Evolutionary Algorithms

Per Kristian Lehre

pkle@imm.dtu.dk

DTU – Technical University of Denmark

IMM – Department of Informatics

9th Estonian Summer School on

Computer and Systems Science,ESSCaSS 2010

Evolutionary Algorithms 1/46

Optimisation

Given a function f:X →R,ﬁnd an x ∈ X

such that f (x) ≥ f (y) for all y ∈ X.

A general problem with lots of applications!

Can be solved eﬃciently in many special cases

mathematical optimisation techniques

optimisation variants of problems in P

Optimisation is thought to be hard in general

approximation algorithms

exact exponential algorithms

problem-independent,randomised search heuristics,

e.g.evolutionary algorithms

Evolutionary Algorithms 2/46

Evolution

Selection

Variation

Evolutionary Algorithms 3/46

Evolutionary Algorithms

Generate the initial population P(0) at random,and set t ←0.

repeat

Evaluate the ﬁtness of each individual in P(t).

Select parents from P(t) based on their ﬁtness.

Obtain population P(t +1) by

applying crossover and mutation to parents.

Set t ←t +1.

until termination criterion satisﬁed.

Basic idea from natural evolution and population genetics.

Survival of the ﬁttest.

Evolutionary Algorithms 4/46

Not A New Idea...

Der Spiegel,November

18th,1964

Evolutionary Strategies (ES),a

type of EAs,invented by Hans-Paul

Schwefel and Ingo Rechenberg at

TUB in 1963.

Optimisation of wing shape using

Konrad Zuse’s Z23.

American and German school ﬁrst

met at PPSN in Dortmund 1990.

Evolutionary Algorithms 5/46

Nature Inspired Optimisation

Problems

Continuous vs.combinatorial

Single- vs multi-objective

Dynamic and stochastic

...

Algorithms

Evolutionary Algorithms

Genetic Algorithms,Evolutionary Strategies,

Genetic Programming,Estimation of Distribution,...

Swarm Optimisation

Ant Colony Optimisation,PSO,Bee hives...

...

Evolutionary Algorithms 6/46

Outline

1

Introduction

Nature Inspired Optimisation

2

Evolutionary Algorithms

Representations

Genetic Operators

Selection Mechanisms

Diversity Mechanisms

Constraint Handling Techniques

3

Runtime Analysis

The Black Box Scenario and No Free Lunch

Runtime of (1+1) EA on OneMax

Overview of Techniques and Results

4

Summary

Evolutionary Algorithms 7/46

A Simple Evolutionary Algorithm

Simple Evolutionary Algorithm

Generate the initial population P(0) at random,and set t ←0.

repeat

Evaluate the ﬁtness of each individual in P(t).

Select parents from P(t) based on their ﬁtness.

Obtain population P(t +1) by

applying crossover and mutation to parents.

Set t ←t +1.

until termination criterion satisﬁed.

Basic idea from natural evolution and population genetics.

Survival of the ﬁttest.

Evolutionary Algorithms 8/46

Representations

Representations of candidate solutions

on which the genetic operators can operate.

Bitstrings commonly used in combinatorial optimisation.

Other representations are possible

in conjunction with specialised genetic operators.

trees,permutations etc.

In general,genotype-phenotype mappings φ:G →P

G set of genotypes/chromosomes

P set of phenotypes/solutions

Fitness function f:P →R

Evolutionary Algorithms 9/46

Locality in Representations [Rothlauf,2006]

Rule of thumb

Small genotypic change → small phenotypic change.

Large genotypic change →large phenotypic change.

Evolutionary Algorithms 10/46

Exploration and Exploitation

Exploration of new parts of search space

Mutation operators

Recombination operators

Exploitation of promising genetic material

Selection mechanism

Evolutionary Algorithms 11/46

Mutation operators for bitstrings

The mutation operator introduces small,

random changes to an individual’s chromosome.

Local Mutation

One randomly chosen bit is ﬂipped.

Global Mutation

Each bit ﬂipped independently with a given probability p

m

,

called the per bit mutation rate,which is often 1/n,

where n is the chromosome length.

Pr [k bits ﬂipped] =

n

k

∙ p

k

m

∙ (1 −p

m

)

n−k

.

Mutation rate

Note the diﬀerence between per bit (gene)

and per chromosome (individual) mutation rates.

Evolutionary Algorithms 12/46

Recombination operators - One point crossover

The recombination operator generates an oﬀspring individual

whose chromosome is composed from the parents’ chromosomes.

Crossover rate

probability of applying crossover to parents

One point crossover between parents x and y

Randomly select a crossover point p in {1,2,...,n}.

Oﬀspring 1 is x

1

∙ ∙ ∙ x

p

∙ y

p+1

∙ ∙ ∙ y

n

.

Oﬀspring 2 is y

1

∙ ∙ ∙ y

p

∙ x

p+1

∙ ∙ ∙ x

n

.

Example

Parent x:101011 | 1010 Oﬀspring 1:101011 | 1110

Parent y:010100 | 1110 Oﬀspring 2:010100 | 1010

Evolutionary Algorithms 13/46

Recombination operators - Multi-point crossover

k-point crossover between parents x and y

Randomly select k crossover points p

1

< ∙ ∙ ∙ < p

k

in {1,2,...,n}.

Oﬀspring 1 is x

1

∙ ∙ ∙ x

p

1

∙ y

p

1

+1

∙ ∙ ∙ y

p

2

∙ x

p

2

+1

∙ ∙ ∙ x

p

3

∙ ∙ ∙ etc.

Oﬀspring 2 is y

1

∙ ∙ ∙ y

p

1

∙ x

p

1

+1

∙ ∙ ∙ x

p

2

∙ y

p

2

+1

∙ ∙ ∙ y

p

3

∙ ∙ ∙ etc.

Example (2-point crossover)

Parent x:101 | 011 | 1010 Oﬀspring 1:101 | 100 | 1010

Parent y:010 | 100 | 1110 Oﬀspring 2:010 | 011 | 1110

Evolutionary Algorithms 14/46

Recombination operators - Uniform crossover

Uniform crossover between parents x and y

Select a bitstring z of length n uniformly at random.

for all i in 1 to n

if z

i

= 1 then bit i in oﬀspring 1 is x

i

else y

i

.

if z

i

= 1 then bit i in oﬀspring 2 is y

i

else x

i

.

Example

z =1010001110

Parent x:1010111010 Oﬀspring 1:1111001010

Parent y:0101001110 Oﬀspring 2:0000111110

Evolutionary Algorithms 15/46

Selection and Reproduction

Selection emphasises the better solutions in a population

One or more copies of good solutions.

Inferior solutions are much less likely to be selected.

Not normally considered a search operator,

but inﬂuences search signiﬁcantly

Selection can be used either before or after search operators.

When selection is used before search operators,the process of

choosing the next generation from the union of all parents

and oﬀspring is sometimes called reproduction.

Generational gap of EA

refers to the overlap (i.e.,individuals that did not go through

any search operators) between the old and new generations.

The two extremes are generational EAs and steady-state EAs.

1-elitism can be regarded as having a generational gap of 1.

Evolutionary Algorithms 16/46

Fitness Proportional Selection

Probability of selecting individual x from population P is

Pr [x] =

f (x)

y∈P

f (y)

.

Use raw ﬁtness in computing selection probabilities.

Does not allow negative ﬁtness values.

Also known as roulette wheel selection.

Weaknesses

Domination of “super individuals” in early generations.

Slow convergence in later generations.

Fitness scaling often used in early days to combat problem

Fitness function f replaced with a scaled ﬁtness function

˜

f.

Evolutionary Algorithms 17/46

Ranking Selection

1

Sort population from best to worst according to ﬁtness:

x

(λ−1)

,x

(λ−2)

,x

(λ−3)

,...,x

(0)

2

Select the γ-ranked individual x

(γ)

with probability Pr [γ],

where Pr [γ] is a ranking function,e.g.

linear ranking

exponential ranking

power ranking

geometric ranking

Evolutionary Algorithms 18/46

Linear ranking

Population size λ,and rank

γ,0 ≤ γ ≤ λ −1,(0 worst)

Linear ranking

Pr

linear

[γ]:=

α +(β −α) ∙

γ

λ−1

λ

where

λ−1

γ=0

Pr

linear

[γ] = 1 implies

α +β = 2 and 1 ≤ β ≤ 2.

In expectation

best individual reproduced β times

worst individual reproduced α times.

γ

0

λ −1

α

β

Rank

Evolutionary Algorithms 19/46

Other ranking functions

Power ranking

Pr

power

[γ]:=

α +(β −α) ∙

γ

λ−1

k

C

,

Geometric ranking

Pr

geom

[γ]:=

α ∙ (1 −α)

λ−1−γ

C

,

Exponential ranking

Pr

exp

[γ]:=

1 −e

−γ

C

,

where C is a normalising factor and 0 < α < β.

Evolutionary Algorithms 20/46

Tournament Selection

Tournament selection with tournament size k

Randomly sample a subset P

of k individuals from population P.

Select the individual in P

with highest ﬁtness.

Often,tournament size k = 2 is used.

Evolutionary Algorithms 21/46

(µ +λ) and (µ,λ) selection

Origins in Evolution Strategies.

(µ +λ)-selection

Parent population of size µ.

Generate λ oﬀspring from randomly chosen parents.

Next population is µ best among parents and oﬀspring.

(µ,λ)-selection (where λ > µ)

Parent population of size µ.

Generate λ oﬀspring from randomly chosen parents.

Next population is µ best among oﬀspring.

Evolutionary Algorithms 22/46

Selection pressure

Degree to which selection emphasises the better individuals.

How can selection pressure be measured and adjusted?

Take-over time τ

∗

[Goldberg and Deb,1991,B¨ack,1994].

1

Initial population with unique ﬁttest individual x

∗

.

2

Apply selection operator repeatedly with no other operators.

3

τ

∗

is number of generations until pop.consists of x

∗

only.

Higher take-over time → lower selection pressure.

Fitness prop.τ

∗

≈

λlnλ

c

assuming ﬁtness f (x) = exp(cx)

Linear ranking τ

∗

≈

2 ln(λ−1)

β−1

1 < β < 2

Tournament τ

∗

≈

lnλ+lnlnλ

lnk

tournament size k

(µ,λ) τ

∗

=

lnλ

ln(λ/µ)

Evolutionary Algorithms 23/46

Diversity Mechanisms

Fitness sharing

g(x):=

f (x)

y,d(x,y)≤σ

s(x,y)

s(x,y):= 1 −

d(x,y)

σ

Crowding

Standard Crowding

Deterministic Crowding

[Sareni and Krahenbuhl,1998]

[Friedrich et al.,2009]

Evolutionary Algorithms 24/46

Constraint Handling Techniques

Constrained optimisation

f:X →R objective function

g

i

:X →R inequality constraint(s)

Maximise f (x) while g

i

(x) ≤ 0.

feasible

infeasible

Penalty approaches:death,static,dynamic,adaptive,...

Multi-objective optimisation

Repair approaches

Decoders

[Coello,2002]

Evolutionary Algorithms 25/46

Analysis of Evolutionary Algorithms

Criteria for evaluating algorithms

1

Correctness

Does the algorithm always give the correct output?

2

Computational Complexity

How much computational resources does

the algorithm require to solve the problem?

Same criteria also applicable to evolutionary algorithms

1

Correctness.

Discover global optimum in ﬁnite time?

2

Computational Complexity.

Time (number of function evaluations)

most relevant computational resource.

Evolutionary Algorithms 26/46

Computational Complexity of EAs

Prediction of resources needed for a given instance.

Usually runtime as function of instance size.

Number of ﬁtness evaluations before ﬁnding optimum.

Exponential runtime =⇒ Ineﬃcient algorithm.

Polynomial runtime =⇒ “Eﬃcient” algorithm.

Evolutionary Algorithms 27/46

Black Box Scenario

Function class F

Photo:E.Gerhard (1846).

f (x

1

),f (x

2

),f (x

3

),...

x

1

,x

2

,x

3

,...

f (x

1

),f (x

2

),f (x

3

),...,f (x

t

)

x

1

,x

2

,x

3

,...,x

t

A

f

Worst case runtime

max

f ∈F

T

A,f

Average case runtime is

f ∈F

Pr [f ] T(A,f )

[Droste et al.,2006]

Evolutionary Algorithms 28/46

Black Box Scenario

Function class F

Photo:E.Gerhard (1846).

f (x

1

),f (x

2

),f (x

3

),...

x

1

,x

2

,x

3

,...

f (x

1

),f (x

2

),f (x

3

),...,f (x

t

)

x

1

,x

2

,x

3

,...,x

t

A

f

Worst case runtime

max

f ∈F

T

A,f

Average case runtime is

f ∈F

Pr [f ] T(A,f )

[Droste et al.,2006]

Evolutionary Algorithms 28/46

Black Box Scenario

Function class F

Photo:E.Gerhard (1846).

f (x

1

),f (x

2

),f (x

3

),...

x

1

,x

2

,x

3

,...

f (x

1

),f (x

2

),f (x

3

),...,f (x

t

)

x

1

,x

2

,x

3

,...,x

t

A

f

Worst case runtime

max

f ∈F

T

A,f

Average case runtime is

f ∈F

Pr [f ] T(A,f )

[Droste et al.,2006]

Evolutionary Algorithms 28/46

No Free Lunch

Theorem

([Wolpert and Macready,1997,Droste et al.,2002b])

Let F be a set of functions f:S →B,

where S and B are ﬁnite sets,B totally ordered.

If F is closed under permutations,

then the average case runtime over F

is the same for all search heuristics.

No search heuristic best on all problems.

Need to consider algorithms on speciﬁc problem classes F.

Function classes closed under permutation are not interesting...

(NB!See [Auger and Teytaud,2008] for continuous spaces.)

Evolutionary Algorithms 29/46

Expected Runtime and Success Probability

The runtime T

A,f

is a random variable.

Expected runtime E[T

A,f

] =

∞

t=1

tPr [T

A,f

= t].

Success probability within t(n) steps Pr [T

A,f

≤ t(n)].

Evolutionary Algorithms 30/46

(1+1) Evolutionary Algorithm

1:Sample x uniformly at random from {0,1}

n

.

2:repeat

3:x

←x.

4:Flip each bit of x

independently with probability 1/n.

5:if f (x

) ≥ f (x) then

6:x ←x

.

7:end if

8:until termination condition met.

Special case of the (µ+λ) EA.

Starting point for rigorous runtime analysis of EAs,e.g.

[M¨uhlenbein,1992,Garnier et al.,1999,Droste et al.,2002a]

Evolutionary Algorithms 31/46

Artiﬁcial Fitness Levels - Upper bounds

Search space partitioned into m subsets A

1

,A

2

,...,A

m

,

with increasing ﬁtness,i.e.f (A

i

) < f (A

j

) for all i < j,

and f (A

m

) optimal.

Fitness

A

1

A

2

A

3

.

.

.

A

m−1

A

m

p

i

:Probability of jumping from

A

i

to any A

j

,i < j.

T

i

:Time to jump from

A

i

to any A

j

,i < j.

Expected runtime

E[T] ≤ E[T

1

+T

2

+∙ ∙ ∙ +T

m

]

= E[T

1

] +E[T

2

] +∙ ∙ ∙ +E[T

m

]

≤ 1/p

1

+1/p

2

+∙ ∙ ∙ 1/p

m

.

Evolutionary Algorithms 32/46

Artiﬁcial Fitness Levels - Upper bound on OneMax

Partitioning of search space in ﬁtness levels

OneMax(x):= x

1

+x

2

+∙ ∙ ∙ +x

n

.

A

i

:all bitstrings with i 0-bits.

p

i

:probability of decreasing number of 1-bits,from within A

i

(at least prob.of ﬂipping one 0-bit,and no other bits)

p

i

≥ i ∙

1

n

∙

1 −

1

n

n−1

≥1/e

≥

i

en

.

Expected runtime

E[T

OneMax

] ≤

n

i=1

1

p

i

=

n

i=1

en

i

= O(n lnn).

Evolutionary Algorithms 33/46

Artiﬁcial Fitness Levels - Upper bound on OneMax

Partitioning of search space in ﬁtness levels

OneMax(x):= x

1

+x

2

+∙ ∙ ∙ +x

n

.

A

i

:all bitstrings with i 0-bits.

p

i

:probability of decreasing number of 1-bits,from within A

i

(at least prob.of ﬂipping one 0-bit,and no other bits)

p

i

≥ i ∙

1

n

∙

1 −

1

n

n−1

≥1/e

≥

i

en

.

Expected runtime

E[T

OneMax

] ≤

n

i=1

1

p

i

=

n

i=1

en

i

= O(n lnn).

Evolutionary Algorithms 33/46

Artiﬁcial Fitness Levels - Exercise

Estimate an upper bound on the expected runtime of (1+1) EA on

LeadingOnes(x):=

n

i=1

i

j=1

x

i

.

x =

Leading 1-bits.

1111111111111111 0

Random bitstring.

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗.

First 0-bit.

Artiﬁcial Fitness Levels,with level given by#leading 1-bits

Probability of increasing p

i

≥ 1/en for all i.

E[T

LeadingOnes

] ≤

n

i=1

1

p

i

= O(n

2

).

Evolutionary Algorithms 34/46

Artiﬁcial Fitness Levels - Exercise

Estimate an upper bound on the expected runtime of (1+1) EA on

LeadingOnes(x):=

n

i=1

i

j=1

x

i

.

x =

Leading 1-bits.

1111111111111111 0

Random bitstring.

∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗∗.

First 0-bit.

Artiﬁcial Fitness Levels,with level given by#leading 1-bits

Probability of increasing p

i

≥ 1/en for all i.

E[T

LeadingOnes

] ≤

n

i=1

1

p

i

= O(n

2

).

Evolutionary Algorithms 34/46

Analytical Tool Box

Artiﬁcial Fitness Levels

[Wegener and Witt,2005]

Markov’s Inequality,Chernoﬀ Bounds

[Motwani and Raghavan,1995]

Typical Runs

Expected Multiplicative Weight Decrease

[Neumann and Wegener,2007]

Drift Analysis [Hajek,1982]

Branching Processes [Lehre,2010]

Yao’s Minimax Principle

[Motwani and Raghavan,1995]

Evolutionary Algorithms 35/46

State of the Art [Oliveto et al.,2007b]

OneMax

(1+1) EA

O(n log n)

[M¨uhlenbein,1992]

(1+λ) EA

O(λn +n log n)

[Jansen et al.,2005]

(µ+1) EA

O(µn +n log n)

[Witt,2006]

1-ANT

O(n

2

) w.h.p.

[Neumann and Witt,2006]

(µ+1) IA

O(µn +n log n)

[Zarges,2009]

Linear Functions

(1+1) EA

Θ(n log n)

[Droste et al.,2002a] and

[He and Yao,2003]

cGA

Θ(n

2+ε

),ε > 0 const.

[Droste,2006]

Max.Matching

(1+1) EA

e

Ω(n)

,PRAS

[Giel and Wegener,2003]

Sorting

(1+1) EA

Θ(n

2

log n)

[Scharnow et al.,2002]

SS Shortest Path

(1+1) EA

O(n

3

log(nw

max

))

[Baswana et al.,2009]

MO (1+1) EA

O(n

3

)

[Scharnow et al.,2002]

MST

(1+1) EA

Θ(m

2

log(nw

max

))

[Neumann and Wegener,2007]

(1+λ) EA

O(nλlog(nw

max

)),λ =

m

2

n

[Neumann and Wegener,2007]

1-ANT

O(mn log(nw

max

))

[Neumann and Witt,2008]

Max.Clique

(1+1) EA

Θ(n

5

)

[Storch,2006]

(rand.planar)

(16n+1) RLS

Θ(n

5/3

)

[Storch,2006]

Eulerian Cycle

(1+1) EA

Θ(m

2

log m)

[Doerr et al.,2007]

Partition

(1+1) EA

PRAS,avg.

[Witt,2005]

Vertex Cover

(1+1) EA

e

Ω(n)

,arb.bad approx.

[Friedrich et al.,2007] and

[Oliveto et al.,2007a]

Set Cover

(1+1) EA

e

Ω(n)

,arb.bad approx.

[Friedrich et al.,2007]

SEMO

Pol.O(log n)-approx.

[Friedrich et al.,2007]

Intersection of

(1+1) EA

1/p-approximation in

[Reichel and Skutella,2008]

p ≥ 3 matroids

O(|E|

p+2

log(|E|w

max

))

UIO/FSM conf.

(1+1) EA

e

Ω(n)

[Lehre and Yao,2007]

Evolutionary Algorithms 36/46

Summary

Evolutionary Algorithms

Representations

Genetic Operators

Selection Mechanisms

Runtime Analysis

No Free Lunch Theorem

Expected Runtime & Success Probability

Evolutionary Algorithms 37/46

References I

Auger,A.and Teytaud,O.(2008).

Continuous lunches are free plus the design of optimal optimization algorithms.

Algorithmica.

B¨ack,T.(1994).

Selective pressure in evolutionary algorithms:A characterization of selection

mechanisms.

In Proceedings of the 1st IEEE Conference on Evolutionary Computation

(CEC 1994),pages 57–62.IEEE Press.

Baswana,S.,Biswas,S.,Doerr,B.,Friedrich,T.,Kurur,P.P.,and Neumann,F.

(2009).

Computing single source shortest paths using single-objective ﬁtness.

In FOGA 09:Proceedings of the tenth ACM SIGEVO workshop on Foundations

of genetic algorithms,pages 59–66,New York,NY,USA.ACM.

Coello,C.C.(2002).

Theoretical and numerical constraint-handling techniques used with evolutionary

algorithms:A survey of the state of the art.

Computer Methods in Applied Mechanics and Engineering,

191(11-12):1245–1287.

Evolutionary Algorithms 38/46

References II

Doerr,B.,Klein,C.,and Storch,T.(2007).

Faster evolutionary algorithms by superior graph representation.

In Proceedings of the 1st IEEE Symposium on Foundations of Computational

Intelligence (FOCI 2007),pages 245–250.

Droste,S.(2006).

A rigorous analysis of the compact genetic algorithm for linear functions.

Natural Computing,5(3):257–283.

Droste,S.,Jansen,T.,and Wegener,I.(2002a).

On the analysis of the (1+1) Evolutionary Algorithm.

Theoretical Computer Science,276:51–81.

Droste,S.,Jansen,T.,and Wegener,I.(2002b).

Optimization with randomized search heuristics–the (a)nﬂ theorem,realistic

scenarios,and diﬃcult functions.

Theoretical Computer Science,287(1):131–144.

Evolutionary Algorithms 39/46

References III

Droste,S.,Jansen,T.,and Wegener,I.(2006).

Upper and lower bounds for randomized search heuristics in black-box

optimization.

Theory of Computing Systems,39(4):525–544.

Friedrich,T.,Hebbinghaus,N.,Neumann,F.,He,J.,and Witt,C.(2007).

Approximating covering problems by randomized search heuristics using

multi-objective models.

In Proceedings of the 9th annual conference on Genetic and evolutionary

computation (GECCO 2007),pages 797–804,New York,NY,USA.ACM Press.

Friedrich,T.,Oliveto,P.S.,Sudholt,D.,and Witt,C.(2009).

Analysis of diversity-preserving mechanisms for global exploration*.

Evol.Comput.,17(4):455–476.

Garnier,J.,Kallel,L.,and Schoenauer,M.(1999).

Rigorous hitting times for binary mutations.

Evolutionary Computation,7(2):173–203.

Evolutionary Algorithms 40/46

References IV

Giel,O.and Wegener,I.(2003).

Evolutionary algorithms and the maximum matching problem.

In Proceedings of the 20th Annual Symposium on Theoretical Aspects of

Computer Science (STACS 2003),pages 415–426.

Goldberg,D.E.and Deb,K.(1991).

A comparative analysis of selection schemes used in genetic algorithms.

In Foundations of Genetic Algorithms,pages 69–93.Morgan Kaufmann.

Hajek,B.(1982).

Hitting-time and occupation-time bounds implied by drift analysis with

applications.

Advances in Applied Probability,14(3):502–525.

He,J.and Yao,X.(2003).

Towards an analytic framework for analysing the computation time of

evolutionary algorithms.

Artiﬁcial Intelligence,145(1-2):59–97.

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References V

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