A Schema-Based Evolutionary Alg’m. for Black-Box Optimization David A. Cape CS 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) = F 0 (u, v, x, z) + F 1 (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 = 8F(1, 1, 0, 1, 0, 1) = 4+1 = 5 F(1, 0, 1, 0, 1, 1) = 1+4 = 5F(1, 0, 0, 0, 0, 1) = 1+1 = 2 F(1, 1, 0, 0, 0, 1) = 0+1 = 1F(1, 1, 1, 1, 0, 1) = 4+0 = 4 F avg (1, #, #, #, #, 1) = [ (1)+4(4)] / 16 =2.5 F avg (1, 1, #, #, #, 1) = [ (4)+2(0)] / 8 = 3.25 F avg (1, 1, #, 1, #, 1) = [8+5+2(4)] / 4 = 5.25 F avg (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 (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?