1. The document proposes and analyzes two distance metrics for the linkage tree genetic algorithm (LTGA): a pairwise metric and a problem-specific metric. 2. Experiments on optimization problems show the pairwise metric significantly improves LTGA scalability. The problem-specific metric, informed by problem structure, yields further speedups on some problems but mixed results on others. 3. Future work aims to design more robust problem-specific metrics and methods to learn metrics from problem instances, improving LTGA performance on complex problems.

1. Pairwise and Problem-Specific Distance Metrics
in 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
pelikan@cs.umsl.edu, mwh308@umsl.edu
2
Utrecht University
Utrecht, The Netherlands
Dirk.Thierens@cs.uu.nl
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

2. Motivation
Linkage learning
Standard crossover often ineffective in presence of epistasis.
Linkage learning aims to learn interactions between problem
variables to ensure that crossover does not disrupt important
partial solutions and it combines them effectively.
Various evolutionary algorithms capable of linkage learning
exist.
This study
Focuses on linkage tree genetic algorithm (LTGA).
Proposes and analyzes two distance metrics in LTGA.
Analyzes LTGA scalability on a large number of problems.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

3. Outline
1. Linkage tree genetic algorithm (LTGA).
2. Distance metrics in LTGA.
3. Experiments.
4. Summary and conclusions.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

4. Linkage Tree
Linkage tree
Leaves are individual variables (string positions).
Each internal node has two subtrees.
Each node represents a subset of variables (descendants).
Descendants of any node form a linkage group.
Linkage groups used as masks in LTGA crossover.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

5. Linkage Tree Genetic Algorithm
LTGA procedure
Starts with a random population.
Initial population may undergo local search.
Each generation performs two rounds of crossover to generate
a new population of the same size.
LTGA crossover
Start with pair (X, Y ) of parents.
For each linkage group [π1 , π2 , . . . , πk ] in T (bottom to top)
Create X and Y by exchanging bits in positions {π1 , . . . , πk }
between X and Y .
If best(X , Y ) is better than best(X, Y ), then replace (X, Y )
with (X , Y ).
The best of the two parents after applying each linkage group
survives to the next population.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

6. Learning Linkage Tree
Learning linkage tree
Start with each variable being a separate linkage group.
Each step merges two closest groups.
Distance of linkage groups based on variation of information.
Each iteration should merge most strongly interacting groups.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

7. Measuring Cluster Distances in LTGA
Distance metric based on variation of information
Distance of clusters Ci and Cj :
H(Ci ) + H(Cj )
D(Ci , Cj ) = 2 −
H(Ci , Cj )
where
H(Ci , Cj ) is the entropy of Ci ∪ Cj
H(Ci ) is the entropy of Ci
H(Cj ) is the entropy of Cj
Bottleneck in learning linkage tree
Most time spent by measuring cluster distances.
Can we alleviate this bottleneck?
We discuss two distance metrics that address this issue.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

8. Pairwise Metric
Pairwise metric
Start by measuring distances between pairs of variables.
Cluster distance computed as average distance between pairs
of variables
1
D (Ci , Cj ) = D(ci , cj )
|Ci | × |Cj |
ci ∈Ci cj ∈Cj
Good news
We only need pairwise statistics.
This results in much faster distance computation.
Surprisingly, this also helps scalability.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

9. Pairwise Metric
Pairwise metric
Start by measuring distances between pairs of variables.
Cluster distance computed as average distance between pairs
of variables
1
D (Ci , Cj ) = D(ci , cj )
|Ci | × |Cj |
ci ∈Ci cj ∈Cj
Good news
We only need pairwise statistics.
This results in much faster distance computation.
Surprisingly, this also helps scalability.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

10. Problem-Specific Metrics
Basic idea
If we could estimate distance of clusters without computing
statistics from current population, we could possibly
save lot of time in learning tree, and
reduce the population sizes and number of generations.
Where to get distances from?
Problem-specific information.
Learning from optimization runs on similar problems.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

11. Additively Decomposable Functions (ADFs)
Additively decomposable function
Additively decomposable function:
m
f (X1 , . . . , Xn ) = fi (Si )
i=1
fi is ith subfunction
Si is subset of variables from {X1 , . . . , Xn }
Variables in located in the same subproblem are expected to
interact more strongly.
Can we use this fact to create a distance metric for LTGA?
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

12. Problem-Specific Metric for ADFs
Distance metric for ADFs
Create graph G = (V, E).
V = {X1 , X2 , . . . , Xn }.
E = {(i, j) : Xi , Xj ∈ Sk }.
Define weight of each edge from E as d(i, j) = 1.
Define li,j the shortest path between i and j.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

13. Problem-Specific Metric for ADFs
Distance metric for ADFs
Use G to compute distances between variables
li,j if a path between Xi and Xj exists
D (Xi , Xj ) =
n otherwise
Cluster distance is defined as an average of pairwise distances
1
D (Ci , Cj ) = D (ci , cj )
|Ci | × |Cj |
ci ∈Ci cj ∈Cj
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

14. Experiments: Test Problems
Problems
Concatenated traps of order k.
Nearest-neighbor NK landscapes with wrap-around
neighborhoods.
2D Ising spin glass.
Why these test problems?
All test problems require linkage learning.
All test problems are nontrivial.
Yet all test problems are solvable in polynomial time.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

15. Experiments: Setup
Test problem parameters, instances
Traps of order k ∈ {5, 6, 7, 8} were tested.
NK landscapes with k = 5 were tested.
For all problems, n was varied.
For NK landscapes and spin glasses, for each n, 1,000
instances were generated and tested.
LTGA setup
Bisection was used to find minimum population size for
convergence to the optimum in 10 out of 10 independent runs.
For traps, bisection is repeated 10 times for each n.
Max. number of generations is set to a sufficiently large value.
Bit-flip local search run on initial population.
Use standard, pairwise, and problem-specific metric.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

16. Results: Pairwise Metric on Trap-5
6
10
1.27
Number of evaluations LTGA (original), O(n )
1.25
LTGA (pairwise), O(n )
5
10
4
10
2 3
10 10
Problem size, n
Pairwise metric allows us to solve much larger problems.
Scalability is slightly improved (surprising).
Results for trap-6 and trap-7 similar.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

17. Results: Pairwise Metric on NK
7 5.14
10 LTGA (original), O(n )
Number of evaluations 3.23
LTGA (pairwise), O(n )
6
10
5
10
4
10
3
10
20 40 60 80 100
Problem size, n
Pairwise metric allows us to solve much larger problems.
Scalability is significantly improved (surprising).
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

18. Results: Pairwise Metric on 2D Spin Glass
7
10
5.38
Number of evaluations LTGA (original), O(n )
3.50
LTGA (pairwise), O(n )
6
10
5
10
4
10
64 100 144 196 256
Problem size, n
Pairwise metric allows us to solve much larger problems.
Scalability is significantly improved (surprising).
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

19. Results: Problem-Specific Metric on Trap-5
6
10
1.25
LTGA (pairwise), O(n )
Number of evaluations 1.26
LTGA (problem), O(n )
5
10
4
10
2 3
10 10
Problem size, n
Problem-specific metric similar to pairwise metric.
CPU slightly decreased though with problem-specific metric.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

20. Results: Problem-Specific Metric on NK
7
10
3.23
Number of evaluations LTGA (pairwise), O(n )
2.87
6 LTGA (problem), O(n )
10
5
10
4
10
3
10
20 40 60 80 100
Problem size, n
Problem-specific metric slightly better than pairwise one.
So problem-specific metric pays off.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

21. Results: Problem-Specific Metric on 2D Spin Glass
8
10
4.05
Number of evaluations LTGA (problem), O(n )
3.50
7 LTGA (pairwise), O(n )
10
6
10
5
10
4
10
64 100 144 196 256
Problem size, n
Problem-specific metric scales worse than pairwise one!
Problem-specific metric is not that great for 2D spin glass.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

22. Conclusions and Future Work
Conclusions
LTGA provides opportunities for efficiency enhancements.
LTGA also provides promising tool for using problem-specific
knowledge and learning from experience whensolving many instances
of similar problems.
Pairwise metric provides important improvement.
Problem-specific metric demonstrates the ability of LTGA to exploit
problem-specific knowledge on additively decomposable functions.
But the results based on problem-specific information are mixed.
Future work
Design more robust and effective problem-specific metrics.
Design methods to learn distance metrics for specific problem classes.
Improve performance of LTGA on problems of complex structure.
Adopt efficiency enhancement techniques for other evolutionary
algorithms to LTGA, including model-directed local search, fitness
modeling, parallelization, and others.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA

23. Acknowledgments
Acknowledgments
NSF; NSF CAREER grant ECS-0547013.
University of Missouri; High Performance Computing
Collaboratory sponsored by Information Technology Services;
Research Award; Research Board.
Martin Pelikan, Mark W. Hauschild, Dirk Thierens Pairwise and Problem-Specific Metrics in LTGA