24-02-2006Siddhartha Shakya1 Estimation Of Distribution Algorithm based on Markov Random Fields Siddhartha Shakya School Of Computing The Robert Gordon

24-02-2006 Siddhartha Shakya 1

Estimation Of Distribution Algorithm based on Markov

Random Fields

Siddhartha ShakyaSchool Of Computing

The Robert Gordon University

http://www.rgu.ac.uk/default.cfm


Outline• From GAs to EDAs

• Probabilistic Graphical Models in EDAs– Bayesian networks

– Markov Random Fields

• Fitness modelling approach to estimating and sampling MRF in EDA– Gibbs distribution, energy function and modelling the

fitness

– Estimating parameters (Fitness modelling approach)

– Sampling MRF (several different approaches)

• Conclusion


Genetic Algorithms (GAs)• Population based optimisation technique

• Based on Darwin's theory of Evolution

• A solution is encoded as a set of symbols known as chromosome

• A population of solution is generated

• Genetic operators are then applied to the population to get next generation that replaces the parent population


Simple GA simulation


GA to EDA


Simple EDA simulation

n

iixpxp

1

)()(

0 1 1 1 1

1 0 1 0 1

0 0 1 0 1

0 1 0 0 0

0 1 1 1 1

1 0 1 0 1

0 0 1 0 1

0 1 0 0 0

0.5 0.5 1.0 1.00.5

0 0 1 1 1

1 1 1 0 1

1 0 1 0 1

0 1 1 1 1

)1( 5 xp


Joint Probability Distribution (JPD)• Solution as a set of

random variables

• Joint probability Distribution (JPD)

• Exponential to the number of variables, therefore not feasible to calculate in most cases

• Needs Simplification!!

},...,,{ 11 nxxxx

),...,,()( 11 nxxxpxp

}1,0{)

arg(2

i

n

xforiesprobabilit

inalmparameters


Factorisation of JPD

• Univariate model: No interaction: Simplest model

• Bivariate model: Pair-wise interaction

• Multivariate Model: interaction of more than two variables

xxxpxp i

n

ii

:)()(1

xxxxxpxp ji

n

iji

,:)|()(1

xxxxxxpxp Ai

n

iAi

,:)|()(1


Typical estimation and sampling of JPD in EDAs

• Learn the interaction between variables in the solution

• Learn the probabilities associated with interacting variables

• This specifies the JPD: p(x)

• Sample the JPD (i.e. learned probabilities)


Probabilistic Graphical Models• Efficient tool to represent the factorisation of

JPD

• Marriage between probability theory and Graph theory

• Consist of Two components

– Structure

– Parameters

• Two types of PGM

– Directed PGM (Bayesian Networks)

– Undirected PGM (Markov Random Field)


Directed PGM (Bayesian networks)• Structure:

Directed Acyclic Graph (DAG)

• Independence relationship:

A variable is conditionally independent of rest of the variables given its parents

• Parameters:

Conditional probabilities

X1 X2

X3

X4 X5

)1|1(),0|1(

),1|0(),0|0(

)1|1(),0|1(

),1|0(),0|0(

)1,1|1(),1,0|1(

),0,1|1(),0,0|1(

),1,1|0(),1,0|0(

),0,1|0(),0,0|0(

)1(),0(

)1(),0(

3535

3535

3434

3434

213213

213213

213213

213213

22

11

xxpxxp

xxpxxp

xxpxxP

xxpxxP

xxxpxxxp

xxxpxxxp

xxxpxxxp

xxxpxxxp

xpxp

xpxp


Bayesian networks• The factorisation of JPD

encoded in terms of conditional probabilities is

• JPD for BN

X1 X2

X3

X4 X5

)1|1(),0|1(

),1|0(),0|0(

)1|1(),0|1(

),1|0(),0|0(

)1,1|1(),1,0|1(

),0,1|1(),0,0|1(

),1,1|0(),1,0|0(

),0,1|0(),0,0|0(

)1(),0(

)1(),0(

3535

3535

3434

3434

213213

213213

213213

213213

22

11

xxpxxp

xxpxxp

xxpxxP

xxpxxP

xxxpxxxp

xxxpxxxp

xxxpxxxp

xxxpxxxp

xpxp

xpxp

n

iiixpxp

1

)|()(

)|()|(),|()()()( 353421321 xxpxxpxxxpxpxpxp


Estimating a Bayesian network• Estimate structure

• Estimate parameters

• This completely specifies the JPD

• JPD can then be Sampled

)1|1(),0|1(

),1|0(),0|0(

)1|1(),0|1(

),1|0(),0|0(

)1,1|1(),1,0|1(

),0,1|1(),0,0|1(

),1,1|0(),1,0|0(

),0,1|0(),0,0|0(

)1(),0(

)1(),0(

3535

3535

3434

3434

213213

213213

213213

213213

22

11

xxpxxp

xxpxxp

xxpxxP

xxpxxP

xxxpxxxp

xxxpxxxp

xxxpxxxp

xxxpxxxp

xpxp

xpxp


X1 X2

X3

X4 X5


BN based EDAs1. Initialise parent solutions

2. Select a set from parent solutions

3. Estimate a BN from selected set

a. Estimate structure

b. Estimate parameters

4. Sample BN to generate new population

5. Replace parents with new set and go to 2 until termination criteria satisfies


How to estimate and sample BN in EDAs

• Estimating structure

– Score + Search techniques

– Conditional independence test

• Estimating parameters

– Trivial in EDAs: Dataset is complete

– Estimate probabilities of parents before child

• Sampling

– Probabilistic Logical Sampling (Sample parents before child)

X1 X2

X3

X4 X5



BN based EDAs• Well established approach in EDAs

BOA, EBNA, LFDA, MIMIC, COMIT, BMDA

References

– Larrañiaga and Lozano 2002

– Pelikan 2002


Markov Random Fields (MRF)• Structure:

Undirected Graph

• Local independence:

A variable is conditionally independent of rest of the variables given its neighbours

• Global Independence:

Two sets of variables are conditionally independent to each other if there is a third set that separates them.

• Parameters:

potential functions defined on the cliques

X1

X3X2

X4 X6X5

),(

),(

),,(

),,(

634

523

4322

3211

xx

xx

xxx

xxx


Markov Random Field• The factorisation of JPD

encoded in terms of potential function over maximal cliques is

• JPD for MRF

),(

),(

),,(

),,(

634

523

4322

3211

xx

xx

xxx

xxx

X1

X3X2

X4 X6X5

functionpartitionais

xxxxxxxxxxZwhere

xxxxxxxxxxZ

xp

x

),(),(),,(),,(,

),(),(),,(),,(1

)(

6352432321

6352432321

m

iic

Zxp

1

)(1

)(


Estimating a Markov Random field• Estimate structure from

data

• Estimate parameters

– Requires potential functions to be numerically defined

• This completely specifies the JPD

• JPD can then be Sampled

– No specific order (not a DAG) so a bit problematic

X1

X3X2

X4 X6X5

),(

),(

),,(

),,(

634

523

4322

3211

xx

xx

xxx

xxx

x

xxxxxxxxxxZwhere

xxxxxxxxxxZ

xp

),(),(),,(),,(,

),(),(),,(),,(1

)(

6352432321

6352432321


MRF in EDA• Has recently been proposed as a

estimation of distribution technique in EDA

• Shakya et al 2004, 2005

• Santana et el 2003, 2005


MRF based EDA1. Initialise parent solutions

2. Select a set from parent solutions

3. Estimate a MRF from selected set

a. Estimate structure

b. Estimate parameters

4. Sample MRF to generate new population

5. Replace parent with new solutions and go to 2 until termination criteria satisfies


How to estimate and sample MRF in EDA

• Learning Structure– Conditional Independence test (MN-EDA, MN-FDA)

– Linkage detection algorithm (LDFA)

• Learning Parameter– Junction tree approach (FDA)

– Junction graph approach (MN-FDA)

– Kikuchi approximation approach (MN-EDA)

– Fitness modelling approach (DEUM)

• Sampling– Probabilistic Logic Sampling (FDA, MN-FDA)

– Probability vector approach (DEUMpv)

– Direct sampling of Gibbs distribution (DEUMd)

– Metropolis sampler (Is-DEUMm)

– Gibbs Sampler (Is-DEUMg, MN-EDA)


Fitness modelling approach• Hamersley Clifford theorem: JPD for

any MRF follows Gibbs distribution

• Energy of Gibbs distribution in terms of potential functions over the cliques

• Assuming probability of solution is proportional to its fitness:

• From (a) and (b) a Model of fitness function - MRF fitness model (MFM) – is derived

)()(/)(

aZ

exp

TxU

m

iicuxU

1

)()(

)()(

)( bZ

xfxp

)())(ln( xUxf

m

iicuxf

1

)())(ln(

,or


MRF fitness Model (MFM)

• Properties:

– Completely specifies the JPD for MRF

– Negative relationship between fitness and Energy i.e. Minimising energy = maximise fitness

• Task:

– Need to find the structure for MRF

– Need to numerically define clique potential function

m

iicuxUxf

1

)()())(ln(


MRF Fitness Model (MFM)• Let us start with simplest model:

univariate model – this eliminates structure learning :)

• For univariate model there will be n singleton clique

• For each singleton clique assign a potential function

• Corresponding MFM

• In terms of Gibbs distribution

iiii xxu )(

X1

X3X2

X4 X6X5

)(

)(

)(

)(

)(

)(

66

55

44

33

22

11

xu

xu

xu

xu

xu

xu

nn xxxxUxf ..)())(ln( 2211

Z

exp

Txm

iii

1

)(


Estimating MRF parameters using MFM

• Each chromosome gives us a linear equation

• Applying it to a set of selected solution gives us a system of linear equations

• Solving it will give us the approximation to the MRF parameters

• Knowing MRF parameters completely specifies JPD

• Next step is to sample the JPD

nn xxxxf ..))(ln( 2211


General DEUM frameworkDistribution Estimation Using MRF algorithm

(DEUM)

1. Initialise parent population P

2. Select set D from P (can use D=P !!)

3. Build a MFM and fit to D to estimate MRF parameters

4. Sample MRF to generate new population

5. Replace P with new population and go to 2 until termination criterion satisfies


How to sample MRF• Probability vector approach

• Direct Sampling of Gibbs Distribution

• Metropolis sampling

• Gibbs sampling


Probability vector approach to sample MRF

• Minimise U(x) to maximise f(x)

• To minimise U(x) Each αixi should be minimum

• This suggests: if αi is negative then corresponding xi

should be positive

• We could get an optimum chromosome for the current population just by looking on α

• However not always the current population contains enough information to generate optimum

• We look on sign of each αi to update a vector of probability

nn xxxxUxf ..)())(ln( 2211


DEUM with probability vector (DEUMpv)


Updating Rule

• Uses sign of a MRF parameter to direct search towards favouring value of respective variable that minimises energy U(x)

• Learning rate controls convergence

);1(0

;)1(0

1

iii

iii

ppthenif

ppthenif

dontoiFor


Simulation of DEUMpv

0 1 1 1 1

1 0 1 0 1

0 0 1 0 1

0 1 0 0 0

0.4 0.6 0.6 0.60.6

0.5 0.5 0.5 0.50.5

4

3

2

1

0 1 1 1 1

1 0 1 0 1

4

3

0 0 1 0 1

0 1 0 0 0

2

1

4.154321

1.154321

0.05 -0.05 -0.625 -0.625-0.05


Results• OneMax Problem


Results• F6 function optimisation


Results• Trap 5 function

• Deceptive problem

• No solution found


Sampling MRF• Probability vector approach

• Direct sampling of Gibbs distribution


• Gibbs sampling


Direct Sampling of Gibbs distribution

• In the probability vector approach, only the sign of MRF parameters has been used

• However, one could directly sample from the Gibbs distribution and make use of the values of MRF parameters

• Also could use the temperature coefficient to manipulate the probabilities



nn

TxU

xxxxUZ

exp

..)()( 2211

)(

TxxUTxxU

TxxU

x

ii ii

i

i

ee

e

xp

xxpxp /)1)((/)1)((

/)1)((

}1,1{

)(

1)()1(

Ti iexp /21

1)1( Ti ie

xp /21

1)1(



• The temperature coefficient has an important role

• Decreasing T will cool probability to either 1 or 0 depending upon sign and value of alpha

• This forms the basis for the DEUM based on direct sampling of Gibbs distribution (DEUMd)

Ti iexp /21

1)1( Ti ie

xp /21

1)1(


DEUM with direct sampling (DEUMd)

1. Generate initial population, P, of size M

2. Select the N fittest solutions, N ≤ M

3. Calculate MRF parameters

4. Generate M new solutions by sampling univariate distribution

5. Replace P by new population and go to 2 until complete

gTe

xpZ

exp

i

m

iii

i

Tx

2

,1

1)1()(

1


DEUMd simulation

4.154321 0 1 1 1 1

1 0 1 0 1

0 0 1 0 1

0 1 0 0 0

4

3

2

1

0 1 1 1 1

1 0 1 0 1

4

3

0 0 1 0 1

0 1 0 0 0

2

1

1.154321

0.05 -0.05 -0.625 -0.625-0.05

iexp i

1

1)1(

0.4 0.6 0.6 0.60.6

0 1 1 1 1

1 0 1 1 1

0 1 1 0 1

0 1 0 1 0

4

4

3

2


Experimental results• OneMax Problem


F6 function optimization


Plateau Problem (n=180)


Checker Board Problem (n=100)


Trap function of order 5 (n=60)


Experimental resultsGA UMDA PBIL DEUMd

Checker Board

Fitness 254.68 ±

(4.39)

233.79 ±

(9.2)

243.5 ±

(8.7)

254.1 ±

(5.17)

Evaluation 427702.2 ±

(1098959.3)

50228.2 ±

(9127)

191476.8 ±

(37866.65)

33994 ±

(13966.75)

Equal-Products

Fitness 211.59 ±

(1058.47)

5.03 ±

(18.29)

9.35 ±

(43.36)

2.14 ±

(6.56)

Evaluation 1000000 ±

(0)

1000000 ±

(0)

1000000 ±

(0)

1000000 ±

(0)

Colville Fitness 0.61 ±

(1.02)

40.62 ±

(102.26)

2.69 ±

(2.54)

0.61 ±

(0.77)

Evaluation 1000000 ±

(0)

62914.56 ±

(6394.58)

1000000 ±

(0)

1000000 ±

(0)

Six Peaks

Fitness 99.1 ±

(9)

98.58 ±

(3.37)

99.81 ±

(1.06)

100 ±

(0)

Evaluation 49506 ±

(4940)

121333.76 ±

(14313.44)

58210 ±

(3659.15)

26539 ±

(1096.45)


Analysis of Results• For Univariate problems (OneMax), given population size of 1.5n,

P=D and T->0, solution was found in single generation

• For problems with low order dependency between variables (Plateau and CheckerBoard), performance was significantly better than that of other Univariate EDAs.

• For the deceptive problems with higher order dependency (Trap function and Six peaks) DEUMd was deceived but by slowing the cooling rate, it was able to find solution for Trap of order 5.

• For the problems where optimum was not known the performance was comparable to that of GA and other EDAs and was better in some cases.


Cost- Benefit Analysis (the cost)• Polynomial cost of estimating the distribution

compared to linear cost of other univariate EDAs

nNnNO

nNNnO

nNnO

)(

)(

)(

2

2

3

• Cost to compute univariate marginal frequency:

)(nNO

• Cost to compute SVD


Cost- Benefit Analysis (the benefit)

• DEUMd can significantly reduce the number of fitness evaluations

• Quality of solution was better for DEUMd than other compared EDAs

• DEUMd should be tried on problems where the increased solution quality outweigh computational cost.





• Gibbs sampling


Example problem: 2D Ising Spin Glass

Given coupling constant J find the value of each spins that minimises H

MRF fitness model


Metropolis Sampler


Difference in Energy


DEUM with Metropolis sampler


Results





• Gibbs sampling


Conditionals from Gibbs distribution

For 2D Ising spin glass problem:


Gibbs Sampler


DEUM with Gibbs sampler


Results


Summary• From GA to EDA

• PGM approach to modelling and sampling distribution in EDA

• DEUM: MRF approach to modelling and sampling

• Learn Structure: No structure learning so far (Fixed models are used)

• Learn Parameter: Fitness modelling approach

• Sample MRF: – Probability vector approach to sample

– Direct sampling of Gibbs distribution

– Metropolis sampler

– Gibbs Sampler

• Results are encouraging and lot more to explore

Documents

24-02-2006Siddhartha Shakya1 Estimation Of Distribution Algorithm based on Markov Random Fields Siddhartha Shakya School Of Computing The Robert Gordon