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

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) =

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

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  1. A Schema-Based Evolutionary Alg’m. for Black-Box Optimization David A. Cape CS 448, Spring 2008 Missouri S & T

  2. 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

  3. Related Work • Model-Building EAs use Estimation of Distribution (EDA) techniques • hBOA • Non-Model-Building EAs • LLGA • mGA • TGA

  4. 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

  5. “Agnostic EA” (AgEA) • Allows ambiguity for each gene • Derived from schema theory • Uses traditional GA (TGA) operators • Duality between monomials and schemata

  6. 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

  7. AgEA vs. TGA on polynomials (Problem difficulty was assessed subjectively)

  8. Conclusion • Novel EA concept based on # alleles • Performs well on some simple problems • Better than competent EAs? hBOA?

  9. Future Work • Comparison to messy GA, LLGA, hBOA • Careful analysis of data • Rigorous statistical tests • Meta-schema theory?

  10. Questions?

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