A Schema-Based Evolutionary Alg’m. for Black-Box Optimization

David A. CapeCS 448, Spring 2008

Missouri S & T

Motivation Arbitrary Additively Decomposable Functions

Example: multivariate polynomial (sum of two 4-bit D-Traps)

F(u, v, w, x, y, z) = F0(u, v, x, z) + F1(u, w, y, z) ={ 3[(1-u)(1-v)(1-x)(1-z)] + 2[u(1-v)(1-x)(1-z) + …]+ 1[uv(1-x)(1-z) + …] + 0[uvx(1-z) + …] + 4uvxz } +{ 3[(1-u)(1-w)(1-y)(1-z)] + 2[u(1-w)(1-y)(1-z) + …] + 1[uw(1-y)(1-z) + …] + 0[uwy(1-z) + …] + 4uwyz }= {5uvxz - u - v - x - z + 3} + {5uwyz - u - w - y - z + 3}

Building Block Hypothesis? F(1, 1, 1, 1, 1, 1) = 4+4 = 8 F(1, 1, 0, 1, 0, 1) = 4+1 = 5 F(1, 0, 1, 0, 1, 1) = 1+4 = 5 F(1, 0, 0, 0, 0, 1) = 1+1 = 2 F(1, 1, 0, 0, 0, 1) = 0+1 = 1 F(1, 1, 1, 1, 0, 1) = 4+0 = 4 Favg(1, #, #, #, #, 1) = [8+5+5+2+4(1)+4(4)] / 16 = 2.5 Favg(1, 1, #, #, #, 1) = [8+5+1+3(4)+2(0)] / 8 = 3.25 Favg(1, 1, #, 1, #, 1) = [8+5+2(4)] / 4 = 5.25 Favg(1, 1, 1, 1, #, 1) = [8+4)] / 2 = 6

Related Work Model-Building EAs use Estimation

of Distribution (EDA) techniques hBOA

Non-Model-Building EAs LLGA mGA TGA

Methodology Goals: Simplicity, generality, efficiency

“Don’t Care” symbols (#) as alleles Mutation from zero or one to # Mutation from # to zero or one Uniform crossover

Nondeterministic Representation Sampling of phenotypes for evaluation Small penalty for each # allele

“Agnostic EA” (AgEA) Allows ambiguity for each gene

Derived from schema theory

Uses traditional GA (TGA) operators

Duality between monomials and schemata

Experimental Design “Arbitrary additively decomposable”

Random multivariate polynomials Sums of trap subfunctions

Not necessarily concatenated Not necessarily adjacent

mGA with default parameters AgEA with equal number of

evaluations

AgEA vs. TGA on polynomials

Time to Find Best Fitness

0

10

20

30

40

50

0 2 4 6 8

Problem Difficulty

Avera

ge G

en

era

tio

ns

AgEA

TGA

(Problem difficulty was assessed subjectively)

Conclusion

Novel EA concept based on # alleles

Performs well on some simple problems

Better than competent EAs? hBOA?

Future Work

Comparison to messy GA, LLGA, hBOA

Careful analysis of data

Rigorous statistical tests

Meta-schema theory?

Questions?