Empirical Analysis of ideal recombination on random decomposable problems

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This paper analyzes the behavior of a selectorecombinative genetic algorithm (GA) with an ideal crossover on a class of random additively decomposable problems (rADPs). Specifically, additively decomposable problems of order k whose subsolution fitnesses are sampled from the standard uniform distribution U[0,1] are analyzed. The scalability of the selectorecombinative GA is investigated for 10,000 rADP instances. The validity of facetwise models in bounding the population size, run duration, and the number of function evaluations required to successfully solve the problems is also verified. Finally, rADP instances that are easiest and most difficult are also investigated.

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Empirical Analysis of ideal recombination on random decomposable problems

  1. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 14. Population Sizing Scalability Gambler’s ruin model bounds population sizing 14
  15. 15. Run Duration Scalability Selection-intensity based run-duration model bounds median convergence time Drift-time model bounds convergence time 15
  16. 16. Number of Function Evaluations Scalability Facetwise models are applicable to rADPs Testing on adversarial problems bounds performance of GAs on rADPs 16
  17. 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. 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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