convergence analysis of canonical genetic algorithm 2010.10.14 chunghsiang, hsueh
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Convergence Analysis of Canonical Genetic Algorithm
2010.10.14ChungHsiang, Hsueh
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Agenda• Introduction
• Markov chain analysis of CGA
• Discussion with respect to the schema theorem
• Conclusion
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Agenda• Introduction
• Markov chain analysis of CGA
• Discussion with respect to the schema theorem
• Conclusion
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Introduction: Gunter Rudolph
• Computational Intelligence Research Group Chair of Algorithm Engineering (LS XI) Department of Computer Science TU Dortmund University
• Associate editor of the IEEE Transactions on EC.
• Editorial board member of the Journal on EC
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Introduction: Markov Chain
Sol: Sol:-> assume = [X Y Z]-> independent of initial distribution!
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Introduction: Markov Chain• Def1: Homogenous Markov Chain
• Def2:Classification of transition matrix• Positive• Nonnegative
• Primitive: • Reducible: • Irreducible• Stochastic: • Stable: if it has identical rows• Column allowable: if it has at least one positive entry in each column
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Agenda• Introduction
• Markov chain analysis of CGA
• Discussion with respect to the schema theorem
• Conclusion
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Describing CGA as A Markov Chain
• Transition matrix • Theorem 3:P=CMS, with
• Convergence of a GA
• Theorem 4: The CGA with parameter ranges as in Theorem 3
does not converge to the global optimum.
n
l
𝜋𝑘(𝑖)
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Theorem 3• Lemma1: Let C,M,S be stochastic matrices, where M is
positive and S is column allowable• ->the product CMS is positive!
• Theorem3: The transition matrix, P = CMS, with and proportional selection, is primitive.
• Proof:C:The crossover operator can be regarded as a random total function whose domain and range are S -> each state of S is mapped probabilistically to another state-> C is stochastic
M:The mutation operator is applied independently to each gene in the population, the probability that state i becomes state j after mutation can be regard to
-> M is positive
S:The probability that the selection does not alter the state generated by mutation can be bounded by:
for all -> S is column allowable
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Theorem 4: CGA does not converge to the global optimum
• Proof:• By Theorem 1• Let be a primitive stochastic matrix. converges as to a positive
stable stochastic matrix
is unique regardless of the initial distribution• Let be any state with and the probability that the GA is in
such a state at step .• →• ->• #Recursive argument?
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Theorem6 & Theorem7• The canonical GA as in Theorem 3 maintaining the best
solution over time after/before selection converges to the global optimum.
• Before proving the theorems…• Theorem 2 & theorem 5 and some adaptation for the Markov
chain description are required…• Theorem 2
• Let P be a reducible stochastic matrix, where C:m*m is a primitive stochastic matrix and R,T. Then
• Theorem 5• In an ergodic Markov chain the expected transition time between
initial state i and any other state j is finite regardless of the state i and j
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Adaption of Markov Chain Description
• 1. Add a super individual which does not take part in the evolutionary process. =>
• 2. It can be accessed by from a population at state I• 3. Make an ergodic Markov chain:
;otherwise
->upgrade matrix
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Adaption of Markov Chain Description(cont.)
• 4.With =
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Theorem 6-Proof• = gathers the transition probabilities for states containing a
global optimal super individual. Since is a primitive stochastic matrix and , Thm2 guarantees that the probability of staying in any non-globally optimal state converges to zero.
• ->
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Theorem 7-Proof
•
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Agenda• Introduction
• Markov chain analysis of CGA
• Discussion with respect to the schema theorem
• Conclusion
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Schema Theorem V.S. Convergence
• The schema theorem states that))(1-m(S, ))
If ))(1-m(, ))
Which does not indicate that the expectation converges to n!-> Lemma 2
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Lemma 2
Proof of (b):
Note: Converse of (b) is not true in general:S={00,01,10,11};g(1,S)=(0,1,1,2); ->
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Agenda• Introduction
• Markov chain analysis of CGA
• Discussion with respect to the schema theorem
• Conclusion
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Conclusion• Convergence to the global optimum is not an inherent
property of the CGA but rather as a consequence of the algorithmic trick of keeping track of the best solution found over time.
• Introducing time varying mutation and selection probabilities may make the Markov process inhomogeneous and reach the global optimum.
• #Introducing time varying mutation alone does not help.• ->Selection operator is the key problem of the CGA.
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Reference• [1]Gunter Rudolph, Convergence Analysis of Canonical Genetic
Algorithms,2002