universal wireless rear speaker kit - rocketfish products

Practical Guidelines for Using EvolutionaryAlgorithms

Darrell WhitleyColorado State AI Lab

Colorado State University

CEC Edinburgh 2005 –1

A Sample Set of Evolutionary Algorithms

• Simple Genetic Algorithm: Holland/Goldberg

• Genitor, Steady-State GAs: Whitley

• CHC: Eshelman

• Evolution Strategies: Schwefel/Rechenburg

• CMA Evolution Strategies:Hansen, Ostermeier

• Parallel Genetic Algorithms

– Island Model Genetic Algorithms

– Cellular Genetic Algorithms

• Other Algorithms:

– Tabu Search

– Pattern Search, Mesh Adaptive Direct Search


The Simple Genetic Algorithm (with Elitism)

RECOMBINATION: Let the following two binary strings represent anencoding of 5 parameters in a parameter optimization problem.

01011 100111101011011 011010001010101

10010 101101100011101 010100101000010

0101110 0111101 0110110 1101000 1010101

10010 010100101000010100111101011011

Which Produces the Offspring

01011 011010001010101101101100011101

1001010 1101100 0111010 1010010 1000010


SIMPLE GENETIC ALGORITHM MODEL

Selection RecombinationCrossover &

Time tintermediate

Time t Time t+1

String 1String 2String 3

String 5String 4

String 5String 3String 2String 2String 1

String nString n

(1&2)(1&2)

(2&3)(2&3)

Child 2Child 1 Child 2Child 1

. . .

.

.

. ... .

.

.

. . .. . .. . .. . .

. . .


5

1

2

37

8

6 4

k

1 2 3 4 5 6 7 8

1 2 3 4 5 6 7 8

Universal Stochastic Sampling, Roulette Wheel Selection


.

.

.

.

.

.

PickBestof 2

intermediateTime tTime t

TOURNAMENT SELECTION


Tournament Selection with Variance Reduction

Assume population size = K.Let πk(i) be the ith element of a random permutation πk.

For i = 1 to K

Compare populaton member ith against member πk(i).Keep the best.

Tournaments

Every individual is in 1 vs 3

EXACTLY 2 Tournaments 2 vs 7

3 vs 6

4 vs 5

5 vs 1

6 vs 2

7 vs 4


Genitor: A “Steady State” GA

• Rank Based Selection

• Two Point Crossover with Reduced Surrogates

• Randomly Choose One Offspring, Mutate

• Insert and Displace Worst

CHC

• Population-elitist selection: Truncation Selection

• Incest Prevention

• HUX

• Restarts


Selection &RecombinationCrossover


.

.

.

. . .

Parent 1Parent 2

Child 1Child 2

String 5String 4Child 2

String n-1


String 5String 4

String n

.

.

.

. . .

. . .

Time t Time t+1

in PopulationInsert

GENITOR MODEL


.

.

.

Child 1Child 2Child 3Child 4

Child n

viaRandomSelectionand HUX

.

.

.

Child lChild k

Best n strings in

P(t) + C(t)

P(t) C(t)

P(t+1)

.

.

.

Parent n

Parent 4Parent 3Parent 2Parent 1

Parent i

Parent j

Parent m

CHC MODEL


1 0 1 1 0 1 0 0 1 0

1 1 0 0 0 1 1 0 0 1

1 0 1 1 0 1 1 0 0 1

1 1 0 0 0 1 0 0 1 0

1 1 0 0 0 1 1 0 0 1

1 0 1 1 0 1 0 0 1 0 1 1 1 0 0 1 1 0 1 0

1 0 0 1 0 1 0 0 0 1

- 1 0 0 - - 1 - 0 1

- 0 1 1 - - 0 - 1 0

HUX

UNIFORM CROSSOVER

ONE POINT CROSSOVER

1 0 0 1 0 1 1 0 0 0

1 1 1 0 0 1 0 0 1 1


EVOLUTION STRATEGIES

• Uses Real-Valued Parameter Representation

• (µ, λ)-selection:λ Offspring replace µ Parents

• (µ + λ)-selection:Truncation Selection

• Self Adaptive Mutation and Rotation

• Blending Recombination

Note when λ > µ we generate extra offspring, then reduce back to µ.


Simple Mutations Correlated Mutation via Rotation


CMA Covariance Matrix Adaptation

Let Z(g+1) be the covariance of the µ best individuals.

Let P(g+1) be the covariance of the evolution path.

The new covariance matrix is:

C(g+1) = (1 − ccov)C(g) + ccov

(

αcovP(g+1) + (1 − αcov)Z(g+1))

Where ccov and αcov are constants that weight each input.


−2 −1.5 −1 −0.5 0 0.5 1 1.5 2−2

−1.5

−1

−0.5

0

0.5

1

1.5

2


Parallel Island Model Genetic Algorithms

ISLAND MODEL WITH MIGRATION



Parallel Cellular Genetic Algorithm

CELLULAR GENETIC ALGORITHM MODEL


Local Quad Search

N P N’a b c c’ b’ a’ x Y z

Q1 Q2 Q3 Q4

“Quad Search” uses only 4 neighbors, and evaluates only 2.On unimodal functions it is proven to converge to optimal

in less than 2L evaluations.


A Hybrid: Genetic Quad Search

We used GENITOR, a steady-state GA, as the genetic algorithm. We usedQuad Search to improve each member of the population.

Function Algorithm Mean σ Solved Evals

CMA-ES -388.0 15.0 0 500K

Quad -434.8 8.4 0 500K

Rana 10-D Genitor -443.4 17.8 0 500K

CHC -495.5 5.5 0 500K

Hybrid Quad -510.3 2.5 26 268K


Ruffled by Ridges:How Evolutionary Algorithms Can Fail

• Direction – coordinate search cannot see improving points that fallbetween axis.

• Precision – increasing precision generally decreases the number of falseoptima.

200 220 240 260 280 300

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The Temperature Inversion Problem

Researchers have created a forward model that relates 43 vertical temperatureprofiles (~x) to 2,000 observed measurements (~y).

• model(~x) −→ ~y

• An analytical inversion of this model is impossible.

• Formulate as an optimization problem:

f(~x) = (~yobs − model(~x))T (~yobs − model(~x))

• Sometimes first order derivatives can be calculated analytically.In the general case, this is impossible.


Why is the temperature problem so hard?

• Ridges in search space.

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200200 220220 220220 240240 240240 260260 260260 280280 280280 300300 300300

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0 10 20 30 40


0 10 20 30 40

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CMA − 162,405 evaluations

0 10 20 30 40

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CHC − 436,212 evaluations


Gray vs Binary vs Real

0001

0011001001100111

1001

10111010

1000

0100

1100

11101111

1101

0101

345

6 7

89

12

15 14

1

2

10 11 13

0000

0

4−bit Gray Encoding

0001

0011001001100111

1001

10111010

1000

0100

1100

11101111

1101

0101

1

2 367

9

1014

12

15

4

11

5

13 8

0000

0

4−bit Binary Encoding

There are good arguments for Gray codes over Binary.But Gray codes are “blind” to ridges.

Sometimes Binary is better.


Gray vs Binary vs Real

Comparing “Real-Valued” and “Bit” representation is much more complexthan most of the literature suggests.

Genetic algorithms at 20 bits of precision can be 10 to 100 times slower toconverge using 20 versus 10 bits of precision.

Low Precision might ”miss” good solutions.But it aids exploration.

High Precision can result in low/slow exploration.


The Testing Problem

1. Test Functions can be TOO EASYE.G. ONEMAX, Sphere Functions

2. Test Functions can be TOO HARDE.G. Random Job Shop Problems, N-K Landscapes

3. Test Functions can be UNREALISTICE.G. Deceptive Functions and Trap Functions(These have theoretical value, but ....)

4. Test Functions can be TOO SPECIALIZEDE.G. MAXSAT: Too many flat plateaus

There are no easy answers.


Parameter Optimization and Test Problems

There are many common test functions. Not all are good test functions.Many are linearly separable:

F (x, y, z) = G(x) + G(y) + G(z)

Pairwise combinations can be used to introduce greater nonlinearity:

F (x, y, z) = G(x, y) + G(y, z) + G(z, x)

where G(x, y) is a nonlinear function of x and y.

Test functions can also be rotated to createnonlinearity.


020

4060

Rosenbrock

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CHC10

CHC20

LS10

LS20

CMA200

CMA500

MADS2n

MADSn+1

GPS2n

GPSn+1

Unifn+1

020

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Rosenbrock Rot

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CHC10

CHC20

LS10

LS20

CMA200

CMA500

MADS2n

MADSn+1

GPS2n

GPSn+1

Unifn+1

Figure 1: The midline=median; the gray box represents 50 percent of evalua-tions. The bars show max and min values, except for outliers (small circles).


010

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CHC10

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CMA200

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MADS2n

MADSn+1

GPS2n

GPSn+1

Unifn+1

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f8f2 Rot

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CHC10

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LS10

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CMA200

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MADS2n

MADSn+1

GPS2n

GPSn+1

Unifn+1

Figure 2: The midline=median; the gray box represents 50 percent of evalua-tions. The bars show max and min values, except for outliers (small circles).


−16000 −10000 −4000

f101

−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000−16000 −10000

CH

C10

CH

C20

LS10LS20

CM

A200

CM

A500

MA

DS

2nM

AD

Sn+

1G

PS

2nG

PS

n+1

Unif

n+1

−12000 −8000

f101 Ro

t

−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000−12000 −8000

CH

C10

CH

C20

LS10LS20

CM

A200

CM

A500

MA

DS

2nM

AD

Sn+

1G

PS

2nG

PS

n+1

Unif

n+1

Figure3:The

midline=m

edian;thegray

boxrepresents50

percentofevalua-tions.The

barsshowm

axand

min

values,exceptforoutliers(smallcircles).

CECEdinburgh

2005–31

−9000 −7000 −5000 −3000

Ran

a

−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000−9000 −7000 −5000

CH

C10

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C20

LS10LS20

CM

A200

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A500

MA

DS

2nM

AD

Sn+

1G

PS

2nG

PS

n+1

Unif

n+1

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a Ro

t

−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000−8000 −6000 −4000

CH

C10

CH

C20

LS10LS20

CM

A200

CM

A500

MA

DS

2nM

AD

Sn+

1G

PS

2nG

PS

n+1

Unif

n+1

Figure4:The

midline=m

edian;thegray

boxrepresents50

percentofevalua-tions.The

barsshowm

axand

min

values,exceptforoutliers(smallcircles).

CECEdinburgh

2005–32

NFL: No Free Lunch

All search algorithms are equivalent when compared over all possible discrete

functions. Wolpert, Macready (1995)

Consider any algorithm Ai applied to function fj .

On(Ai, fj) outputs the order in which Ai visits the elements in the codomainof fj . Resampling is ignored. For every pair of algorithms Ak and Ai and for

any function fj , there exist a function fl such that

On(Ai, fj) ≡ On(Ak, fl)

Consider a “BestFirst” versus a “WorstFirst” local search with restarts. Forevery j there exists an l such that

On(BestF irst, fj) ≡ On(WorstF irst, fl)


Theorem:NFL holds for a set of functions IFF

the set of functions form a permutation set.

The “Permutation Set” is the closure of a setof functions with respect to a permutation operator.

(Schmacher, Vose and Whitley–GECCO 2001).

F1: A B C F1: 0 0 0 1

F2: A C B F2: 0 0 1 0

F3: B A C F3: 0 1 0 0

F4: B C A F4: 1 0 0 0

F5: C A B

F6: C B A


POSSIBLE POSSIBLE

ALGORITHMS FUNCTIONS

A1: 1 2 3 F1: A B C

A2: 1 3 2 F2: A C B

A3: 2 1 3 F3: B A C

A4: 2 3 1 F4: B C A

A5: 3 1 2 F5: C A B

A6: 3 2 1 F6: C B A


QUESTION:

How should we evaluate search algorithms?

Let β represent a set of benchmarks.P (β) is the permutation closure over β.

If algorithm S is better than algorithm T on β...Then T is better than S on P (β) − β.

This is True in the aggregate, but not on average.


EXPERIMENT

Let’s evolve Lisp Programs ala Genetic Programming.But let’s try different evolutionary algorithms.

The GAs use mutation and tree-based crossover. The ESs use mutation only.

We will evolve

• 11-Multiplexer

• Pole Balancing with Cart Centering

• Symbolic Regression

We will compare

• A Generational GA, Popsize = 1000, Tournament Size 2 and 7

• A Steady-State GA, Popsize = 1000, Tournament Size 2 and 7

• Evolution Strategies: (1+1)ES, (50+250)ES, (100,500)ES, (200,1000)ES


Pole Balancing

0 10 20 30 40 50

0.2

0.4

0.6

0.8

1.0

Evaluations (x 1000)

Bes

t fitn

ess

gen_t2_1000 gen_t7_1000 ss_t2_1000 ss_t7_1000

0 10 20 30 40 50

0.2

0.4

0.6

0.8

1.0


Bes

t fitn

ess

es1+10 es50+250 es100+500 es200+1000


Symbolic Regression: x6− 2x4 + x2

0 20 40 60 80 100

0.0

0.5

1.0

1.5

2.0

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3.0


Bes

t fitn

ess

gen_t2_1000 gen_t7_1000 ss_t2_1000 ss_t7_1000

0 20 40 60 80 100

0.0

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1.0

1.5

2.0

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3.0

Evaluations (x 1000)B

est f

itnes

s

es1+10 es50+250 es100+500 es200+1000


The 11 Multiplexer

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t fitn

ess

gen_t2_1000 gen_t7_1000 ss_t2_1000 ss_t7_1000

0 20 40 60 80 100

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Bes

t fitn

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es1+10 es50+250 es100+500 es200+1000


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