1. FDA- A scalable evolutionary algorithm for the optimization of ADFs By Hossein Momeni

2. Page 2 Outline Factorization Theorem FDA Analysis of FDA for large populations Boltzmann and Truncation selections Finite and critical population Numerical results LFDA Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

3. Page 3 Introduction In a deceptive function the global optimum x=(1,…,1) is isolated. Neighbors of the second best fitness value x=(0,…,0) have large fitness value GAs are deceived by the fitness distribution Most Gas will convergence to x=(0,…,0) Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

4. Page 4 Solutions Mathematical methods are suitable to optimize deceptive functions Consider additively decomposed functions (ADF) S j are non-overlapping substrings of X with k elements This class of functions is of great theoretical and practical importance Optimization of an arbitrary in this space is NP complete Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

5. Page 5 ADFs Optimization Approaches Adaptive recombination Explicit detection of relations (kargupta&Goldberg, 97) Dependency trees(Baluja&Davies, 97) Bivariate marginal distributions (pelikan&Muhleinbein,98) Estimation of Distributions(Muhlenbein et all,1997) Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

6. Page 6 ADF Definition: An additively decomposed function (ADF) is defined by: For theoretical analysis, use Boltzmann Distribution Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

7. Page 7 Gibbs or Boltzmann distribution Definition: The Gibbs or Boltzmann distribution of a function f is defined for u>=1 by is partition function larger function value f(x) and larger p(x) Such a search distribution is suitable for an optimization problem exponential computation Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

8. Page 8 Reduce of B.D. computation Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47 1) Approximate the Boltzmann distribution (simulated Annealing) 2)Look for ADFs with distribution computation in Polynomial time factorize distribution into a product of marginal and conditional probabilities (used by FDA)

9. Page 9 Input sets for Factorization theorem Definition: if S={s 1,s 2, …, s l } for i=1, 2,…, l then In the decomposable graphs theory: d i histories b i residuals c i separators Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

10. Page 10 Factorization Theorem Theorem1: Let p(x) be a Boltzmann distribution on X If then Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

11. Page 11 FDA r S0: set t=0, generate (1-r)*N>>o point randomly and r*N points (Equation 16) S1: selection S2: Compute using selected points S3: Generate a new population S4: If termination criteria is met, Finish S5: Add the best point of previous generation to generated points (elitist) S6: Set t=t+1, Go to Step2 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

12. Page 12 Analysis of Factorization Algorithm The computational Complexity depends on the factorization and population size N Number of function evaluations: FE=GEN e *N GEN e is the number of generation till Convergence p(x,t+1)=p(x,t) The computational Complexity of computing N new search points is The Computational Complexity of computing probability is Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

13. Page 13 Analysis of … (Contd) Computation of FDA depends on: 1)Number of decomposition functions (l) 2)Size of the defining sets (s i ) 3)Size of selected point (M) An infinite population is needed to exactly computation Should use a minimal population size N * in a numerical efficient FDA Computation of N * is a difficult problem for any search method using a population of points Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

14. Page 14 FDA-FAC S0: set i=1, is non-linear sub-function S1: compute S2: Select s k which has maximal overlap with and S3: if no set is found go to step 5 S4: Setif i<L go Step1 S5: Compute the factorization using Eq. 6 with sets Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

15. Page 15 Generation of Initial Population Normally the initial population is generated randomly with ADF, initial point can be generated with this information. Generate subsets with high local fitness values Distribution is an approximation of Conditional probabilities are computed using local fitness functions Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

16. Page 16 Generation of Initial Population…. The larger u, the steeper distribution if u=1 the distribution is uniform. if function Onemax(n)=∑x i then FDA computes span=1 and u=10 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

17. Page 17 Generation of Initial Population…. if function Onemax(n)=∑x i then FDA computes span=1 and u=10 There will be 10 times more 1s than 0s in the initial population Such an initial population might not give a B.D. Only half of the population is generated by this method Other half is generated randomly Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

18. Page 18 Convergence of FDA If points are selected base on Bol. Distribution convergence of FDA is proved. The distribution p s of selected points is given by: If p(x,t) is B.D. then p s (x,t) is B.D. FDA computes new search points according to Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

19. Page 19 Theorem2 : If the initial points are distributed according to with u>=1, then for FDA the distribution at generation is given by with Tip: B. Selection with fixed basis v>1 defines an annealing schedule with that t is number of generation Theorem3 remains valid for any annealing schedule with Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

20. Page 20 Theorem 3(Convergence): Let be the set of optima, then base on Theorem 2 : FDA with B. selection is exact simulated annealing algorithm. simulated annealing is controlled by 2 parameters: N(T) and annealing schedule N can be called population size Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

21. Page 21 Truncation Selection Vs B. selection Numerically truncation selection is easier to implement With truncation threshold ד the best ד*N individual are selected. Conditional probabilities of selected point is: Based on factorization theorem to generate new search points : Problem: After Truncation selection the distribution is not B.D. therefore: With this inequalitythat this makes a convergence proof difficult. Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

22. Page 22 Theoretical Analysis for Infinite populations For analysis two linear function will be investigated: OneMax has (n+1) different fitness value which are multinomial D. Int has 2 n different fitness value. For ADFs the multinomial distribution is typical The distribution generated by Int is more special Both functions is linear, therefore can use following factorization: Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

23. Page 23 Theorem 4 For B. selection with basis v the probabilities distribution for OneMax is given by: Number of generations to generate the optimum is given by: Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

24. Page 24 Theorem 5 For Truncation selection ד with selection intensity I ד the marginal probability p(t) obeys for OneMax The approximate solution of this equation is : Where The number of generations till convergence is given by: Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

25. Page 25 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

26. Page 26 Comparison Truncation & B. selection T.S. need more number of generation to convergence than B.S. GEN e is of order for B.S. and for T.S. is If basis v is small (e.g. v=1.2) T.S. convergence is faster Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

27. Page 27 B.S. with fixed v gives an annealing schedule of Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

28. Page 28 FDA with truncation selection generates a B.D. with annealing schedule The annealing schedule depends on the average fitness and the variance of the population. Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

29. Page 29 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

30. Page 30 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47

31. Page 31 Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47 For Int the B.D. is concentrated around the optimum The selected population has a small diversity In finite population this cause a problem, some genes will get fixed to wrong alleles

32. Page 32 Analysis of FDA for Finite Populations Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47 In finite population, convergence of FDA can be Probabilistic

33. Page 33 Analysis of FDA for Finite Populations Factorized Distributed Algorithm Iran University of Science and Technology November 2006 Of 47 Cumulative fixation probability for Int(16) Truncation Selection vs. Boltzmann selection with v=1.01