A new crossover technique in Genetic Programming

Janet Clegg

Intelligent Systems Group

Electronics Department

This presentation

Describe basic evolutionary optimisation

Overview of failed attempts at crossover methods

Describe the new crossover technique

Results from testing on two regression problems

Evolutionary optimisation

Start by choosing a set of random trial solutions

(population)

Each trial solution is evaluated

(fitness / cost)

1.2 0.9 0.8 1.4 0.2 0.3 2.1 2.3 0.3 0.9 1.3 1.0

0.8 2.4 0.4 1.7 1.5 0.6 1.3 0.8 2.5 0.7 1.4 2.1

Parent selection

0.9 2.1 1.3 0.7

1.4 0.6 1.7 1.5

2.1

1.7

Select mother

Select father

Perform crossover

child 1 child 2

Mutation

Probability of mutation small (say 0.1)

This provides a new population of solutions –

the next generation

Repeat generation after generation

1 select parents

2 perform crossover

3 mutate

until converged

Genetic Algorithms (GA) –

Optimises some quantity by varying parameter values which have numerical values

Genetic Programming (GP) –

Optimises some quantity by varying parameters which are functions / parts of computer code / circuit components

Two types of evolutionary optimisation

Representation

Representation

Traditional GA’s - binary representation

e.g. 1011100001111

Floating point GA – performs better than binary

e.g. 7.2674554

Genetic Program (GP)Nodes represent functions whose inputs are below the branches attached to the node

Some crossover methods

Crossover in a binary GA

Mother

1 0 0 0 0 0 1 0 = 130

Father

0 1 1 1 1 0 1 0 = 122

Child 1

1 1 1 1 1 0 1 0 = 250

Child 2

0 0 0 0 0 0 1 0 = 2

Min parameter value

Max parameter value

Mother father

Offspring chosen as random point between mother and father

Crossover in a floating point GA

Traditional method of crossover in a GP

mother father

Child 1 Child 2

Motivation for this work

Tree crossover in a GP does not always perform well

Angeline and Luke and Spencer compared:-

(1) performance of tree crossover

(2) simple mutation of the branches

difference in performance was statistically insignificant

Consequently some people implement GP’s with no crossover - mutation only

Motivation for this work

In a GP many people do not use crossover so mutation is the more important operator

In a GA the crossover operator contributes a great deal to its performance - mutation is a secondary operator

AIM:- find a crossover technique in GP which works as well as the crossover in a GA

Cartesian Genetic Programming

Cartesian Genetic Programming (CGP)

Julian Miller introduced CGP

Replaces tree representation with directed graphs – represented by a string of integers

The CGP representation will be explained within the first test problem

CGP uses mutation only – no crossover

First simple test problem

A simple regression problem:-

Finding the function to best fit data taken from

122 xx

The GP should find this exact function as the optimal fit

The traditional GP method for this problem

Set of functions and terminals

Functions Terminals

+ 1

- x

*

*

x x x 1

-

*

(1-x) * (x*x)

Initial population created by randomly choosing functions and terminals within the tree structures

Crossover by randomly swapping sub-branches of the parent trees

mother father

Child 1 Child 2

Set of functions – each function identified by an integer

Functions Integer representation

+ 0- 1

* 2

Set of terminals – each identified by an integer

Terminals Integer representation

1 0x 1

CGP representation

Creating the initial population

2

First integer is random choice of function 0 (+), 1 (-), or 2 (*)

0 1

Second two integers are random choice of terminals 0 (1) or 1 (x)

0

2

1 1

Next integers are random choice of inputs for the function from the set

0 (1) 1 (x) or node 2

3

Creating the initial population

2 0 1 0

2

1 1

random choice of inputs from

0 1 2 3 4 5 Terminals nodes

3

1 3 1 0 2 3

4 5

2 4 1

6

random choice for output from

0 1 2 3 4 5 6 Terminals all nodes

5output

2 0 1 0 1 1 1 3 1 0 2 3 2 4 1 5 2 3 4 5 6 output

output

5

+

x x x 1

*

+

32

(1*x) +(x+x) = 3x

122 xx

Run the CGP with test data taken from the function

Population size 30

28 offspring created at each generation

Mutation only to begin with

Fitness is the sum of squared differences between data and function

Result

0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

2 3 4 5 6 output

xxx 1*)1(

5

+

x 1 x +

*

+

23

1 x

2

= 122 xx

Any two runs of a GP (or GA) will not be exactly the same

To analyse the convergence of the GP we need lots of runs

All the following graphs depict the average convergence out of 4000 runs

Statistical analysis of GP

Introduce crossover

Introducing some Crossover

Parent 1

0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

Parent 2

2 0 1 0 1 1 1 3 1 0 2 3 2 4 1 5

Child 1

0 0 1 2 2 1 1 3 1 0 2 3 2 4 1 5

Child 2

2 0 1 0 1 1 1 2 2 0 3 2 0 5 1 5

Pick random crossover point

GP with and without crossover

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100

Without crossoverWith crossover

-350

-300

-250

-200

-150

-100

-50

0 200 400 600 800 1000

No CrossoverWith Crossover

GA with and without crossover

Parent 1

0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

Parent 2

2 0 1 0 1 1 1 3 1 0 2 3 2 4 1 5

Child 1

0 0 1 2 2 1 1 3 1 0 2 3 2 4 1 5

Child 2

2 0 1 0 1 1 1 2 2 0 3 2 0 5 1 5

Random crossover point but must be between the nodes

0 0 1 1 1 2 2 3 2 2 4 1 0 3 5 6 2 3 4 5 6 output

+-

*

x + * x

1 x- +

x +

1 x

1 x

6

3

2

5

4

3

2

2

GP crossover at nodes

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100

Without crossoverWith crossover

Crossover at nodes

Parent 1

0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

Parent 2

2 0 1 0 1 1 1 3 1 0 2 3 2 4 1 5

Child 1

0 0 1 2 2 1 1 3 1 0 2 3 2 4 1 5

Child 2

2 0 1 0 1 1 1 2 2 0 3 2 0 5 1 5

Pick a random node along the string and swap this single node

Crossover only one node

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100

Without crossoverWith crossover

Crossover only one node

Parent 1

0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

Parent 2

2 0 1 0 1 1 1 3 1 0 2 3 2 4 1 5

Child 1

2 0 1 0 2 1 1 2 1 0 2 2 2 5 1 5

Child 2

0 0 1 2 1 1 1 3 2 0 3 3 0 4 1 5

Each integer in child randomly takes value from either mother or father

Random swap crossover

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100

Without crossoverWith crossover

Random swap crossover

Comparison with random search

GP with no crossover performs better than any of the trial crossover here

How much better than a completely random search is it?

The only means it will improve on a random search are by

parent selection

mutation

Comparison with a random search

0

1

2

3

4

5

6

7

8

9

0 10 20 30 40 50 60 70 80 90 100

Without crossoverRandom search

Comparison with a random search

0

0.001

0.002

0.003

0.004

0.005

0.006

0.007

0.008

0.009

0.01

30 35 40 45 50 55 60 65 70 75 80

Without crossoverRandom search

GP converges in 58 generations

Random search 73 generations

GA performance compared with a completely random search

GA tested on a large problem –

A random search would have involved searching through

150,000,000 data points

The GA reached the solution after testing

27,000 data points

( average convergence of 5000 GA runs)

Probability of random search reaching solution in 27,000 trials is 0.00018 !!!!

Why does GP tree crossover not always work well?

f1

f2 f3

f5f4

x8x7x6x5x4x3x2x1

f6 f7

f1 { f2 [ f4( x1,x2 ), f5( x3,x4 ) ] , f3 [ f6( x5,x6 ), f7( x7,x8 ) ] }

g1 { g2 [ g4( y1,y2 ), g5( y3,y4 ) ] , g3 [ g6( y5,y6 ), g7( y7,y8 ) ] }

f1 { f2 [ f4( x1,x2 ), f5( x3,x4 ) ] , f3 [ f6( x5,x6 ), g7( y7,y8 ) ] }

x2x1

f

g

g( x1 ) = f( x2 )

Good!

x2x1

f

gg( x2 )

f( x1 )

Good!

f( x1 )

f( x2 )

g( x2 )

g( x1 )

0

0.5

1 0

0.5

10.6

0.65

0.7

0.75

0.8

0.85

0.9

0.951

1.05

1.1

1.15

Suppose we are trying to find a function to fit a set of data looking like this

exp( -(0.5-x)*(0.5-x)-(0.5-y)*(0.5-y) )

0

0.5

1 0

0.5

1

0.60.65

0.70.750.8

0.850.9

0.951

1.051.1

1.15

1/( 0.85+( (0.5-x)*(0.5-x)+(0.5-y)*(0.5-y) )**0.5 )

Suppose we have 2 parents which fit the data fairly well

22 5.0)5.0(

1 xye

/

1

+

xy

^ ^

- - 22

0.50.5

exp

Choose crossover point

/

1

2

^

+

+

yx

^ ^

- -2

0.50.5

0.5

0.85 22 5.05.085.0

1

yx

Choose crossover point

exp( -(0.5-x)*(0.5-x)-(0.5-y)*(0.5-y) )

0

0.5

1 0

0.5

1

0.6

0.7

0.8

0.9

1

1.1

1.2

1/( 0.85+( (0.5-x)*(0.5-x)+(0.5-y)*(0.5-y) )**0.5 )1/(0.85+((x-0.5)**2+(x-0.5)**2)**0.5)

0

0.5

1 0

0.5

1

0.4

0.5

0.6

0.7

0.8

0.9

1

1.1

1.2

Introducing the new technique

Based on Julian Miller’s

Cartesian Genetic Program (CGP)

The CGP representation is changed

Integer values are replaced by floating point values

Crossover is performed as in a floating point GA

CGP representation0 0 1 2 2 1 1 2 2 0 3 2 0 5 1 5

1 2 3 4 5 output

New representation – replace integers with floating point variables

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16

Where the xi lie in a defined range, say 1,0ix

2 3 4 5 6 output

Interpretation of the new representation

For the variables xi which represent choice of function

If the set of functions is

+ - *

0 0.33 0.66 1

+ - *

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16

1 2 3 4 5 output

0 0.33 0.66 1

1 x node 1

0 0.2 0.4 1

1 node 1 x node 2 node 3

0.6 0.8

The crossover operator

Crossover is performed as in a floating point GA

Two parents, p1 and p2 are chosen

Offspring o1 and o2 are created by

Uniformly generated random number is chosen 0 < ri < 1

oi = p1 + ri (p2 − p1) when p1 < p2

Min parameter value

Max parameter value

Mother father

Offspring chosen as random point between mother and father

Crossover in the new representation

Why is this crossover likely to work better than tree crossover?

x8

f1

f2 f3

f5f4

x7x6x5x4x3x2x1

f6 f7

f1 { f2 [ f4( x1,x2 ), f5( x3,x4 ) ] , f3 [ f6( x5,x6 ), f7( x7,x8 ) ] }

g1 { g2 [ g4( y1,y2 ), g5( y3,y4 ) ] , g3 [ g6( y5,y6 ), g7( y7,y8 ) ] }

f1 { f2 [ f4( x1,x2 ), f5( x3,x4 ) ] , f3 [ f6( x5,x6 ), g7( y7,y8 ) ] }

Mathematical interpretation of tree crossover

1321 ,..., nxxxf

x1 x2 x3 x4 x5 x6 x7 x8 x9 x10 x11 x12 x13 x14 x15 x16

Mathematical interpretation of the new method

1 2 3 4 5 output

The fitness can be thought of as a function of these 16 variables, say

1621 ,...,, xxxf

and the optimisation becomes that of finding the values of these 16 variables which give the best fitness

The new crossover guides each variable towards its optimal value in a continuous manner

The new technique has been tested on two problems

Two regression problems studied by Koza

246 2 xxx

xxx 35 2

(1)

(2)

Test data – 50 points in the interval [-1,1]

Fitness is the sum of the absolute errors over 50 points

Population size 50 – 48 offspring each generation

Tournament selection used to select parents

Various rates of crossover tested 0% 25% 50% 75%

Number of nodes in representation = 10

The following results are based on the average convergence of the new method out of 1000 runs

Considered converged when the absolute error is less than 0.01 at all of the 50 data points (same as Koza)

Results based on

(1) average convergence graphs

(2) the average number of generations to converge

(3) Koza’s computational effort figure

Statistical analysis of new method

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Average convergence for 246 2 xxx

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

200 300 400 500 600 700 800 900 1000

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Convergence for latter generations

Introduce variable crossover

At generation number 1

crossover performed 90% of the time

Rate of crossover linearly decreases until

Generation number 180

crossover is 0%

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Variable crossover

Variable crossover

Average number of generations to converge and computational effort

Percentage crossover

Average number of

generations to converge

Koza’s computational

effort

0% 168 30,00025% 84 9,00050% 57 8,00075% 71 6,000

Variable crossover

47 10,000

1

2

3

4

5

6

7

8

9

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Average convergence for xxx 35 2

0

1

2

3

4

5

6

7

8

9

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Variable crossover

Variable crossover

Percentage crossover

Average number of

generations to converge

Koza’s computational

effort

0% 516 44,00025% 735 24,00050% 691 14,00075% 655 11,000

Variable crossover

278 13,000

Average number of generations to converge and computational effort

0

1000

2000

3000

4000

5000

6000

7000

8000

9000

0 20 40 60 80 100

Num

ber o

f gen

erat

ions

to c

onve

rge

Run number

Problem given in Equation 4Problem given in Equation 5

Number of generations to converge for both problems over 100 runs

0

1

2

3

4

5

6

7

8

9

0 100 200 300 400 500 600 700 800 900 1000

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Variable crossover

Average convergence ignoring runs which take over 1000 generations to converge

The new technique has reduced the average number of generations to converge

From 168 down to 47 for the first problem tested

From 516 down to 278 for the second problem

Conclusions

When crossover is 0% this new method is equivalent to the traditional CGP - mutation only

The computational effort figures for 0% crossover here are similar to those reported for the traditional CGP

Although a larger mutation rate and population size have been used here

Conclusions

Future work

Investigate the effects of varying the GP parameters

population size

mutation rate

selection strategies

Test the new technique on other problems

larger problems

other types of problems

Thankyou for listening!

0

0.5

1

1.5

2

2.5

3

3.5

4

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Variable crossover

Average convergence for

using 50 nodes instead of 10

246 2 xxx

Average number of generations to converge and computational effort

Percentage crossover

Average number of

generations to converge

Koza’s computational

effort

0% 78 18,000

25% 85 13,000

50% 71 11,000

75% 104 13,000

Variable crossover

45 14,000

0

1

2

3

4

5

6

7

8

9

0 50 100 150 200 250 300 350 400 450 500

Cos

t fun

ctio

n

Generation number

Crossover 0%Crossover 25%Crossover 50%Crossover 75%

Variable crossover

xxx 35 2Average convergence for

using 50 nodes instead of 10

Average number of generations to converge and computational effort

Percentage crossover

Average number of

generations to converge

Koza’s computational

effort

0% 131 18,000

25% 193 17,000

50% 224 12,000

75% 152 19,000

Variable crossover

58 16,000