Hybridization of global and local search algorithms is a well-established technique for enhancing the efficiency of search algorithms. Hybridizing estimation of distribution algorithms (EDAs) has been repeatedly shown to produce better performance than either the global or local search algorithm alone. The hierarchical Bayesian optimization algorithm (hBOA) is an advanced EDA which has previously been shown to benefit from hybridization with a local searcher. This paper examines the effects of combining hBOA with a deterministic hill climber (DHC). Experiments reveal that allowing DHC to find the local optima makes model building and decision making much easier for hBOA. This reduces the minimum population size required to find the global optimum, which substantially improves overall performance.

1. Effects of a Deterministic Hill Climber on hBOA
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg
Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)
University of Missouri, St. Louis, MO
http://medal.cs.umsl.edu/
err26c@umsl.edu pelikan@cs.umsl.edu
Illinois Genetic Algorithms Laboratory
University of Illinois at Urbana-Champaign, Urbana, IL
http://www-illigal.ge.uiuc.edu/ deg@uiuc.edu
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

2. Background & Motivation
Hybrid Estimation of Distribution Algorithms (EDAs)
For very large or complex problems, scalability may not be
sufficient; additional efficiency enhancement may be required
Hybridization is a popular method of efficiency enhancement
for EDAs and other GEAs
A hybrid combines a local search algorithm with a global
searcher such as an EDA
Hybrid hBOA
The hierarchical Bayesian optimization algorithm (hBOA) is
frequently hybridized
This study focuses on hBOA combined with a deterministic
hill climber (DHC)
Examine the effects of varying the parameters of DHC
Study the effects of variance reduction on model building
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

3. Outline
1. Algorithms
2. Goals & Methodology
3. Test Problems
4. Results
5. Discussion of Results
6. Summary and conclusions.
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

4. Algorithms
Hierarchical Bayesian Optimization Algorithm (hBOA)
Randomly initialize a population of binary strings
Select promising solutions from the population
Build a Bayesian network from the selected solutions
Generate new solutions from the constructed model
Incorporate new solutions into the population using restricted
tournament replacement
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

5. Algorithms
Deterministic Hill Climber (DHC)
Flip one bit in the string that will produce the most fitness
improvement
Repeat until a specified quality has been reached, a maximum
number of flips has been performed, or no more improvement
is possible
hBOA has frequently been combined with DHC to improve
performance (e.g., Pelikan & Goldberg, 2003; Pelikan &
Hartmann, 2006; Pelikan, Katzgraber, & Kobe, 2008)
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

6. Goals & Methodology
Experiment Goals
Examine the effects of DHC on performance of hBOA
Determine optimal settings for DHC
Methodology
Tune the frequency of local search, i.e. the proportion of
strings in the population on which DHC operates
Tune the duration of local search, i.e. the maximum number
of flips per string
Compare
Population size
Number of generations until optimum
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

7. Test Problems
Test Problems
Trap-5
2-D Ising Spin Glass
3-CNF MAXSAT
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

8. Test Problems
Trap-5
Concatenation of non-overlapping subproblems of 5 bits
The fitness of each subproblem is determined as
5 if u = 5,
ftrap5 (u) = ,
4−u otherwise
where u is the number of ones in the subproblem
All subproblems are added together to determine the string’s
fitness
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

9. Test Problems
Trap-5
Properties
Separable, non-overlapping subproblems
Subproblems are fully deceptive
Difficult for local search, simple GA
Parameters
Problem sizes of 100-350 bits
Population size determined empirically using bisection method
10 independent bisections of 10 runs each
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

10. Test Problems
2-D Ising Spin Glass
Spins are arranged on a 2-D grid
Each spin si gets a value from +1, −1
Each edge between neighboring spins si and sj is assigned a
real value Ji,j
The task is to find a configuration of spins to minimize the
energy, specified as
n
H(s) = − Jij si sj
i,j=0
Spins are represented as bits in the binary string, 1 for +1, 0
for −1
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

11. Test Problems
2-D Ising Spin Glass
Properties
More complex fitness landscape than trap
Not fully decomposable into subproblems of bounded order,
substructures overlap
Solvable in polynomial time
Parameters
Problem sizes 8x8, 10x10, 12x12, 14x14, 16x16 (64 - 256 bits)
1000 instances
For each instance, Jij randomly set to +1 or −1
Population size determined using bisection method, 10
successful runs per instance
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

12. Test Problems
MAXSAT
Instances consist of 3-CNF formulas: conjunctions of
disjunctions of 3 literals
The task is to find an assignment of Boolean variables to
satisfy the maximum number of clauses
Bits in the string represent the variables, with 1 representing
true and 0 representing false
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

13. Test Problems
MAXSAT
Properties
Complex fitness landscape
Not fully decomposable into subproblems of bounded order,
substructures overlap
Not solvable in polynomial time
Parameters
Problem sizes of 50 and 75 variables
100 randomly-generated instances
Population size determined using bisection method, 10
successful runs per instance
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

14. Experimental Results
Trap-5 (350 bits)
Reduction factor, speed-up: ratio of values without DHC to
those with DHC
60 1.8
Population Size Reduction Factor
1.7
50
Generations Speed-Up
1.6
40 1.5
1.4
30 1.3
1.2
20
1.1
10 1
0.9
0 0.8
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1 0.2 0.6 1
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

15. Experimental Results
Trap-5 (350 bits)
100 7
90
6
Evaluations Speed-Up
80
Flips per Evaluation
70 5
60 4
50
40 3
30 2
20
1
10
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1 0.2 0.6 1
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

16. Experimental Results
Trap-5
100000 100
No DHC No DHC
With DHC With DHC
Population Size
Generations
10000
10
1000
100 1
100 150 200 250 300 350 100 150 200 250 300 350
Problem size (# of bits) Problem size (# of bits)
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

17. Experimental Results
Ising Spin Glass (16x16, 256 bits)
16 4.5
Population Reduction Factor
14 4
Generation Speed-up
12
3.5
10
3
8
2.5
6
2
4
2 1.5
0 1
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1.0 0.2 0.6 1.0
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

18. Experimental Results
Ising Spin Glass (16x16, 256 bits)
70 9
60 8
7
Evaluation Speedup
Flips per Evaluation
50
6
40 5
30 4
3
20
2
10 1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1.0 0.2 0.6 1.0
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

19. Experimental Results
Ising Spin Glass
10000 100
No DHC No DHC
With DHC With DHC
Population Size
Generations
1000
10
100
10 1
64 100 144 196 256 64 100 144 196 256
Problem size (# of bits) Problem size (# of bits)
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

20. Experimental Results
MAXSAT (75 bits)
Population Reduction Factor (original/reduced)
120 2.8
2.6
100 2.4
Generation Speed-up
2.2
80 2
1.8
60
1.6
40 1.4
1.2
20 1
0.8
0 0.6
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1.0 0.2 0.6 1.0
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

21. Experimental Results
MAXSAT (75 bits)
300 6
250 5
Evaluation speedups
Flips per Evaluation
200 4
150 3
100 2
50 1
0 0
0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Max. Flips (% of Prob. Size) Max. Flips (% of Prob. Size)
pdhc pdhc
0.1 0.5 0.9 0.1 0.5 0.9
0.2 0.6 1.0 0.2 0.6 1.0
0.3 0.7 0.3 0.7
0.4 0.8 0.4 0.8
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

22. Discussion of Results
Effects of DHC
DHC improved performance on each test problem, for all sizes
Substantial benefits were obtained despite the fact that
DHC by itself performs poorly on each of these problems
Benefits of DHC are mainly due to variance reduction
Reduced variance makes model building easier
Also makes decision making easier
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

23. Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5
Model quality
Without DHC, the perfect model for trap-5 requires
connections between each variable within the partition, 10
links in total
With DHC, each partition contains either 11111 or 00000, so
fewer dependencies (4) are required
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

24. Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5 (200 bits)
Model quality: Proportion of correct dependencies
Without DHC, much larger populations are required to obtain
the 10 necessary dependencies
With DHC, smaller populations are required to find the 4
necessary dependencies
First generation Middle of the run
proportion of correct dependencies
proportion of correct dependencies
1 1
0.9 No DHC 0.9 No DHC, 10th generation
0.8 With DHC 0.8 With DHC, 5th generation
0.7 0.7
0.6 0.6
0.5 0.5
0.4 0.4
0.3 0.3
0.2 0.2
0.1 0.1
0 0
50
10
20
40
80
16
32 0
64 0
12 0
25 00
51 00
10 00
20 40
40 80
50
10
20
40
80
16
32 0
64 0
12 0
25 00
51 00
10 00
20 40
0
0
0
0
0
0
0
8
6
2
2
4 0
96 0
0
0
0
0
0
0
0
8
6
2
2
48 0
00
00
Population Size Population Size
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

25. Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5 (200 bits)
Model quality: proportion of incorrect dependencies
Without DHC, very many incorrect (spurious) dependencies
are discovered for all but very large populations
With DHC, very few spurious dependencies are discovered,
except for very large populations
First generation Middle of the run
proportion of incorrect dependencies
proportion of incorrect dependencies
0.035 0.04
No DHC 0.035 With DHC, 5th generation
No DHC, 10th
0.03 With DHC generation
0.025 0.03
0.025
0.02
0.02
0.015
0.015
0.01 0.01
0.005 0.005
0 0
50
10
200
400
800
160
3200
6400
1200
25800
51600
10200
20240
40480
50
10
20
40
80
16
32 0
64 0
12 0
25 00
51 00
10 00
20 40
96 0
0
0
0
0
0
0
0
8
6
2
2
48 0
00
00
0
Population Size Population Size
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

26. Summary & Conclusions
DHC’s Effects on hBOA
DHC significantly improves the performance of hBOA on each
of the test problems
In these experiments the maximum improvement was attained
by allowing DHC to operate on the full population and run
until it found the local optimum
DHC helps by reducing the variance
Model building is easier
Decision making is also easier
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA

27. Acknowledgments
Acknowledgments
NSF; NSF CAREER grant ECS-0547013.
U.S. Air Force, AFOSR; FA9550-06-1-0096.
University of Missouri; High Performance Computing
Collaboratory sponsored by Information Technology Services;
Research Award; Research Board.
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA