advanced neighborhoods and problem difficulty measures

Motivation Outline NK Landscapes Algorithms Problem difficulty measures Experiments Conclusions

Advanced Neighborhoods andProblem Difficulty Measures

M. Hauschild1 M. Pelikan1

1Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)Department of Mathematics and Computer Science

University of Missouri - St. Louis

Genetic and Evolutionary Computation Conference, 2011

M. Hauschild and M. Pelikan University of Missouri - St. Louis

Advanced Neighborhoods and Problem Difficulty Measures


Motivation

Important to understand problem difficulty.Identifying more difficult problems.Useful to help choose the algorithm.

Many difficulty measures have been proposed.For advanced EAs, measures often do not correlate withactual difficulty.Why?




Motivation

Type of neighborhoods explored.Hamming distance is often used.Assumes bit flip neighborhood.For advanced EAs, hamming distance might not be a goodmeasure of distance.

Extend difficulty measures to nontrivial neighborhoods.Should more closely correlate with advanced EAs

Test these new difficulty measures.




Outline

NK landscapes.Algorithms.

GA.Ideal ECGA.hBOA.Deterministic Hill-Climber.

Problem difficulty measures.Correlation length.Fitness distance correlation.Neighborhood structures

Experiments

Conclusions and future work.




NK Landscapes

Popular test function developed by Kaufmann (1989).

Gives a model of a tunable rugged landscape.An NK fitness landscape is defined by

Number of bits, n.Number of neighbors per bit, k .Set of k neighbors

∏(Xi ) for i-th bit, Xi .

Subfunction fi defining contribution of Xi and∏

(Xi ).

The objective function fnk to maximize is defined as

fnk (X0, . . . , Xn−1) =

n−1∑

i=0

fi(Xi ,∏

(Xi)) (1)




NK landscapes

Nearest neighbor NK landscapes.Bits are arranged in a circle.Neighbors of each bit restricted to the following k bits.Parameter step ∈ {1, 2, . . . , k + 1} used to control overlap.

We examine step = k + 1, fully separable.Enforces strict neighborhoods

Bit positions shuffled randomly to increase difficulty.




Genetic Algorithm

Binary tournament selectionUniform crossover

Probability of crossover, pc = 0.6

Bit flip mutationProbability of mutation, pm = 1/n




Ideal Extended Compact GA (ECGA)

Harik, 1999.

Treats groups of string positions as single variables.We consider ideal version of ECGA.

Perfect model built from problem instance.Gives us idealized operator.

Partitions contain k + 1 bits.




hierarchical Bayesian Optimization Algorithm (hBOA)

Pelikan, Goldberg, and Cantú-Paz; 2001Uses Bayesian network with local structures to modelsolutions

Acyclic directed GraphString positions are the nodesEdges represent conditional dependenciesWhere there is no edge, implicit independence




Deterministic Hill-Climber

Deterministic hill climber (DHC) used for all runsPerforms single-bit changes that lead to maximumimprovement.Stops when no single-bit change leads to improvement.




Fitness landscapes

Components of a fitness landscape.A set S of admissible solutions.Fitness function f that assigns a value to each solution in S.Distance measure d defines distance between solutions inS.

Distance measured(x , y) defines steps to get from x to y .This is what we want to look at.

Hamming distance is standard distance measureDoesn’t correspond to advanced EAs




Correlation length

Measures ruggedness of landscape.

Quantifies strength of relationship between fitness valuesafter taking s steps from a solution x .Simple functions

Fitness changes gradually with distance.

Difficult functionsFitness can change drastically.




Correlation length

Random walk of m − 1 stepsDistance measure

Fitness values of {ft}t=1...m

Random walk correlation function ρ(s) for gap s:

ρ(s) =1

σ2F (m − s)

m−s∑

t=1

(ft − f̄ )(ft+s − f̄ ), (2)

Correlation length:

l = −1

ln(|ρ(1)|), (3)

Smaller correlation length, harder the problem.




Fitness distance correlation (FDC)

Consider a set of solutions and their distance from a globaloptimum.

FDC quantifies strength of relationship between fitnessand distance to nearest global optimum.Simple functions

Higher fitness values should be closer to optimumGoing to optimum directly should be easy

Difficult functionsSolutions close to optimum might have low fitnessNot easy to go directly to optimum




Fitness distance correlation (FDC)

Consider n solutions.Fitness values: F = {f1, f2, . . . , fn}.Distances from optimum: D = {d1, d2, ..., dn}

Important statisticsStandard deviation of F ,D is σF , σD

Covariance of F and D is:

cFD =1n

n∑

i=1

(fi − f̄ )(di − d̄), (4)

Fitness distance correlation:

τ =cFD

σFσD(5)

FDC takes values from [-1, 1].Smaller the values of τ , easier the maximization problem.Consider only local optima due to DHC.




Advanced Neighborhoods

How do we determine if solutions are close to each other?The neighborhood around each solution.

Correlation lengthBit flip neighborhood

Could be misleading for advanced EAs

Consider two additional neighborhood operatorsFixed partitionRandom partition





Fixed partitionSolutions that can be reached by all possible permutationsof strongly linked bits.For separable NK landscapes, neighborhood is k + 1 bits insize.k bits that are all nearest neighbors to the same bit.

Random partitionSolutions reachable by permuting x random bits.For this work, x = k + 1To compare against tightly correlated neighborhoods.





Fitness distance correlationHamming distance

Could be misleading for advanced EAs

Partition distanceMeasures distance by amount of differing partitions instrings.Total distance is sum of number of different partitions theyhave.

Partitions considered as a single variable.




Experimental Setup

NK landscapesProblem size of n = 120.Two step sizes considered, step ∈ {4, 6}.Corresponding k were considered, k ∈ {3, 5}.1000 instances for each combination of n and step.

DHC used on all solutions during evaluations.Difficulty of instances ranked by number of DHC flips.




Experimental Setup

Correlation lengthFor each instance, 100 random walks of 1000 steps eachRepeated for all 3 neighborhood step operatorsUsed to estimate correlation length

Fitness distance correlationFor each instance, 100 samples of 1000 solutions eachRepeated for both neighborhood operatorsUsed to estimate fitness distance correlation




Correlation length results

NK instances solved with GA of n = 120, k = 3desc. of DHC steps for correlation length correlation length correlation lengthinstances GA using bit flip using random partition using set partition10% easiest 3130.5 27.722(1) 13.882(1) 29.415(5)25% easiest 3577.6 27.668(3) 13.851(3) 29.424(1)50% easiest 4165 27.674(2) 13.855(2) 29.420(3)all instances 5655.3 27.603(4) 13.818(4) 29.417(4)50% hardest 7145.6 27.531(5) 13.781(5) 29.413(6)25% hardest 8545.6 27.519(6) 13.770(6) 29.407(7)10% hardest 10556 27.479(7) 13.740(7) 29.422(2)

Bit flip and random partition show strong relationship.

Set partition seems to have no relation to actual difficulty.





NK instances solved with hBOA of n = 120, k = 3desc. of DHC steps for correlation length correlation length correlation lengthinstances hBOA using bit flip using random partition using set partition10% easiest 4371.6 27.817(1) 13.922(1) 29.417(4)25% easiest 4645.4 27.719(2) 13.874(2) 29.419(2)50% easiest 4890.6 27.685(3) 13.855(3) 29.414(6)all instances 5540.4 27.603(4) 13.818(4) 29.417(5)50% hardest 6190.3 27.520(6) 13.781(6) 29.419(1)25% hardest 6798 27.504(7) 13.777(7) 29.418(3)10% hardest 7828.5 27.556(5) 13.800(5) 29.412(7)

Bit flip and random partition show weak relationship.

Set partition seems to have no relation to actual difficulty.





NK instances solved with ECGAperfect of n = 120, k = 3desc. of DHC steps for correlation length correlation length correlation lengthinstances ECGAperfect using bit flip using random partition using set partition10% easiest 1487.8 27.581(4) 13.804(5) 29.427(1)25% easiest 1569.9 27.556(7) 13.794(7) 29.426(2)50% easiest 1669.3 27.581(5) 13.805(4) 29.426(3)all instances 1873 27.603(3) 13.818(3) 29.417(5)50% hardest 2076.7 27.624(2) 13.831(2) 29.407(7)25% hardest 2201.5 27.577(6) 13.803(6) 29.411(6)10% hardest 2316.3 27.745(1) 13.875(1) 29.421(4)

Bit flip and random partition show no relation to actualdifficulty.

Set partition is no better.





NK instances solved with GA of n = 120, k = 5desc. of DHC steps for correlation length correlation length correlation lengthinstances GA using bit flip using random partition using set partition10% easiest 13328 19.139(2) 6.399(1) 19.435(7)25% easiest 16877 19.134(4) 6.395(2) 19.448(1)50% easiest 21211 19.128(5) 6.395(3) 19.446(5)all instances 33457 19.137(3) 6.395(4) 19.447(4)50% hardest 45703 19.145(1) 6.394(5) 19.448(2)25% hardest 56967 19.124(6) 6.387(6) 19.447(3)10% hardest 71993 19.073(7) 6.373(7) 19.438(6)

Bit flip and set partition show no relationship.

Random partition correctly ranks the instances.





NK instances solved with hBOA of n = 120, k = 5desc. of DHC steps for correlation length correlation length correlation lengthinstances hBOA using bit flip using random partition using set partition10% easiest 7482.9 19.207(1) 6.416(1) 19.444(5)25% easiest 7945.3 19.203(2) 6.414(2) 19.440(7)50% easiest 8399.8 19.179(3) 6.408(3) 19.447(1)all instances 9311.6 19.137(4) 6.395(4) 19.447(2)50% hardest 10223 19.095(6) 6.381(6) 19.447(3)25% hardest 10900 19.066(7) 6.374(7) 19.444(4)10% hardest 11806 19.119(5) 6.390(5) 19.441(6)

Bit flip and random partition show a weak relationship.

Set partition is ineffective.





NK instances solved with ECGAperfect of n = 120, k = 5desc. of DHC steps for correlation length correlation length correlation lengthinstances ECGAperfect using bit flip using random partition using set partition10% easiest 2128.3 19.157(2) 6.396(3) 19.437(7)25% easiest 2273.6 19.129(5) 6.390(5) 19.443(6)50% easiest 2428.9 19.112(7) 6.386(6) 19.445(5)all instances 2770.3 19.137(4) 6.395(4) 19.447(4)50% hardest 3111.7 19.162(1) 6.403(1) 19.449(3)25% hardest 3415.5 19.153(3) 6.399(2) 19.452(2)10% hardest 3869.8 19.121(6) 6.383(7) 19.452(1)

Bit flip and random partition are ineffective.

Set partition shows an inverse relationship.




Fitness distance correlation results

NK instances solved with GA of n = 120, k = 3

desc. of DHC for FDC FDCinstances GA bit distance partition10% easiest 3130.5 -0.65915(7) -0.67115(6)25% easiest 3577.6 -0.65378(6) -0.67190(7)50% easiest 4165 -0.64500(5) -0.66583(5)all instances 5655.3 -0.62917(4) -0.65771(4)50% hardest 7145.6 -0.61334(3) -0.64959(3)25% hardest 8545.6 -0.60603(2) -0.64778(2)10% hardest 10556 -0.59750(1) -0.64301(1)

Bit distance correctly ranks instances.

Partition distance also shows strong relationship.





NK instances solved with hBOA of n = 120, k = 3

desc. of DHC for FDC FDCinstances hBOA bit distance partition10% easiest 4371.6 -0.65842(7) -0.68031(7)25% easiest 4645.4 -0.64346(6) -0.67007(6)50% easiest 4890.6 -0.63894(5) -0.66447(5)all instances 5540.4 -0.62917(4) -0.65771(4)50% hardest 6190.3 -0.61940(3) -0.65095(3)25% hardest 6798 -0.61141(2) -0.64588(2)10% hardest 7828.5 -0.60218(1) -0.64098(1)

Bit distance correctly ranks instances.

Partition distance correctly ranks instances.





NK instances solved with ECGAperfect of n = 120, k = 3

desc. of DHC for FDC FDCinstances ECGAp bit distance partition10% easiest 1487.8 -0.64460(7) -0.67060(7)25% easiest 1569.9 -0.63868(6) -0.66548(6)50% easiest 1669.3 -0.63145(5) -0.66049(5)all instances 1873 -0.62917(3) -0.65771(3)50% hardest 2076.7 -0.62689(2) -0.65493(2)25% hardest 2201.5 -0.62623(1) -0.65466(1)10% hardest 2316.3 -0.63102(4) -0.65841(4)

Both unable to differentiate between the harder instances.





NK instances solved with GA of n = 120, k = 5

desc. of DHC for FDC FDCinstances GA bit dist. partition10% easiest 13328 -0.40403(7) -0.44834(7)25% easiest 16877 -0.39321(6) -0.44414(6)50% easiest 21211 -0.38170(5) -0.44055(5)all instances 33457 -0.36075(4) -0.43371(4)50% hardest 45703 -0.33981(3) -0.42687(3)25% hardest 56967 -0.32773(2) -0.42079(2)10% hardest 71993 -0.30975(1) -0.41416(1)

FDC with both distance measures able to correctly rank allinstances.





NK instances solved with hBOA of n = 120, k = 5

desc. of DHC for FDC FDCinstances hBOA bit distance partition10% easiest 7482.9 -0.37490(7) -0.45194(7)25% easiest 7945.3 -0.37354(6) -0.44697(6)50% easiest 8399.8 -0.36649(5) -0.43962(5)all instances 9311.6 -0.36075(4) -0.43371(4)50% hardest 10223 -0.35501(3) -0.42780(3)25% hardest 10900 -0.35022(2) -0.42226(2)10% hardest 11806 -0.34699(1) -0.41737(1)






NK instances solved with ECGAperfect of n = 120, k = 5

desc. of DHC for FDC FDCinstances ECGAp bit distance partition10% easiest 2128.3 -0.36917(7) -0.443970(7)25% easiest 2273.6 -0.36651(6) -0.438740(6)50% easiest 2428.9 -0.36419(5) -0.435860(5)all instances 2770.3 -0.36075(4) -0.433710(4)50% hardest 3111.7 -0.35731(3) -0.431560(3)25% hardest 3415.5 -0.35088(2) -0.426320(2)10% hardest 3869.8 -0.34526(1) -0.421470(1)





Conclusions

Modified difficulty measures.Take into account more complex neighborhoods.

Different difficulty measures used on NK landscapes.Compared to actual computation requirements.

GA, ECGAperfect , hBOA.

Correlation length.Hamming distance initially worked for GA, hBOA.

Correlation weaker as problems got harder.

Advanced neighborhoods did not improve results.




Conclusions

Fitness distance correlation.Noisy results for easier problems.Was able to rank difficulty on harder problems.Advanced neighborhoods did not improve results.

Future WorkLook at additional measures of problem difficulty.

Signal-to-noise ratio.Scaling.Fluctuating crosstalk.

Develop new neighborhood operators.




Any Questions?



advanced neighborhoods and problem difficulty measures

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