modeling xcs in class imbalances: population sizing and parameter settings

Modeling XCS in Class Imbalances: Population Size

and Parameter Settings

Albert Orriols-Puig1,2 David E. Goldberg2

Kumara Sastry2 Ester Bernadó-Mansilla1

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

2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering

University of Illinois at Urbana Champaign

Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 2GECCO’07

Framework

Domain Learner Datamodel

Information basedon experience

Knowledgeextraction

Consisting of

Examples

Counter-examples

In real-world domains, typically:Higher 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 illnessesDetection of oil spills in satellite images

New instance

Predicted Output


Framework

Do learners suffer from class imbalances?

Learner Minimize theglobal error

TrainingSet

examplesnumbererrorsnumerrorsnum

error cc 21 .. +=Biased towards

the overwhelmed class

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

And what about incremental learning?– Sampling instances of the minority class less frequently


Aim

Facetwise analysis of XCS for class imbalances

How can XCS create rules of the minority class

When XCS will remove these rules

Population size bound with respect to the imbalance ratio

Until which imbalance ratio would XCS be able to learn from the minority class?


Outline

1. Description of XCS

2. Facetwise Analysis

3. Design of test Problems

4. XCS on the one-bit Problem

5. Analysis of Deviations

6. Results

7. Conclusions

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


Description of XCS

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 exp4 C A P ε F num as ts exp5 C A P ε F num as ts exp6 C A P ε F num as ts exp

…

Population [P]

Environment

Problem instance

Match set generation

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 exp6 C A P ε F num as ts exp

…

Match Set [M]


…

Action Set [A]

c1 c2 … cn

Prediction Array

Genetic Algorithm

Selection, Reproduction, Mutation

Deletion

Selectedaction

ClassifierParameters

Update

REWARD1000/0

In single-step tasks:

Random Action

Minorityclass instance


…

Population [P] Match set generation


…

Match Set [M]

Starved niches

Majorityclass instance


…

Population [P] Match set generation


…

Match Set [M]

Nourished niches

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



Outline






6. Results

7. Conclusions


Facetwise Analysis

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

Derive a bound on the population size to guarantee that XCS will learn starved nichesDepart from theory developed for 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

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

Approach to LCS Analysis and Design.



Facetwise Analysis

Assumptions– Problems consisting of n classes

– One class sampled with a lower frequency: minority class

– Probability of sampling an instance of the minority class:

ir11 Ps(min)+

=

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



Facetwise Analysis

Facetwise Analysis– Population initialization

– Generation of correct representatives of starved niches

– Time of extinction of these correct classifiers

– Population size bound



Population Initialization

Covering procedure– 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 least,

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

Population size

Population specificity

Initially 1 – P#



Population Initialization1. Description of XCS2. Facetwise Analysis3. Design of test Problems4. XCS on the one-bit Problem5. Analysis of Deviations6. Results7. Conclusions

Probability to apply covering on the first minority class instance

l = 20


Facetwise Analysis







Creation of Representatives of Starved Niches

Assumptions– 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?– In the population there are:

• Representative of nourished niches• Overgeneral classifiers

– Specifying correctly all the bits of the schema that represents the starved niche



Creation of Representatives ofStarved Niches

Summing up, time to get the first representative of a starved niche

Time to extinction


n: number of classes

μ: Mutation probability

km: Order of the schema


Facetwise Analysis







Bounding the Population Size

Population size bound to guarantee that there will be representatives of starved niches

– Require that:

– Bound:



μ: Mutation probability

km: Order of the schema


Bounding the Population Size

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

– Population size bound:



ir: Imbalance ratio


Outline






6. Results

7. Conclusions


Design of Test Problems

One-bit problem

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

Parity problem

– The k bits of parity form a single building block

Undersampling instances of the class labeled as 1

000110 :0 Value of the left-most bit

Condition length (l)

ir11 Ps(min)+

=

01001010


Condition length (l)

:1 Number of 1 mod 2

Relevantbits ( k)


Outline






6. Results

7. Conclusions


XCS on the one-bit Problem

XCS configuration

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

TP rate * TN rate > 95%

– Results are averages over 25 seeds

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



XCS on the one-bit Problem

N remains constant up to ir = 64

N increases linearly from ir=64 to ir=256

N increases exponentially fromir=256 to ir=1024

Higher ir could not be solved



Outline






6. Results

7. Conclusions


Analysis of the Deviations

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

belong to nourished niches. – These new promising rules inherit parameters from these classifiers. This is

specially delicate for the action set size (as).

– Approach: initialize as=1.

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

before receiving the first negative reward– Approach: set θsub>ir

Stabilizing the population before testing– Overgeneral classifiers poorly evaluated– Approach: introduce some extra runs at the end of learning with the GA switched

off.



Outline






6. Results

7. Conclusions


XCS+PCM in the one-bit Problem

N remains constant up to ir = 128

For higher ir, N slightly increases

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



XCS+PCM in the Parity Problem

Building blocks of size 3 need to be processed

Empirical results agree with thetheory

Population size bound to guaranteethat a representative of the nichewill receive a genetic event



Outline






6. Results

7. Conclusions


Conclusions and Further Work

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:

– as initialization

– Subsumption

– Stabilization of the population

XCS really robust to class imbalances

Further analysis of the covering operator


Modeling XCS in Class Imbalances: Population Size

and Parameter Settings

Albert Orriols-Puig1,2 David E. Goldberg2

Kumara Sastry2 Ester Bernadó-Mansilla1

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

2Illinois Genetic Algorithms LaboratoryDepartment of Industrial and Enterprise Systems Engineering

University of Illinois at Urbana Champaign


Motivation

And what about incremental learning?

Sampling instances of the minority class less frequently

This influences the mechanisms of XCS (Orriols & Bernadó, 2006)



Niched Mutation vs. Free Mutation– 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

classifiers that belong to nourished niches

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

– Approach: initialize as=1.




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

reward before receiving the first negative reward

– Approach: set θsub>ir

Stabilizing the population before testing– Overgeneral classifiers poorly evaluated

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

We gather all these little tweaks in XCS+PMC


modeling xcs in class imbalances: population sizing and parameter settings

Technology

ts exp5c

ts exp6c

ts exp3c

ts expselection

ts exp2c

ts exp4c

ts exp1c

ts expselected