modelling the initialisation stage of the alkr representation for discrete domains and gabil...

Modelling the Initialisation Stage of the ALKRRepresentation for Discrete Domains and

GABIL Encoding

María A. Franco, Natalio Krasnogor, Jaume Bacardit

University of Nottingham, UK.ASAP Research Group,

School of Computer Sciencemxf@cs.nott.ac.uk

July 14, 2011

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 1 / 25

Problem definition

BioHEL[Bacardit et al., 2009a] is a Genetic Based MachineLearning (GBML) designed to cope with large scaledatasets[Bacardit et al., 2009b].

I Iterative Rule Learning approachI Attribute List Knowledge Representation (ALKR)I ILAS Windowing schemeI Default ruleI Smart initialisation mechanisms (covering)I GPU-based evaluation process

ProblemThe system obtains good results [Stout et al., 2008], but we do nothave a formal understanding of why, when and how this happens.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 2 / 25

Problem definition

BioHEL[Bacardit et al., 2009a] is a Genetic Based MachineLearning (GBML) designed to cope with large scaledatasets[Bacardit et al., 2009b].

I Iterative Rule Learning approachI Attribute List Knowledge Representation (ALKR)I ILAS Windowing schemeI Default ruleI Smart initialisation mechanisms (covering)I GPU-based evaluation process

ProblemThe system obtains good results [Stout et al., 2008], but we do nothave a formal understanding of why, when and how this happens.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 2 / 25

What is the aim of this work?

The aim of this work is to model the initialisation stage of the BioHELsystem and calculate the probability of having a good initialpopulation. Two conditions should be meet[Goldberg, 2002]:

A good individual exists in an initial population (building blocks)The initial population covers the whole search space

BackgroundThese probabilities are also know as schema and covering bound.This have already being determined for XCS and the ternaryrepresentation {1,0,#} by [Butz, 2006].

ProblemModels need to be adapted for our ALKR+GABIL representation.Moreover, we want to model the impact of the BioHEL mechanismsthat are relevant in initialisation: covering and default rule.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 3 / 25

What is the aim of this work?

The aim of this work is to model the initialisation stage of the BioHELsystem and calculate the probability of having a good initialpopulation. Two conditions should be meet[Goldberg, 2002]:

A good individual exists in an initial population (building blocks)The initial population covers the whole search space

BackgroundThese probabilities are also know as schema and covering bound.This have already being determined for XCS and the ternaryrepresentation {1,0,#} by [Butz, 2006].

ProblemModels need to be adapted for our ALKR+GABIL representation.Moreover, we want to model the impact of the BioHEL mechanismsthat are relevant in initialisation: covering and default rule.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 3 / 25

What is the aim of this work?

The aim of this work is to model the initialisation stage of the BioHELsystem and calculate the probability of having a good initialpopulation. Two conditions should be meet[Goldberg, 2002]:

A good individual exists in an initial population (building blocks)The initial population covers the whole search space

BackgroundThese probabilities are also know as schema and covering bound.This have already being determined for XCS and the ternaryrepresentation {1,0,#} by [Butz, 2006].

ProblemModels need to be adapted for our ALKR+GABIL representation.Moreover, we want to model the impact of the BioHEL mechanismsthat are relevant in initialisation: covering and default rule.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 3 / 25

1 BackgroundGABIL RepresentationAttribute List Knowledge Representation (ALKR)

2 Probabilistic modelsInitial considerationsSchema boundHow does the overlapping affects?Covering bound

3 Generalised model for x-ary attributesSchema and Covering bound

4 Conclusions and Further Work

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 4 / 25

How does GABIL works?

The GABIL representation[Jong and Spears, 1991] is used insideALKR to represent nominal attributes.

ExampleF1 ={A,B,C} F2={O,P} F3={W,Z,X,Y}

F1 F2 F3100 01 1101

F1 is A ∧ F2 is P ∧ (F3 is W ∨ F3 is Z ∨ F3 is Y)

In GABIL, when initialising the attribute values we set the bit to 1 withprobability p and to 0 with probability 1− p

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 5 / 25

How does GABIL works?

The GABIL representation[Jong and Spears, 1991] is used insideALKR to represent nominal attributes.

ExampleF1 ={A,B,C} F2={O,P} F3={W,Z,X,Y}

F1 F2 F3100 01 1101

F1 is A ∧ F2 is P ∧ (F3 is W ∨ F3 is Z ∨ F3 is Y)

In GABIL, when initialising the attribute values we set the bit to 1 withprobability p and to 0 with probability 1− p

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 5 / 25

How does Attribute List Knowledge Representation works?

ALKR Classifier Example

numAtt

predicates

class

whichAtt

3

0

0.70.5

1

0.3

offsetPred 0

How do we select the attributes in the list?

ld =

{1 d <= ExpAttsExpAtts

d d > ExpAtts

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 6 / 25

Initial considerations for the probabilistic models

Mechanisms involved in initialisation

CoveringDefault Rule

⇒We have to consider 4initialisation scenarios

Types of attributesFully mapped attributesPartially mapped attributes.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 7 / 25

Initial considerations for the probabilistic models

Mechanisms involved in initialisation

CoveringDefault Rule

⇒We have to consider 4initialisation scenarios

Types of attributesFully mapped attributesPartially mapped attributes.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 7 / 25

Initial considerations for the probabilistic models

Mechanisms involved in initialisation

CoveringDefault Rule

⇒We have to consider 4initialisation scenarios

Types of attributesFully mapped attributesPartially mapped attributes.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 7 / 25

Schema bound

ProblemWe want to calculate the probability of having good classifiers orrepresentatives in an initial population. Classifiers that do not makemistakes, since they represent correctly all the specified bits in anoriginal problem rule.

ExampleConsidering the rule #10#1 with 3 values specified (k=3), the followingclassifiers are representatives: 110*1, 11011, 010*1.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 8 / 25

Schema bound

ProblemWe want to calculate the probability of having good classifiers orrepresentatives in an initial population. Classifiers that do not makemistakes, since they represent correctly all the specified bits in anoriginal problem rule.

ExampleConsidering the rule #10#1 with 3 values specified (k=3), the followingclassifiers are representatives: 110*1, 11011, 010*1.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 8 / 25

Schema bound

QuestionWhat is the probability of obtaining a representative with at least kvalues specified?

To become a representative the rule should:1 Specify at least k attributes correctly.2 The rest of the attributes should not have all 0’s.

P(rep) =2kf (ldp(1−p))k(1−ld(1−p)2)d−k

where kf is the number of fully map attributes

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 9 / 25

Schema bound

QuestionWhat is the probability of obtaining a representative with at least kvalues specified?

To become a representative the rule should:1 Specify at least k attributes correctly.2 The rest of the attributes should not have all 0’s.

Without using any of the mechanisms:

P(rep) =2kf (ldp(1−p))k(1−ld(1−p)2)d−k

n

where kf is the number of fully map attributes

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 9 / 25

Schema bound

QuestionWhat is the probability of obtaining a representative with at least kvalues specified?

To become a representative the rule should:1 Specify at least k attributes correctly.2 The rest of the attributes should not have all 0’s.

Using default rule:

P(rep) =2kf (ldp(1−p))k(1−ld(1−p)2)d−k

n−1

where kf is the number of fully map attributes

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 9 / 25

Schema bound

QuestionWhat happens when we use covering?

1 We sample an instance with uniform probabilities for all classes.2 We set the bits corresponding to the instance values to 1.

I It is not possible to have all 0’s anymore.

P(rep) = m (ld (1− p))k

where m is the number of classes mapped by the problem rules

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 10 / 25

Schema bound

QuestionWhat happens when we use covering?

1 We sample an instance with uniform probabilities for all classes.2 We set the bits corresponding to the instance values to 1.

I It is not possible to have all 0’s anymore.

P(rep) = mn (ld (1− p))k

where m is the number of classes mapped by the problem rules

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 10 / 25

Schema bound

QuestionWhat happens when we use covering and default rule?

1 We sample an instance with uniform probabilities for all classes.2 We set the bits corresponding to the instance values to 1.

I It is not possible to have all 0’s anymore.

P(rep) = mn−1 (ld (1− p))k

where m is the number of classes mapped by the problem rules

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 10 / 25

Problems used for model validation

Binary and Ternary Multiplexer problemsI k address bitsI 2k string bits (3k for ternary case)

k-Disjuntive Normal Functions[Butz and Pelikan, 2006, Franco et al., 2010].

I r disjunctive termsI d possible attributesI k represented attributes in each term

Example kDNF: d = 10, k = 3, r = 3

(¬x1 ∧ x5 ∧ x7) ∨ (x1 ∧ ¬x2 ∧ x8) ∨ (x4 ∧ ¬x5 ∧ ¬x9)

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 11 / 25

Problems used for model validation

Binary and Ternary Multiplexer problemsI k address bitsI 2k string bits (3k for ternary case)

k-Disjuntive Normal Functions[Butz and Pelikan, 2006, Franco et al., 2010].

I r disjunctive termsI d possible attributesI k represented attributes in each term

Example kDNF: d = 10, k = 3, r = 3

(¬x1 ∧ x5 ∧ x7) ∨ (x1 ∧ ¬x2 ∧ x8) ∨ (x4 ∧ ¬x5 ∧ ¬x9)

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 11 / 25

Schema bound - Model validation

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0 2 4 6 8 10

P(r

ep)

k - Number of Attributes

Teoretical p=0.75Empirical p=0.75

Teoretical p=0.50Empirical p=0.50

Teoretical p=0.25Empirical p=0.25

(a) MUX - No Covering

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

P(r

ep)

k - Number of Attributes

Teoretical p=0.75Empirical p=0.75

Teoretical p=0.50Empirical p=0.50

Teoretical p=0.25Empirical p=0.25

(b) MUX- Covering

0

0.02

0.04

0.06

0.08

0.1

0 2 4 6 8 10

P(r

ep)

k - Number of attributes especified

Empirical p=0.75Teoretical p=0.75Empirical p=0.50

Teoretical p=0.50Empirical p=0.25

Teoretical p=0.25Empirical NoDef p=0.75

Teoretical NoDef p=0.75Empirical NoDef p=0.50

Teoretical NoDef p=0.50Empirical NoDef p=0.25

Teoretical NoDef p=0.25

(c) kDNF - No Covering

0

0.2

0.4

0.6

0.8

1

0 2 4 6 8 10

P(r

ep)

k - Number of attributes especified

Empirical p=0.75Teoretical p=0.75Empirical p=0.50

Teoretical p=0.50Empirical p=0.25

Teoretical p=0.25Empirical NoDef p=0.75

Teoretical NoDef p=0.75Empirical NoDef p=0.50

Teoretical NoDef p=0.50Empirical NoDef p=0.25

Teoretical NoDef p=0.25

(d) kDNF - Covering

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 12 / 25

What have we calculated so far?

These models so far only hold for:

Problems withno-overlapping

Problems that have justone rule

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 13 / 25

What happens here?

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 14 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

r

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

r

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

?

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

?

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

?

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

?

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

How does the overlapping affects the probability of arepresentative?

P(niche) =P(rep)

2k(1− (1− 2−k)r)

1?

=ExamplesNiche (EN)

ExamplesCovered (EC)

EC = 2d(

1−(1− 2−k)r

)EN =

2d

2k

P′(rep) = 1− (1− P(niche))r

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 15 / 25

Validation of models considering overlapping

0 2 4 6 8 10 1

5

25

0 0.2 0.4 0.6 0.8

1

P(rep)Teoretical

Empirical r=1Empirical r=5

Empirical r=10Empirical r=20Empirical r=40

Atts esp (k)# of rules

P(rep)

(e) Base Case

0 2 4 6 8 10 1

5

25

0 0.2 0.4 0.6 0.8

1

P(rep)Teoretical

Empirical r=1Empirical r=5

Empirical r=10Empirical r=20Empirical r=40

Atts esp (k)# of rules

P(rep)

(f) Covering and Default Class

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 16 / 25

Covering bound

ProblemHow can we calculate the probability of covering the whole searchspace?

We need to calculate the probability of matching an instance

Base case P(match) = (1− ld + ldp)d

Covering case P(match) =(

1− ld + ld(

1+p2

))d

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound

ProblemHow can we calculate the probability of covering the whole searchspace?

We need to calculate the probability of matching an instance

Base case P(match) = (1− ld + ldp)d

Covering case P(match) =(

1− ld + ld(

1+p2

))d

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound

ProblemHow can we calculate the probability of covering the whole searchspace?

We need to calculate the probability of matching an instance

Base case P(match) = (1− ld + ldp)d

Covering case P(match) =(

1− ld + ld(

1+p2

))d

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound - Model validation

(g) No covering

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20

P(m

atch

)

k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

(h) Covering

0

0.2

0.4

0.6

0.8

1

0 5 10 15 20P

(mat

ch)

k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 18 / 25

What happens with x-ary attributes?

What happens when the problem is not binary but has more than 2values per attribute?

Generalised models for x-ary attributesWhere t is the number of values per attribute and e is the number ofactive bits per attribute.

Example 1: 101|110|011:0⇒ t=3 e=2Example 2: 001|100|010:1⇒ t=3 e=1

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 19 / 25

What happens with x-ary attributes?

What happens when the problem is not binary but has more than 2values per attribute?

Generalised models for x-ary attributesWhere t is the number of values per attribute and e is the number ofactive bits per attribute.

Example 1: 101|110|011:0⇒ t=3 e=2Example 2: 001|100|010:1⇒ t=3 e=1

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 19 / 25

Generalised model for x-ary attributes

Schema bound

Base case P(rep) =tkf (ldpe(1−p)t−e)k(1−ld(1−p)t)d−k

n

Covering case P(rep) = mn

(ldpe−1 (1− p)t−e−1

)k

Covering bound

Base case P(match) = (1− ld + ldp)d

Covering case P(match) =(

1− ld + ld(

1+(t−1)pt

))d

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 20 / 25

Generalised model for x-ary attributes

Schema bound with Default Rule

Base case P(rep) =tkf (ldpe(1−p)t−e)k(1−ld(1−p)t)d−k

n−1

Covering case P(rep) = mn−1

(ldpe−1 (1− p)t−e−1

)k

Covering bound

Base case P(match) = (1− ld + ldp)d

Covering case P(match) =(

1− ld + ld(

1+(t−1)pt

))d

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 20 / 25

Generalised model for x-ary attributes

Schema bound validation (with ternary multiplexer problems)

(i) No covering

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

1 2 3 4 5 6

P(r

ep)

k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

(j) Covering

0

0.1

0.2

0.3

0.4

0.5

0.6

1 2 3 4 5 6P

(rep

)k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

≈ 5 times more probability of generatinga good individual when using covering

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 21 / 25

Generalised model for x-ary attributes

Covering bound validation (with ternary multiplexer problems)

(k) No covering

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14

P(m

atch

)

k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

(l) Covering

0

0.2

0.4

0.6

0.8

1

2 4 6 8 10 12 14P

(mat

ch)

k - Number of Attributes

Empirical p=0.75Model p=0.75

Empirical p=0.50Model p=0.50

Empirical p=0.25Model p=0.25

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 22 / 25

Conclusions

The presented models explains what is the probability of havinga good initial population in BioHEL considering de ALKRrepresentation and other initialisation mechanisms.We also presented a generalisation of the model for x-aryattributes and adjusted the probability for problems withoverlapping.These models explain the benefits of BioHEL initialisationmechanisms giving a further understanding of how the BioHELsystem works.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 23 / 25

Further Work

Simplify the current models to make them less dependent onproblem parameters not known beforehand.Model the reproductive opportunity and learning time of BioHEL.Derive boundaries for the population size and other user-definedparameters in BioHEL.

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 24 / 25

Modelling the Initialisation Stage of the ALKRRepresentation for Discrete Domains and

GABIL Encoding

María A. Franco, Natalio Krasnogor, Jaume Bacardit

University of Nottingham, UK.ASAP Research Group,

School of Computer Sciencemxf@cs.nott.ac.uk

July 14, 2011

Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 25 / 25

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Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 25 / 25