Intelligent Bias of Network Structures in the Hierarchical BOA

Motivation Outline hBOA Biasing Experiments Conclusions

Intelligent Bias of Network Structures in theHierarchical BOA

M. Hauschild1 M. Pelikan1

1Missouri 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


Motivation

In optimization, always looking to solve harder problemshBOA can solve a broad class of problems robustly andfast

Scalability isn’t always enough

Much work has been done in speeding up hBOASporadic Model-BuildingParallelizationOthers




Motivation

Each run of an EDA leaves us with a tremendous amountof information

The algorithm decomposes the problem for usLeft with a series of models

Methods have been developed to exploit this informationRequire hand-inspectionVery sensitive to parameters

Wanted to develop a method that is less sensitive toparameters




Outline

hBOABiasing hBOA

Structural Priors in Bayesian NetworksSplit Probability MatrixSPM-based Bias

Test ProblemsExperiments

Trap-52D Ising Spin Glasses

Conclusions




hierarchical Bayesian Optimization Algorithm (hBOA)

Pelikan, Goldberg, and Cantú-Paz; 2001Uses Bayesian network with local structures to modelsolutions

Acyclic directed GraphString positions are the nodesEdges represent conditional dependenciesWhere there is no edge, implicit independence

Niching to maintain diversity




hBOA

Two ComponentsStructure

Edges determine dependenciesMajority of time spent here

ParametersConditional probabilities depending on parentsExample - p(Accident|Wet Road, Speed)

Network built greedily, one edge at a time

Metric punishes complexity




hBOA




Biasing




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 theproblem size, d is maximum number of splits, and e is themaximum generation

S stores, for each possible pair of decision variables, theconditional probability of a split between them (by gen.)

In our sampling we use a threshold of 90% for e




SPM-Based Bias

No splits One split

node i

node

j

100 200 300 400

100

200

300

400 0

0.1

0.2

0.3

0.4

0.5

2 4 6

1

2

3

4

5

6

node i

node

j

100 200 300 400

100

200

300

400 0

0.2

0.4

0.6

0.8

1

2 4 6

1

2

3

4

5

6




SPM-Based Bias

Want to define our own prior probability

Prior probability of network structure:

p(B|ξ) =

n∏

i=1

p(Ti). (3)

For a particular decision tree Ti , p(Ti) is given by:

p(Ti) =∏

j 6=i

qκ

i ,j ,k(i ,j), (4)

where qi ,j ,k(i ,j) denotes the probability that there are atleast k(i , j) splits on Xj in decision trees for Xi . κ is used totune the effect of prior information.




SPM-Based Bias

Consider evaluation of split on Xj in Ti given k − 1 splits

Gains in log-likelihood after a split without considering priorinformation:

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




Trap-5

Partition binary string into disjoint groups of 5 bits

trap5(ones) =

{

5 if ones = 54 − ones otherwise

, (7)

Total fitness is sum of single traps

Global Optimum: String 1111...1

Local Optimum: 00000 in any partition




2D Ising Spin Glass

Origin in physicsSpins arranged on a 2D gridEach spin sj can have two values: +1 or -1Each connection i , j has a weight Jij . Set of weightsspecifies one instance.Energy is given by...

E(C) =∑

〈i ,j〉

siJi ,jsj , (8)




2D Ising Spin Glass

Problem is to find the values of the spins so energy isminimizedVery hard for most optimization techniques

Extremely large number of local optimaDecomposition of bounded order is insufficientSolvable in polynomial time by analytical techniques

hBOA has been shown emperically to solve it in polynomialtime

A deterministic hill-climber(DHC) is used to improve thequality of evaluated solutions




Experiments on Trap-5

Need to learn SPM from sample

Show effects of SPM using various κ

Problem sizes from n = 50 to n = 175SPM learned from 10 bisection runs of 10 runs each

Used to bias model building in another 10 bisection runsThreshold of 90%

Varied κ from 0.05 to 3




Speedups on Trap-5, κ = 1

Execution Speedup

50 75 100 125 150 1751

2

3

4

5

6

7

Exe

cutio

n T

ime

Spe

edup

Problem Size

Evaluation Speedup

50 75 100 125 150 1751

2

3

4

Eva

luat

ion

Spe

edup

Problem Size

Reduction in Bits Examined

50 75 100 125 150 1750

20

40

60

80

Red

uctio

n F

acto

r

Problem Size




Effects of κ on Trap-5 of n = 100

Execution Time

0.05 1 2 30

5

10

15

Exe

cutio

n T

ime

κ

Evaluations

0.05 1 2 30

5

10

15x 104

Eva

luat

ions

κ

Bits Examined

0.25 1 2 30

5

10x 107

Bits

Exa

min

ed

κ




Experiments on 2D Ising Spin Glass

Need to learn SPM from sample

Show effects of SPM using various κ

100 instances of 3 different sizesCross-validation

SPM learned from 90 instances, used to solve remaining 10Repeated 10 timesThreshold of 90%

Varied κ from 0.05 to 3




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 1.520 × 20 1.42 0.96 1.8424 × 24 1.56 0.98 2.03




Effects of κ on 2D Ising spin glass

16 × 16

0.05 1 2 30

2

4

6

8

Exe

cutio

n T

ime

κ

20 × 20

0.05 1 2 35

10

20

30

40

Exe

cutio

n T

ime

κ

24 × 24

0.05 1 2 30

50

100

150

200

Exe

cutio

n T

ime

κ





16 × 16

0.05 1 2 32000

3000

4000

5000

Eva

luat

ions

κ

20 × 20

0.05 1 2 33000

4000

5000

6000

7000

8000

Eva

luat

ions

κ

24 × 24

0.05 1 2 30.5

1

1.5

2x 104

Eva

luat

ions

κ





16 × 16

0.05 1 2 30.5

1

1.5

2x 108

Bits

Exa

min

ed

κ

20 × 20

0.05 1 2 32

4

6

8x 108

Bits

Exa

min

ed

κ

24 × 24

0.05 1 2 31

2

3

4x 109

Bits

Exa

min

ed

κ





κ that led to maximum speedup

size κ Exec. speedup Eval. Speedup Bits Exam16 × 16 0.75 1.24 0.96 1.6620 × 20 1.25 1.44 0.94 1.8524 × 24 1 1.56 0.98 2.03




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 2DIsing spin glasses

This is only one way

Can be extended to many other problems




Conclusions

Efficiency enhancements work together

Parallelization 50Hybridization 2Soft bias from past runs 1.5Evaluation Relaxation 1.1Total 165




Any Questions?



Intelligent Bias of Network Structures in the Hierarchical BOA

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