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Notes on the Distinction of Gaussian and Cauchy Mutations

This presentation by Dr. Kuo-Torng Lan from Takming University of Science and Technology delves into the distinctions between Gaussian and Cauchy mutations in the context of evolutionary computation. It covers key concepts such as mutation step sizes, population dynamics, and strategies for escaping local optima. Dr. Lan presents analyses and simulation results using benchmark functions like the Ackley and modified Schaffer functions, showcasing the advantages of Cauchy mutations in optimizing evolutionary algorithms. The findings highlight the effectiveness of Cauchy mutations in converging towards global optima.

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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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