Convergence Analysis of Canonical Genetic Algorithm

1. Convergence Analysis of Canonical Genetic Algorithm 2010.10.14 ChungHsiang, Hsueh

2. Agenda • Introduction • Markov chain analysis of CGA • Discussion with respect to the schema theorem • Conclusion

3. Agenda • Introduction • Markov chain analysis of CGA • Discussion with respect to the schema theorem • Conclusion

4. 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

5. Introduction: Markov Chain Sol: Sol: -> assume = [X Y Z] -> independent of initial distribution!

6. 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

7. Agenda • Introduction • Markov chain analysis of CGA • Discussion with respect to the schema theorem • Conclusion

8. 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

9. Theorem 3 • Lemma1: Let C,M,S be stochastic matrices, where M is positive and S is columnallowable • ->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

10. 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?

11. Theorem6 & Theorem7 • The canonical GA as in Theorem 3 maintaining the best solution over timeafter/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

12. 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

13. Adaption of Markov Chain Description(cont.) • 4. With =

14. 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. • ->

15. Theorem 7-Proof

16. Agenda • Introduction • Markov chain analysis of CGA • Discussion with respect to the schema theorem • Conclusion

17. 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

18. 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); ->

19. Agenda • Introduction • Markov chain analysis of CGA • Discussion with respect to the schema theorem • Conclusion

20. 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.

21. Reference • [1]Gunter Rudolph, Convergence Analysis of Canonical Genetic Algorithms,2002