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Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
Spurious Dependencies and EDA Scalability
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Spurious Dependencies and EDA Scalability

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More on this work can be found in the technical report: …

More on this work can be found in the technical report:
http://medal.cs.umsl.edu/show_abstract.php?type=tr&number=2010002

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  • 1. Spurious Dependencies and EDA Scalability Elizabeth Radetic and Martin Pelikan Missouri Estimation of Distribution Algorithms Laboratory (MEDAL) University of Missouri, St. Louis, MO http://medal.cs.umsl.edu/ pelikan@cs.umsl.edu Download MEDAL Report No. 2010002 http://medal.cs.umsl.edu/files/2010002.pdf Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 2. Motivation Estimation of distribution algorithms (EDAs) Replace standard crossover and mutation by building a probabilistic model of selected solutions, and sampling the probabilistic model to generate new solutions. Can solve many problems intractable with standard EAs. Model accuracy It is important that the EDA model is accurate. Types of inaccuracies for dependency-based models Missing dependencies. Spurious, unnecessary dependencies. Most prior work focused on missing dependencies. This study Focus on effects of spurious dependencies. Theoretical study for population sizing. Empirical study for the number of generations. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 3. Outline 1. Model accuracy. 2. Spurious dependencies Model for spurious dependencies. Effects on population sizing. Effects on the number of generations. 3. Experiments. 4. Conclusions and future work. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 4. Dependency-Based Probabilistic Models in EDAs Dependency-based probabilistic models Encode dependencies and independencies between variables. Dependency structure decomposes the problem. Subproblems should be of bounded order. Examples Marginal product models. Bayesian networks. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 5. Marginal Product Model Beyond Pairwise Dependencies: ECGA Variables are divided into linkage groups. Defines problem decomposition into separable subproblems. !  Extended Compact GA (ECGA) (Harik, 1999). Distribution of each group encoded by probability table. We Consider groups of string positions.solutions. !  assume binary representation of candidate String Model !!! Martin Pelikan, Probabilistic Model-Building GAs 32 Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 6. Model Accuracy Types of inaccuracies Missing dependencies. Spurious, unnecessary dependencies. Example: Trap-5 n/5 ftrap5 (X1 , . . . , Xn ) = i=1 trap5 (X5i−4 + X5i−3 + X5i−2 + X5i−1 + X5i ) 5 if u = 5 trap5 (u) = 4−u otherwise Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 7. Onemax Model of Spurious Dependencies Onemax is the sum of bits in the binary string n onemax(X1 , . . . , Xn ) = i=1 Xi Perfect and spurious models for onemax Perfect model assumes no dependence at all. Spurious model assumes linkage groups of order kspurious > 1. Parameter kspurious controls order of spurious dependencies. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 8. Effects of Spurious Models on EDA Performance Two main effects of spurious dependencies Population size. Number of generations. Population sizing decomposition Population size requirements should increase Effects depend on learning, but sometimes substantial. Number of generations Number of generations may decrease due to weaker variation. Effects not expected substantial. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 9. EDA Population Sizing and Spurious Dependencies Population sizing decomposition Initial supply Initial population is random. Ensure sufficient supply of partial solutions for each group. Decision making Decision making between partial solutions is stochastic. Ensure that best partial solution wins in each group. Model building Ensure accurate enough models to find the optimum. The reason for spurious dependencies, not the effect. Focus in this work Initial supply. Decision making. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 10. Population Sizing: Initial Supply Initial supply for perfect model (Goldberg et al., 2001) N = 2 ln 2m Initial supply for arbitrary kspurious n N = 2kspurious kspurious ln 2 + ln kspurious Initial-supply population increase factor n kspurious ln 2 + ln kspurious γis = 2kspurious −1 ln 2 + ln n Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 11. Population Sizing: Decision Making Decision making for perfect model (Harik et al., 1997) 1 N = − ln α π(n − 1) 2 Decision making for arbitrary kspurious N = −2kspurious −2 ln α π(n − 1) Decision-making population increase factor γdm = 2kspurious Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 12. Number of Generations Effects of spurious dependencies on number of generations Spurious dependencies weaken the mixing. This reduces the effects of variation. This should reduce the number of generations until convergence (assuming a large enough population). No theoretical model as of now. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 13. Description of Experiments Operators Binary tournament selection without replacement. Three replacement types Full replacement. Elitist replacement (50% worst are replaced). Restricted tournament replacement (niching). Models with various levels of spurious linkage. Parameters Optimal population size obtained by bisection. Runs stop when a solution close enough to the optimum is reached (allow one linkage group to end up incorrect). Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 14. Population Size (Full Replacement) Population size ratio 1000 Gambler’s ruin Gambler’s ruin Initial supply 16 Population size Initial supply 800 Experiment Experiment 12 600 400 8 200 4 0 1 1.5 2 2.5 3 3.5 4 4.5 5 1 2 3 4 5 Spurious linkage group size Spurious linkage group size (a) Population size (b) Population size ratio owth of the population size with respect spurious is exponential. a problem Increase of population size with k to the group size for side shows the actual population sizes compared to the theoretical mo Theory provides a conservative bound. d side shows the ratio of the population sizes with spurious linkage and th spurious linkage. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability 500
  • 15. 1 1.5 2 2.51.5 3.52.5 1.5 3.5 2.5 4.5 3.5 4 4.5 5 2 1 1 3 2 1 4.5 2 4 3 5 4 3 5 1 3 2 1 4 3 2 5 4 3 5 4 5 Population Size group sizegroup size group size Strategies) linkage group size (All Replacement linkage group sizegroup size Spurious linkage linkage Spurious Spurious linkage Spurious Spurious linkage Spurious (a) Population size Population size(b) Population size ratio (a) Population size (a) (b) Population size ratio (b) Population size ratio wth of theGrowth population size with respect respect sizethe size forsize for 300 of 300 Figure 2: population size with respect to the group group group a problem bits. 2: Growth of the of the population size with to the to for a problem of a problem The left-hand side the actual population sizes compared to the theoretical model, mod t-hand side the actual population sizes compared to the theoretical model, whereas side shows shows shows the actual population sizes compared to the theoretical whe the right-hand sidethe ratiopopulation sizes with spurious linkagelinkage linkage and the t-hand side the ratio of the ofratiopopulation sizes with spurious and the population side shows shows shows the the of the population sizes with spurious and the popula sizes with no spurious linkage. th no spurious linkage. purious linkage. Full replacement Elitist replacement RTR 500 500 500 1200 size 1200 size blem Problem Problem size 1000 Problem size 1000 size 1000 Problem Problem size Problem size Problem size Problem size 00 300 300 300 300 300 300 300 300 Population size Population size Population size Population size Population size Population size Population size 1000 1000 400 400 400 40 240 240 800 800 240 800 240 240 240 240 240 800 80 800 180 180 180 180 180 300 180 300 180 300 180 20 120 120 600 600 120 600 120 120 120 120 120 600 600 60 60 60 60 60 60 400 400 400 200 60 200 60 200 60 400 400 200 200 200 200 200 100 100 100 0 0 0 0 0 0 0 0 5 2 2.51.5 3.52.5 1.5 3.5 2.5 4.5 3.5 4 4.51 51.5 2 2.51.5 3.52.5 1.5 3.5 2.5 4.5 3.5 4 4.5 1.5 2 2.51.5 3.52.51.5 3.52.5 4.53 1 3 2 1 4.5 2 4 3 5 4 3 5 1 3 2 1 4.5 2 4 3 5 4 3 5 1 5 1 3 2 1 4 4.52 5 4 3 3 up size (bits per group)per group) per group)Group size (bits per group)per group) per group) Group sizeGroup size (bits (bits Group sizeGroup size (bits (bits Group size (bits per group)per group) Group sizeGroup size (bits (bits (a) Full replacement replacementFull replacement(b) Elitist replacement (a) (b) Elitist replacement (b) Elitist replacement (c) RTR(c) RTR(c) RTR gure Figure theGrowth population size with respect respect spurious linkage linkage size. Growth of 3: population size with respect to the spurious linkage group size. grou 3: Growth of the of the population size with to the to the spurious group Increase of population size with kspurious similar in all cases. ows the averageaverage number of spurious linkage groups (groups at leastat leasteach p 1(a) shows the number of spurious linkage groups (groups at size of size each prob- for average number of spurious linkage groups (groups of size of least 2) for 2) for 2) results results resultsthe number of the number of such groups increases approximately liw . The indicate that indicate that suchof such groups increases approximately linearly em size. The indicate that the number groups increases approximately linearly with m size. Figure 1(b) the average size ofaverage spurious linkage linkage groups. For size, pr Figure 1(b) shows shows the average spurious linkage groups.groups. For each problem problem size. Figure 1(b) shows the size of size of spurious For each problem each rious linkagelinkage linkage groups is close to two, indicating thatlinkage linkage groups w the spurious groups groups is close to two, indicating that linkage groups groups were cre of size of spurious is close to two, indicating that larger larger larger were created Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 16. Number of Generations (All Replacement Strategies) Full replacement Elitist replacement RTR 1e+07 1e+07 1e+07 Number of generations Number of generations Number of generations Number of generations Number of generations Number of generations Number of generations 80 80 Problem size Problem size 80 Problem size 80 80 Problem size Problem size Problem size Problem size Problem size Problem siz 120 70 300 70 300 120 12070 300 120 70 300 70 300 120 120 1e+06 1e+06 300 1e+06 300 300 240 60 60240 60 6060 240 240 60 60240 60 60 100000 240 240 240 60 60 100000 100000 180 180 180 180 180 180 180 180 50 50 50 50 50 10000 120 10000 120 10000 120 40 40 40 40 40 60 60 60 1000 1000 1000 30 30 30 30 30 20 20 20 20 20 100 100 100 10 10 10 10 10 10 10 10 2 2.5 1.5 3.51 41.5 2 52.5 3 3.5 4 4.51.5 1 3 2 2.5 4.5 3.5 4 4.5 5 3 1 5 2 2.5 1.5 3.51 41.5 2 52.5 3 3.5 4 4.5 5 1.5 2 2.5 3 3.5 4 1 1.5 2 2.5 1 3 2 2.5 4.5 3.5 4 4.5 5 3 1 1 1.5 2 2.5 3 5 4.5 3.5 4 4 p SizeGroup Size (bits per group)per group) (bits per group) Size (bits Group Group size Group size (bits per group)per group) Group size Group size (bits per gro (bits per group) size (bits Group (bits per group) size (b Group replacement Full replacement Elitist replacement (a) Full (a) replacement (b) (b) Elitist replacement (b) Elitist replacement (c) RTR (c) RTR(c) RTR owthGrowthFullthe numberreplacement respect respectrespect spurious linkagesize. e Figure the Growth of the number of generations with spurious linkage group linkage 4: of 4: number of generations with with to the to the to the spurious group of and elitist of generations Number of generations slightly decreases with kspurious . 240 240 Niching (restricted tournament replacement) 20000 20000 20000 200 200 200 er of generations er of generations er of generations er of evaluations er of evaluations er of evaluations 220 220 180 180 180 18000 18000 18000 pulation size 200 200 Number of 160generations dramatically increases! 160 160 16000 16000 16000 180 180 140 140 Full repl. Full repl. Full repl. 14000 140 14000 Full repl. Full repl. Full 14000 160 Full 160 Full repl. Full repl. 120 120 Elitist repl. Elitist repl.Elitist repl. Elitist repl. Elitist repl.Elitist repl. 12000 120 12000 12000 RTR repl. RTR repl. RTR 140 Elitist 140 Elitist repl.Elitist repl. 100 repl. 100RTR repl. RTR repl. RTR repl. 10000 100 10000 10000 120 RTR 120 RTR repl. RTRPelikan 80 Elizabeth Radetic and Martin repl. repl. 80 80 Spurious Dependencies and EDA Scalability 8000 8000 8000
  • 17. Spurious Linkage in Multivariate EDAs Experiment Use optimal population size in ECGA. Observe spurious dependencies in actual models. Avg. number of groups > 1 140 Avg. size of groups > 1 Replacement 2.05 Replacement 1.8 Replacement Average group size 120 RTR 1.75 RTR 2.045 RTR 100 Elitist Elitist 1.7 Elitist Full 2.04 Full Full 80 1.65 2.035 1.6 60 2.03 1.55 40 2.025 1.5 20 2.02 1.45 0 2.015 1.4 50 100 150 200 250 300 50 100 150 200 250 300 50 100 150 200 250 300 Problem size (number of bits) Problem size (number of bits) Problem size (number of bits) (a) Number of spurious linkage (b) Avg. size of spurious linkage (c) Average linkage group size groups groups Figure 1: The average number of spurious linkage groups (groups of size ≥ 2), the average size of linkage groups of size ≥ 2, and the average linkage group size (including all linkage groups) for ECGA on onemax. Three replacement strategies are considered: full replacement, elitist replace- ment and RTR. For each problem size and replacement strategy, the results represent an average over 100 runs (10 bisections of 10 runs each). Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 18. Conclusions and Future Work Conclusions Population size increases exponentially with kspurious . Number of generations mostly unaffected. But for niching, the number of generations skyrocks! Spurious dependencies should not be ignored. Future work From our model to multivariate EDAs In most EDAs population sizing driven by model building. Almost always the models contain spurious dependencies. How do the models interact? Dramatic increase in the number of generations with niching Explain why. Propose ways to deal with it. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability
  • 19. Acknowledgments Acknowledgments NSF; NSF CAREER grant ECS-0547013. University of Missouri; High Performance Computing Collaboratory sponsored by Information Technology Services; Research Award; Research Board. Elizabeth Radetic and Martin Pelikan Spurious Dependencies and EDA Scalability

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