Pairwise and Problem-Specific Distance Metrics in the Linkage Tree Genetic Algorithm

Pairwise and Problem-Specific Distance Metricsin the Linkage Tree Genetic Algorithm

Martin Pelikan1, Mark W. Hauschild1, Dirk Thierens2

1 Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)University of Missouri, St. Louis, MO

[email protected], [email protected]

2 Utrecht UniversityUtrecht, The [email protected]

Download MEDAL Report No. 2011001

http://medal.cs.umsl.edu/files/2011001.pdf

Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

http://medal.cs.umsl.edu/files/2011001.pdf

Motivation

Linkage learning

I Standard crossover often ineffective in presence of epistasis.I Linkage learning aims to learn interactions between problem

variables to ensure that crossover does not disrupt importantpartial solutions and it combines them effectively.

I Various evolutionary algorithms capable of linkage learningexist.

This study

I Focuses on linkage tree genetic algorithm (LTGA).I Proposes and analyzes two distance metrics in LTGA.I Analyzes LTGA scalability on a large number of problems.


Outline

1. Linkage tree genetic algorithm (LTGA).

2. Distance metrics in LTGA.

3. Experiments.

4. Summary and conclusions.


Linkage Tree

Linkage treeI Leaves are individual variables (string positions).I Each internal node has two subtrees.I Each node represents a subset of variables (descendants).I Descendants of any node form a linkage group.I Linkage groups used as masks in LTGA crossover.


Linkage Tree Genetic Algorithm

LTGA procedure

I Starts with a random population.I Initial population may undergo local search.I Each generation performs two rounds of crossover to generate

a new population of the same size.

LTGA crossover

I Start with pair (X, Y ) of parents.I For each linkage group [π1, π2, . . . , πk] in T (bottom to top)

I Create X ′ and Y ′ by exchanging bits in positions {π1, . . . , πk}between X and Y .

I If best(X ′, Y ′) is better than best(X, Y ), then replace (X, Y )with (X ′, Y ′).

I The best of the two parents after applying each linkage groupsurvives to the next population.


Learning Linkage Tree

Learning linkage treeI Start with each variable being a separate linkage group.I Each step merges two closest groups.I Distance of linkage groups based on variation of information.I Each iteration should merge most strongly interacting groups.


Measuring Cluster Distances in LTGA

Distance metric based on variation of informationI Distance of clusters Ci and Cj :

D(Ci, Cj) = 2− H(Ci) + H(Cj)

H(Ci, Cj)

whereI H(Ci, Cj) is the entropy of Ci ∪ Cj

I H(Ci) is the entropy of Ci

I H(Cj) is the entropy of Cj

Bottleneck in learning linkage treeI Most time spent by measuring cluster distances.I Can we alleviate this bottleneck?I We discuss two distance metrics that address this issue.


Pairwise Metric

Pairwise metric

I Start by measuring distances between pairs of variables.I Cluster distance computed as average distance between pairs

of variables

D′(Ci, Cj) =1

|Ci| × |Cj |∑

ci∈Ci

∑cj∈Cj

D(ci, cj)

Good news

I We only need pairwise statistics.I This results in much faster distance computation.I Surprisingly, this also helps scalability.


Problem-Specific Metrics

Basic idea

I If we could estimate distance of clusters without computingstatistics from current population, we could possibly

I save lot of time in learning tree, andI reduce the population sizes and number of generations.

Where to get distances from?

I Problem-specific information.I Learning from optimization runs on similar problems.


Additively Decomposable Functions (ADFs)

Additively decomposable function

I Additively decomposable function:

f(X1, . . . , Xn) =m∑

i=1

fi(Si)

I fi is ith subfunctionI Si is subset of variables from {X1, . . . , Xn}

I Variables in located in the same subproblem are expected tointeract more strongly.

I Can we use this fact to create a distance metric for LTGA?


Problem-Specific Metric for ADFs

Distance metric for ADFs

I Create graph G = (V,E).I V = {X1, X2, . . . , Xn}.I E = {(i, j) : Xi, Xj ∈ Sk}.I Define weight of each edge from E as d(i, j) = 1.I Define li,j the shortest path between i and j.


Problem-Specific Metric for ADFs

Distance metric for ADFs

I Use G to compute distances between variables

D′′(Xi, Xj) =

{li,j if a path between Xi and Xj existsn otherwise

I Cluster distance is defined as an average of pairwise distances

D′′(Ci, Cj) =1

|Ci| × |Cj |∑

ci∈Ci

∑cj∈Cj

D′′(ci, cj)


Experiments: Test Problems

Problems

I Concatenated traps of order k.I Nearest-neighbor NK landscapes with wrap-around

neighborhoods.I 2D Ising spin glass.

Why these test problems?

I All test problems require linkage learning.I All test problems are nontrivial.I Yet all test problems are solvable in polynomial time.


Experiments: Setup

Test problem parameters, instances

I Traps of order k ∈ {5, 6, 7, 8} were tested.I NK landscapes with k = 5 were tested.I For all problems, n was varied.I For NK landscapes and spin glasses, for each n, 1,000

instances were generated and tested.

LTGA setup

I Bisection was used to find minimum population size forconvergence to the optimum in 10 out of 10 independent runs.

I For traps, bisection is repeated 10 times for each n.I Max. number of generations is set to a sufficiently large value.I Bit-flip local search run on initial population.I Use standard, pairwise, and problem-specific metric.


Results: Pairwise Metric on Trap-5

102

103

104

105

106

Problem size, n

Num

ber

of e

valu

atio

ns

LTGA (original), O(n1.27)

LTGA (pairwise), O(n1.25)

I Pairwise metric allows us to solve much larger problems.I Scalability is slightly improved (surprising).I Results for trap-6 and trap-7 similar.


Results: Pairwise Metric on NK

20 40 60 80 10010

3

104

105

106

107

Problem size, n

Num

ber

of e

valu

atio

ns



I Pairwise metric allows us to solve much larger problems.I Scalability is significantly improved (surprising).


Results: Pairwise Metric on 2D Spin Glass

64 100 144 196 25610

4

105

106

107

Problem size, n

Num

ber

of e

valu

atio

ns



I Pairwise metric allows us to solve much larger problems.I Scalability is significantly improved (surprising).


Results: Problem-Specific Metric on Trap-5

102

103

104

105

106

Problem size, n

Num

ber

of e

valu

atio

ns


LTGA (problem), O(n1.26)

I Problem-specific metric similar to pairwise metric.I CPU slightly decreased though with problem-specific metric.


Results: Problem-Specific Metric on NK

20 40 60 80 10010

3

104

105

106

107

Problem size, n

Num

ber

of e

valu

atio

ns



I Problem-specific metric slightly better than pairwise one.I So problem-specific metric pays off.


Results: Problem-Specific Metric on 2D Spin Glass

64 100 144 196 25610

4

105

106

107

108

Problem size, n

Num

ber

of e

valu

atio

ns



I Problem-specific metric scales worse than pairwise one!I Problem-specific metric is not that great for 2D spin glass.


Conclusions and Future Work

Conclusions

I LTGA provides opportunities for efficiency enhancements.I LTGA also provides promising tool for using problem-specific

knowledge and learning from experience whensolving many instancesof similar problems.

I Pairwise metric provides important improvement.I Problem-specific metric demonstrates the ability of LTGA to exploit

problem-specific knowledge on additively decomposable functions.I But the results based on problem-specific information are mixed.

Future work

I Design more robust and effective problem-specific metrics.I Design methods to learn distance metrics for specific problem classes.I Improve performance of LTGA on problems of complex structure.I Adopt efficiency enhancement techniques for other evolutionary

algorithms to LTGA, including model-directed local search, fitnessmodeling, parallelization, and others.


Acknowledgments

Acknowledgments

I NSF; NSF CAREER grant ECS-0547013.

I University of Missouri; High Performance ComputingCollaboratory sponsored by Information Technology Services;Research Award; Research Board.


Pairwise and Problem-Specific Distance Metrics in the Linkage Tree Genetic Algorithm

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