Empirical Analysis of Ideal Crossover on Random Additively Decomposable Problems

Kumara Sastry1, Martin Pelikan2, David E. Goldberg1

1Illinois Genetic Algorithms Laboratory (IlliGAL)University of Illinois at Urbana-Champaign, Urbana, IL 618012Missouri Estimation of Distribution Algorithm Lab (MEDAL)

University of Missouri at St. Louis, St. Louis, MO

ksastry@uiuc.edu, pelikan@cs.umsl.edu, deg@uiuc.eduhttp://www.illigal.uiuc.edu, http://medal.cs.umsl.edu

Supported by AFOSR FA9550-06-1-0096, NSF DMR 03-25939, and CAREER ECS-0547013. Computational results obtained using CSE’s Turing cluster.

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RoadmapAdversarial test problem design

Random additively decomposable problems

Ideal crossover

Scalability of selectorecombinative GAsPopulation sizing and Run duration

Experimental Procedure

Key Results

Summary and Conclusions

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Adversarial Test Problem Design

Test systems on boundary of design envelopeCommon approach in designing complex systems

GAs are complex systems [Goldberg, 2002]

GA design envelope characterized by different dimensions of problem difficulty

Thwart the mechanism of GAs to the extreme

P

Fluctuating

Deception NoiseScaling R

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Random Additively Decomposable Problem

Focus on nearly decomposable problems [Simon, 1960]

Three desired featuresScalability: Able to control problem size and difficultyKnown optimum: Allows comparison of different solversEasy problem instance generation

rADP fitness function:

Si represents variable subset for ith subproblemEach subset consists of k bitsgi is the fitness of the ith subproblemgi is sampled from uniform distribution U[0,1]

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Ideal Crossover: Exchange Building Blocks

Population sizing and run duration models assume good exchange of building blocks

Simulate what we ideally want to achieve with model-building GAs

For example, extended compact GA [Harik, 1999]

Ideal recombination operatorEffectively exchange building blocksDon’t disrupt any building block

Uniform building-block-wise crossoverExchange BBs with probability 0.5 BBs #1 and #3 exchanged

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Purpose: Analyze Ideal Crossover on rADPs

Analyze behavior of selectorecombinative GAs on rADPs

Verify the validity of lessons learned from adversarial test problems

Expand the pool of test problems

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Selectorecombinative GA Population Sizing

Gambler’s ruin model[Harik, et al, 1997]

Combines decision making and supply models

Additive Gaussian noise with variance σ2

N

Noise-to-fitness variance ratio

Error toleranceSignal-to-Noise ratio # Competing sub-components

# Components (# BBs)

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GA Run Duration (Selection)

Selection-Intensity based model [Bulmer, 1980; Mühlenbein & Schlierkamp-Voosen, 1993; Thierens & Goldberg, 1994; Bäck, 1994; Miller & Goldberg, 1995 & 1996]

Derived for the OneMax problemApplicable to additively-separable problems [Miller, 1997]

Selection Intensity

Problem size (m·k )

[Miller & Goldberg, 1995; Sastry & Goldberg, 2002]

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GA Run Duration (Drift)

Accumulation of stochastic errors due to finite population

Proportion of competing sub-solutions change due to drift

Drift time [Goldberg & Segrest, 1987]:

Substituting population sizing bound

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Signal-to-Noise Ratio for rADPs

Signal d is the fitness difference between best and second best sub-solutions

jth order statistic follows a Beta distribution with α = j and β = 2k-j+1

Probability density function (p.d.f) of d:

p.d.f. of sub-solution fitness variance approximation

E[1/d] = 2k and E[σ2BB] ≈ 1/12

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Assumptions and Experimental SetupNon-overlapping sub-problems

Identical sub-problems across different partitionsg1 = g2 = … = gm

Selectorecombinative GABinary tournament selection

10,000 random problem instancesm = 5 – 50, k = 3, 4, and 5

Minimum population size determined by bisection methodPopulation correctly converges to at least m-1 out of m BBs in 49 out of 50 independent runsAveraged over 30 bisection runsResults averaged over 1,500 GA runs

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Population Sizing & Run Duration Histograms

Tail increases with m0.15-0.59% of rADP instances require # evals greater than 3σ from the median

Population sizem = 50

Run durationm = 50

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Easy and Hard Problem Instances

Sorted subsolution index

Subs

olut

ion

fitn

ess

Sorted subsolution index

Subs

olut

ion

fitn

ess

Hard instance

Easy instance

Min signalMax noise

Max signalMin noise

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Population Sizing Scalability

Gambler’s ruin model bounds population sizing

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Run Duration Scalability

Selection-intensity based run-duration model bounds median convergence time

Drift-time model bounds convergence time

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Number of Function Evaluations Scalability

Facetwise models are applicable to rADPsTesting on adversarial problems bounds performance of GAs on rADPs

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Easy and Hard Scalable Problem Instances

Easy instances have large signal differenceHard instances have very small signal difference

Easy ScalableInstances

Hard ScalableInstances

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Summary and Conclusions

Empirically analyzed behavior of selectorecombinative GA with ideal crossover:

Class of random additively decomposable problemsSub-solution fitness sampled from uniform distribution

Verified applicability of facetwise models:Developed based on adversarial problemsGA scales subquadratically with problem size

Analyzed easy and hard problem instances:Easy problem instances have large signal, small variance.Hard problem instances have small signal, large variance