A new crossover technique in Genetic Programming

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A new crossover technique in Genetic Programming
Janet Clegg Intelligent Systems Group Electronics
Department
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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
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Evolutionary optimisation
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Start by choosing a set of random trial
solutions (population)
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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
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Parent selection
Select mother
2.1
0.9 2.1 1.3 0.7
Select father
1.7
1.4 0.6 1.7 1.5
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Perform crossover
child 1
child 2
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Mutation
Probability of mutation small (say 0.1)
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This provides a new population of solutions
the next generation Repeat
generation after generation 1 select parents
2 perform crossover 3 mutate until
converged
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Two types of evolutionary optimisation
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
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Representation
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Representation
Traditional GAs - 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
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Some crossover methods
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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
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Crossover in a floating point GA
Mother father
Min parameter value
Max parameter value
Offspring chosen as random point between mother
and father
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Traditional method of crossover in a GP
mother
father
Child 2
Child 1
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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 GPs with no crossover - mutation only
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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
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Cartesian Genetic Programming
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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
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First simple test problem A simple regression
problem- Finding the function to best fit data
taken from
The GP should find this exact function as the
optimal fit
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The traditional GP method for this problem Set
of functions and terminals
Functions Terminals
1
- x

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Initial population created by randomly choosing
functions and terminals within the tree structures
(1-x) (xx)
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Crossover by randomly swapping sub-branches of
the parent trees
mother
father
Child 2
Child 1
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CGP representation
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 0
x 1
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Creating the initial population
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0 1
0
1 1
2
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First integer is random choice of function 0 (),
1 (-), or 2 ()
Second two integers are random choice of
terminals 0 (1) or 1 (x)
Next integers are random choice of inputs for the
function from the set 0 (1) 1 (x) or node 2
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Creating the initial population
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0 1
0
1 1
1 3 1 0 2 3
2
4 1
5
output
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5
6
2
3
random choice of inputs from 0 1
2 3 4 5 Terminals
nodes
random choice for output from 0 1
2 3 4 5 6 Terminals
all nodes
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2 0 1 0 1 1 1 3 1
0 2 3 2 4 1
5
2 3
4 5 6
output
output
5
3
2
(1x) (xx) 3x
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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
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Result 0 0 1 2 2 1 1
2 2 0 3 2 0 5 1
5
2 3
4 5 6
output

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Statistical analysis of GP
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
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Introduce crossover
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Introducing some Crossover
Pick random crossover point
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
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GP with and without crossover
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GA with and without crossover
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Random crossover point but must be between the
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
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0 0 1 1 1 2 2 3 2
2 4 1 0 3 5
6
2 3
4 5 6
output
6
3
5
4
2
3
2
2
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GP crossover at nodes
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Pick a random node along the string and swap this
single 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 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
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Crossover only one node
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Each integer in child randomly takes value from
either mother or father
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
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Random swap crossover
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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
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Comparison with a random search
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Comparison with a random search
GP converges in 58 generations Random search 73
generations
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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
!!!!
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Why does GP tree crossover not always work well?
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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 )

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g( x1 ) f( x2 )
g
f
Good!
x1
x2
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f( x2 )
g( x1 )
g( x2 )
f( x1 )
g
g( x2 )
f
Good!
x1
x2
f( x1 )
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Suppose we are trying to find a function to fit a
set of data looking like this
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Suppose we have 2 parents which fit the data
fairly well
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Choose crossover point
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Choose crossover point
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Introducing the new technique
Based on Julian Millers 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
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CGP representation 0 0 1 2 2 1
1 2 2 0 3 2
0 5 1 5
2 3
4 5 6
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
1 2
3 4 5
output
Where the xi lie in a defined range, say
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Interpretation of the new representation For the
variables xi which represent choice of
function If the set of functions is
-
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x1 x2 x3 x4 x5 x6 x7 x8 x9
x10 x11 x12 x13 x14 x15
x16
1 2
3 4 5
output
node 1
x
0.66
0
0.33
1
node 3
1
0
0.2
0.4
0.6
0.8
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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 lt ri lt 1 oi
p1 ri (p2 - p1) when p1 lt
p2
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Crossover in the new representation
Mother father
Min parameter value
Max parameter value
Offspring chosen as random point between mother
and father
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Why is this crossover likely to work better than
tree crossover?
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Mathematical interpretation of tree crossover
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 )

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Mathematical interpretation of the new method
x1 x2 x3 x4 x5 x6 x7 x8 x9
x10 x11 x12 x13 x14 x15
x16
1 2
3 4 5
output
The fitness can be thought of as a function of
these 16 variables, say
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
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The new technique has been tested on two
problems Two regression problems studied by Koza
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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
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Statistical analysis of new method
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) Kozas
computational effort figure
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Average convergence for
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Convergence for latter generations
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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
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Variable crossover
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Average number of generations to converge and
computational effort
Percentage crossover Average number of generations to converge Kozas computational effort
0 168 30,000
25 84 9,000
50 57 8,000
75 71 6,000
Variable crossover 47 10,000
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Average convergence for
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Variable crossover
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Average number of generations to converge and
computational effort
Percentage crossover Average number of generations to converge Kozas computational effort
0 516 44,000
25 735 24,000
50 691 14,000
75 655 11,000
Variable crossover 278 13,000
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Number of generations to converge for both
problems over 100 runs
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Average convergence ignoring runs which take over
1000 generations to converge
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Conclusions
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
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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
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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
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Thankyou for listening!
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Average convergence for using
50 nodes instead of 10
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Average number of generations to converge and
computational effort
Percentage crossover Average number of generations to converge Kozas computational effort
0 78 18,000
25 85 13,000
50 71 11,000
75 104 13,000
Variable crossover 45 14,000
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Average convergence for using 50
nodes instead of 10
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Average number of generations to converge and
computational effort
Percentage crossover Average number of generations to converge Kozas computational effort
0 131 18,000
25 193 17,000
50 224 12,000
75 152 19,000
Variable crossover 58 16,000