The hierarchical Bayesian optimization algorithm (hBOA) can solve nearly decomposable and hierarchical problems of bounded difficulty in a robust and scalable manner by building and sampling probabilistic models of promising solutions. This paper analyzes probabilistic models in hBOA on two common test problems: concatenated traps and 2D Ising spin glasses with periodic boundary conditions. We argue that although Bayesian networks with local structures can encode complex probability distributions, analyzing these models in hBOA is relatively straightforward and the results of such analyses may provide practitioners with useful information about their problems. The results show that the probabilistic models in hBOA closely correspond to the structure of the underlying optimization problem, the models do not change significantly in subsequent iterations of BOA, and creating adequate probabilistic models by hand is not straightforward even with complete knowledge of the optimization problem.

1. Analyzing Probabilistic Models
in Hierarchical BOA
on Traps and Spin Glasses
Mark Hauschild, Martin Pelikan,
Claudio F. Lima, Kumara Sastry

2. Motivation
Background
hBOA can solve difficult problems scalably and reliably
Much empirical work has been done
efficiency enhancement (EE)
scalability
Relatively little studying of the models themselves
Purpose
Understand models in hBOA
Design problem-specific EE
Design automated EE techniques
Educate researchers about their problems

3. Outline
Hierarchical BOA (hBOA)
Models in hBOA
Experiments
Trap 5
2d Ising Spin Glass
Conclusions
Future Work

4. Hierarchical Bayesian
Optimization Algorithm (hBOA)
Pelikan, Goldberg and Cantu-Paz (2001)
Uses Bayesian network with local structures
to model promising solutions
Bayesian network
Acyclic directed Graph
String positions are the nodes
Edges represent conditional dependencies
Where there is no edge, implicit independence
Sampled to create new population each gen.
Niching to maintain diversity

5. hBOA
Bayesian
Current New
network
Selection
population population

6. Bayesian Network
Structure
Edges determine
dependence
Parameters
p(Burglary|Alarm)

7. What are we looking for?
Accuracy
Do the models accurately represent the
underlying problem structure
Jij
Effects on scalability
Using models to understand problems
Dynamics
How the models change over time
Efficiency Enhancement

8. Test Problems
Selecting our test problems
Have to know what a good model looks like
Non-trivial
Easily scaled up
Diverse
Test Problems Selected
Trap-5
2D Ising Spin Glass

9. Trap-5
Partition binary string into disjoint
groups of 5 bits
5 if ones = 5
trap (ones) =
4 ones otherwise
Total fitness is sum of single traps
Optimum: String 1111…1

10. Perfect Model for Trap-5
Each 5-bit partition should be fully (or close to fully)
connected (10 edges)
Necessary dependencies
Bits between partitions should be independent
Unnecessary dependencies

11. Trap-5 Experiments
Problem sizes 15-210
30 runs for each size
Bisection was used to determine
population size

12. Trap-5 Results
Necessary and unnecessary dependencies w.r.t problem size
Three snapshots taken in each run (first, middle, last gen.)
Average of the 30 runs for each problem size

13. Trap-5 Results
a) Middle snapshot b) End snapshot
Ratio of unnecessary dependencies to total
dependencies as problem size increases

14. Trap-5 Results
300 600
Total Total
+ New Dep + New Dep
250 500
− Old Dep − Old Dep
Dependencies
Dependencies
200 400
150 300
100 200
50 100
0 0
1 2 3 4 5 6 7 8 9 10 11 12 13 5 10 15 20 25
Generations Generations
a) N = 100 bits. b) N = 200 bits.
Model dynamics for two different problem sizes.
Dependencies were counted as the same even if their
direction changed.

15. 2D Ising Spin Glass
Origin in physics
Spins arranged on a 2D
grid (periodic boundary
conditions)
Each spin (sj) can have
two values: +1 or -1 E = E (c ) si J i , j s j
Set of connection weights i, j
Jij specifies an instance.

16. 2D Ising Spin Glass
Very hard for most optimization techniques
Extremely large number of local optima
Any decomposition of bounded order is insufficient
The problem is solvable in polynomial time by
analytical techniques
hBOA does solve it in polynomial time
A deterministic hill-climber (DHC) is used to
improve the quality of all evaluated solutions
Single bit flips until no improvements

17. Perfect Model for Spin-Glass
Not sure what the perfect model is
To some degree, all positions are
dependent on all others
If we consider all, it will not scale
We would expect the interactions
should decrease in magnitude as their
distance increases

18. Spin Glass Experiments
100 different instances of size 10x10 to 20x20
Only graphs of 20x20 are shown
With and without DHC
5 runs of each instance
Scaling as we restrict models
Restrict models to short order dependencies

19. Spin Glass Results
800 800
Dependencies
Dependencies
600 600
400 400
200 200
0 0
0 10 20 0 10 20
Distance Distance
a) First generation b) Second snapshot
800 800
Dependencies
Dependencies
600 600
400 400
200 200
0 0
0 10 20 0 10 20
Distance Distance
c) Third snapshot d) Last generation

20. Spin Glass Results (No DHC)
800 800
Dependencies
Dependencies
600 600
400 400
200 200
0 0
0 10 20 0 10 20
Distance Distance
a) First generation b) Second snapshot
800 800
Dependencies
Dependencies
600 600
400 400
200 200
0 0
0 10 20 0 10 20
Distance Distance
c) Third snapshot d) Last generation

21. Spin Glass Results
1800 800
Total Total
+ New Dep + New Dep
1500
− Old Dep − Old Dep
600
Dependencies
Dependencies
1200
900 400
600
200
300
0 0
1 2 3 4 5 6 7 8 9 101112131415 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Generations Generations
a) All dependencies b) Neighbor dependencies
One example instance (rest similar)
Dependencies were counted the same even if their
direction changed

22. Restricting hBOA models
5
10
stock
d(2)
Fitness Evaluations
d(1)
4
10
3
10
100 144 196 256 324400 625 900
Problem Size
Evaluations varying by different restrictions
Experiments without DHC had similar scaling

23. Conclusions
Studying hBOA models is challenging but
possible
hBOA generates accurate models quickly
Models reveal a lot of information about the
problem
Models do not change rapidly over time
Information about models can help us design
EE

24. Future Work
Do model analysis on other challenging
problems
Artificial
Real world
Develop techniques to automatically
exploit this information for EE

25. Any Questions?