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A comparative analysis of selection schemes used in genetic algorithms

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

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A comparative analysis of selection schemes used in genetic algorithms

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

  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: • p(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 * p(i,t) • m(i,t+1) = m(i,t) * f(i)/f(avg,t) • P(i,t+1) = P(i,t) * f(i)/f(avg,t) Genetic Algorithms

  8. Solution to diff eq • m(i,t+1) = m(i,t) * f(i) •  divide by population size • P(i,t+1) = f(i) m(i,t)/∑m(j,t+1) • P(i,t+1) = f(i) m(i,t)/∑f(j)m(j,t) • P(i,t+1) = f(i) P(i,t)/∑f(j)P(j,t) • Note: m(i,t) = m(i,0) f(i)^t • P(i,t) = f(i)t P(i,0)/∑f(j)t P(j,0) Genetic Algorithms

  9. Graph of Eqn, implementation Convergence behavior Genetic Algorithms

  10. Takeover time • 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 Genetic Algorithms

  11. 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

  12. 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

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

  14. 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

  15. 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

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

  17. Binary Tournament • Tournament size = 3 Genetic Algorithms

  18. Tournaments Genetic Algorithms

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

  20. Genitor Genetic Algorithms

  21. Growth Comparison Genetic Algorithms

  22. Takeover time comparison Genetic Algorithms

  23. Time complexity Genetic Algorithms

  24. Conclusions • The paper provides a framework for comparing selection operators • Implications for genetic search • The models provide us with theory necessary to compare selection methods and • Balance growth ration (quick convergence) with higher crossover/mutation (more exploration) Genetic Algorithms

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