DMR 03-25939, and CAREER ECS-0547013. Computational results obtained using CSE’s Turing cluster.
- 2. Roadmap Adversarial test problem design Random additively decomposable problems Ideal crossover Scalability of selectorecombinative GAs Population sizing and Run duration Experimental Procedure Key Results Summary and Conclusions 2
- 3. Adversarial Test Problem Design Test systems on boundary of design envelope Common approach in designing complex systems GAs are complex systems [Goldberg, 2002] GA design envelope characterized by different dimensions of problem difficulty Thwart the mechanism of GAs to the extreme Fluctuating R P Noise Deception Scaling 3
- 4. Random Additively Decomposable Problem Focus on nearly decomposable problems [Simon, 1960] Three desired features Scalability: Able to control problem size and difficulty Known optimum: Allows comparison of different solvers Easy problem instance generation rADP fitness function: Si represents variable subset for ith subproblem Each subset consists of k bits gi is the fitness of the ith subproblem gi is sampled from uniform distribution U[0,1] 4
- 5. Ideal Crossover: Exchange Building Blocks Population sizing and run duration models assume good exchange of building blocks Simulate what we ideally want to achieve with model- building GAs For example, extended compact GA [Harik, 1999] Ideal recombination operator Effectively exchange building blocks Don’t disrupt any building block Uniform building-block-wise crossover BBs #1 and #3 exchanged Exchange BBs with probability 0.5 5
- 6. Purpose: Analyze Ideal Crossover on rADPs Analyze behavior of selectorecombinative GAs on rADPs Verify the validity of lessons learned from adversarial test problems Expand the pool of test problems 6
- 7. Selectorecombinative GA Population Sizing Noise-to-fitness variance ratio Error tolerance # Components (# BBs) Signal-to-Noise ratio # Competing sub-components Gambler’s ruin model [Harik, et al, 1997] Combines decision making and supply models Additive Gaussian noise with variance σ2N 7
- 8. GA Run Duration (Selection) Selection-Intensity based model [Bulmer, 1980; Mühlenbein & Schlierkamp-Voosen, 1993; Thierens & Goldberg, 1994; Bäck, 1994; Miller & Goldberg, 1995 & 1996] Derived for the OneMax problem Applicable to additively-separable problems [Miller, 1997] Problem size (m·k ) Selection Intensity [Miller & Goldberg, 1995; Sastry & Goldberg, 2002] 8
- 9. GA Run Duration (Drift) Accumulation of stochastic errors due to finite population Proportion of competing sub-solutions change due to drift Drift time [Goldberg & Segrest, 1987]: Substituting population sizing bound 9
- 10. Signal-to-Noise Ratio for rADPs Signal d is the fitness difference between best and second best sub-solutions jth order statistic follows a Beta distribution with α = j and β = 2k-j+1 Probability density function (p.d.f) of d: p.d.f. of sub-solution fitness variance approximation E[1/d] = 2k and E[σ2BB] ≈ 1/12 10
- 11. Assumptions and Experimental Setup Non-overlapping sub-problems Identical sub-problems across different partitions g1 = g2 = … = gm Selectorecombinative GA Binary tournament selection 10,000 random problem instances m = 5 – 50, k = 3, 4, and 5 Minimum population size determined by bisection method Population correctly converges to at least m-1 out of m BBs in 49 out of 50 independent runs Averaged over 30 bisection runs Results averaged over 1,500 GA runs 11
- 12. Population Sizing & Run Duration Histograms Population size Run duration m = 50 m = 50 Tail increases with m 0.15-0.59% of rADP instances require # evals greater than 3σ from the median 12
- 13. Easy and Hard Problem Instances Hard instance Subsolution fitness Min signal Max noise Sorted subsolution index Easy instance Subsolution fitness Max signal Min noise Sorted subsolution index 13
- 14. Population Sizing Scalability Gambler’s ruin model bounds population sizing 14
- 15. Run Duration Scalability Selection-intensity based run-duration model bounds median convergence time Drift-time model bounds convergence time 15
- 16. Number of Function Evaluations Scalability Facetwise models are applicable to rADPs Testing on adversarial problems bounds performance of GAs on rADPs 16
- 17. Easy and Hard Scalable Problem Instances Easy Scalable Hard Scalable Instances Instances Easy instances have large signal difference Hard instances have very small signal difference 17
- 18. Summary and Conclusions Empirically analyzed behavior of selectorecombinative GA with ideal crossover: Class of random additively decomposable problems Sub-solution fitness sampled from uniform distribution Verified applicability of facetwise models: Developed based on adversarial problems GA scales subquadratically with problem size Analyzed easy and hard problem instances: Easy problem instances have large signal, small variance. Hard problem instances have small signal, large variance 18

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