gecco'2007: modeling xcs in class imbalances: population size and parameter settings

Modeling XCS in Class Modeling XCS in Class Imbalances: Population Size Imbalances: Population Size

and Parameter Settingsg

Albert Orriols-Puig1,2 David E. Goldberg2

Kumara Sastry2 Ester Bernadó-Mansilla1Kumara Sastry Ester Bernadó Mansilla

1Research Group in Intelligent SystemsEnginyeria i Arquitectura La Salle, Ramon Llull UniversityEnginyeria i Arquitectura La Salle, Ramon Llull University

2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering

University of Illinois at Urbana ChampaignUniversity of Illinois at Urbana Champaign

Framework

New instance

Learner M d l

Information basedon experience

Knowledgeextraction

New instance

Domain Learner Model

Consisting

Examples

Counter-examples

Predicted Output

ofCou te e a p es

In real-world domains, typically:, yp yHigher cost to obtain examples of the concept to be learntSo, distribution of examples in the training dataset is usually imbalanced

Applications:Fraud detectionMedical diagnosis of rare illnesses

Slide 2GRSI Enginyeria i Arquitectura la Salle

Medical diagnosis of rare illnessesDetection of oil spills in satellite images

Framework

Do learners suffer from class imbalances?– Methods that do global optimization

L Minimize theTrainingLearner Minimize the

global errorSet

examplesnumbererrorsnumerrorsnum

error cc 21 .. +=Biased towards

the overwhelmed class

Maximization of the overwhelmed class accuracy,in detriment of the minority class.


Motivation

And what about incremental learning?

Sampling instances of the minority class less frequently

Rules that match instances of the minority class poorly activated

Rules of the minority class would receive less genetic opportunities (Orriols & Bernadó, 2006)


Aim

Facetwise analysis of XCS for class imbalances

Impact of class imbalances on the initialization process

How can XCS create rules of the minority class if the covering process failsg p

Population size bound with respect to the imbalance ratioPopulation size bound with respect to the imbalance ratio

U til hi h i b l ti ld XCS b bl t lUntil which imbalance ratio would XCS be able to learn from the minority class?


Outline1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

1. Description of XCS

2 Facetwise Analysis2. Facetwise Analysis

3. Design of test Problems

4. XCS on the one-bit Problem

5 A l i f D i ti5. Analysis of Deviations

6. Results

7. Conclusions


Description of XCS1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsp

In single-step tasks:

6. Results7. Conclusions

Environment

g p

Problem instance

1 C A P ε F num as ts exp3 C A P ε F num as ts exp5 C A P ε F num as ts exp

Match Set [M]Selected

action

Minorityclass instance


Match Set [M]Majorityclass instance


Match Set [M]


Population [P] Match set generation

5 C A P ε F num as ts exp6 C A P ε F num as ts exp

…Prediction Array

REWARD1000/01 C A P ε F num as ts exp




…1 C A P ε F num as ts exp2 C A P ε F num as ts exp3 C A P ε F num as ts exp



…


…

A ti S t [A]

c1 c2 … cn

Random Action


… Starved niches


… Nourished niches


C

Action Set [A]

Selection, Reproduction, Mutation

Deletion ClassifierParameters

Update6 C A P ε F num as ts exp…Genetic Algorithm

Update

Problem niche: the schema defines the relevant


Problem niche: the schema defines the relevant attributes for a particular problem niche. Eg: 10**1*

Outline






6. Results

7. Conclusions


Facetwise Analysis1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsy6. Results7. Conclusions

Study XCS capabilities to provide representatives of starved niches:– Population initialization– Generation of correct representatives of starved niches– Time of extinction of these correct classifiers

Derive a bound on the population sizeDerive a bound on the population sizeDepart from theory developed for XCS

– (Butz, Kovacs, Lanzi, Wilson,04): Model of generalization pressures of XCS(Butz, Kovacs, Lanzi, Wilson,04): Model of generalization pressures of XCS – (Butz, Goldberg & Lanzi, 04): Learning time bound – (Butz, Goldberg, Lanzi & Sastry, 07): Population size bound to guarantee niche

supportsupport– (Butz, 2006): Rule-Based Evolutionary Online Learning Systems: A Principled

Approach to LCS Analysis and Design.



Assumptions– Problems consisting of n classesProblems consisting of n classes

– One class sampled with a lower frequency: minority class

classminority theof instances num.classminority theother than classany of instances num.ir =

– Probability of sampling an instance of the minority class:

i11Ps(min) =

i1irPs(maj) =

ir1( )

+ ir1( j)

+



Facetwise AnalysisPopulation initialization– Population initialization

– Generation of correct representatives of starved niches

– Time of extinction of these correct classifiers

– Population size bound


Population Initialization1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsp6. Results7. Conclusions

Covering procedure– Covering: Generalize over the input with probability P#Covering: Generalize over the input with probability P#

– P# needs to satisfy the covering challenge (Butz et al., 01)

Would I trigger covering on minority class instances?– Probability that one instance is covered by at leastProbability that one instance is covered, by, at least,

one rule is (Butz et. al, 01):Inputlength

Population specificity

Initially 1 – P#

Population size

y #


Population Initialization1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsp6. Results7. Conclusions


Creation of Representatives of Starved Niches

1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of DeviationsStarved Niches 6. Results7. Conclusions

AssumptionsCovering has not provided any representative of starved niches– Covering has not provided any representative of starved niches

– Simplified model: only consider mutation in our model.

How can we generate representative of starved niches?– Specifying correctly all the bits of the schema that represents theSpecifying correctly all the bits of the schema that represents the

starved niche


Creation of Representatives of Starved Niches


Possible cases:– Sample a minority class instance

ir11 Ps(min)+

=

• Activate a niche of the minority class μ: Mutation probability

K : Order of the schema

• Activate a niche of another class

Km: Order of the schema

– Sample a majority class instanceir1

ir Ps(maj)+

=

• Activate a niche of the minority class

• Activate a niche of another class


Creation of Representatives ofStarved Niches


Summing up, time to get the first representative of a starved nichesta ed c e

n: number of classes

μ: Mutation probability

Km: Order of the schema

It increases:Linearly with the number of classes

Exponentially with the order of the schemaExponentially with the order of the schema

It does not depend on the imbalance ratio


Bounding the Population Size1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsg p6. Results7. Conclusions

Time to extinction

– Consider random deletion:



Population size bound to guarantee that there will be representatives of starved nichesep ese tat es o sta ed c es

– Require that:

– Bound:



Population size bound to guarantee that representatives of starved niches will receive a genetic opportunity:– Consider θGA = 0

– We require that the best representative of a starved niche receive a genetic event before being removed

– Time to receive the first genetic event



Population size bound to guarantee that representatives of starved niches will receive a genetic opportunity:o sta ed c es ece e a ge et c oppo tu ty

The population size to guarantee that the best representatives of starve niches will receive at least one genetic opportunity g pp y

increases linearly with the imbalance ratio


Outline






6. Results

7. Conclusions


Design of Test Problems1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsg6. Results7. Conclusions

One-bit problem

000110 :0 Value of the left-most bit

Condition length (l)

– Only two schemas of order one: 0***** and 1*****

– Undersampling instances of the class labeled as 1

ir11 Ps(min)+

=


Design of Test Problems1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsg6. Results7. Conclusions

Parity problemCondition

01001010

Condition length (l)

:1 Number of 1 mod 2

Relevant

– The k bits of parity form a single building block

bits ( k)

– Undersampling instances of the class labeled as 1

1ir1

1 Ps(min)+

=


Outline






6. Results

7. Conclusions


XCS on the one-bit Problem1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

XCS configuration

α=0.1, ν=5, ε0=1, θGA=25, χ=0.8, μ=0.4, θdel=20, θsub=200, δ=0.1, P#=0.6selection=tournament, mutation=niched, [A]sub=false, N = 10,000 ir

Evaluation of the results:Evaluation of the results:– Minimum population size to achieve:

TP rate * TN rate > 95%TP rate TN rate > 95%

R lt 25 d– Results are averages over 25 seeds


XCS on the one-bit Problem1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

N remains constant up to ir = 64

N increases linearly from ir=64 to ir=256

N increases exponentially fromp yir=256 to ir=1024

Higher ir could not be solved


Outline






6. Results

7. Conclusions


Analysis of the Deviations1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsy6. Results7. Conclusions

Niched Mutation vs. Free Mutation– Classifiers can only be created if minority class instances are sampled– Classifiers can only be created if minority class instances are sampled

Inheritance Error of Classifiers’ Parameters– New promising representatives of starved niches are created from

l ifi th t b l t i h d i hclassifiers that belong to nourished niches

– These new promising rules inherit parameters from these classifiers. This is specially delicate for the action set size (as)This is specially delicate for the action set size (as).

– Approach: initialize as=1.


Analysis of the Deviations1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsy6. Results7. Conclusions

Subsumption– An overgeneral classifier of the majority class may receive ir positive– An overgeneral classifier of the majority class may receive ir positive

reward before receiving the first negative reward

– Approach: set θsub>irpp sub

Stabilizing the population before testingStabilizing the population before testing– Overgeneral classifiers poorly evaluated

Approach: introduce some extra runs at the end of learning with the GA– Approach: introduce some extra runs at the end of learning with the GA switched off.

We gather all these little tweaks in XCS+PMC


Outline






6. Results

7. Conclusions


XCS+PCM in the one-bit Problem1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

N remains constant up to ir = 128

F hi h i N li htl iFor higher ir, N slightly increases

We only have to guarantee that aWe only have to guarantee that a representative of the starved niche will be created


XCS+PCM in the Parity Problem1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviationsy6. Results7. Conclusions

Building blocks of size 3 need to be processed

Empirical results agree with thetheory

P l ti i b d t tPopulation size bound to guaranteethat a representative of the nichewill receive a genetic event


Outline






6. Results

7. Conclusions


Conclusions and Further Work1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

We derived models that analyzed the representatives of starved niches provided by covering and mutation

A population size bound was derived

We saw that the empirical observations met the theory if four aspects were considered:

– Type of mutation

– as initialization

– Subsumption

– Stabilization of the populationStabilization of the population

Further analysis of the covering operator


Modeling XCS in Class Modeling XCS in Class Imbalances: Population Size Imbalances: Population Size

and Parameter Settingsg

Albert Orriols-Puig1,2 David E. Goldberg2

Kumara Sastry2 Ester Bernadó-Mansilla1Kumara Sastry Ester Bernadó Mansilla

1Research Group in Intelligent SystemsEnginyeria i Arquitectura La Salle, Ramon Llull UniversityEnginyeria i Arquitectura La Salle, Ramon Llull University

2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering

University of Illinois at Urbana ChampaignUniversity of Illinois at Urbana Champaign

gecco'2007: modeling xcs in class imbalances: population size and parameter settings

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