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

Local optimumLocal optimum

Global optimum

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

g e n e r a t i o n t + 1

t g e n e r a t i o n δ

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

0

5

10

15

20

25

30

35

40

45

1 101 201 301 401 501 601 701 801 901

Generations

Fitn

ess

valu

e

step size=0.4step size=0.6step size=0.8

16

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

0

5

10

15

20

25

30

35

40

45

1 101 201 301 401 501 601 701 801 901

Generations

Fitn

ess

valu

e

step size=0.4

step size=0.6

step size=0.8

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

0

1

2

3

4

5

6

7

1 101 201 301 401 501 601 701 801 901

Generations

Fitn

ess

valu

e

Gaussian

Cauchy

20

III. Simulation Results

• Benchmark function 2: modified Schaffer function

0

1

2

3

4

5

6

7

1 101 201 301 401 501 601 701 801 901

Generations

Fitn

ess

valu

e

Gaussian

Cauchy

24

III. Simulation Results

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

G ss s s s

( )( )( . )( . )( . )

1

1 1 0 2 1 0 05 1 0 01

G(s)+

Vin Vout

Rule-Based Controller

e

ddt

VK

c

p

25

III. Simulation Results

• DC motor control: (K. T. Lan, “Design a rule-based controller ...)

The chromosome (i.e. control table)

26

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

0

0.5

1

1.5

2

2.5

0.1 0.6 1.1 1.6 2.1 2.6 3.1

mutation step size

Ris

eTim

e(R

2) EA(Gauss, avg)

EA(Gauss, best)

EA(Cauchy, avg)

EA(Cauchy, best)

27

III. Simulation Results

• 2D fractal pattern Design : (K. T. Lan, et al., “Design a 2D f

ractal pattern by using the evolutionary computation,” TAAI 2006, in Chinese.)

28

III. Simulation Results

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

The chromosome (i.e. 2D pattern)

29

III. Simulation Results

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

30

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.

31

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: 10x10x10No. of Reef: 60Near optimal design: FD= 2.3843

32

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: 12x12x12No. of Reef: 94Near optimal design: FD=2.4055

33

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

34

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.