Modelling the Initialisation Stage of the ALKR
Representation for Discrete Domains and
GABIL Encoding
María A. Franco, Natalio Krasnogor, Jaume Bacardit
University of Nottingham, UK.
ASAP Research Group,
School of Computer Science
mxf@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 Machine
Learning (GBML) designed to cope with large scale
datasets[Bacardit et al., 2009b].
Iterative Rule Learning approach
Attribute List Knowledge Representation (ALKR)
ILAS Windowing scheme
Default rule
Smart initialisation mechanisms (covering)
GPU-based evaluation process
Problem
The system obtains good results [Stout et al., 2008], but we do not
have 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 Machine
Learning (GBML) designed to cope with large scale
datasets[Bacardit et al., 2009b].
Iterative Rule Learning approach
Attribute List Knowledge Representation (ALKR)
ILAS Windowing scheme
Default rule
Smart initialisation mechanisms (covering)
GPU-based evaluation process
Problem
The system obtains good results [Stout et al., 2008], but we do not
have 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 BioHEL
system and calculate the probability of having a good initial
population. 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
Background
These probabilities are also know as schema and covering bound.
This have already being determined for XCS and the ternary
representation {1,0,#} by [Butz, 2006].
Problem
Models need to be adapted for our ALKR+GABIL representation.
Moreover, we want to model the impact of the BioHEL mechanisms
that 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 BioHEL
system and calculate the probability of having a good initial
population. 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
Background
These probabilities are also know as schema and covering bound.
This have already being determined for XCS and the ternary
representation {1,0,#} by [Butz, 2006].
Problem
Models need to be adapted for our ALKR+GABIL representation.
Moreover, we want to model the impact of the BioHEL mechanisms
that 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 BioHEL
system and calculate the probability of having a good initial
population. 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
Background
These probabilities are also know as schema and covering bound.
This have already being determined for XCS and the ternary
representation {1,0,#} by [Butz, 2006].
Problem
Models need to be adapted for our ALKR+GABIL representation.
Moreover, we want to model the impact of the BioHEL mechanisms
that 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 Background
GABIL Representation
Attribute List Knowledge Representation (ALKR)
2 Probabilistic models
Initial considerations
Schema bound
How does the overlapping affects?
Covering bound
3 Generalised model for x-ary attributes
Schema 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 inside
ALKR to represent nominal attributes.
Example
F1 ={A,B,C} F2={O,P} F3={W,Z,X,Y}
F1 F2 F3
100 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 with
probability 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 inside
ALKR to represent nominal attributes.
Example
F1 ={A,B,C} F2={O,P} F3={W,Z,X,Y}
F1 F2 F3
100 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 with
probability 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 3
whichAtt 0
predicates 0.5 0.7 0.3
offsetPred 0
class 1
How do we select the attributes in the list?
1 d <= ExpAtts
ld = ExpAtts
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
Covering ⇒ We have to consider 4
initialisation scenarios
Default Rule
Types of attributes
Fully mapped attributes
Partially 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
Covering ⇒ We have to consider 4
initialisation scenarios
Default Rule
Types of attributes
Fully mapped attributes
Partially 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
Covering ⇒ We have to consider 4
initialisation scenarios
Default Rule
Types of attributes
Fully mapped attributes
Partially mapped attributes.
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 7 / 25

Schema bound
Problem
We want to calculate the probability of having good classifiers or
representatives in an initial population. Classifiers that do not make
mistakes, since they represent correctly all the specified bits in an
original problem rule.
Example
Considering the rule #10#1 with 3 values specified (k=3), the following
classifiers 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
Problem
We want to calculate the probability of having good classifiers or
representatives in an initial population. Classifiers that do not make
mistakes, since they represent correctly all the specified bits in an
original problem rule.
Example
Considering the rule #10#1 with 3 values specified (k=3), the following
classifiers 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
Question
What is the probability of obtaining a representative with at least k
values 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.
k d−k
2 f (ld p(1−p))k (1−ld (1−p)2 )
P(rep) =
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
Question
What is the probability of obtaining a representative with at least k
values 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:
k d−k
2 f (ld p(1−p))k (1−ld (1−p)2 )
P(rep) = 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
Question
What is the probability of obtaining a representative with at least k
values 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:
k d−k
2 f (ld p(1−p))k (1−ld (1−p)2 )
P(rep) = 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
Question
What 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.
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
Question
What 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.
It is not possible to have all 0’s anymore.
P(rep) = m
n (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
Question
What 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.
It is not possible to have all 0’s anymore.
P(rep) = m
n−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 problems
k address bits
2k string bits (3k for ternary case)
k-Disjuntive Normal Functions
[Butz and Pelikan, 2006, Franco et al., 2010].
r disjunctive terms
d possible attributes
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 problems
k address bits
2k string bits (3k for ternary case)
k-Disjuntive Normal Functions
[Butz and Pelikan, 2006, Franco et al., 2010].
r disjunctive terms
d possible attributes
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
0.8 1
Teoretical p=0.75 Teoretical p=0.75
0.7 Empirical p=0.75 Empirical p=0.75
Teoretical p=0.50 Teoretical p=0.50
Empirical p=0.50 0.8 Empirical p=0.50
0.6
Teoretical p=0.25 Teoretical p=0.25
Empirical p=0.25 Empirical p=0.25
0.5 0.6
P(rep)
P(rep)
0.4
0.3 0.4
0.2
0.2
0.1
0 0
0 2 4 6 8 10 0 2 4 6 8 10
k - Number of Attributes k - Number of Attributes
(a) MUX - No Covering (b) MUX- Covering
0.1 1
Empirical p=0.75 Empirical p=0.75
Teoretical p=0.75 Teoretical p=0.75
Empirical p=0.50 Empirical p=0.50
0.08 Teoretical p=0.50 0.8 Teoretical p=0.50
Empirical p=0.25 Empirical p=0.25
Teoretical p=0.25 Teoretical p=0.25
0.06 Empirical NoDef p=0.75 0.6 Empirical NoDef p=0.75
P(rep)
P(rep)
Teoretical NoDef p=0.75 Teoretical NoDef p=0.75
Empirical NoDef p=0.50 Empirical NoDef p=0.50
0.04 Teoretical NoDef p=0.50 0.4 Teoretical NoDef p=0.50
Empirical NoDef p=0.25 Empirical NoDef p=0.25
Teoretical NoDef p=0.25 Teoretical NoDef p=0.25
0.02 0.2
0 0
0 2 4 6 8 10 0 2 4 6 8 10
k - Number of attributes especified k - Number of attributes especified
(c) kDNF - No Covering (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 with Problems that have just
no-overlapping one 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 a
representative?
P(rep)
P(niche) =
r
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
r
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
?
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
?
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
?
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
?
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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 a
representative?
P(rep)
P(niche) =
2k 1 − (1 − 2−k )r
1 ExamplesNiche (EN)
=
? ExamplesCovered (EC)
r
EC = 2d 1 − 1 − 2−k
2d
EN =
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
Teoretical Teoretical
P(rep) Empirical r=1 P(rep) Empirical r=1
Empirical r=5 Empirical r=5
1 Empirical r=10 1 Empirical r=10
Empirical r=20 Empirical r=20
0.8 Empirical r=40 0.8 Empirical r=40
0.6 0.6
0.4 0.4
0.2 0.2
0 0
25 25
0 2 5 0 2 5
4 # of rules 4 # of rules
Atts esp (k) 6 8 Atts esp (k) 6 8
10 1 10 1
(e) Base Case (f) Covering and Default Class
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 16 / 25

Covering bound
Problem
How can we calculate the probability of covering the whole search
space?
We need to calculate the probability of matching an instance
d
Base case P(match) = (1 − ld + ld p)
d
1+p
Covering case P(match) = 1 − ld + ld 2
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound
Problem
How can we calculate the probability of covering the whole search
space?
We need to calculate the probability of matching an instance
d
Base case P(match) = (1 − ld + ld p)
d
1+p
Covering case P(match) = 1 − ld + ld 2
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound
Problem
How can we calculate the probability of covering the whole search
space?
We need to calculate the probability of matching an instance
d
Base case P(match) = (1 − ld + ld p)
d
1+p
Covering case P(match) = 1 − ld + ld 2
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 17 / 25

Covering bound - Model validation
(g) No covering (h) Covering
1 1
Empirical p=0.75 Empirical p=0.75
Model p=0.75 Model p=0.75
Empirical p=0.50 Empirical p=0.50
0.8 Model p=0.50 0.8 Model p=0.50
Empirical p=0.25 Empirical p=0.25
Model p=0.25 Model p=0.25
0.6 0.6
P(match)
P(match)
0.4 0.4
0.2 0.2
0 0
0 5 10 15 20 0 5 10 15 20
k - Number of Attributes k - Number of Attributes
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 2
values per attribute?
Generalised models for x-ary attributes
Where t is the number of values per attribute and e is the number of
active bits per attribute.
Example 1: 101|110|011:0 ⇒ t=3 e=2
Example 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 2
values per attribute?
Generalised models for x-ary attributes
Where t is the number of values per attribute and e is the number of
active bits per attribute.
Example 1: 101|110|011:0 ⇒ t=3 e=2
Example 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
k d−k
tkf (ld pe (1−p)t−e ) (1−ld (1−p)t )
Base case P(rep) = n
k
m t−e−1
Covering case P(rep) = n ld pe−1 (1 − p)
Covering bound
d
Base case P(match) = (1 − ld + ld p)
d
1+(t−1)p
Covering case P(match) = 1 − ld + ld t
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
k d−k
tkf (ld pe (1−p)t−e ) (1−ld (1−p)t )
Base case P(rep) = n−1
k
m t−e−1
Covering case P(rep) = n−1 ld pe−1 (1 − p)
Covering bound
d
Base case P(match) = (1 − ld + ld p)
d
1+(t−1)p
Covering case P(match) = 1 − ld + ld t
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 (j) Covering
0.16 0.6
Empirical p=0.75 Empirical p=0.75
0.14 Model p=0.75 Model p=0.75
Empirical p=0.50 0.5 Empirical p=0.50
0.12 Model p=0.50 Model p=0.50
Empirical p=0.25 Empirical p=0.25
Model p=0.25 0.4 Model p=0.25
0.1
P(rep)
P(rep)
0.08 0.3
0.06
0.2
0.04
0.1
0.02
0 0
1 2 3 4 5 6 1 2 3 4 5 6
k - Number of Attributes k - Number of Attributes
≈ 5 times more probability of generating
a 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 (l) Covering
1 1
Empirical p=0.75 Empirical p=0.75
Model p=0.75 Model p=0.75
Empirical p=0.50 Empirical p=0.50
0.8 Model p=0.50 0.8 Model p=0.50
Empirical p=0.25 Empirical p=0.25
Model p=0.25 Model p=0.25
0.6 0.6
P(match)
P(match)
0.4 0.4
0.2 0.2
0 0
2 4 6 8 10 12 14 2 4 6 8 10 12 14
k - Number of Attributes k - Number of Attributes
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 having
a good initial population in BioHEL considering de ALKR
representation and other initialisation mechanisms.
We also presented a generalisation of the model for x-ary
attributes and adjusted the probability for problems with
overlapping.
These models explain the benefits of BioHEL initialisation
mechanisms giving a further understanding of how the BioHEL
system 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 on
problem parameters not known beforehand.
Model the reproductive opportunity and learning time of BioHEL.
Derive boundaries for the population size and other user-defined
parameters in BioHEL.
Franco et al. (University of Nottingham) Modelling Initialisation using ALKR+GABIL July 14, 2011 24 / 25

Modelling the Initialisation Stage of the ALKR
Representation for Discrete Domains and
GABIL Encoding
María A. Franco, Natalio Krasnogor, Jaume Bacardit
University of Nottingham, UK.
ASAP Research Group,
School of Computer Science
mxf@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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