network crossover performance on nk landscapes and deceptive problems

Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions

Network Crossover Performance on NKLandscapes and Deceptive Problems

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, 2010

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

Network Crossover Performance on NK Landscapes and Deceptive Problems


Motivation

Always looking to solve difficult problems with GAs.In a scalable and robust manner.Must respect linkage between bits.

Most common variation operators do not do this.Uniform, two-point crossover.

One solution is linkage-learning GAs.EDAs respect linkages.Come at the cost of model-building.




Motivation

Often have prior information about a problem.Graph-based problems.EDA models on similar problems.

What is the best way to exploit this information?Bias EDA model building.Sample directly from a network model.Modify the crossover operator itself.

Test this operator against an EDA.




Outline

Network crossoverAlgorithms

GAhBOADeterministic Hill-Climber

Test ProblemsExperiments

Trap-5NK Landscapes

Conclusions




Network Crossover

Two-parent crossovers start by creating a binary mask.What bits to exchange and what to keep the same.

Uniform crossover sets the bits randomly.How to create a mask to respect linkages?

Start with a matrix G specifying strongest linkages.




Network Crossover

G is often not hard to obtain.Graph problems have this implicitly.MAXSAT and other problems also easy.Trial runs of EDAs.

Only requires strongest connections.Does not require perfect knowledge.




Network Crossover

To build the maskChoose a random bitRandomized breadth-first search to expand maskRepeat until mask is complete

Stop when mask size is n/2.

Bits close in G less likely to be disrupted.

Bits far from each other more likely to be disrupted.




Network Crossover




Genetic Algorithm

Three crossover operators used.Network crossoverTwo-pointUniform

Probability of crossover, pc = 0.6

Probability of mutation, pm = 1/n




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 maximumperformanceStops when no single-bit change leads to improvement

Originally considered not using DHCDramatically improved performance




Trap-5

Partition binary string into disjoint groups of 5 bits

trap5(ones) =

{

5 if ones = 54 − ones otherwise

, (1)

Total fitness is sum of single traps

Global Optimum: String 1111...1

Local Optimum: 00000 in any partition

G has all bits in the same partition connected




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)) (2)




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.

For step = 1, maximum overlap.For step = k + 1, fully separable.

Bit positions shuffled randomly to increase difficulty.




NK landscapes

Unrestricted NK landscapes.NP-complete for k > 1Branch and bound algorithm used to find optima.

Nearest neighbor NK landscapes.Polynomial solvability.Dynamic programming used to find optima.

G connects all neighboring bits.




Experimental Setup

Trap-5Problem sizes from n = 100 to n = 300.Bisection used, 10 out of 10 independent runs.10 independent bisection runs performed.Some experiments cut short at extreme problem sizes.

Unrestricted NK landscapesProblem sizes of n ∈ {20, 22, . . . , 38}.k = 51000 random problem instances for each setting.




Experimental Setup

Nearest neighbor NK landscapesProblem sizes of n ∈ {30, 60, . . . , 210}.Two step sizes considered, step ∈ {1, 5}.k = 51000 instances for each combination of n, k , step.




Experimental Setup

Two replacement techniques considered.Restricted Tournament Replacement(RTR)

Niching, replaces similar solutions.Window size set to w = min{n, N/5}.

ElitismKeeps a portion of the best individuals each generation.50% of the most fit individuals kept.

Examined three measuresEvaluationsLocal search stepsExecution time




Trap-5

Evaluations, RTR

100 150 200 250 300

105

Problem Size

Eva

luat

ions

netxuniformhboa2−point

DHC flips, RTR

100 150 200 250 300

105

1010

Problem Size

Num

ber

of fl

ips


Execution Time, RTR

100 150 200 250 300

100

Problem Size

Exe

cutio

n T

ime





Trap-5

Evaluations, elitism

100 150 200 250 300

105

Problem Size

Eva

luat

ions

netxuniformhboa

DHC flips, elitism

100 150 200 250 300

105

Problem Size

Num

ber

of fl

ips

netxuniformhboa

Execution Time, elitism

100 150 200 250 300

100

Problem Size

Exe

cutio

n T

ime

netxuniformhboa




Nearest neighbor NK, step = 5

Evaluations, RTR

30 60 90 120150 210

104

Problem Size

Eva

luat

ions

netxuniformhboa2p

DHC flips, RTR

30 60 90 120150 210

105

Problem Size

Num

ber

of fl

ips

netxuniformhboa2p

Execution Time, RTR

30 60 90 120150 210

100

Problem Size

Exe

cutio

n T

ime

netxuniformhboa2p





Evaluations, RTR

30 60 90 120150 210

104

Problem Size

Eva

luat

ions

netxuniformhboa2p

DHC flips, RTR

30 60 90 120150 210

105

Problem Size

Num

ber

of fl

ips

netxuniformhboa2p

Execution Time, RTR

30 60 90 120150 210

100

Problem Size

Exe

cutio

n T

ime

netxuniformhboa2p






30 60 90 120150 210

105

Problem Size

Eva

luat

ions

netxuniformhboa2p

DHC flips, elitism

30 60 90 120150 210

105

Problem Size

Num

ber

of fl

ips

netxuniformhboa2p


30 60 90 120150 210

100

Problem Size

Exe

cutio

n T

ime

netxuniformhboa2p




Nearest neighbor NK by difficulty

n = 120, step = 5 n = 120, step = 1

0 0.5 10.2

0.4

0.6

0.8

1

Percent easiest netx instances

Num

ber

of fl

ips/

mea

n

netxuniformhboa

0 0.5 10.2

0.4

0.6

0.8

1


Num

ber

of fl

ips/

mea

n

netxuniformhboa




Unrestricted NK landscapes

Evaluations, RTR

20 22 26 30 34 3810

2

Problem Size

Eva

luat

ions

netxuniformhboa2p

DHC flips, RTR

20 22 26 30 34 38

103

Problem Size

Num

ber

of fl

ips

netxuniformhboa2p

Execution Time, RTR

20 22 26 30 34 38

10−2

Problem Size

Exe

cutio

n T

ime

netxuniformhboa2p




Unrestricted NK landscapes


20 22 26 30 34 3810

2

Problem Size

Eva

luat

ions

netxuniformhboa2p

DHC flips, elitism

20 22 26 30 34 38

103

Problem Size

Num

ber

of fl

ips

netxuniformhboa2p


20 22 26 30 34 38

10−2

Problem Size

Exe

cutio

n T

ime

netxuniformhboa2p




Unrestricted NK by difficulty

n = 38, RTR n = 38, elitism

0 0.5 10.2

0.4

0.6

0.8

1


Num

ber

of fl

ips/

mea

n

netxuniformhboa2p

0 0.5 10.2

0.4

0.6

0.8

1


Num

ber

of fl

ips/

mea

n

netxuniformhboa2p




Conclusions

Compared GA with network crossover againstGA with uniform and two-point crossover.hBOA, a state of the art EDA.

On nearest neighbor NK landscapes and trap5Network crossover had the best execution time through allsettings.Niching with RTR outperformed elitism.hBOA had the least variance in instance difficulty.

On unrestricted NK landscapesResults less clear.hBOA had the best scalability.RTR and elitism results were mixed.




Conclusions

Future WorkTest on more diverse problems.Use trial runs of an EDA to learn the crossover network.Test other network based crossovers.Test against a version of hBOA that takes into accountproblem structure




Any Questions?



network crossover performance on nk landscapes and deceptive problems

Documents

uniform crossover

network model

graph problems

difcult problems

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nk landscapes conclusions