This document describes research on intelligently biasing the network structures learned by the hierarchical Bayesian Optimization Algorithm (hBOA) to improve its performance. The researchers developed a method called split probability matrix (SPM) biasing, which uses prior information stored in an SPM to bias hBOA's network building process towards structures with a certain number of splits. They tested SPM biasing on two problems: Trap-5 and 2D Ising spin glasses. The experiments showed that SPM biasing significantly reduced hBOA's execution time, number of evaluations, and bits examined on both problems, with the best speedups achieved around a tuning parameter κ value of 1.

1. Motivation Outline hBOA Biasing Experiments Conclusions
Intelligent Bias of Network Structures in the
Hierarchical BOA
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, 2009
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

2. Motivation Outline hBOA Biasing Experiments Conclusions
Motivation
In optimization, always looking to solve harder problems
hBOA can solve a broad class of problems robustly and
fast
Scalability isn’t always enough
Much work has been done in speeding up hBOA
Sporadic Model-Building
Parallelization
Others
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

3. Motivation Outline hBOA Biasing Experiments Conclusions
Motivation
Each run of an EDA leaves us with a tremendous amount
of information
The algorithm decomposes the problem for us
Left with a series of models
Methods have been developed to exploit this information
Require hand-inspection
Very sensitive to parameters
Wanted to develop a method that is less sensitive to
parameters
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

4. Motivation Outline hBOA Biasing Experiments Conclusions
Outline
hBOA
Biasing hBOA
Structural Priors in Bayesian Networks
Split Probability Matrix
SPM-based Bias
Test Problems
Experiments
Trap-5
2D Ising Spin Glasses
Conclusions
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

5. Motivation Outline hBOA Biasing 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
Niching to maintain diversity
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

6. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
Two Components
Structure
Edges determine dependencies
Majority of time spent here
Parameters
Conditional probabilities depending on parents
Example - p(Accident|Wet Road, Speed)
Network built greedily, one edge at a time
Metric punishes complexity
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

7. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

8. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

9. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

10. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

11. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

12. Motivation Outline hBOA Biasing Experiments Conclusions
hBOA
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

13. Motivation Outline hBOA Biasing Experiments Conclusions
Structural Priors
Bayesian-Dirichlet metric for network B and data set D with
prior knowledge ξ is
p(B|ξ)p(D|B, ξ)
p(B|D, ξ) = · (1)
p(D|ξ)
where p(B|ξ) is the prior probability of network structure.
Bias towards simpler models is given by
p(B|ξ) = c2−0.5( i |Li |)log2 N
, (2)
where N is the population and i |Li | is the number of
leaves.
Want to modify this based on prior information
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

14. Motivation Outline hBOA Biasing Experiments Conclusions
Biasing
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

15. Motivation Outline hBOA Biasing Experiments Conclusions
Split Probability Matrix
Lets bias towards same number of splits
Use split probability matrix to store our prior knowledge
4-dimensional matrix of size n × n × d × e where n is the
problem size, d is maximum number of splits, and e is the
maximum generation
S stores, for each possible pair of decision variables, the
conditional probability of a split between them (by gen.)
In our sampling we use a threshold of 90% for e
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

16. Motivation Outline hBOA Biasing Experiments Conclusions
SPM-Based Bias
No splits One split
1
1 0.5 1
2 2 0.8
100 3 100 3
4
5
0.4 4
5
node j
node j
6
2 4 6
6
2 4 6
0.6
200 0.3 200
0.4
0.2
300 300 0.2
0.1
400 0 400 0
100 200 300 400 100 200 300 400
node i node i
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

17. Motivation Outline hBOA Biasing Experiments Conclusions
SPM-Based Bias
Want to define our own prior probability
Prior probability of network structure:
n
p(B|ξ) = p(Ti ). (3)
i=1
For a particular decision tree Ti , p(Ti ) is given by:
p(Ti ) = κ
qi,j,k (i,j), (4)
j=i
where qi,j,k (i,j) denotes the probability that there are at
least k(i, j) splits on Xj in decision trees for Xi . κ is used to
tune the effect of prior information.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

18. Motivation Outline hBOA Biasing Experiments Conclusions
SPM-Based Bias
Consider evaluation of split on Xj in Ti given k − 1 splits
Gains in log-likelihood after a split without considering prior
information:
δi,j = log2 p(D|B , ξ) − log2 p(D|B, ξ) − 0.5log2N. (5)
where B is the network before the split and B is after.
SPM used to compute gains after a split:
δi,j = log2 p(D|B , ξ) − log2 p(D|B, ξ) + κ log2 Si,j,k (i,j),g (6)
This bias can still be overcome
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

19. Motivation Outline hBOA Biasing Experiments Conclusions
Trap-5
Partition binary string into disjoint groups of 5 bits
5 if ones = 5
trap5 (ones) = , (7)
4 − ones otherwise
Total fitness is sum of single traps
Global Optimum: String 1111...1
Local Optimum: 00000 in any partition
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

20. Motivation Outline hBOA Biasing Experiments Conclusions
2D Ising Spin Glass
Origin in physics
Spins arranged on a 2D grid
Each spin sj can have two values: +1 or -1
Each connection i, j has a weight Jij . Set of weights
specifies one instance.
Energy is given by...
E (C) = si Ji,j sj , (8)
i,j
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

21. Motivation Outline hBOA Biasing Experiments Conclusions
2D Ising Spin Glass
Problem is to find the values of the spins so energy is
minimized
Very hard for most optimization techniques
Extremely large number of local optima
Decomposition of bounded order is insufficient
Solvable in polynomial time by analytical techniques
hBOA has been shown emperically to solve it in polynomial
time
A deterministic hill-climber(DHC) is used to improve the
quality of evaluated solutions
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

22. Motivation Outline hBOA Biasing Experiments Conclusions
Experiments on Trap-5
Need to learn SPM from sample
Show effects of SPM using various κ
Problem sizes from n = 50 to n = 175
SPM learned from 10 bisection runs of 10 runs each
Used to bias model building in another 10 bisection runs
Threshold of 90%
Varied κ from 0.05 to 3
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

23. Motivation Outline hBOA Biasing Experiments Conclusions
Speedups on Trap-5, κ = 1
Execution Speedup Evaluation Speedup
Execution Time Speedup
7 4
Evaluation Speedup
6
5 3
4
3 2
2
1 1
50 75 100 125 150 175 50 75 100 125 150 175
Problem Size Problem Size
Reduction in Bits Examined
80
Reduction Factor
60
40
20
0
50 75 100 125 150 175
Problem Size
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

24. Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on Trap-5 of n = 100
Execution Time Evaluations
4
15 x 10
15
Execution Time
Evaluations
10 10
5 5
0 0
0.05 1 2 3 0.05 1 2 3
κ κ
Bits Examined
7
x 10
10
Bits Examined
5
0
0.25 1 2 3
κ
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

25. Motivation Outline hBOA Biasing Experiments Conclusions
Experiments on 2D Ising Spin Glass
Need to learn SPM from sample
Show effects of SPM using various κ
100 instances of 3 different sizes
Cross-validation
SPM learned from 90 instances, used to solve remaining 10
Repeated 10 times
Threshold of 90%
Varied κ from 0.05 to 3
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

26. Motivation Outline hBOA Biasing Experiments Conclusions
Speedups on 2D Ising spin glass
Speedups obtained using SPM bias where κ = 1
size Exec. speedup Eval. Speedup Bits Exam.
16 × 16 1.16 0.87 1.5
20 × 20 1.42 0.96 1.84
24 × 24 1.56 0.98 2.03
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

27. Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
16 × 16 20 × 20
8 40
Execution Time
Execution Time
6 30
4
20
2
10
0 5
0.05 1 2 3 0.05 1 2 3
κ κ
24 × 24
200
Execution Time
150
100
50
0
0.05 1 2 3
κ
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

28. Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
16 × 16 20 × 20
5000 8000
7000
Evaluations
Evaluations
4000
6000
5000
3000
4000
2000 3000
0.05 1 2 3 0.05 1 2 3
κ κ
24 × 24
4
x 10
2
Evaluations
1.5
1
0.5
0.05 1 2 3
κ
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

29. Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
16 × 16 20 × 20
8 8
x 10 x 10
2 8
Bits Examined
Bits Examined
1.5 6
1 4
0.5 2
0.05 1 2 3 0.05 1 2 3
κ κ
24 × 24
9
x 10
4
Bits Examined
3
2
1
0.05 1 2 3
κ
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

30. Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
κ that led to maximum speedup
size κ Exec. speedup Eval. Speedup Bits Exam
16 × 16 0.75 1.24 0.96 1.66
20 × 20 1.25 1.44 0.94 1.85
24 × 24 1 1.56 0.98 2.03
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

31. Motivation Outline hBOA Biasing Experiments Conclusions
Conclusions
Unlike many EAs, we are left with a series of models
Many ways to try and exploit this information
Proposed a method to bias network structure in hBOA
Led to speedups from 3.5-6 on Trap-5 and up to 1.5 on 2D
Ising spin glasses
This is only one way
Can be extended to many other problems
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

32. Motivation Outline hBOA Biasing Experiments Conclusions
Conclusions
Efficiency enhancements work together
Parallelization 50
Hybridization 2
Soft bias from past runs 1.5
Evaluation Relaxation 1.1
Total 165
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA

33. Motivation Outline hBOA Biasing Experiments Conclusions
Any Questions?
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA