Theory’for’Heuris-c’Op-mizaon’ Connued:’’’ Gene-c’Algorithms’ ·...
Transcript of Theory’for’Heuris-c’Op-mizaon’ Connued:’’’ Gene-c’Algorithms’ ·...
Theory for Heuris-c Op-miza-on Con-nued:
Gene-c Algorithms
There is a handout, which is pages 28-‐33 in Goldberg’s book on Gene-c
Algorithms
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Next Topic is Theory for Gene-c Algorithms
• A basic defini-on in Gene-c Algorithms is the SCHEMA.
• A schema is a set of genes that make up a par-al solu-on to our op-miza-on problem.
• Plural of schema is “schemata”. • Schemata defining subsets of similar chromosomes. • We denote a building block as {1*0***} where the * indicates it can be either 0 or 1.
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Let H1, H2, H3, H4, be four schemata represented by the following strings
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Average fitness and Disrup-on from Crossover • The average fitness is f(H) • For example: (Previous slide)
• During crossover, a schema may be cut (disrupted), which occurs when a crossover point is selcted within it defining length.
• What affects the probability of disrupDon??
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Order and Length of Schema • The order of a schema H denoted by o(H) is the number of non-‐* symbols it contains
• The defining length denoted by δ(H) is the distance from the first non* to the last non*posi-on.
• So for *1*0*110, the order o(H) is 5 and its defining length δ(H) is 6 (=8-‐2)
• During a crossover a schema may be cut, so that is why the GA theory depends on schema.
• The symbols in the decision variable are some-mes called “metasymbols”, which for binary strings are 0’s and 1’s.
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Effect of Reproduc-on Without Crossover on Schema
• Let S[j] be the jth individual in the gene-c algorithm popula-on and assume j=1,…,M.
• Let f , j=1,…,M be the fitness of S[j]. • Let Avgf be the average fitness in the whole popula-on. (Avgf is
called f in text.)
Avgf= • The probability of S[j] being selected as a parent (roulege wheel) is
f/Avgf • Let m(Hi, g) be the expected number of Hi schemas present in the
GA popula-on in the gth genera-on. So what is the expected value of m(Hi, g)? (see board)
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Individual and Average Fitness • M= number of individual strings in popula-on • S[j]= an individual string in the popula-on, j=1,…,M
• f is the fitness of S[j] • avg f = average fitness of S[j] =
• The probability of S[j] being selected as a parent (roulege wheel) is f/Avgf
• P(t)= popula-on in t itera-on ( P(t))={S[j], j=1,…,M}, a set)
• f(H)= average fitness of all S[j] in schema H.
• m(H,t)= expected number of S[j] contained in schema H in itera-on t
Defini-ons • Probability a string S[j] from P(t) is selected to be a parent (by roulege) is
• The expected value of m(H,t+1) is
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Effect of Crossover
• Does the Schema structure tell us something about the likelihood that the Schema will survive cross over?
• Example: H1=[11****] of length 6 with defining length
• Where can the crossover occur and not disrupt the schema (i.e. the parent’s por-on is s-ll consistent with H1)? Let ps(Hi) be the probability of survival of the schema through one itera-on or opera-on.
• What is the probability that the schema is disrupted? 10
A general Rule for ps -‐-‐Disrup-on of Schema
• Consider H2=[1***0*] with (H2)=4.
• There are 5 cross over points. How many of them destroy the schema?
• What is ps(H2)?
• How do we express this if we know one crossover is done?
• How do we express this if there is a probability pc that one cross over is done?
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Average fitness and Disrup-on from Crossover • The average fitness is f(H) = • For example
• During crossover, a schema may be cut (disrupted), which occurs when a crossover point is selcted within it defining length.
• What affects the probability of disrupDon??
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Probability of Schema DisrupBon: Consider two schema and their defining length δ and order:
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Effect of Crossover
• Does the Schema structure tell us something about the likelihood that the Schema will survive cross over?
• Example: H1=[11****] of length 6 with defining length
• Where can the crossover occur and not disrupt the schema (i.e. the parent’s por-on is s-ll consistent with H1)?
• Let ps(Hi) be the probability of survival of the schema through one itera-on or opera-on.
• What is the probability that the schema is disrupted?
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Defini-ons • M= number of individual strings • S[j]= an individual string in the popula-on, j=1,…,M (book is S[j]=p[j]) • H= a schema • f & f = f fitness of S[j] and f = • f(H)= average fitness of all S[j] in schema H. • P(t)= popula-on in t itera-on ( P(t))={S[j], j=1,…,M}, a set) • m(H,t)= number of S[j] contained in schema H in itera-on t • Probability a string S[j] from P(t) is selected to be a parent (by roulege) is
• The expected value of m(H,t+1) is
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Effect of Crossover • Does the Schema structure tell us something about the
likelihood that the Schema will survive cross over? • Example: H1=[11****] of length 6 with defining length
• Where can the crossover occur and not disrupt the schema (i.e. the parent’s por-on is s-ll consistent with H1)? Let ps(Hi) be the probability of survival of the schema through one itera-on or opera-on.
• What is the probability that the schema is disrupted?
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Effect of Muta-on,
• What determines if a schema H survives mutaBon? • For example, what if H1= [10*******1***1]. How is it
disturbed by muta-on?
• Let pm, be the probability that one bit is mutated.
• Then the probability that schema H1 survives muta-on is
• What is the combined effect of Crossover Muta-on and Reproduc-on?
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Surviving Muta-om
• For a schema to survive, none of the o(H) fixed bit posi-ons must be affected by muta-on.
• Let pm be the probability of muta-on. • Then probability of a single bit not being mutated is 1-‐ pm
• Then the probability of x in H not being mutated out of H is (1-‐ pm )o(H)
Surviving Muta-on
Assume there exists an U>1 for all itera-ons t up to K such that At >U
What happens as t increases? • Now we replace the At expression by U in the inequality so then
• m(H,t+1)> m(H,t)U> m(H,t-‐1)*U2 etc. so • Therefore, • m(H,t+1) > m(H,0) U t+1 • Since U> 1 this means that up to K itera-ons the expected number of members in the popula-ons will increase.
• If no such U exists, then the schema H will gradually die out.
Schema Theorem:
• Schema Theorem: Highly fit, short-‐defining length schemata are most likely undisturbed and are propagated from generaDon to generaDon. These schemata receive exponenDally increasing number of trials in subsequent generaDons.
• This “Theorem is controversial. What do you think about it?
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So is this a Theorem?
• No this is not a rigorous proof and in fact people have developed counter examples that show that GA can converge to the wrong solu-on.
• What is wrong with it? (My idea)
Recall assump-on: there exists an U>1 for all itera-ons t up to K such that At
>U
What happens with convergence? What is the difference between f(H,t) and Avgf(t)? Will there always be a U>1 for which all t, At >U?
Problems with “theorem”
• Hence it is not possible that there exists a U>1 for all t.
• Thus, just because an author calls something a “theorem” does not mean it is mathema-cally true.
• In this case it is more of an explana-on of why things work for a finite number of evalua-ons rather than a “mathema-cal” theorem.