The document describes experiments comparing a genetic algorithm using network crossover to hierarchical Bayesian optimization and deterministic hill climbing on deceptive and NK landscape test problems. The experiments tested different problem sizes, replacement techniques, and landscape parameters. Results showed that network crossover outperformed other operators on evaluations, local search steps, and runtime on difficult problem instances by better respecting linkages between variables.

1. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover Performance on NK
Landscapes and Deceptive Problems
M. Hauschild1 M. Pelikan1
1 Missouri 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

2. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

3. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

4. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Outline
Network crossover
Algorithms
GA
hBOA
Deterministic Hill-Climber
Test Problems
Experiments
Trap-5
NK Landscapes
Conclusions
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

5. Motivation Outline Network Crossover Algorithms Test Problems Experiments 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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

6. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

7. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
To build the mask
Choose a random bit
Randomized breadth-first search to expand mask
Repeat 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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

8. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

9. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

10. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

11. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

12. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

13. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Network Crossover
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

14. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Genetic Algorithm
Three crossover operators used.
Network crossover
Two-point
Uniform
Probability of crossover, pc = 0.6
Probability of mutation, pm = 1/n
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

15. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
hierarchical Bayesian Optimization Algorithm (hBOA)
Pelikan, Goldberg, and Cantú-Paz; 2001
Uses Bayesian network with local structures to model
solutions
Acyclic directed Graph
String positions are the nodes
Edges represent conditional dependencies
Where there is no edge, implicit independence
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

16. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Deterministic Hill-Climber
Deterministic hill climber (DHC) used for all runs
Performs single-bit changes that lead to maximum
performance
Stops when no single-bit change leads to improvement
Originally considered not using DHC
Dramatically improved performance
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

17. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Trap-5
Partition binary string into disjoint groups of 5 bits
5 if ones = 5
trap5 (ones) = , (1)
4 − ones otherwise
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

18. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
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
n−1
fnk (X0 , . . . , Xn−1 ) = fi (Xi , (Xi )) (2)
i=0
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

19. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

20. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
NK landscapes
Unrestricted NK landscapes.
NP-complete for k > 1
Branch and bound algorithm used to find optima.
Nearest neighbor NK landscapes.
Polynomial solvability.
Dynamic programming used to find optima.
G connects all neighboring bits.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

21. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Experimental Setup
Trap-5
Problem 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 landscapes
Problem sizes of n ∈ {20, 22, . . . , 38}.
k =5
1000 random problem instances for each setting.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

22. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Experimental Setup
Nearest neighbor NK landscapes
Problem sizes of n ∈ {30, 60, . . . , 210}.
Two step sizes considered, step ∈ {1, 5}.
k =5
1000 instances for each combination of n, k , step.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

23. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Experimental Setup
Two replacement techniques considered.
Restricted Tournament Replacement(RTR)
Niching, replaces similar solutions.
Window size set to w = min{n, N/5}.
Elitism
Keeps a portion of the best individuals each generation.
50% of the most fit individuals kept.
Examined three measures
Evaluations
Local search steps
Execution time
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

24. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Trap-5
Evaluations, RTR DHC flips, RTR
10
netx 10 netx
uniform
uniform
Number of flips
Evaluations
hboa
hboa
2−point
2−point
5
10
5
10
100 150 200 250 300 100 150 200 250 300
Problem Size Problem Size
Execution Time, RTR
netx
Execution Time
uniform
hboa
2−point
0
10
100 150 200 250 300
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

25. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Trap-5
Evaluations, elitism DHC flips, elitism
netx netx
Number of flips
uniform uniform
Evaluations
hboa hboa
5
10
5
10
100 150 200 250 300 100 150 200 250 300
Problem Size Problem Size
Execution Time, elitism
netx
Execution Time
uniform
hboa
0
10
100 150 200 250 300
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

26. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK, step = 5
Evaluations, RTR DHC flips, RTR
netx netx
Number of flips
uniform uniform
Evaluations
5
4 hboa 10 hboa
10
2p 2p
30 60 90 120 150 210 30 60 90 120 150 210
Problem Size Problem Size
Execution Time, RTR
netx
Execution Time
uniform
0 hboa
10 2p
30 60 90 120 150 210
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

27. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK, step = 5
Evaluations, RTR DHC flips, RTR
netx netx
Number of flips
uniform uniform
Evaluations
5
4 hboa 10 hboa
10
2p 2p
30 60 90 120 150 210 30 60 90 120 150 210
Problem Size Problem Size
Execution Time, RTR
netx
Execution Time
uniform
0 hboa
10 2p
30 60 90 120 150 210
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

28. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK, step = 1
Evaluations, RTR DHC flips, RTR
netx netx
Number of flips
uniform uniform
Evaluations
4 hboa 5
10 hboa
10 2p 2p
30 60 90 120 150 210 30 60 90 120 150 210
Problem Size Problem Size
Execution Time, RTR
netx
Execution Time
uniform
hboa
2p
0
10
30 60 90 120 150 210
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

29. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK, step = 5
Evaluations, elitism DHC flips, elitism
netx netx
Number of flips
uniform uniform
Evaluations
hboa hboa
5
10 2p 2p
5
10
30 60 90 120 150 210 30 60 90 120 150 210
Problem Size Problem Size
Execution Time, elitism
netx
Execution Time
uniform
hboa
2p
0
10
30 60 90 120 150 210
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

30. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK, step = 1
Evaluations, elitism DHC flips, elitism
netx netx
Number of flips
uniform uniform
Evaluations
5
10 hboa hboa
2p 5 2p
10
30 60 90 120 150 210 30 60 90 120 150 210
Problem Size Problem Size
Execution Time, elitism
netx
Execution Time
uniform
hboa
2p
0
10
30 60 90 120 150 210
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

31. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Nearest neighbor NK by difficulty
n = 120, step = 5 n = 120, step = 1
1 1
Number of flips/mean
Number of flips/mean
0.8 0.8
0.6 0.6
netx netx
0.4 uniform 0.4 uniform
hboa hboa
0.2 0.2
0 0.5 1 0 0.5 1
Percent easiest netx instances Percent easiest netx instances
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

32. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Unrestricted NK landscapes
Evaluations, RTR DHC flips, RTR
netx netx
Number of flips
uniform uniform
Evaluations
hboa hboa
2p 3 2p
10
2
10
20 22 26 30 34 38 20 22 26 30 34 38
Problem Size Problem Size
Execution Time, RTR
netx
Execution Time
uniform
hboa
−2 2p
10
20 22 26 30 34 38
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

33. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Unrestricted NK landscapes
Evaluations, elitism DHC flips, elitism
netx netx
Number of flips
uniform uniform
Evaluations
hboa hboa
2p 2p
3
10
2
10
20 22 26 30 34 38 20 22 26 30 34 38
Problem Size Problem Size
Execution Time, elitism
netx
Execution Time
uniform
hboa
−2 2p
10
20 22 26 30 34 38
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

34. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Unrestricted NK by difficulty
n = 38, RTR n = 38, elitism
1 1
Number of flips/mean
Number of flips/mean
netx netx
0.8 uniform 0.8 uniform
hboa hboa
0.6 2p 0.6 2p
0.4 0.4
0.2 0.2
0 0.5 1 0 0.5 1
Percent easiest netx instances Percent easiest netx instances
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

35. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Conclusions
Compared GA with network crossover against
GA with uniform and two-point crossover.
hBOA, a state of the art EDA.
On nearest neighbor NK landscapes and trap5
Network crossover had the best execution time through all
settings.
Niching with RTR outperformed elitism.
hBOA had the least variance in instance difficulty.
On unrestricted NK landscapes
Results less clear.
hBOA had the best scalability.
RTR and elitism results were mixed.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

36. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Conclusions
Future Work
Test 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 account
problem structure
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems

37. Motivation Outline Network Crossover Algorithms Test Problems Experiments Conclusions
Any Questions?
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Network Crossover Performance on NK Landscapes and Deceptive Problems