Genetic Algorithms and Genetic Programming
Evolutionary Computation
1. Computational procedures patterned after biological evolution
2. Search procedure that probabilistically applies search operators to a set of points in the search space.
Biological Evolution
• Lamarck and others– Species “transmute” over time
• Darwin and Wallace– Consistent heritable variation among individuals in the
population– Natural selection of the fittest
• Mendel and genetics– A mechanism for inheriting traits– genotype -> phenotype mapping
Biological Terminology• gene
• functional entity that codes for a specific feature e.g. eye color• set of possible alleles
• allele• value of a gene e.g. blue, green, brown• codes for a specific variation of the gene/feature
• locus• position of a gene on the chromosome
• genome• set of all genes that define a species• the genome of a specific individual is called genotype• the genome of a living organism is composed of several chromosomes
• population• set of competing genomes/individuals
Genotype versus Phenotype
• genotype• blue print that contains the information to construct an organism e.g. human DNA• genetic operators such as mutation and recombination modify the genotype during reproduction• genotype of an individual is immutable
(no Lamarckian evolution)• phenotype
• physical make-up of an organism• selection operates on phenotypes (Darwin’s principle : “survival of the fittest”
The Genetic Algorithm
• Directed search algorithms based on the mechanics of biological evolution
• Developed by John Holland, University of Michigan (1970’s)– To understand the adaptive processes of
natural systems– To design artificial systems software that
retains the robustness of natural systems
The Genetic Algorithm (cont.)
• Provide efficient, effective techniques for optimization and machine learning applications
• Widely-used today in business, scientific and engineering circles
Components of a GA
A problem to solve, and ...• Encoding technique (gene, chromosome)
• Initialization procedure (creation)
• Evaluation function (environment)
• Selection of parents (reproduction)
• Genetic operators (mutation, recombination)
• Parameter settings (practice and art)
Genotype Operators
• recombination (crossover)• combines two parent genotypes into a new offspring• generates new variants by mixing existing genetic material• stochastic selection among parent genes
• mutation• random alteration of genes• maintain genetic diversity
• in genetic algorithms crossover is the major operator whereas mutation only plays a minor role
Genotype space = {0,1}L
Phenotype space
Encoding (representation)
Decoding(inverse representation)
011101001
010001001
10010010
10010001
Representation
Simple Genetic Algorithm{
initialize population;
evaluate population;
while TerminationCriteriaNotSatisfied{
select parents for reproduction;
perform recombination and mutation;
evaluate population;}
}
The GA Cycle of Reproduction
reproduction
population evaluation
modification
discard
deleted members
parents
children
modifiedchildren
evaluated children
Population
Chromosomes could be:– Bit strings (0101 ... 1100)– Real numbers (43.2 -33.1 ... 0.0 89.2) – Permutations of element (E11 E3 E7 ... E1 E15)– Lists of rules (R1 R2 R3 ... R22 R23)– Program elements (genetic programming)– ... any data structure ...
population
Reproduction
reproduction
population
parents
children
Parents are selected at random with selection chances biased in relation to chromosome evaluations.
Chromosome Modification
modificationchildren
• Modifications are stochastically triggered• Operator types are:
– Mutation– Crossover (recombination)
modified children
The simple GA
• Has been subject of many (early) studies– still often used as benchmark for novel GAs
• Shows many shortcomings, e.g.– Representation is too restrictive– Mutation & crossovers only applicable for bit-string &
integer representations– Selection mechanism sensitive for converging
populations with close fitness values– Generational population model (step 5 in SGA repr.
cycle) can be improved with explicit survivor selection
Representing Hypotheses
Represent:(Outlook = Overcast v Rain) ^ (Wind = Strong)by
Outlook Wind011 10
Represent:IF Wind = Strong THEN PlayTennis = yesby
Outlook Wind PlayTennis011 10 10
SGA operators: 1-point crossover
• Choose a random point on the two parents• Split parents at this crossover point• Create children by exchanging tails
• Pc typically in range (0.6, 0.9)
SGA operators: mutation
• Alter each gene independently with a probability pm
• pm is called the mutation rate– Typically between 1/pop_size and 1/ chromosome_length
Alternative Crossover Operators
• Performance with 1 Point Crossover depends on the order that variables occur in the representation
– more likely to keep together genes that are near each other
– Can never keep together genes from opposite ends of string
– This is known as Positional Bias
– Can be exploited if we know about the structure of our problem, but this is not usually the case
n-point crossover
• Choose n random crossover points• Split along those points• Glue parts, alternating between parents• Generalisation of 1 point (still some positional bias)
Uniform crossover
• Assign 'heads' to one parent, 'tails' to the other• Flip a coin for each gene of the first child• Make an inverse copy of the gene for the second child• Inheritance is independent of position
Order 1 crossover
• Idea is to preserve relative order that elements occur• Informal procedure:
1. Choose an arbitrary part from the first parent
2. Copy this part to the first child
3. Copy the numbers that are not in the first part, to the first child:
• starting right from cut point of the copied part, • using the order of the second parent • and wrapping around at the end
4. Analogous for the second child, with parent roles reversed
Order 1 crossover example• Copy randomly selected set from first parent
• Copy rest from second parent in order 1,9,3,8,2
Crossover OR mutation?
• Decade long debate: which one is better / necessary / main-background
• Answer (at least, rather wide agreement):– it depends on the problem, but– in general, it is good to have both– both have another role– mutation-only-EA is possible, xover-only-EA would not work
Evaluation
• The evaluator decodes a chromosome and assigns it a fitness measure
• The evaluator is the only link between a classical GA and the problem it is solving
evaluation
evaluatedchildren
modifiedchildren
Selection Schemes• stochastic sampling
• roulette wheel selection• spin wheel N times
• stochastic universal sampling• roulette wheel selection• single spin, wheel has N equally spaced markers
• tournament selection• choose k candidates at random with uniform probability• pick best one for reproduction• expected number of offspring best : k , average k ½ k-1 , worst k 1/N k-1
Replacement Schemes• generational replacement
• entire population is replaced each generation• non-overlapping population
100100100111001
SelectionCrossoverMutation
101110100001011
• steady state replacement• a single individual (worst, random) is replaced by one offspring• overlapping population
100100100111001
SelectionCrossoverMutation
10010010011100101110
Deletion
• Generational GA:entire populations replaced with each iteration
• Steady-state GA:a few members replaced each generation
population
discard
discarded members
An Abstract Example
Distribution of Individuals in Generation 0
Distribution of Individuals in Generation N
A Simple Example
The Traveling Salesman Problem:
Find a tour of a given set of cities so that – each city is visited only once– the total distance traveled is minimized
Representation
Representation is an ordered list of city
numbers known as an order-based GA.
1) London 3) Dunedin 5) Beijing 7) Tokyo
2) Venice 4) Singapore 6) Phoenix 8) Victoria
CityList1 (3 5 7 2 1 6 4 8)
CityList2 (2 5 7 6 8 1 3 4)
Crossover combines inversion and
recombination:
* *
Parent1 (3 5 7 2 1 6 4 8)
Parent2 (2 5 7 6 8 1 3 4)
Child (5 8 7 2 1 6 3 4)
This operator is called the Order1 crossover.
Crossover
Mutation involves reordering of the list:
* *
Before: (5 8 7 2 1 6 3 4)
After: (5 8 6 2 1 7 3 4)
Mutation
TSP Example: 30 Cities
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
x
y
Solution i (Distance = 941)
TSP30 (Performance = 941)
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
x
y
Solution j(Distance = 800)44626967786462544250404038213567606040425099
TSP30 (Performance = 800)
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
x
y
Solution k(Distance = 652)
TSP30 (Performance = 652)
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
x
y
Best Solution (Distance = 420)42383526213532
73846445860697678716967628494
TSP30 Solution (Performance = 420)
0
20
40
60
80
100
120
0 10 20 30 40 50 60 70 80 90 100
x
y
Overview of Performance
TSP30 - Overview of Performance
0
200
400
600
800
1000
1200
1400
1600
1800
1 3 5 7 9 11 13 15 17 19 21 23 25 27 29 31
Generations (1000)
Distance
BestWorstAverage
Population Models
• SGA uses a Generational model:– each individual survives for exactly one generation– the entire set of parents is replaced by the offspring
• At the other end of the scale are Steady-State models:– one offspring is generated per generation,– one member of population replaced,
• Generation Gap – the proportion of the population replaced– 1.0 for GGA, 1/pop_size for SSGA
GABIL [de Jong et al, 1993]
Learn disjunctive set of propositional rules; competitive with C4.5Fitness
Fitness(h) = (correct(h))2
Representation
IF a1=T ^ a2=F THEN c=T; IF a2=T THEN c=Frepresented by:
a1 a2 c a1 a2 c10 01 1 11 10 0
Genetic operators???– want variable length rule sets– want only well-formed bit-string hypotheses
Crossover with variable-length bit-stringsStart with:
a1 a2 c a1 a2 c
h1 10 01 1 11 10 0
h1 01 11 0 10 01 0
• choose crossover points for h1, e.g. after bits 1, 8• now restrict points in h2 to those that produce bitstrings with well-
defined semantics, e.g. 〈 1,3 〉 , 〈 1,8 〉 , 〈 6,8 〉 .if we choose 〈 1,3 〉 , the results is:
a1 a2 c
h3: 11 10 0
a1 a2 c a1 a2 c a1 a2 c
h4: 00 01 1 11 11 0 10 01 0
GABIL extensions
Add new genetic operators, also applied probabilistically:
1. AddAlternative: generalise constraint on a1 by changing a 0 to 1
2. DropCondition: generalise constraint on ai by changing every 0 to 1.
And, add new field to bitstring to determine whether to allow these:
a1 a2 c a1 a2 c AA DC 01 11 0 10 01 0 1 0
so now the learning strategy also evolves.
GABIL Results
• Performance of GABIL comparable to symbolic rule/tree learning methods C4.5, ID5R, AQ14.
• Average performance on a set of 12 synthetic problems:– GABIL without AA and DC operators: 92.1%
accuracy– GABIL with AA and DC operators 95.2% accuracy– symbolic learning methods ranged from 91.2% to
96.6%
Extensions to the Simple GA
Encoding schemes• gray encoding• messy genetic algorithms
Replacement schemes• generational replacement• steady state replacement
Fitness scaling• linear scaling• - truncation• ranking
Selection schemes• stochastic sampling• tournament selection
• automatic generation of computer programs by means of natural evolution see Koza 1999• programs are represented by a parse tree (LISP expression)• tree nodes correspond to functions : - arithmetic functions {+,-,*,/} - logarithmic functions {sin,exp}• leaf nodes correspond to terminals : - input variables {X1, X2, X3} - constants {0.1, 0.2, 0.5 }
Genetic Programming
+
*
X3X2
X1
tree is parsed from left to right: (+ X1 (* X2 X3)) X1+(X2*X3)
Genetic Programming : Crossover
+
*
X3X2
X1
-
/
X1-
X2
X3X2
parent A parent B
+ *
X3X2X1
-
/
X1-X2
X3X2
offspring A offspring B
Block-Stacking ProblemNUAL S I R V E
RSAL
NIVE
U
• objective: place the blocks in the correct order such that the stack forms the word universal• functions: set of actions, logical operators, do-until loop• terminals: set of sensors that indicate top block on stack, next suitable block on table etc.• each program tree is tested on 166 different initial configurations• fitness: #configurations for which the stack was correct after program execution
stacktable
Block-Stacking Problem
NUAL S I R V E
CSNN
Sensors:• CS: current stack, name of the top block of the stack• TB: top correct block, name of the topmost block on the stack such that it and all blocks underneath are in correct order• NN: next block needed, name of the block needed above TB
TB
Block-Stacking Problem
NUAL I R V ES
MS NN
Functions:• MS(X): move block X to the top of the stack, return value X• MT(X): moves the block on top of the stack to the table if X is anywhere in the stack returns X• DU(exp1, exp2): execute exp1 until the predicate exp2 becomes true• NOT(exp1) : negation of expression exp1• EQ(exp1, exp2) : returns true if exp1 and exp2 are equal
SMS TB
N
Block-Stacking Problem
N
UAL I R V ES
NNCSTB
• (EQ (MS NN) (EQ (MS NN) (MS NN))) move next needed block to the stack three times in a row (succeeds with a stack VERSAL and U N I on the table• (DU (MS NN) (NOT NN)) move next needed block to the stack until no more blocks are needed• (EQ (DU (MT CS) (NOT CS)) (DU (MS NN) (NOT NN))) empty the stack on the table and then build the stack in the correct order• (EQ (DU (MT CS) (EQ (CS TB))) (DU (MS NN) (NOT NN)))
Learned Program
• Trained to fit 166 test problems
• Using population of 300 programs, found this after 10 generations:
(EQ (DU (MT CS)(NOT CS))(DU (MS NN)(NOT NN)))
Genetic Programming
• More interesting example: design electronic filter circuits– individual are programs that transform beginning cct to final
cct, by adding/subtracting components and connections.– Use population of 640,000 run on 64 node parallel processor– Discover circuits competitive with best human desgn
GP for classifying images
• Teller & Veloso, 1997• Fitness: based on coverage & accuracy• Representation:
– Primitives include Add, Sub, Mult, Div, Not, Max, Min, Read, Write, If-Then-Else, Either, Pixel, Least, Most, Ave, Variance, Difference, Mini, Library
– Mini refers to a local subroutine that is separately co-evolved– Library refers to a global library subroutine (evolved by
selecting the most useful minis)• Genetic operators:
– Crossover mutation– Create “mating pools” and use rank proportionate
reproduction.
Biological Evolution
• Lamarck (19th century)– believed individual genetic makeup was
altered by lifetime experience– but current evidence contradicts this view
• What is the impact of individual learning on population evolution?
Baldwin Effect
• Assume– individual learning has no direct influence on individual DNA– But ability to learn reduces need to “hard wire” traitsin DNA
• Then– Ability of individuals to learn will support more diverse gene
pool• because learning allow individuals with various “hard
wired” traits to be successful– More diverse gene pool will support faster evolution of gene
pool.– > individual learning (indirectly) increases rate of evolution.
Baldwin Effect
Plausible example:1. New predator appears in the environment2. Individual who can learn (to avoid it) will
be selected3. Increase in learning individuals will
support more diverse gene pool4. resulting in faster evolution5. possibly resulting in new non-learned
traits such as instinctive fear of predator
Computer experiements on Baldwin Effect• Evolve simple neural nets
– [Hinton & Nowlan, 1987]– some network weights fixed during lifetime, others trainable– genetic makeup determines which are fixed and their weight
values
• Results:– With no individual learning, population failed to improve over
time– when individual learning allowed
• early generations: population contained many individuals with many trainable weights
• later generations: higher fitness, while number of trainable weights decreased.
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