Download Selection and Recombination

Selection and
Recombination
Temi avanzati di Intelligenza Artificiale - Lecture 4
Prof. Vincenzo Cutello
Department of Mathematics and Computer Science
University of Catania
Selection and Recombination
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1. Review of the previous lecture
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(a) How a simple evolutionary algorithm works
(b) Some crossover and mutation operators
2. Selection and reproduction
3. Recombination
4. A sample algorithm
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Selection and Reproduction
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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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Fitness Proportional Selection
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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.
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Fitness Scaling
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Simple scaling
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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
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The ith individual's fitness is defined as:
 f original(t )  ( f * (t )  c f (t )), if f scaled (t )  0
f scaled (t )  
0,
otherwise
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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.
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Power scaling
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The ith individual's fitness is defined as:
fscaled(t) = (foriginal(t)) k
where k > 0.
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Exponential scaling
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The ith individual's fitness is defined as:
fscaled (t) = exp( foriginal (t)/T )
where T > 0 is the temperature, approaching zero.
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Ranking
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Linear ranking
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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:
Plinear (i) 
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  [rank (i) /(   1)](    )
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Where  () indicates the expected number of offspring produced by the
worst (best) individuals. Note that:  1
P
i 0
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linear
(i )  1
Hence  + = 2. So 1  2 and  = 2- .
Power ranking
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C below is a normalising factor. 0<  < .
Ppower (i) 
  [rank (i) /(   1)]k (    )
C
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Geometric ranking:
Pgeom (i) 
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 (1   )  1rank(i )
C
Exponential ranking:
1  e  rank(i )
Pexp (i) 
C
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Xin Yao ranking:
PXinYao (i ) 
i 1
 1
 ( j  1)
j 0
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Recombination for Real-valued
Representation
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Discrete Recombination
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does not change actual (gene) values. Very similar to the
crossover operators on binary strings.
Intermediate Recombination
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does change actual (gene) values. Usually based on some
kind of average/mixture among multiple parents.
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Discrete Recombination
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Multi-point Recombination
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Global Discrete Recombination
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Similar to that for the binary representation.
Similar to uniform crossover for the binary
representation.
Geometric explanation.
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Intermediate Recombination
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With two parents
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Given x1 and x2:
xi  x1i  (1   ) x2i ,  [0,1]
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With more parents
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Given x1 and x2:
xi  1 x1i   2 x2i  ...
where

i
1
i
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Other Recombination
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Heuristic Recombination
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Assume x2 is no worse than x1:
x   ( x2  x1 )  x2
where u is a uniformly distributed random number in [0,1].
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Simplex Recombination
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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 .
x  c  (c  x2 )
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Geometric Recombination
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Can be generalised to multiple parents.
x  ( x11x21 , x12 x22 ,...)
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Quadratic Recombination
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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.
x4, j
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2
2
2
2
2
2
(
x

x
)
f
(
x
)

(
x

x
)
f
(
x
)

(
x

x
1 2, j 3, j
1
3, j
1, j
2
1, j
2, j ) f ( x3 )
 
2 ( x2, j  x3, j ) f ( x1 )  ( x3, j  x1, j ) f ( x2 )  ( x1, j  x2, j ) f ( x3 )
One offspring is generated from three parents.
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What Does Quadratic Recombination Mean
Note that we are minimising “fitness” here
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A Hybrid EA With Local Search
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1. Initialize  individuals at random.
2. Perform local search from each individual.
3. REPEAT
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(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.
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Local Search With Random Memorising
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Store best solutions in a memory.
Retrieve a random one (old best) when a new best solution is
found.
Search along the direction of oldnew, i.e., the direction of:
xnew  xold
s :
xnew  xold
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Experimental Studies
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18 multimodal benchmark functions.
Population size 30
Maximum function evaluation 500,000.
50 independent runs for each function.
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Benchmark Functions (f8 - f25 )
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Results on f8-f13
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Results on f8-f13 with population size 50 and 60
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Results on f14-f25
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Results on f16-f23 with population size 10
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Summary
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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.
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References
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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.
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