Convergence Analysis of Canonical Genetic Algorithm

2010.10.14ChungHsiang, Hsueh

Agenda• Introduction

• Markov chain analysis of CGA

• Discussion with respect to the schema theorem

• Conclusion

Agenda• Introduction

• Markov chain analysis of CGA

• Discussion with respect to the schema theorem

• Conclusion

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

Introduction: Markov Chain

Sol: Sol:-> assume = [X Y Z]-> independent of initial distribution!

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

Agenda• Introduction

• Markov chain analysis of CGA

• Discussion with respect to the schema theorem

• Conclusion

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)

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

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?

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

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

Adaption of Markov Chain Description(cont.)

• 4.With =

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.

• ->

Theorem 7-Proof

•

Agenda• Introduction

• Markov chain analysis of CGA

• Discussion with respect to the schema theorem

• Conclusion

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

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); ->

Agenda• Introduction

• Markov chain analysis of CGA

• Discussion with respect to the schema theorem

• Conclusion

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.

Reference• [1]Gunter Rudolph, Convergence Analysis of Canonical Genetic

Algorithms,2002