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 Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
Outline
hBOABiasing hBOA
Structural Priors in Bayesian NetworksSplit Probability MatrixSPM-based Bias
Test ProblemsExperiments
Trap-52D Ising Spin Glasses
Conclusions
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments 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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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
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
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
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
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
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
Motivation Outline hBOA Biasing Experiments Conclusions
Structural Priors
Bayesian-Dirichlet metric for network B and data set D withprior knowledge ξ is
p(B|D, ξ) =p(B|ξ)p(D|B, ξ)
p(D|ξ)· (1)
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 |)log2N , (2)
where N is the population and∑
i |Li | is the number ofleaves.
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
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
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 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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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.
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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 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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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)
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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 = 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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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 sizesCross-validation
SPM learned from 90 instances, used to solve remaining 10Repeated 10 timesThreshold 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
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 1.520 × 20 1.42 0.96 1.8424 × 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
Motivation Outline hBOA Biasing Experiments Conclusions
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
Motivation Outline hBOA Biasing Experiments Conclusions
Effects of κ on 2D Ising spin glass
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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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 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
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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 2DIsing 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
Motivation Outline hBOA Biasing Experiments Conclusions
Conclusions
Efficiency enhancements work together
Parallelization 50Hybridization 2Soft bias from past runs 1.5Evaluation Relaxation 1.1Total 165
M. Hauschild and M. Pelikan University of Missouri - St. Louis
Intelligent Bias of Network Structures in the Hierarchical BOA
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