Using Quotient Graphs to Model Neutrality in Evolutionary Search

Dominic WilsonDevinder Kaur

University of Toledo

Quotient Models and Graphs: Are widely applicable.

Binary and non-binary Genetic Algorithms Grammatical Evolution Cartesian Genetic Programming

Quotient Models and Graphs: Can explain why performance improvements

are usually smaller for later generations of evolution (e.g. ONEMAX);

Generations

Per

form

ance

Quotient Models and Graphs: Can explain the change of the location of a

steady state population with mutation rate;

J. Richter, A. Wright and J. Paxton. "Exploration of Population Fixed Points Versus Mutation Rates for Functions of Unitation", GECCO-2004.

Quotient Models and Graphs: Exact Markov models; Reduce the degrees of freedom needed

for modeling; Show aspects of evolutionary search that

are not obvious (e.g. correlated mutational drives).

Can track population movements on complex landscapes;

Why Models? To understand and explain the complex dynamics of

Evolutionary Computing systems; Examples of models:

Schema. (J. H. Holland. Adaptation in Natural and Artificial Systems. University of Michigan Press, 1975.)

Predicates. (M.D. Vose, “Generalizing the notion of schema in genetic algorithms. “,Artificial Intelligence, 50 1991.)

Formae. (N. J. Radcliffe. “Equivalence class analysis of genetic algorithms.” Complex Systems, 5(2),1991.)

Unitation Functions. (J. E. Rowe, “Population fixed-points for functions of unitation,” FOGA 5, 1999.)

Model Similarities Schemata, Predicates, Formae and Unitation

Functions are defined based on subsets of the genotype space.

They are oblivious of the genotype-to phenotype map.

Quotient Models and Graphs Quotient models are formed by grouping

subsets of the genotype space that have the same fitness and search behavior. They are therefore aware of the structure of the genotype-to-phenotype map.

Quotient graphs visually portray quotient models. They consist of nodes that have the same fitness and search behavior, connected by directed arcs.

Content Create an example quotient model.

Show how quotient models can be used to explain evolutionary search behavior.

Example Genotype to Fitness Map3 bit Strings

XFitness

F000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

0, 111

,ii

XF

X otherwise

F is like ONEMAX

except for string “111”

Example Map on a Cube3 bit Strings

XFitness

F000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Fitness Distribution on Mutation

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Each string with only one bit set to “1” has the same neighborhood!

They also have the same fitness.

Fitness Distribution on Mutation

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Fitness Distribution on Mutation

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Fitness Distribution on Mutation

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

String with fitness “0” do not have the same neighborhood!

Quotient Graph

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Quotient Graph

Quotient Graph

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Represents the same neighborhood information as the cube

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Quotient Graph

Correlated mutational drives

Quotient Graph

8 nodes 4 nodes

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Larger Quotient Graphs

8 bit ONEMAX

256 9nodes nodes

2 1n nodes n nodes

n bit ONEMAX

StringFitness Map as Linear Map

Strings X

Fitness

F000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

01 0 0 0 0 0 0 0

T

X

and

50 0 0 0 0 1 0 0

T

X

01 0 0

T

F

10 1 0

T

F

, and 20 0 1

T

F

F XAF: Fitness

X: String

A: String to fitness map (linear operator)

Mapping3 bit

Strings (X)

Fitness

F

000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

0

0

00 1 0 0 0 0 0 0 1

00 0 1 1 0 1 0 0 0 *

01 0 0 0 1 0 1 1 0

1

0

0

2 5

F AX

,

1,

0i j

if string j maps to fitness iA

otherwise

Mutation

33

322

232

3223

)1(

..

..

..

)1()1()1(

)1()1()1(

...)1()1()1(

111

010

001

000

111...010001000

M

Bit mutation probability: Mutation rate matrix: M

Probability distribution of fitness on mutation

AMXX

X: Current String;

MX: Probability distribution of string after mutation;

AMX: Probability distribution of string fitness after mutation

AM

Search distribution

3 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3

3 3

0 (1 ) 3 (1 )

1 (1 ) (1 ) (1 ) 2 (1 )

2 (1 ) (1 ) (1 ) 2 (1 )

3 (1 ) (1 ) 2 (1 )

4 (1 ) (1 ) (1 ) 2 (1 )

5 (1 ) (1 ) 2 (1 )

6 (1 ) (1 ) 2 (

7 (1 )

T

2

3 2

3 2

3 2

3 2

3 2

2 3 2

2 2

3 (1 )

2 (1 )

2 (1 )

(1 ) 2 (1 )

2 (1 )

(1 ) 2 (1 )

1 ) (1 ) 2 (1 )

3 (1 ) 3 (1 )

3 8 3 8 8 8by by byA M

Search distribution

3 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3 2

2 2 3

3 3

0 (1 ) 3 (1 )

1 (1 ) (1 ) (1 ) 2 (1 )

2 (1 ) (1 ) (1 ) 2 (1 )

3 (1 ) (1 ) 2 (1 )

4 (1 ) (1 ) (1 ) 2 (1 )

5 (1 ) (1 ) 2 (1 )

6 (1 ) (1 ) 2 (

7 (1 )

T

2

3 2

3 2

3 2

3 2

3 2

2 3 2

2 2

3 (1 )

2 (1 )

2 (1 )

(1 ) 2 (1 )

2 (1 )

(1 ) 2 (1 )

1 ) (1 ) 2 (1 )

3 (1 ) 3 (1 )

Probability distribution of string fitness after mutation

Rows 1, 2 and 4 are identical;

Rows 3, 5 and 6 are identical;

:

Example Map on a Cube3 bit Strings

XFitness

F000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Quotient Graph

000

(0)

001

(1)

100

(1)

011

(2)

110

(2)

111

(0)101

(2)010

(1)

Quotient Graph

Quotient sets

0 1 2 3 4 5 6 7

[0 ] 1 0 0 0 0 0 0 0

[0 ] 0 0 0 0 0 0 0 1

[1] 0 1 1 0 1 0 0 0

[2] 0 0 0 1 0 1 1 0

a

bQ

quotient set assignment matrix:

3 bit Strings X

Fitness

F000 0

001 1

010 1

011 2

100 1

101 2

110 2

111 0

One set for each color.

Quotient model

[1]

[0a]

[2]

[0b]

0 1 2 3 4 5 6 7

[0 ] 1 0 0 0 0 0 0 0

[0 ] 0 0 0 0 0 0 0 1

[1] 0 1 1 0 1 0 0 0

[2] 0 0 0 1 0 1 1 0

a

bQ

Quotient Mutation Rate Matrix

MQX QMX

.

1( )T TM QMQ QQ

Mutation rate matrix: MQuotient mutation rate matrix: M

Quotient assignment matrix: Q

Quotient Mutation Rate Matrix

3 3 2 2

3 3 2 2

2 2 3 2 3 2

2 2 3 2 3 2

[0 ] [0 ] [1] [2]

[0 ] (1 ) 3 (1 ) 3 (1 )

[0 ] (1 ) 3 (1 ) 3 (1 )

[1] (1 ) (1 ) (1 ) 2 (1 ) 2 (1 )

[2] (1 ) (1 ) 2 (1 ) (1 ) 2 (1 )

a b

a

bM

Quotient mutation rate matrix: M

Quotient Graph of 4 bit ONEMAX with neutral layer of fitness 3

[3a]

[4]

[1]

[2]

[0]

[3e]

[3f]

[3c]

[3d]

[3b]

[3a]

[4]

[1]

[2]

[0]

[3e]

[3f]

[3c]

[3d]

[3b]

Fitness Drives Correlated mutational Drives

E. Galvan-Lopez , R. Poli, “An Empirical Investigation of How and Why Neutrality Affects Evolutionary Search” GECCO’06.

Example Quotient Graphs