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Some Theory

Some Theory. Contents. Takeover time selection only Goldberg & Deb (1998) Statistical Dynamics of the Royal Road GA mutation + selection Nimwegen, Crurchfield and Mitchell (1999). Takeover time.

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Some Theory

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  1. Some Theory MKI44: EAs

  2. Contents • Takeover timeselection onlyGoldberg & Deb (1998) • Statistical Dynamics of the Royal Road GAmutation + selectionNimwegen, Crurchfield and Mitchell (1999) MKI44: EAs

  3. Takeover time • Takeover timeselection onlyD.E. Goldberg & K. Deb (1991). A comparative analysis of selection schemes used in genetic algorithms. In G. Rawlins (ed.), Foundations of Genetic Algorithms. Morgan Kaufman, San Francisco, pp. 69-93 MKI44: EAs

  4. Selection • Fitness proportionate reproduction • Rank-based selection • Tournament selection • Generational • Steady state MKI44: EAs

  5. Selection • Birth, life, and death:m number of individualsi class of individuals with fitness fit timeb bornd death MKI44: EAs

  6. Selection • Fitness proportionate reproduction, nonoverlapping generations: Probability of selection MKI44: EAs

  7. Selection • Let the population grow according to the uncoupled growth equations:Solution for the proportions: MKI44: EAs

  8. Fitness • Suppose (i = 1/n, 2/n, …, 1) • Approximate (n large; p0(x) = 1): • Best x = 1: MKI44: EAs

  9. Fitness • Best x = 1: • Takeover when • Takeover time t*: MKI44: EAs

  10. Fitness • Suppose (i = 1/n, 2/n, …, 1) • Same analysis • Takeover time t* (comparable) MKI44: EAs

  11. Overview MKI44: EAs

  12. Macroscopic behavior • Statistical Dynamics of the Royal Road GAmutation + selectionE. van Nimwegen, J.P. Crutcheld & M. Mitchell (1999), Statistical Dynamics of the Royal Road Genetic Algorithm, Theoretical Computer Science, 229:41-102 MKI44: EAs

  13. Royal Road fitness functions • Bit strings of length L = NK • N blocks of length K • Fitness of a block is 1 if all bits are 1 and else 0 • 0  f(s)  N MKI44: EAs

  14. Genetic Algorithm • Randomly initialize population of size M of strings of length L • Evaluate fitness f(s) • Fitness proportional select M strings with replacement • Mutate each gene in all strings with probability q • Go to step 2 MKI44: EAs

  15. Genetic Algorithm • Four parameters:N, K, q and M • Experiments: L = NK = 60 • Average fitness (solid lines) and best fitness (diamonds->thick solid lines) of 9 runs MKI44: EAs

  16. MKI44: EAs

  17. Experiments • Run d clearly shows epochal dynamics. • Experimentally, the length of the epochs varies greatly from run to run, but fitness levels are the same • Reproducible macroscopic feature of the dynamics MKI44: EAs

  18. Mathematical Analysis • Assumptions • Maximum entropy assumtion: all microscopic states consistent with a macroscopic state are equally likely • Fitness distribution Pr(f) • Construct generation operator G, that in the limit of infinite populations, deterministically describes the action of the GA on fitness distributions MKI44: EAs

  19. Mathematical Analysis • State of population given by distribution(P0(t), P1(t),…,Pf(t),…,PN(t)), where Pf(t) is the fraction of individuals in the population with fitness f at time t • N+1 dimensional vector • Average fitness • State space MKI44: EAs

  20. Mathematical Analysis • State space • Size • M, selection operator S, mutation operator M, G=MS MKI44: EAs

  21. Mathematical Analysis • Probability As to transform an unaligned block into an aligned block • If bits in block are independent (lowerbound): • Upperbound: only one bit is missing, so • Used: As = A MKI44: EAs

  22. Mutation operator • Probability Dto destroy an aligned block • Probability Mij that mutation turns a string with fitness j (j aligned blocks) into a string with fitness i: MKI44: EAs

  23. Mutation operator • Fitness distribution after mutation: • M is a stochastic matrix: MKI44: EAs

  24. Selection operator • Selection operator is a diagonal matrix MKI44: EAs

  25. Generation operator • Combine M and S • Do some matrix operations, like diagonalizing G for different values of t, results in fitness estimations, fitting well to the experimental fitnesses(averaged over 20 runs with parameter values N=3, K=4, q = 0.01, M = 104) MKI44: EAs

  26. Generation operator MKI44: EAs

  27. GA Dynamics in Fitness Distribution Space MKI44: EAs

  28. Finite Population Dynamics MKI44: EAs

  29. Predicted Epoch levels MKI44: EAs

  30. Predicted Epoch levels • Some experiments with crossover show the same epoch fitness levels • The paper also looks at the stability MKI44: EAs

  31. Crossover? • T.Jansen and I. Wegener (2002), The Analysis of Evolutionary Algorithms-A Proof That Crossover Really Can Help, Algorithmica 34:47-66 MKI44: EAs

  32. Take Home Message • Eas are complex systems • Microscopic behavior versus macroscopic behavior • Difficult, interesting, important MKI44: EAs

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