Evolution strategies (ES)

Chapter 4

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Evolution strategies

     Overview of theoretical aspects Algorithm – The general scheme – Representation and operators Example Properties Applications

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES quick overview (I)

 Developed: Germany in the 1970’s  Early names: Ingo Rechenberg, Hans-Paul Schwefel and and Peter Bienert (1965), TU Berlin   In the beginning,

maxima ESs were not devised to compute minima or

of real-valued static functions with fixed numbers of variables and without noise during their evaluation.

fore as

Rather

, they came to the

a set of rules for the automatic design and analysis of consecutive experiments with stepwise variable adjustments

driving a suitably flexible object / system into its optimal state in spite of environmental noise.

Search

strategy –

Concurrent,

guided by

absolute quality

of individuals

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES quick overview (II)

 Typically

applied to

: – application concerning

shape optimization

: a slender 3D body in a wind tunnel flow into a shape with minimal drag per volume.

– – – numerical optimisation; continuous parameter optimisation computational fluid dynamics: the design of a 3D convergent divergent hot water flashing nozzle.

 ESs are

closer to Larmackian evolution

(which states that acquired characteristics can be passed on to offspring).

 The difference between GA and ES is the

Representation

and

Survival selection

mechanism, that imply survival in the new population of part from the old population

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES quick overview (III)

      Attributed

features

: – fast – good optimizer for real-valued optimisation (real-valued vectors are used to represent individuals) – relatively much theory Strong emphasis on

mutation

for creating offspring Mutation is implemented by adding some random noise drawn from Gaussian distribution Mutation parameters are changed during a run of the algorithm In the ES the

control parameter are included in the chromosomes

and co-evolve with the solutions.

Special: –

self-adaptation of (mutation) parameters standard

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES Algorithm - The general scheme

  An Example Evolution Strategy Procedure ES{ t = 0; Initialize P(t); Evaluate P(t); While (Not Done) { Parents(t) = Select_Parents(P(t)); Offspring(t) = Procreate(Parents(t)); Evaluate(Offspring(t)); P(t+1)= Select_Survivors(P(t),Offspring(t)); t = t + 1; } The

differences between GA and ES

consists in

representation

and

survivors selection

(in the new population will survive the best of parents and offspring unlike generational genetic algorithms where children replaced the parents).

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES technical summary tableau

Representation Recombination Mutation Parent selection Survivor selection Specialty Real-valued vectors Encoding also the mutation rate Discrete or intermediary Gaussian perturbation Uniform random (  ,  ) or (  +  ) Self-adaptation of mutation step sizes

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Evolution Strategies

 There are basically 4 types of ESs – – The Simple

(1+1)-ES

(In this strategy the aspect of

collective learning in a population is missing

.

The population is composed of a single individual

).

The

(



+1)-ES

(The first multimember ES. 

offspring

)

parents give birth to 1

For the next two ESs

 –

parents give birth to

The

(



+



)-ES. P(t+1) = Best

 

offspring

of the



+



individuals

– The

(



,



)-ES

.

P(t+1) = Best



of the



offspring.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

(1+1) - Evolution Strategies (

two membered Evolution Strategy

)

    Before the (1+1)-ES there were no more than two rules: –

1.

Change all variables at a time, mostly slightly and at random.

–

2.

If the new set of variables does not diminish the goodness of the device, keep it, otherwise return to the old status.

The Simple

(1+1)-ES

(In this strategy the aspect of collective learning in a population is missing.

The population is composed of a single individual

).

(1+1)-ES is a

stochastic optimization method having similarities with Simulated Annealing

.

Represents a local search strategy that perform the current solution exploitation.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

(1+1) - Evolution Strategies features

    the

convergence velocity

, the expected distance traveled into the useful direction per iteration, is inversely proportional to the number of variables of the objective function; linear convergence order can be achieved if (or mean step-size or

standard deviation the mutation strength

of each component of the normally distributed mutation vector) is adjusted to the proper order of magnitude, permanently; the optimal mutation strength corresponds to a certain

probability

that

success is independent of the dimension of the search space and is the range of one fifth

for both model functions (

sphere

model and

corridor

model).

the

convergence (velocity) rate

of a ES (1 +1) is defined as the ratio of the Euclidean Distance (ED) traveled towards the optimal point and the number of generations required for running this distance.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Introductory example

  Task: minimise f : R n  R Algorithm: “two-membered ES” using – Vectors from R n directly as chromosomes – Population size 1 – Only mutation creating one child – Greedy selection

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Standard deviation. Normal distribution

    Consider X =  variable.

x 1 , x 2 , …,x n 

n

-dimensional random The mean (

μ

) M(X)=(x 1 + x 2 , +…+x n The

square of standard deviation

)/n.

(also called

variance

):  2 = M(X-M(X)) 2 =  (x k - M(X)) 2 / n

Normal distribution

: N(

μ

,  ) = The distribution with

μ

= 0 and

σ

2 = 1 is called the

standard normal

.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Illustration of normal distribution

http://fooplot.com/

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Introductory example: pseudocode

Minimization problem

       Set t = 0 Create initial point x t =  x 1 t ,…,x n t  REPEAT UNTIL (

TERMIN.COND

satisfied) DO Draw z y i t = x i t i from a normal distribution for all i = 1,…,n + z i or y i t = x i t + N(0,  ) IF f(x t ) < f(y t ) THEN x t+1 = x t ELSE x t+1 = y t endIF – Set t = t+1 endDO

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Introductory example: mutation mechanism

     z values drawn from normal distribution N(

μ

,  ) – – Mean

μ

is set to 0 Standard deviation  is called the mutation step size  is varied on the fly by the “1/5 success rule”: This rule resets  – – –    =  =  =  • / c if

P

s c if

P

s if

P

s after every k iterations by > 1/5 (Foot of big hill  increase σ) < 1/5 (Near the top of the hill  = 1/5 decrease σ) where

P

s is the % of successful mutations (those in which the

child is fitter than parents

), 0.8  c  1, usualy c=0.817

Mutation rule for object variables

x (

x i t

)

is

additive

, while the mutation rule for dispersion (  ) is

multiplicative

.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

The Rechenberg’s 1/5

th

- succes rule

• The

1/5 th rule of success

is a mechanism that ensures efficient heuristic search with the price of decreased robustness.

• • The

ratio of successful mutations and other mutations must be the fifth (1/5)

.

IF this ratio is greater than 1/5 the dispersion must be increased

(accelerates convergence)

.

ELSE

•

IF this ratio is less than 1/5 the dispersion must be decreased.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

The implementation of the Rechenberg’s

1/5

th

-rule

1. perform the

(

1 + 1

)

-ES for a number

G

of generations: − keep

σ

constant during this period − count the number

Gs

of successful mutations during this period 2. determine an estimate of the

success probability P

s by

P

s :=

Gs/G

3. change

σ

according to

σ

:=

σ / c,

if

P

s

>

1

/

5

σ

:=

σ

·

c,

if

P

s

<

1

/

5

σ

:=

σ,

if

P

s = 1

/

5 4. goto 1.

The

optimal value of the factor c depends on the objective function to be optimized

,

the dimensionality N of the search space

, and on the large

N

≥ 30,

G

=

N

number G

. If

N

is sufficiently is a reasonable choice. Under this condition Schwefel (1975) recommended using 0

.

85 ≤

c <

1.

Since we are not finding better solutions, we have reached the top of the hill.

Rechenberg’s 1/5 rule

reduces the standard deviation

σ

 in the case that the system was not very successful in finding better solutions.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Another historical example: the jet nozzle experiment

Task: to optimize the shape of a jet nozzle Approach: random mutations to shape + selection Initial shape Final shape

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Another historical example: the jet nozzle experiment cont’d

In order to be able to vary the length of the nozzle and the position of its throat, gene duplication and gene deletion was mimicked to evolve even the number of variables, i.e., the nozzle diameters at fixed distances.

The perhaps optimal, at least unexpectedly good and so far best-known shape of the nozzle was counter-intuitively strange

, and it took a while, until the one-component two-phase supersonic flow phenomena far from thermodynamic equilibrium, involved in achieving such good result, were understood.

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

The disadvantages of (1+1)-ES

• Fragile nature of the search point by point based on the 1/5 successful rule may lead to stagnation in a local minimum point.

• Dispersion (step size) is the same for each dimension (coordinate) within search space.

• Does not use recombination; it is not using a real population • There is

no mechanism to allow individual adjustment of stride for each coordinate axis

of the search space. The lack of such a mechanism is that the procedure will

optimum point

.

move slowly to the

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

(



+



), (



,



) - (multi membered Evolution Strategies)



parents give birth to



offspring

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Representation

   Chromosomes consist of three parts: – – Object variables: x 1 ,…,x n Strategy parameters:   Mutation step sizes:  1 ,…,  n  Rotation angles:  1 ,…,  n  Not every component is always present Full size:  x 1 ,…,x n ,  1 ,…,  n ,  1 ,…,  k   where k = n(n-1)/2 (no. of i,j pairs)

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutation

    Main mechanism: changing value by adding random noise drawn from normal distribution x’ i = x i + N(0,  ) Key idea: – –   is part of the chromosome  x 1 ,…,x n ,   is also mutated into  ’ (see later how) Thus: mutation step size  is coevolving with the solution x

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutate



first

 Net mutation effect:  x,     x’,  ’    Order is important: – – first   then x   ’ (see later how) x’ = x + N(0,  ’) Rationale: new  x’ ,  ’  is evaluated twice – – Primary: x’ is good if f(x’) is good Secondary:  ’ is good if the x’ it created is good  Reversing mutation order this would not work

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutation case 1: Uncorrelated mutation with one

      Chromosomes:  x 1 ,…,x n ,    ’ =  • exp(  • N(0,1)) x’ i = x i +  ’ • N(0,1) Typically the “learning rate”   1/ n ½ And we have a boundary rule  ’ <  0   ’ =  0

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutants with equal likelihood

Circle: mutants having the same chance to be created

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutation case 2: Uncorrelated mutation with n



’s

      Chromosomes:   i ’ =  i • exp(  ’ • x 1 ,…,x n ,  1 ,…,  n N(0,1) +  • N i (0,1)) x’ i = x i +  i ’ • N i (0,1)  Two learning rate parmeters: –  ’ overall learning rate –    coordinate wise learning rate 1/(2 n) ½ And  i ’ <  0 and   i   ’ =  0 1/(2 n ½ ) ½

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutants with equal likelihood

Ellipse: mutants having the same chance to be created

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutation case 3: Correlated mutations

  Chromosomes:  x 1 ,…,x n ,  1 ,…,  n ,  1 ,…,  k  where k = n • (n-1)/2  and the covariance matrix C is defined as: – c ii =  i 2  – – c ij = 0 if i and j are not correlated c ij = ½ • (  i 2  j 2 ) • tan(2  ij ) if i and j are correlated Note the numbering / indices of the  ‘s

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Correlated mutations cont’d

The mutation mechanism is then:    i ’  j ’ =  i • =  j exp( +  •  ’ • N(0,1) + N (0,1)  • N i (0,1))    

x

’ =

x

– –   +

N

(

0,C’

)

x C’

stands for the vector  x 1 ,…,x n is the covariance matrix

C

 after mutation of the  1/(2 n) ½ and   1/(2 n ½ ) ½ and   5 °  i ’ <  0   i ’ =  0 and |  ’ j | >    ’ j =  ’ j - 2  sign(  j ’ ) values

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Mutants with equal likelihood

Ellipse: mutants having the same chance to be created

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Recombination

 Creates one child  Acts per variable / position by either – Averaging parental values, or – Selecting one of the parental values  From two or more parents by either: – Using two selected parents to make a child – Selecting two parents for each position anew

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Names of recombinations

z i = (x i + y i )/2 z i is x i or y i chosen randomly Two fixed parents Two parents selected for each i Local intermediary Local discrete Global intermediary Global discrete

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Parent selection

 Parents are selected by uniform random distribution whenever an operator needs one/some   Thus: ES parent selection is unbiased - every individual has the same probability to be selected Note that in ES “parent” means a population member (in GA’s: a population member selected to undergo variation)

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Survivor selection

  Applied after creating  children from the  parents by mutation and recombination Deterministically chops off the “bad stuff”  Basis of selection is either: – – The set of children only: (  ,  )-selection The set of parents and children: (  +  )-selection

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Survivor selection cont’d

    (  +  )-selection is an elitist strategy (  ,  ) selection can “forget” Often (  ,  )-selection is preferred for: – Better in leaving local optima – – Better in following moving optima Using the + strategy bad  values can survive in  x,  if their host x is very fit too long Selective pressure in ES is very high (   7 •  is the common setting)

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Self-adaptation illustrated

 Given a dynamically changing fitness landscape (optimum location shifted every 200 generations)  Self-adaptive ES is able to – follow the optimum and – adjust the mutation step size after every shift !

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Self adaptation illustrated cont’d

Changes in the fitness values (left) and the mutation step sizes (right)

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Prerequisites for self-adaptation

     > 1 to carry different strategies  >  to generate offspring surplus

Not “too” strong selection, e.g.,

 

7 •

 (  ,  )-selection to get rid of misadapted  ‘s  Mixing strategy parameters by (intermediary) recombination on them

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

ES Applications:

 Lens shape optimization required to Light refraction  Distribution of fluid in a blood network  Brachystochrone curve  Solving the Rubik's Cube

A.E. Eiben and J.E. Smith, Introduction to Evolutionary Computing Evolution Strategies

Example application: the Ackley function (B äck et al ’93)

 

f

The Ackley function (here used with n =30): (

x

)   20  exp     0 .

2 1

n

Evolution strategy: 

i n

  1

x i

2     exp    1

n i n

  1 cos( 2 

x i

)     20 

e

– – – – Representation:   -30 < x i < 30 (coincidence of 30’s!) 30 step sizes (30,200) selection Termination : after 200000 fitness evaluations Results: average best solution is 7.48 • 10 –8 (very good)