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Unformatted text preview: Lecture 16: 03/26/2007 Recall:
Standard (Simple) Genetic Algorithm (SGA) o Fixedsize populations (n)
0 Individuals in population are fixedlength (L) bit strings
o Exogenous fitness function—each bit string x has a fitness f(x)—assume that f(x)>0, all x P(t)>P(t+1) (populations) [that is P(t) evolves to P(t+1)] via: 1. Selection of parents for P(t+1) from P(t)
2. Generation of offspring from selected parents Talked briefly about various selection methods (fitnessprop, tournament, ranked, elitist.) Generation of
offspring via crossover mutation. Crossover probability pc and mutation probability pm—pc“.75, pm“.01
and also different kinds of crossover (onepoint, twopoint, or uniform.) Do some backofenvelope calculation regarding SGA with fitnessproportional selection with onept.
crossover (w/ prob. PC), bitwise mutation (pm). What is fitnessprop. selection? A probabilistic selection method that gives every individual a nonzero
probability of being selected as a parent (ie. don’t just pick the best!) and doesn’t guarantee ﬂ individual is a parent. To get the nparents, draw from P9t); draw an individual xePsel(x) proportional to
f(x) with replacement. @: there is only one way to do this— ( ) = 2(—)() all xeP(t)
E () Easy to prove: want psel(x)=Cof(x) for some Co, for all x. For this to be a probability distribution,
need: => 22500 Note: 1. Roulettewheel selection is another name for this. Think of a roulette wheel with
circumferencez E ( ) ( ). Assign each xeP(t) an arc length=f(x). 2. Say M>O; define a new fitness function g(x)=f(x) + M. g(x) defines same ordering
offitness of individuals; however, for each xeP(t), U: () ()+ — ( ) = — Ze()() [ZE()()]+
Underf Under g As M gets larger, "underg” distribution >uniform—ie, psel(x)~>1/n, all x Moral: M>°o => "bleach out” selection pressure. Question: if using fitness proportional selection as above, given xeP(t), what is E(# of times x gets picked
as a parent for P(t+1))? (Emphasize: two copies of same string are different x’s.) Say draw parents 1,..., n; let Note: =f(x)/avg fitness in P(t)
In particular, if f(x)>avg fitness(P(t)), then E(# times x picked >1.) Alternative approach to calculating expected value above: you're picking n parents, for each k, the probability that x pts selected k times, is: ( )* (1 — ( )) =>E(# xpicks)=2 ( )(1 — ( D HW: reconcile these formulas; also look at a variation on fitness proportional selection—"expected III number contro
Question: what if P(t) contains m copies of some string x? Then E(# times that string appears among parents)=m*E(#times individual x is a parent) 2%
12mm 3 m relatively arge—> even if f(x) smallish or mediocre, "expect" x to appear among parents. Question: Given any subset Q subset P(t), what is: o Prob{some member of Q appears among parents for P(t_1)}?
o E(# times a member of Q appears among parents}? Prob{some member of Q picked MW}=—ZZ E (8)
e () Hence, using "Zitype argument,” find that E(# of Qelements among parents for 2e () (+1): *Ze()() 126 () lZe()() Where m=size of Q=Q f average fitness of strings in Q is bigger then the average fitness of strings in P(t), then “expect” at least
Q members of Qto appear among parents. Won’t necessarily see "all members of Q” represented—Q might contain some lowf strings, e.g. Specialize this result>Schema Theorem. Early motivation (Holland): biological evolutions "seems" to build viable genotypes from "lowerorder
blocks.” Maybe GAS do this, or can be made/designed to do this, or "generically" will do this, or something. Schema Theorem is a case of previous result where Q is set of representatives of some schema in P(t).
Recall from the book that a schema H is a set of strings, where some bits are specified and some are free.
Eg 1*0*={1000, 1001, 11000, 1101} [L=4] For any schema H, can compute its static average fitness: Where H =size of H Holland’s wishful thinking. Statis BuildingBlock Hypothesis: SGA will evolve populations where "highly
fit schemas” have lots of representatives; start with short defining —length schemas, which, via crossover, assemble into longer ones, etc. Recall that defining length of a schema is distance between left most + right most defined bits. "Highly
fit schemas.” Holland meant in terms of static average fitness. Holland used multiarmed bandit (slot
machines with different probabilities of winning, want to maximize revenue, want to find best machine,
but don’t want to move around too much) technique on this—analysis flawed because used static average H—fitness’s instead of actual average fitness’s of Hn ( ) ...
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 Spring '07
 DELCHAMPS
 Algorithms

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