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Comparative Analysis of Genetic Algorithm Selection Schemes

This paper defines and compares four selection schemes in genetic algorithms, presenting a technique for modeling and testing their performance. It analyzes convergence time, growth ratios, and practical applications. Many claims of superiority are examined, providing valuable insights for the practical application of genetic algorithms.

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Comparative Analysis of Genetic Algorithm Selection Schemes

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  1. A comparative analysis of selection schemes used in genetic algorithms Genetic Algorithms David E. Goldberg Kalyanmoy Deb

  2. What is the paper about? • Defines and compare four selection schemes • Presents a technique for comparisons: • Produce a difference/differential equation modeling the selection scheme • Test computer implementation against diff. equation model • Defines criteria for comparison: • Convergence time • Schema growth ratios • Conclusions: practical applications of analysis Genetic Algorithms

  3. Where are we now? • Many papers claim the superiority of this or that selection scheme • But many of these claims are based on limited (and uncontrolled experiments). • Little analysis has been done • This paper attempts to provide the needed analysis Genetic Algorithms

  4. What selection strategies? • Proportionate reproduction • Ranking selection • Tournament selection • Genitor (“steady state”) selection Genetic Algorithms

  5. Birth, life, and death • m(i, t+1) = m(i, t) + m(i, t, b) – m(i,t,d) • Ex: in non-overlapping population models: • m(i,t+1) = m(i,t,b) ; m(i,t,d) = m(i,t) • We can also do proportions: • P(i,t+1) = P(i,t) + P(i,t,b) – P(i,t,d) Genetic Algorithms

  6. Proportionate Reproduction • Probability of selection: • prob(i,t) = f(i)/∑m(j,t) f(j) • Various methods for implementation: • Roulette wheel • Stochastic remainder • Stochastic universal Genetic Algorithms

  7. How many in next generation? • m(i,t+1) = m(i,t) * n * prob(i,t) • m(i,t+1) = m(i,t) * f(i)/f(avg,t) • Proportion(i,t+1) = Proportion(i,t) * f(i)/f(avg,t) Genetic Algorithms

  8. Graph of Eqn, implementation Genetic Algorithms Convergence behavior

  9. How many individuals between specified values of x in objective function f(x)? Let p0(x) be uniform, integral  1 Consider f(x) = xc and f(x) = ecx Limits x and x – 1/n Takeover time Genetic Algorithms

  10. Behavior of f(x) = x^c Integrate with limits x & x – 1/n x = 1 is best, x = 0 is worst individual Compare theory and experiment for f(x) = x Genetic Algorithms

  11. Takeover time for f(x) = x^c Solving for t and approximating Setting x = 1, we get proportion of best individual Setting P = n-1/n, we calculate when population contains n-1 best individuals Thus the takeover time for a polynomially distributed objective function is O(nlogn) Genetic Algorithms

  12. Takeover time for f(x) = e^cx The takeover time for a polynomially and exponentially distributed objective function is O(nlogn) Genetic Algorithms

  13. Time complexity of Proportionate Reproduction • Roulette Wheel • O(n2) or O(nlogn) with binary search • Stochastic remainder selection • floor(f(i)/favg) number of copies • Remainder = flip(fractional(f(i)/favg)) • O(n) without replacement or O(n2) with Genetic Algorithms

  14. Ranking • Sort from best to worst • Create a transformation function called an assignment function that converts a rank to an equivalent “fitness” • assignFunction(rank) • Proportionate reproduction on assignFunction(rank) Genetic Algorithms

  15. Tournament Selection • Binary • N-ary • Randomly choose N individuals from population • Select best for further processing Genetic Algorithms

  16. Binary Tournament • Tournament size = 3 Genetic Algorithms

  17. Tournaments Genetic Algorithms

  18. Genitor • Choose an offspring based on ranking • Replace worst individual in population Genetic Algorithms

  19. Genitor Genetic Algorithms

  20. Growth Comparison Genetic Algorithms

  21. Takeover time comparison Genetic Algorithms

  22. Time complexity Genetic Algorithms

  23. Conclusions • The paper provides a framework for comparing selection operators • Compares selection “pressure” for each type of selection • Introduces the concept of takeover time to help us understand the exploration/exploitation tradeoff • Provides takeover time estimates for different types of selection • Implications for genetic search • The models provide us with theory necessary to compare selection methods and • Balance growth ratio (quick convergence) with higher crossover/mutation (more exploration) Genetic Algorithms

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