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Modeling XCS in Class
Imbalances: Population Size
  and Parameter Settings
                       g

    Albert Orriols-Puig1,2                      David E. Goldberg2
     Kumara Sastry2                           Ester Bernadó-Mansilla1
                                                    Bernadó Mansilla

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

                 2Illinois
                         Genetic Algorithms Laboratory
       Department of Industrial and Enterprise Systems Engineering
               University of Illinois at Urbana Champaign
Framework

                                                                                                New instance

                         Information based                                        Knowledge
                           on experience                                          extraction

                                                    Learner                                      Model
                                                                                                 Mdl
            Domain


                                                                                               Predicted Output
                            Examples

           Consisting       Cou te e a p es
                            Counter-examples
              of


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

       Applications:
           Fraud detection
           Medical diagnosis of rare illnesses
           Detection of oil spills in satellite images
                                             Enginyeria i Arquitectura la Salle                                   Slide 2
GRSI
Framework

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


              Training                                          Minimize the
                               Learner
                               L
                Set                                             global error




                                                        num. errorsc1 + num. errorsc 2
                                             error =
           Biased towards
                                                                number examples
       the overwhelmed class




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




                                  Enginyeria i Arquitectura la Salle                     Slide 3
GRSI
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)




                              Enginyeria i Arquitectura la Salle    Slide 4
GRSI
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 fails
                 gp


          Population size bound with respect to the imbalance ratio


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


                             Enginyeria i Arquitectura la Salle       Slide 5
GRSI
1. Description of XCS
                                                                 2. Facetwise Analysis
                                                                 3. Design of test Problems

       Outline                                                   4. XCS on the one-bit Problem
                                                                 5. Analysis of Deviations
                                                                 6. Results
                                                                 7. Conclusions




        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle                       Slide 6
GRSI
1. Description of XCS
                                                                                                                                         2. Facetwise Analysis
                                                                                                                                         3. Design of test Problems

       Description of XCS
             p                                                                                                                           4. XCS on the one-bit Problem
                                                                                                                                         5. Analysis of Deviations
                                                                                                                                         6. Results
                                                                                                                                         7. Conclusions


       In single-step tasks:
             g      p

                                                                        Environment

                                                                Match Set [M]
                                                                Match Set [M]
            Problem
          Minority
          Majority
       classinstance
             instance
                                                             1C     A   PεF   num as ts exp
                                                             1C     A   PεF   num as ts exp
                                                                                                                              Selected
                                                             3C     A   PεF   num as ts exp
                                                             3C     A   PεF   num as ts exp
                                                                                                                               action
                                                             5C     A   PεF   num as ts exp
                                                             5C     A   PεF   num as ts exp
         Population [P]
         Population [P]                                      6C     A   PεF   num as ts exp
                                                             6C     A   PεF   num as ts exp
                                           Match set
                                          Match set
                                                                                                                                                           REWARD
                                                                            …
                                                                            …
                                          generation
                                          generation
        1C   A   PεF    num as ts exp
        1C   A   PεF    num as ts exp
                                                                                                                             Prediction Array               1000/0
        2C   A   PεF    num as ts exp
        2C   A   PεF    num as ts exp
        3C   A   PεF    num as ts exp
        3C   A   PεF    num as ts exp
                                                                                                                                         …
                                                                                                                              c1 c2            cn
        4C   A   PεF    num as ts exp
        4C   A   PεF    num as ts exp
        5C   A   PεF    num as ts exp
        5C   A   PεF    num as ts exp
        6C   A   PεF    num as ts exp
        6C   A   PεF    num as ts exp                                                                                  Random Action
                                                            Nourished niches
                                                             Starved niches
                   …
                   …
                                                                                            Action S t
                                                                                            A ti Set [A]
                                                                                         1C        A   PεF   num as ts exp
                               Deletion
                                                                                                                                             Classifier
                                                                                         3C        A   PεF   num as ts exp
                                                       Selection, Reproduction,
                                                                                                                                            Parameters
                                                               Mutation
                                                                                         5C        A   PεF   num as ts exp
                                                                                                                                              Update
                                                                                         6C        A   PεF   num as ts exp
                                                                                                         …
                                        Genetic Algorithm


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

                                                              Enginyeria i Arquitectura la Salle                                                             Slide 7
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 8
GRSI
1. Description of XCS
                                                                                     2. Facetwise Analysis
                                                                                     3. Design of test Problems

       Facetwise Analysis
                     y                                                               4. XCS on the one-bit Problem
                                                                                     5. Analysis of Deviations
                                                                                     6. Results
                                                                                     7. 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 size
        Depart 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.




                                          Enginyeria i Arquitectura la Salle                             Slide 9
GRSI
1. Description of XCS
                                                                             2. Facetwise Analysis
                                                                             3. Design of test Problems

       Facetwise Analysis
                     y                                                       4. XCS on the one-bit Problem
                                                                             5. Analysis of Deviations
                                                                             6. Results
                                                                             7. Conclusions




        Assumptions
         – Problems consisting of n classes
         – One class sampled with a lower frequency: minority class

                 num. instances of any class other than the minority class
            ir =
                           num. instances of the minority class

         – Probability of sampling an instance of the minority class:


                           1                                     ir
                 Ps(min) =                             Ps(maj) =
                   (   )                                 ( j)
                          1+ i                                  1+ i
                             ir                                    ir



                                     Enginyeria i Arquitectura la Salle                         Slide 10
GRSI
1. Description of XCS
                                                                     2. Facetwise Analysis
                                                                     3. Design of test Problems

       Facetwise Analysis
                     y                                               4. XCS on the one-bit Problem
                                                                     5. Analysis of Deviations
                                                                     6. Results
                                                                     7. Conclusions




        Facetwise Analysis
         – Population initialization
         – Generation of correct representatives of starved niches
         – Time of extinction of these correct classifiers
         – Population size bound




                                Enginyeria i Arquitectura la Salle                      Slide 11
GRSI
1. Description of XCS
                                                                                             2. Facetwise Analysis
                                                                                             3. Design of test Problems

       Population Initialization
         p                                                                                   4. XCS on the one-bit Problem
                                                                                             5. Analysis of Deviations
                                                                                             6. Results
                                                                                             7. Conclusions




        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
                                           covered, by, least,
          one rule is (Butz et. al, 01):        Population Input
                                                                   specificity    length
                                                                 Initially 1 – P#
                                                                         y

                                                                                           Population size




                                  Enginyeria i Arquitectura la Salle                                            Slide 12
GRSI
1. Description of XCS
                                                             2. Facetwise Analysis
                                                             3. Design of test Problems

       Population Initialization
         p                                                   4. XCS on the one-bit Problem
                                                             5. Analysis of Deviations
                                                             6. Results
                                                             7. Conclusions




                        Enginyeria i Arquitectura la Salle                      Slide 13
GRSI
1. Description of XCS
                                                                     2. Facetwise Analysis
                                                                     3. Design of test Problems

       Facetwise Analysis
                     y                                               4. XCS on the one-bit Problem
                                                                     5. Analysis of Deviations
                                                                     6. Results
                                                                     7. Conclusions




        Facetwise Analysis
         – Population initialization
         – Generation of correct representatives of starved niches
         – Time of extinction of these correct classifiers
         – Population size bound




                                Enginyeria i Arquitectura la Salle                      Slide 14
GRSI
1. Description of XCS

       Creation of Representatives of
                                                                         2. Facetwise Analysis
                                                                         3. Design of test Problems
                                                                         4. XCS on the one-bit Problem

       Starved Niches                                                    5. Analysis of Deviations
                                                                         6. Results
                                                                         7. Conclusions




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




                                    Enginyeria i Arquitectura la Salle                      Slide 15
GRSI
1. Description of XCS

       Creation of Representatives of
                                                                                                   2. Facetwise Analysis
                                                                                                   3. Design of test Problems
                                                                                                   4. XCS on the one-bit Problem

       Starved Niches                                                                              5. Analysis of Deviations
                                                                                                   6. Results
                                                                                                   7. Conclusions




        Possible cases:
                                                                                       1
                                                                            Ps(min) =
         – Sample a minority class instance
                                                                                     1 + ir

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

             • Activate a niche of another class


                                                                                       ir
                                                                            Ps(maj) =
         – Sample a majority class instance
                                                                                     1 + ir

             • Activate a niche of the minority class


             • Activate a niche of another class




                                       Enginyeria i Arquitectura la Salle                                             Slide 16
GRSI
1. Description of XCS

       Creation of Representatives of
                                                                               2. Facetwise Analysis
                                                                               3. Design of test Problems
                                                                               4. XCS on the one-bit Problem

       Starved Niches                                                          5. Analysis of Deviations
                                                                               6. Results
                                                                               7. Conclusions




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


                                                                        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 schema
                    It does not depend on the imbalance ratio




                                   Enginyeria i Arquitectura la Salle                             Slide 17
GRSI
1. Description of XCS
                                                                     2. Facetwise Analysis
                                                                     3. Design of test Problems

       Facetwise Analysis
                     y                                               4. XCS on the one-bit Problem
                                                                     5. Analysis of Deviations
                                                                     6. Results
                                                                     7. Conclusions




        Facetwise Analysis
         – Population initialization
         – Generation of correct representatives of starved niches
         – Time of extinction of these correct classifiers
         – Population size bound




                                Enginyeria i Arquitectura la Salle                      Slide 18
GRSI
1. Description of XCS
                                                                    2. Facetwise Analysis
                                                                    3. Design of test Problems

       Bounding the Population Size
              g       p                                             4. XCS on the one-bit Problem
                                                                    5. Analysis of Deviations
                                                                    6. Results
                                                                    7. Conclusions




        Time to extinction

         – Consider random deletion:




                               Enginyeria i Arquitectura la Salle                      Slide 19
GRSI
1. Description of XCS
                                                                     2. Facetwise Analysis
                                                                     3. Design of test Problems

       Facetwise Analysis
                     y                                               4. XCS on the one-bit Problem
                                                                     5. Analysis of Deviations
                                                                     6. Results
                                                                     7. Conclusions




        Facetwise Analysis
         – Population initialization
         – Generation of correct representatives of starved niches
         – Time of extinction of these correct classifiers
         – Population size bound




                                Enginyeria i Arquitectura la Salle                      Slide 20
GRSI
1. Description of XCS
                                                                  2. Facetwise Analysis
                                                                  3. Design of test Problems

       Bounding the Population Size
              g       p                                           4. XCS on the one-bit Problem
                                                                  5. Analysis of Deviations
                                                                  6. Results
                                                                  7. Conclusions




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

         – Require that:


         – Bound:




                             Enginyeria i Arquitectura la Salle                      Slide 21
GRSI
1. Description of XCS
                                                                         2. Facetwise Analysis
                                                                         3. Design of test Problems

       Bounding the Population Size
              g       p                                                  4. XCS on the one-bit Problem
                                                                         5. Analysis of Deviations
                                                                         6. Results
                                                                         7. Conclusions




        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




                                    Enginyeria i Arquitectura la Salle                      Slide 22
GRSI
1. Description of XCS
                                                                       2. Facetwise Analysis
                                                                       3. Design of test Problems

       Bounding the Population Size
              g       p                                                4. XCS on the one-bit Problem
                                                                       5. Analysis of Deviations
                                                                       6. Results
                                                                       7. Conclusions




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




                          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




                              Enginyeria i Arquitectura la Salle                          Slide 23
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 24
GRSI
1. Description of XCS
                                                                                          2. Facetwise Analysis
                                                                                          3. Design of test Problems

       Design of Test Problems
           g                                                                              4. XCS on the one-bit Problem
                                                                                          5. Analysis of Deviations
                                                                                          6. Results
                                                                                          7. Conclusions




        One-bit problem

                        Condition
                         length (l)
                       000110 :0                             Value of the left-most bit



         – Only two schemas of order one: 0***** and 1*****
         – Undersampling instances of the class labeled as 1


                                           1
                                Ps(min) =
                                         1 + ir




                                      Enginyeria i Arquitectura la Salle                                     Slide 25
GRSI
1. Description of XCS
                                                                                                     2. Facetwise Analysis
                                                                                                     3. Design of test Problems

       Design of Test Problems
           g                                                                                         4. XCS on the one-bit Problem
                                                                                                     5. Analysis of Deviations
                                                                                                     6. Results
                                                                                                     7. Conclusions




        Parity problem
                              Condition
                               length (l)
                                                                                 Number of 1 mod 2
                            01001010 :1
                           Relevant
                           bits ( k)


         – The k bits of parity form a single building block
         – Undersampling instances of the class labeled as 1

                                           1
                                Ps(min) =
                                         1 + ir




                                            Enginyeria i Arquitectura la Salle                                          Slide 26
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 27
GRSI
1. Description of XCS
                                                                         2. Facetwise Analysis
                                                                         3. Design of test Problems

       XCS on the one-bit Problem                                        4. XCS on the one-bit Problem
                                                                         5. Analysis of Deviations
                                                                         6. Results
                                                                         7. 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#=0.6
           selection=tournament, mutation=niched, [A]sub=false, N = 10,000 ir



        Evaluation of the results:
         – Minimum population size to achieve:
                                TP rate * TN rate > 95%


         –R
          Results are averages over 25 seeds
              lt                          d



                                    Enginyeria i Arquitectura la Salle                      Slide 28
GRSI
1. Description of XCS
                                                                           2. Facetwise Analysis
                                                                           3. Design of test Problems

       XCS on the one-bit Problem                                          4. XCS on the one-bit Problem
                                                                           5. Analysis of Deviations
                                                                           6. Results
                                                                           7. Conclusions




                                                           N remains constant up to ir = 64

                                                           N increases linearly from ir=64
                                                           to ir=256

                                                           N increases exponentially from
                                                                          p        y
                                                           ir=256 to ir=1024

                                                           Higher ir could not be solved




                      Enginyeria i Arquitectura la Salle                                      Slide 29
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 30
GRSI
1. Description of XCS
                                                                         2. Facetwise Analysis
                                                                         3. Design of test Problems

       Analysis of the Deviations
           y                                                             4. XCS on the one-bit Problem
                                                                         5. Analysis of Deviations
                                                                         6. Results
                                                                         7. Conclusions




        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 th t b l
            l   ifi    that belong t nourished niches
                                   to    ihd ih
         – These new promising rules inherit parameters from these classifiers.
           This is specially delicate for the action set size (as)
                                                              (as).
         – Approach: initialize as=1.


                                    Enginyeria i Arquitectura la Salle                      Slide 31
GRSI
1. Description of XCS
                                                                          2. Facetwise Analysis
                                                                          3. Design of test Problems

       Analysis of the Deviations
           y                                                              4. XCS on the one-bit Problem
                                                                          5. Analysis of Deviations
                                                                          6. Results
                                                                          7. Conclusions




        Subsumption
         – An overgeneral classifier of the majority class may receive ir positive
           reward before receiving the first negative reward
         – Approach: set θsub>ir
            pp


        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


                                    Enginyeria i Arquitectura la Salle                       Slide 32
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 33
GRSI
1. Description of XCS
                                                                             2. Facetwise Analysis
                                                                             3. Design of test Problems

       XCS+PCM in the one-bit Problem                                        4. XCS on the one-bit Problem
                                                                             5. Analysis of Deviations
                                                                             6. Results
                                                                             7. Conclusions




                                                             N remains constant up to ir = 128

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




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




                      Enginyeria i Arquitectura la Salle                                        Slide 34
GRSI
1. Description of XCS
                                                                           2. Facetwise Analysis
                                                                           3. Design of test Problems

       XCS+PCM in the Parity Problem
                           y                                               4. XCS on the one-bit Problem
                                                                           5. Analysis of Deviations
                                                                           6. Results
                                                                           7. Conclusions




                                                           Building blocks of size 3 need to
                                                           be processed

                                                           Empirical results agree with the
                                                           theory




                                                           Population size bound to guarantee
                                                           P     l ti    ib      dt          t
                                                           that a representative of the niche
                                                           will receive a genetic event




                      Enginyeria i Arquitectura la Salle                                      Slide 35
GRSI
Outline

        1. Description of XCS
        2.
        2 Facetwise Analysis
        3. Design of test Problems
        4. XCS on the one-bit Problem
        5. Analysis f D i ti
        5 A l i of Deviations
        6. Results
        7. Conclusions




                            Enginyeria i Arquitectura la Salle   Slide 36
GRSI
1. Description of XCS
                                                                         2. Facetwise Analysis
                                                                         3. Design of test Problems

       Conclusions and Further Work                                      4. XCS on the one-bit Problem
                                                                         5. Analysis of Deviations
                                                                         6. Results
                                                                         7. 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 population

        Further analysis of the covering operator


                                    Enginyeria i Arquitectura la Salle                      Slide 37
GRSI
Modeling XCS in Class
Imbalances: Population Size
  and Parameter Settings
                       g

    Albert Orriols-Puig1,2                     David E. Goldberg2
          Kumara Sastry2                Ester Bernadó-Mansilla1
                                              Bernadó Mansilla

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

                 2Illinois
                        Genetic Algorithms Laboratory
      Department of Industrial and Enterprise Systems Engineering
              University of Illinois at Urbana Champaign

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GECCO'2007: Modeling XCS in Class Imbalances: Population Size and Parameter Settings

  • 1. Modeling XCS in Class Imbalances: Population Size and Parameter Settings g Albert Orriols-Puig1,2 David E. Goldberg2 Kumara Sastry2 Ester Bernadó-Mansilla1 Bernadó Mansilla 1Research Group in Intelligent Systems Enginyeria i Arquitectura La Salle, Ramon Llull University 2Illinois Genetic Algorithms Laboratory Department of Industrial and Enterprise Systems Engineering University of Illinois at Urbana Champaign
  • 2. Framework New instance Information based Knowledge on experience extraction Learner Model Mdl Domain Predicted Output Examples Consisting Cou te e a p es Counter-examples of , yp y In real-world domains, typically: Higher cost to obtain examples of the concept to be learnt So, distribution of examples in the training dataset is usually imbalanced Applications: Fraud detection Medical diagnosis of rare illnesses Detection of oil spills in satellite images Enginyeria i Arquitectura la Salle Slide 2 GRSI
  • 3. Framework Do learners suffer from class imbalances? – Methods that do global optimization Training Minimize the Learner L Set global error num. errorsc1 + num. errorsc 2 error = Biased towards number examples the overwhelmed class Maximization of the overwhelmed class accuracy, in detriment of the minority class. Enginyeria i Arquitectura la Salle Slide 3 GRSI
  • 4. 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) Enginyeria i Arquitectura la Salle Slide 4 GRSI
  • 5. 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 fails gp Population size bound with respect to the imbalance ratio U til which imbalance ratio would XCS be able to learn Until hi h i b l ti ld b bl t l from the minority class? Enginyeria i Arquitectura la Salle Slide 5 GRSI
  • 6. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Outline 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 6 GRSI
  • 7. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Description of XCS p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions In single-step tasks: g p Environment Match Set [M] Match Set [M] Problem Minority Majority classinstance instance 1C A PεF num as ts exp 1C A PεF num as ts exp Selected 3C A PεF num as ts exp 3C A PεF num as ts exp action 5C A PεF num as ts exp 5C A PεF num as ts exp Population [P] Population [P] 6C A PεF num as ts exp 6C A PεF num as ts exp Match set Match set REWARD … … generation generation 1C A PεF num as ts exp 1C A PεF num as ts exp Prediction Array 1000/0 2C A PεF num as ts exp 2C A PεF num as ts exp 3C A PεF num as ts exp 3C A PεF num as ts exp … c1 c2 cn 4C A PεF num as ts exp 4C A PεF num as ts exp 5C A PεF num as ts exp 5C A PεF num as ts exp 6C A PεF num as ts exp 6C A PεF num as ts exp Random Action Nourished niches Starved niches … … Action S t A ti Set [A] 1C A PεF num as ts exp Deletion Classifier 3C A PεF num as ts exp Selection, Reproduction, Parameters Mutation 5C A PεF num as ts exp Update 6C A PεF num as ts exp … Genetic Algorithm Problem niche: the schema defines the relevant attributes for a particular problem niche. Eg: 10**1* Enginyeria i Arquitectura la Salle Slide 7 GRSI
  • 8. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 8 GRSI
  • 9. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. 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 size Depart 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. Enginyeria i Arquitectura la Salle Slide 9 GRSI
  • 10. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Assumptions – Problems consisting of n classes – One class sampled with a lower frequency: minority class num. instances of any class other than the minority class ir = num. instances of the minority class – Probability of sampling an instance of the minority class: 1 ir Ps(min) = Ps(maj) = ( ) ( j) 1+ i 1+ i ir ir Enginyeria i Arquitectura la Salle Slide 10 GRSI
  • 11. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Facetwise Analysis – Population initialization – Generation of correct representatives of starved niches – Time of extinction of these correct classifiers – Population size bound Enginyeria i Arquitectura la Salle Slide 11 GRSI
  • 12. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Population Initialization p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions 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 covered, by, least, one rule is (Butz et. al, 01): Population Input specificity length Initially 1 – P# y Population size Enginyeria i Arquitectura la Salle Slide 12 GRSI
  • 13. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Population Initialization p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 13 GRSI
  • 14. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Facetwise Analysis – Population initialization – Generation of correct representatives of starved niches – Time of extinction of these correct classifiers – Population size bound Enginyeria i Arquitectura la Salle Slide 14 GRSI
  • 15. 1. Description of XCS Creation of Representatives of 2. Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem Starved Niches 5. Analysis of Deviations 6. Results 7. Conclusions 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? – Specifying correctly all the bits of the schema that represents the starved niche Enginyeria i Arquitectura la Salle Slide 15 GRSI
  • 16. 1. Description of XCS Creation of Representatives of 2. Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem Starved Niches 5. Analysis of Deviations 6. Results 7. Conclusions Possible cases: 1 Ps(min) = – Sample a minority class instance 1 + ir • Activate a niche of the minority class μ: Mutation probability Km: Order of the schema • Activate a niche of another class ir Ps(maj) = – Sample a majority class instance 1 + ir • Activate a niche of the minority class • Activate a niche of another class Enginyeria i Arquitectura la Salle Slide 16 GRSI
  • 17. 1. Description of XCS Creation of Representatives of 2. Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem Starved Niches 5. Analysis of Deviations 6. Results 7. Conclusions Summing up, time to get the first representative of a sta ed c e starved niche 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 schema It does not depend on the imbalance ratio Enginyeria i Arquitectura la Salle Slide 17 GRSI
  • 18. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Facetwise Analysis – Population initialization – Generation of correct representatives of starved niches – Time of extinction of these correct classifiers – Population size bound Enginyeria i Arquitectura la Salle Slide 18 GRSI
  • 19. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Bounding the Population Size g p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Time to extinction – Consider random deletion: Enginyeria i Arquitectura la Salle Slide 19 GRSI
  • 20. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Facetwise Analysis y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Facetwise Analysis – Population initialization – Generation of correct representatives of starved niches – Time of extinction of these correct classifiers – Population size bound Enginyeria i Arquitectura la Salle Slide 20 GRSI
  • 21. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Bounding the Population Size g p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Population size bound to guarantee that there will be representatives o sta ed niches ep ese tat es of starved c es – Require that: – Bound: Enginyeria i Arquitectura la Salle Slide 21 GRSI
  • 22. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Bounding the Population Size g p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions 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 Enginyeria i Arquitectura la Salle Slide 22 GRSI
  • 23. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Bounding the Population Size g p 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Population size bound to guarantee that representatives o sta ed c es of starved niches will receive a ge et c oppo tu ty ece e genetic opportunity: 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 Enginyeria i Arquitectura la Salle Slide 23 GRSI
  • 24. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 24 GRSI
  • 25. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Design of Test Problems g 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions One-bit problem Condition length (l) 000110 :0 Value of the left-most bit – Only two schemas of order one: 0***** and 1***** – Undersampling instances of the class labeled as 1 1 Ps(min) = 1 + ir Enginyeria i Arquitectura la Salle Slide 25 GRSI
  • 26. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Design of Test Problems g 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Parity problem Condition length (l) Number of 1 mod 2 01001010 :1 Relevant bits ( k) – The k bits of parity form a single building block – Undersampling instances of the class labeled as 1 1 Ps(min) = 1 + ir Enginyeria i Arquitectura la Salle Slide 26 GRSI
  • 27. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 27 GRSI
  • 28. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems XCS on the one-bit Problem 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. 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#=0.6 selection=tournament, mutation=niched, [A]sub=false, N = 10,000 ir Evaluation of the results: – Minimum population size to achieve: TP rate * TN rate > 95% –R Results are averages over 25 seeds lt d Enginyeria i Arquitectura la Salle Slide 28 GRSI
  • 29. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems XCS on the one-bit Problem 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions N remains constant up to ir = 64 N increases linearly from ir=64 to ir=256 N increases exponentially from p y ir=256 to ir=1024 Higher ir could not be solved Enginyeria i Arquitectura la Salle Slide 29 GRSI
  • 30. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 30 GRSI
  • 31. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Analysis of the Deviations y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions 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 th t b l l ifi that belong t nourished niches to ihd ih – These new promising rules inherit parameters from these classifiers. This is specially delicate for the action set size (as) (as). – Approach: initialize as=1. Enginyeria i Arquitectura la Salle Slide 31 GRSI
  • 32. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Analysis of the Deviations y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Subsumption – An overgeneral classifier of the majority class may receive ir positive reward before receiving the first negative reward – Approach: set θsub>ir pp 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 Enginyeria i Arquitectura la Salle Slide 32 GRSI
  • 33. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 33 GRSI
  • 34. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems XCS+PCM in the one-bit Problem 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions N remains constant up to ir = 128 For hi h i F higher ir, N slightly increases li htl i We only have to guarantee that a representative of the starved niche will be created Enginyeria i Arquitectura la Salle Slide 34 GRSI
  • 35. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems XCS+PCM in the Parity Problem y 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. Conclusions Building blocks of size 3 need to be processed Empirical results agree with the theory Population size bound to guarantee P l ti ib dt t that a representative of the niche will receive a genetic event Enginyeria i Arquitectura la Salle Slide 35 GRSI
  • 36. Outline 1. Description of XCS 2. 2 Facetwise Analysis 3. Design of test Problems 4. XCS on the one-bit Problem 5. Analysis f D i ti 5 A l i of Deviations 6. Results 7. Conclusions Enginyeria i Arquitectura la Salle Slide 36 GRSI
  • 37. 1. Description of XCS 2. Facetwise Analysis 3. Design of test Problems Conclusions and Further Work 4. XCS on the one-bit Problem 5. Analysis of Deviations 6. Results 7. 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 population Further analysis of the covering operator Enginyeria i Arquitectura la Salle Slide 37 GRSI
  • 38. Modeling XCS in Class Imbalances: Population Size and Parameter Settings g Albert Orriols-Puig1,2 David E. Goldberg2 Kumara Sastry2 Ester Bernadó-Mansilla1 Bernadó Mansilla 1Research Group in Intelligent Systems Enginyeria i Arquitectura La Salle, Ramon Llull University 2Illinois Genetic Algorithms Laboratory Department of Industrial and Enterprise Systems Engineering University of Illinois at Urbana Champaign