using previous models to bias structural learning in the hierarchical boa
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Motivation Outline hBOA Problems Biasing Experiments Conclusions
Using Previous Models to Bias StructuralLearning in the Hierarchical BOA
M. Hauschild1 M. Pelikan1 K. Sastry2 D.E. Goldberg2
1Missouri Estimation of Distribution Algorithms Laboratory (MEDAL)Department of Mathematics and Computer Science
University of Missouri - St. Louis
2Illinois Genetic Algorithms Laboratory (ILLiGAL)Department of Industrial and Enterprise Systems Engineering
University of Illinois at Urbana-Champaign
Genetic and Evolutionary Computation Conference, 2008
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems 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, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Motivation
Each run of hBOA leaves us with a tremendous amount ofinformation
The algorithm decomposes the problem for usThis information we get “for free”
Use this knowledge to bias our model-building on similarproblems
Exploit this free informationFocus our model-based search to promising regions
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Outline
hBOA
Test Problems
Biasing Model-BuildingExperiments
Probability Coincidence Matrix (PCM)Distance-based bias
Conclusions
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems 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, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems 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, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Test Problems
Test ProblemsConcatenated Trap of Order 52D Ising Spin GlassMAXSAT
ExperimentsBiasing using PCMBiasing using Distance metric
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Trap-5
Partition binary string into disjoint groups of 5 bits
trap5(ones) =
{
5 if ones = 54 − ones otherwise
, (1)
Total fitness is sum of single traps
Optimum: String 1111...1
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
2D Ising Spin Glass
Origin in physics
Spins arranged on a 2D grid
Each spin sj can have two values: +1 or -1
Each connection i , j has a weight Jij . Set of weightsspecifies one instance.
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
2D Ising Spin Glass
Energy is given by...
E(C) =∑
〈i ,j〉
siJi ,jsj , (2)
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 solves it in polynomial timeA deterministic hill-climber(DHC) is used to improve thequality of evaluated solutions
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
MAXSAT
Find the maximum satisfiable clauses in a propositionallogic formulaUsually expressed in k-CNF form
Logical and of clausesLogical or of literalsClauses of at most length k
Example of a 3-CNF with 4 propositions
(X4 ∨ X3) ∧ (X1 ∨ ¬X2) ∧ (¬X4 ∨ X2 ∨ X3) (3)
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
MAXSAT
Can be mapped to many other problems
NP-Complete when k > 1Not much structure in general case
Combined graph-coloringCombined depending on parameter pRegular ring lattice (1-p edges)Random graph (p edges)
Test under various amounts of structure
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Biasing Model Building
Exploit information from previous runs to simplify MBTwo approaches
Bias MB using a Probability Coincidence Matrix (PCM)Directly gathered from previous runs
Bias MB using a distance thresholdPrior knowledge of the problem
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Biasing using PCM
When hBOA solves a problem, left with a series of modelsCompute a PCM from these models
Pij is probability of an edge between i and jIf Pij=.25, 25% of models contain an edge between i and j
Do not consider when an edge occursSome runs more strongly represented
Good and bad
We can now use PCM to restrictPij < pmin
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Biasing using PCM
Trap-5 10x10 Ising spin glass
node i
node
j
10 20 30 40 50
10
20
30
40
50 0
0.5
1
2 4 6 8 10 12
2468
1012
node i
node
j
20 40 60 80 100
20
40
60
80
100 0
0.2
0.4
0.6
2 4 6 8 10 12
2468
1012
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Biasing using distance metric
On other problems, define a distance metricVector Q such that Qi is proportion of dependencies at idistance or shorterQmax = 1Corresponds to strength of dependency between variablesMAXSAT, MVC, IsingCan now use Q to restrict
Qd < qmin
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Biasing Model Building
Benefits (of good bias)Model-building becomes faster
Less dependencies to examine
Increase effectiveness of searchLess spurious dependencies
Negatives (of poor bias)Slower convergenceLarger populationFailure to solve the problem
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
PCM on Ising Spin Glass
Need to learn PCM from sample
Show for some pmin, the effects
100 instances of 4 different sizesCross-validation
PCM built from 90 instances, used to solve remaining 10Repeated 10 times
Varied pmin from 0 to maximum value based oncomputational time
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
PCM on Ising Spin Glass
20x20
0 5 100
1
2
3
4
5
Minimum edge percentage allowed
Exe
cutio
n T
ime
Spe
edup
28x28
0 0.5 1 1.5 20
1
2
3
4
5
Minimum edge percentage allowed
Exe
cutio
n T
ime
Spe
edup
24x24
0 1 2 30
1
2
3
4
5
Minimum edge percentage allowed
Exe
cutio
n T
ime
Spe
edup
32x32
0 0.5 1 1.5 20
1
2
3
4
5
Minimum edge percentage allowed
Exe
cutio
n T
ime
Spe
edup
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
PCM on Ising Spin Glass
20x20
0 5 100
5
10
15
20
25
30
35
Minimum edge percent allowed
Red
uctio
n F
acto
r
28x28
0 0.5 1 1.5 20
5
10
15
20
25
30
35
Minimum edge percent allowed
Red
uctio
n F
acto
r
24x24
0 1 2 30
5
10
15
20
25
30
35
Minimum edge percent allowed
Red
uctio
n F
acto
r
32x32
0 0.5 1 1.5 20
5
10
15
20
25
30
35
Minimum edge percent allowed
Red
uctio
n F
acto
r
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on Ising Spin Glass
100 instances of 4 different sizesRestrict dependencies by maximum distance allowed
From 1 to half the maximumOmitted results for too severe restrictions
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on Ising Spin Glass
16x16
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
← 1
← 2 ← 3
← 4
← 5
← 6← 7
← 8
16 →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
24x24
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
← 2
← 3
← 4← 5
← 6← 7
← 8
← 9← 10
← 11← 12
24 →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
20x20
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
← 1
← 2
← 3
← 4← 5
← 6
← 7← 8
← 9← 10
20 →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
28x28
0.2 0.4 0.6 0.8 10
1
2
3
4
5
6
← 2
← 3
← 4← 5← 6
← 7← 8
← 9← 10← 11
← 12← 13
← 14
28 →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on Ising Spin Glass
16x16
0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
35
← 1
← 2
← 3
← 4
← 5← 6← 7← 8 ← 16
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
24x24
0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
35
← 2
← 3
← 4
← 5
← 6← 7
← 8← 9← 10← 11← 12 ← 24
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
20x20
0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
35
← 1
← 2
← 3
← 4
← 5← 6
← 7← 8← 9← 10 ← 20
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
28x28
0.2 0.4 0.6 0.8 10
5
10
15
20
25
30
35
← 2
← 3
← 4
← 5
← 6
← 7← 8
← 9← 10← 11← 12← 13← 14← 28
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on MAXSAT
Must define a distance metricCreate a graph for each MAXSAT instance
Propositions in same clause are connectedDistance between any other propositions is shortest path inthis graph
Combined graph-coloring, 3 levels of structurep = 1, 2−4, 2−8
100 instances of each type
500 propositions, 3100 clauses
Distances from 1 to the maximum
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on MAXSAT
p=1
0.98 0.99 10
0.5
1
1.5
2
4 →
5 →All →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
p=2−4
0.7 0.8 0.9 10
0.5
1
1.5
2
2.5
3
← 3
← 4
← 5
All →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
p=2−8
0.4 0.6 0.8 10
0.5
1
1.5
2
2.5
3
← 3← 4
← 5
← 6← 7
All →
Original Ratio of Total Dependencies
Exe
cutio
n T
ime
Spe
edup
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Distance Bias on MAXSAT
p=1
0.98 0.99 10
0.5
1
1.5
24 →
5 →All →
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
p=2−4
0.7 0.8 0.9 10
2
4
6
8
10 ← 3
← 4
← 5
All →
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
p=2−8
0.4 0.6 0.8 10
2
4
6
8
10
← 3
← 4
← 5
← 6← 7
All →
Original Ratio of Total Dependencies
Red
uctio
n F
acto
r
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Conclusions
We took some first steps in exploiting previous runs tospeed up new runsLeads to speedups from 4-5 on Ising spin glasses to1.5-2.5 on MAXSATCan be extended to many other problemsNext step is to try and automate processEfficiency enhancements work together
Parallelization 50Hybridization 2Learning from past runs 2Evaluation Relaxation 1.1Total 220
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
Motivation Outline hBOA Problems Biasing Experiments Conclusions
Any Questions?
M. Hauschild, M. Pelikan, K. Sastry and D. E. Goldberg Universities of Missouri - St. Louis and Illinois at Urbana-Champaign
Using Previous Models to Bias hBOA
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