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Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello Department of Mathematics and Computer Science University of Catania

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Selection and Recombination - Lecture 43 Selection and Reproduction  Selection is not normally regarded as a search operator although it does in 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.  The generational gap of an 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.  Elitism = copying the best individual to the next generation.

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Page 1: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4

Prof. Vincenzo Cutello

Department of Mathematics and Computer Science University of Catania

Page 2: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 2

Selection and Recombination 1. Review of the previous lecture

(a) How a simple evolutionary algorithm works (b) Some crossover and mutation operators

2. Selection and reproduction 3. Recombination 4. A sample algorithm

Page 3: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 3

Selection and Reproduction Selection is not normally regarded as a search operator

although it does in 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.

The generational gap of an 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.

Elitism = copying the best individual to the next generation.

Page 4: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 4

Fitness Proportional Selection Also known as roulette wheel selection. Use raw fitness in computing selection probabilities.

Does not allow negative fitness values. Weakness: Domination of super individuals in early

generations and slow convergence in later generations.

Fitness scaling has often been used in early days to combat the two problems.

Page 5: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 5

Fitness Scaling Simple scaling

The ith individual's fitness is defined as: f scaled (t) = f original (t) - f worst (t);

where t is the generation number and f worst (t) the fitness of the worst individual so far.

Sigma scaling The ith individual's fitness is defined as:

where c is a constant, e.g., 2, f*(t) is the average fitness in the current population, and f (t) is the standard deviation of the fitness in the current population.

otherwise,0

0)( if)),()(()()(

* tftctftftf scaledforiginal

scaled

Page 6: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 6

Power scaling The ith individual's fitness is defined as:

fscaled(t) = (foriginal(t)) k

where k > 0. Exponential scaling

The ith individual's fitness is defined as: fscaled (t) = exp( foriginal (t)/T )

where T > 0 is the temperature, approaching zero.

Page 7: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 7

Ranking Linear ranking

Assume that the population size is , rank 0 indicates the worst individual and rank -1 the best. The probability of selecting the ith individual is:

Where () indicates the expected number of offspring produced by the worst (best) individuals. Note that:

Hence + = 2. So 1 2 and = 2- . Power ranking

C below is a normalising factor. 0< < .

))](1/()([)(

irankiPlinear

1

0

1)(

ilinear iP

CirankiP

k

power)()]1/()([)(

Page 8: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 8

Geometric ranking:

Exponential ranking:

Xin Yao ranking:

CiP

irank

geom

)(1)1()(

CeiP

irank )(

exp1)(

1

0

)1(

1)(

j

XinYao

j

iiP

Page 9: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 9

Recombination for Real-valued Representation Discrete Recombination

does not change actual (gene) values. Very similar to the crossover operators on binary strings.

Intermediate Recombination does change actual (gene) values. Usually based on some

kind of average/mixture among multiple parents.

Page 10: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 10

Discrete Recombination Multi-point Recombination

Similar to that for the binary representation. Global Discrete Recombination

Similar to uniform crossover for the binary representation.

Geometric explanation.

Page 11: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 11

Intermediate Recombination With two parents

Given x1 and x2:

With more parents Given x1 and x2:

where

]1,0[,)1( 21 iii xxx

...2211 iii xxx

i

i 1

Page 12: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 12

Other Recombination Heuristic Recombination

Assume x2 is no worse than x1:

where u is a uniformly distributed random number in [0,1]. Simplex Recombination

Randomly select a group (> 2) of parents. Assume x1 is the best individual and x2 the worst in the group. Compute the centroid, c, of the group without x2 .

Geometric Recombination Can be generalised to multiple parents.

212 )( xxxx

)( 2xccx

,...),( 22122111 xxxxx

Page 13: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 13

Quadratic Recombination Let xij be the j-th component of the vectors xi, i 1,2,3, j

1,…,n, where n is the dimensionality. We approximate the position of P4 using the quadratic interpolation method as follow.

One offspring is generated from three parents.

)()()()()()()()()()()()(

21

3,2,12,1,31,3,2

32,2

2,12

2,1

2,31

2,3

2,2

,4 xfxxxfxxxfxxxfxxxfxxxfxx

xjjjjjj

jjjjjjj

Page 14: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 14

What Does Quadratic Recombination Mean

Note that we are minimising “fitness” here

Page 15: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 15

A Hybrid EA With Local Search 1. Initialize individuals at random. 2. Perform local search from each individual. 3. REPEAT

(a) Generate 3 points P1 , P2 , P3 by global discrete recombination. (b) Perform a quadratic approximation using P1 , P2 , P3 to produce a

point P4 . (c) Perform a local search from P4 and update P4 with the search

result. (Sounds Lamarkian.) (d) Place P1 , P2 , P3 , P4 into the population and do a ( + 4)

truncation selection. 4. UNTIL termination criteria are met.

Page 16: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 16

Local Search With Random Memorising Store best solutions in a memory. Retrieve a random one (old best) when a new best solution is

found. Search along the direction of oldnew, i.e., the direction of:

oldnew

oldnew

xxxxs

:

Page 17: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 17

Experimental Studies 18 multimodal benchmark functions. Population size 30 Maximum function evaluation 500,000. 50 independent runs for each function.

Page 18: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 18

Benchmark Functions (f8 - f25 )

Page 19: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 19

Results on f8-f13

Page 20: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 20

Results on f8-f13 with population size 50 and 60

Page 21: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 21

Results on f14-f25

Page 22: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 22

Results on f16-f23 with population size 10

Page 23: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 23

Summary Different problems require different search

operators and selection schemes. There is no universally best one.

Using real vectors is usually more appropriate than binary strings for function optimisation.

Many search operators are heuristics-based. Domain knowledge can often be incorporated into search operators and representation.

Page 24: Selection and Recombination Temi avanzati di Intelligenza Artificiale - Lecture 4 Prof. Vincenzo Cutello…

Selection and Recombination - Lecture 4 24

References T. Bäck, D. B. Fogel, and Z. Michalewicz (eds.),

Handbook of Evolutionary Computation, IOP Publ. Co. & Oxford University Press, 1997. Part C. (in the School library)

K.-H. Liang, X. Yao and C. S. Newton, “Combining landscape approximation and local search in global optimization" Proc. of the 1999 Congress on Evolutionary Computation, Vol. 2, IEEE Press, Piscataway, NJ, USA, pp.1514-1520, July 1999.