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Notes on the Distinction of Gaussian and Cauchy Mutations

Notes on the Distinction of Gaussian and Cauchy Mutations. Speaker ： Kuo-Torng, Lan. Ph. D. Takming Univ. of Science and Technology. I. Introduction II. Analyses of Two Mutations III. Simulation Results IV. Conclusions. I. Introduction. Rank or Roulette-wheel selection?

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Notes on the Distinction of Gaussian and Cauchy Mutations

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  1. Notes on the Distinction of Gaussian and Cauchy Mutations Speaker：Kuo-Torng, Lan. Ph. D. Takming Univ. of Science and Technology

  2. I. Introduction II. Analyses of Two Mutations III. Simulation Results IV. Conclusions

  3. I. Introduction • Rank or Roulette-wheel selection? • Gaussian or Cauchy mutation? • Population size? Mutation step size? … • escaping local optima & converging to the global optimum

  4. I. Introduction Individuals: walk randomly Population: go toward the local(global) optimum

  5. II. Analyses of Two Mutations • Assume the dimension of the individual is 1. • Assume the mutation step size is • The mutation is

  6. II. Analyses of Two Mutations • And X is a random variable with the Gaussian distribution. Its pdf is • And X is a random variable with the Cauchy distribution. Its pdf is

  7. II. Analyses of Two Mutations • Condition 1: Local Escape on Valley landscape

  8. II. Analyses of Two Mutations • Condition 1: Local Escape on Valley landscape For GMO: For CMO:

  9. II. Analyses of Two Mutations • Condition 2: Local Convergence on hill landscape

  10. II. Analyses of Two Mutations • Condition 2: Local Convergence on hill landscape For GMO: For CMO:

  11. II. Analyses of Two Mutations

  12. III. Simulation Results • Benchmark function 1: Ackey function • Benchmark function 2: modified Schaffer function • DC motor control(2005) • 2D fractal pattern Design(2006) • 3D fractal pattern Design(2008)

  13. III. Simulation Results • Benchmark function 1: Ackey function

  14. III. Simulation Results • Benchmark function 1: Ackey function

  15. III. Simulation Results • Benchmark function 1: Ackey function - by Gaussian mutation

  16. III. Simulation Results • Benchmark function 1: Ackey function - by Cauchy mutation

  17. III. Simulation Results • Benchmark function 2: modified Schaffer function

  18. III. Simulation Results • Benchmark function 2: modified Schaffer function

  19. III. Simulation Results • Benchmark function 2: modified Schaffer function

  20. III. Simulation Results • Benchmark function 2: modified Schaffer function

  21. III. Simulation Results • DC motor control: (K. T. Lan,“Design a rule-based controller for DC servo-motor Control byevolutionary computation,” TAAI 2005, in Chinese.)

  22. III. Simulation Results • DC motor control: (K. T. Lan,“Design a rule-based controller ...) The chromosome (i.e. control table)

  23. III. Simulation Results • DC motor control: (K. T. Lan,“Design a rule-based controller …,” )

  24. III. Simulation Results • 2D fractal pattern Design: (K. T. Lan,et al.,“Design a 2D fractal pattern by using the evolutionary computation,” TAAI 2006, in Chinese.)

  25. III. Simulation Results • 2D fractal pattern Design: (K. T. Lan,et al.,“Design a ...) The chromosome (i.e. 2D pattern)

  26. III. Simulation Results • 2D fractal pattern Design: (K. T. Lan,et al.,“Design a ...)

  27. III. Simulation Results • 3D fractal pattern Design: (K. T. Lan,et al.,“The problems for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.) The Cauchy mutation is predominant to Gaussian.

  28. III. Simulation Results • 3D fractal pattern Design: (K. T. Lan,et al.,“The problems for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.) Searching space: 10x10x10 No. of Reef: 60 Near optimal design: FD= 2.3843

  29. III. Simulation Results • 3D fractal pattern Design: (K. T. Lan,et al.,“The problems for design a 3D fractal pattern by using the evolutionary computation,” TAAI 2008, in Chinese.) Searching space: 12x12x12 No. of Reef: 94 Near optimal design: FD=2.4055

  30. IV. Conclusions • A larger mutation step size can lead population to escape local optima and tend towards the global optimum • A smaller mutation step size can finely tune the population • Cauchy mutation possesses more power in escaping local optima

  31. IV. Conclusions • For local convergence, the Cauchy technique is nearly equal to the Gaussian after evolving more generations. • Therefore, Cauchy mutation is suggested to avoid the dilemma problem and achieve the acceptable performance for evolutionary computation. Thanks for your kindly attention.

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