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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.
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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, MOhttp://medal.cs.umsl.edu/
err26c@umsl.edu pelikan@cs.umsl.edu
Illinois Genetic Algorithms Laboratory
University of Illinois at Urbana-Champaign, Urbana, ILhttp://www-illigal.ge.uiuc.edu/ deg@uiuc.edu
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Background & Motivation
Hybrid Estimation of Distribution Algorithms (EDAs)
I For very large or complex problems, scalability may not besufficient; additional efficiency enhancement may be required
I Hybridization is a popular method of efficiency enhancementfor EDAs and other GEAs
I A hybrid combines a local search algorithm with a globalsearcher such as an EDA
Hybrid hBOA
I The hierarchical Bayesian optimization algorithm (hBOA) isfrequently hybridized
I This study focuses on hBOA combined with a deterministichill climber (DHC)
I Examine the effects of varying the parameters of DHCI 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
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
Algorithms
Hierarchical Bayesian Optimization Algorithm (hBOA)
I Randomly initialize a population of binary strings
I Select promising solutions from the population
I Build a Bayesian network from the selected solutions
I Generate new solutions from the constructed model
I Incorporate new solutions into the population using restrictedtournament replacement
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Algorithms
Deterministic Hill Climber (DHC)
I Flip one bit in the string that will produce the most fitnessimprovement
I Repeat until a specified quality has been reached, a maximumnumber of flips has been performed, or no more improvementis possible
I hBOA has frequently been combined with DHC to improveperformance (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
Goals & Methodology
Experiment Goals
I Examine the effects of DHC on performance of hBOA
I Determine optimal settings for DHC
Methodology
I Tune the frequency of local search, i.e. the proportion ofstrings in the population on which DHC operates
I Tune the duration of local search, i.e. the maximum numberof flips per string
I CompareI Population sizeI Number of generations until optimum
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
Test Problems
I Trap-5
I 2-D Ising Spin Glass
I 3-CNF MAXSAT
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
Trap-5
I Concatenation of non-overlapping subproblems of 5 bits
I The fitness of each subproblem is determined as
ftrap5(u) =
{
5 if u = 5,
4 − u otherwise,
where u is the number of ones in the subproblem
I All subproblems are added together to determine the string’sfitness
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
Trap-5
I PropertiesI Separable, non-overlapping subproblemsI Subproblems are fully deceptiveI Difficult for local search, simple GA
I ParametersI Problem sizes of 100-350 bitsI Population size determined empirically using bisection method
I 10 independent bisections of 10 runs each
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
2-D Ising Spin Glass
I Spins are arranged on a 2-D grid
I Each spin si gets a value from +1,−1
I Each edge between neighboring spins si and sj is assigned areal value Ji,j
I The task is to find a configuration of spins to minimize theenergy, specified as
H(s) = −
n∑
i,j=0
Jijsisj
I Spins are represented as bits in the binary string, 1 for +1, 0for −1
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
2-D Ising Spin Glass
I PropertiesI More complex fitness landscape than trapI Not fully decomposable into subproblems of bounded order,
substructures overlapI Solvable in polynomial time
I ParametersI Problem sizes 8x8, 10x10, 12x12, 14x14, 16x16 (64 - 256 bits)I 1000 instances
I For each instance, Jij randomly set to +1 or −1
I Population size determined using bisection method, 10successful runs per instance
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
MAXSAT
I Instances consist of 3-CNF formulas: conjunctions ofdisjunctions of 3 literals
I The task is to find an assignment of Boolean variables tosatisfy the maximum number of clauses
I Bits in the string represent the variables, with 1 representingtrue and 0 representing false
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Test Problems
MAXSAT
I PropertiesI Complex fitness landscapeI Not fully decomposable into subproblems of bounded order,
substructures overlapI Not solvable in polynomial time
I ParametersI Problem sizes of 50 and 75 variablesI 100 randomly-generated instancesI 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
Experimental Results
Trap-5 (350 bits)
I Reduction factor, speed-up: ratio of values without DHC tothose with DHC
0
10
20
30
40
50
60
0 0.2 0.4 0.6 0.8 1
Pop
ulat
ion
Siz
e R
educ
tion
Fac
tor
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
0 0.2 0.4 0.6 0.8 1G
ener
atio
ns S
peed
-Up
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
Trap-5 (350 bits)
0
10
20
30
40
50
60
70
80
90
100
0 0.2 0.4 0.6 0.8 1
Eva
luat
ions
Spe
ed-U
p
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91
0
1
2
3
4
5
6
7
0 0.2 0.4 0.6 0.8 1
Flip
s pe
r E
valu
atio
nMax. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
Trap-5
100
1000
10000
100000
350 300 250 200 150 100
Pop
ulat
ion
Siz
e
Problem size (# of bits)
No DHCWith DHC
1
10
100
350 300 250 200 150 100G
ener
atio
nsProblem size (# of bits)
No DHCWith DHC
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
Ising Spin Glass (16x16, 256 bits)
0
2
4
6
8
10
12
14
16
0 0.2 0.4 0.6 0.8 1
Pop
ulat
ion
Red
uctio
n F
acto
r
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
1
1.5
2
2.5
3
3.5
4
4.5
0 0.2 0.4 0.6 0.8 1
Gen
erat
ion
Spe
ed-u
pMax. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
Ising Spin Glass (16x16, 256 bits)
0
10
20
30
40
50
60
70
0 0.2 0.4 0.6 0.8 1
Eva
luat
ion
Spe
edup
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
0
1
2
3
4
5
6
7
8
9
0 0.2 0.4 0.6 0.8 1
Flip
s pe
r E
valu
atio
nMax. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
Ising Spin Glass
10
100
1000
10000
256 196 144 100 64
Pop
ulat
ion
Siz
e
Problem size (# of bits)
No DHCWith DHC
1
10
100
256 196 144 100 64G
ener
atio
nsProblem size (# of bits)
No DHCWith DHC
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
MAXSAT (75 bits)
0
20
40
60
80
100
120
0 0.2 0.4 0.6 0.8 1
Pop
ulat
ion
Red
uctio
n F
acto
r (o
rigin
al/r
educ
ed)
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
0.6 0.8
1 1.2 1.4 1.6 1.8
2 2.2 2.4 2.6 2.8
0 0.2 0.4 0.6 0.8 1
Gen
erat
ion
Spe
ed-u
pMax. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Experimental Results
MAXSAT (75 bits)
0
50
100
150
200
250
300
0 0.2 0.4 0.6 0.8 1
Eva
luat
ion
spee
dups
Max. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
0
1
2
3
4
5
6
0 0.2 0.4 0.6 0.8 1
Flip
s pe
r E
valu
atio
nMax. Flips (% of Prob. Size)
pdhc0.10.20.30.4
0.50.60.70.8
0.91.0
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Discussion of Results
Effects of DHC
I DHC improved performance on each test problem, for all sizes
I Substantial benefits were obtained despite the fact that
DHC by itself performs poorly on each of these problems
I Benefits of DHC are mainly due to variance reductionI Reduced variance makes model building easierI Also makes decision making easier
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5
I Model qualityI Without DHC, the perfect model for trap-5 requires
connections between each variable within the partition, 10links in total
I With DHC, each partition contains either 11111 or 00000, sofewer dependencies (4) are required
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5 (200 bits)
I Model quality: Proportion of correct dependenciesI Without DHC, much larger populations are required to obtain
the 10 necessary dependenciesI With DHC, smaller populations are required to find the 4
necessary dependencies
First generation Middle of the run
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
50 100
200 400
800 1600
3200 6400
12800
25600
51200
102400
204800
409600
prop
ortio
n of
cor
rect
dep
ende
ncie
s
Population Size
No DHCWith DHC
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
1
50 100
200 400
800 1600
3200 6400
12800
25600
51200
102400
204800
prop
ortio
n of
cor
rect
dep
ende
ncie
s
Population Size
No DHC, 10th generationWith DHC, 5th generation
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Discussion of Results
Effects of DHC on Model Building in hBOA: Trap-5 (200 bits)
I Model quality: proportion of incorrect dependenciesI Without DHC, very many incorrect (spurious) dependencies
are discovered for all but very large populationsI With DHC, very few spurious dependencies are discovered,
except for very large populations
First generation Middle of the run
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
50 100
200 400
800 1600
3200 6400
12800
25600
51200
102400
204800
409600
prop
ortio
n of
inco
rrec
t dep
ende
ncie
s
Population Size
No DHCWith DHC
0
0.005
0.01
0.015
0.02
0.025
0.03
0.035
0.04
50 100
200 400
800 1600
3200 6400
12800
25600
51200
102400
204800
prop
ortio
n of
inco
rrec
t dep
ende
ncie
s
Population Size
No DHC, 10th generationWith DHC, 5th generation
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Summary & Conclusions
DHC’s Effects on hBOA
I DHC significantly improves the performance of hBOA on eachof the test problems
I In these experiments the maximum improvement was attainedby allowing DHC to operate on the full population and rununtil it found the local optimum
I DHC helps by reducing the varianceI Model building is easierI Decision making is also easier
Elizabeth Radetic, Martin Pelikan, and David E. Goldberg Effects of a Deterministic Hill Climber on hBOA
Acknowledgments
Acknowledgments
I NSF; NSF CAREER grant ECS-0547013.
I U.S. Air Force, AFOSR; FA9550-06-1-0096.
I University of Missouri; High Performance ComputingCollaboratory 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
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