quotient models and graphs:
DESCRIPTION
Using Quotient Graphs to Model Neutrality in Evolutionary Search Dominic Wilson Devinder Kaur University of Toledo. Quotient Models and Graphs:. Are widely applicable. Binary and non-binary Genetic Algorithms Grammatical Evolution Cartesian Genetic Programming. Performance. Generations. - PowerPoint PPT PresentationTRANSCRIPT
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