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Substructural Surrogates for Learning
Decomposable Classification Problems:
  Implementation and First Results


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

               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
Motivation


           Nearly decomposable functions in engineering systems
               • (Simon,69; Gibson,79; Goldberg,02)


           Design decomposition principle in:

           – Genetic Algorithms (GAs)
               • (Goldberg,02; Pelikan,05; Pelikan, Sastry & Cantú-Paz, 06)
           – Genetic Programming (GPs)
               • (Sastry & Goldberg, 03)
           – Learning Classifier Systems (LCS)
               • (Butz, Pelikan, Llorà & Goldberg, 06)
               • (Llorà, Sastry, Goldberg & de la Ossa, 06)


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


           Our approach: build surrogates for classification.
           – Probabilistic models     Induce the form of a surrogate
           – Regression     Estimate the coefficients of the surrogate
           – Use the surrogate to classify new examples


           Surrogates used for efficiency enhancement of:
           – GAs:
               • (Sastry, Pelikan & Goldberg, 2004)
               • (Pelikan & Sastry, 2004)
               • (Sastry, Lima & Goldberg, 2006)
           – LCSs:
               • (Llorà, Sastry, Yu & Goldberg, 2007)


                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 3
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 4
IWLCS’07
1. Methodology
                                                                                                                    2. Implementation
                                                                                                                    3. Test Problems
     Overview of the Methodology                                                                                    3. Results
                                                                                                                    4. Discussion
                                                                                                                    5. Conclusions




             Structural                                Surrogate                                             Classification
            Model Layer                               Model Layer                                            Model Layer




                                                •Infer structural form                                   •Use the surrogate
           • Extract dependencies
                                                 and the coefficients                                     function to predict the
            between attributes
                                                 based on the                                             class of new
           • Expressed as:
                                                 structural model                                         examples
              • Linkage Groups
              • Matrices                        (Sastry, Lima & Goldberg, 2006)
              • Bayesian Networks
           • Example: x-or



                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                          Slide 5
IWLCS’07
1. Methodology
                                                                                                               2. Implementation
                                                                                                               3. Test Problems
     Surrogate Model Layer                                                                                     3. Results
                                                                                                               4. Discussion
                                                                                                               5. Conclusions




           Input: Database of labeled examples with nominal attributes


           Substructure identified by the structural model layer
           – Example: [0 2] [1]

           Surrogate form induced from the structural model:

                  f(x0, x1, x2) = cI + c0x0 + c1x1 + c2x2 + c0,2x0x2


           Coefficients of surrogates: linear regression methods



                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 6
IWLCS’07
1. Methodology
                                                                                                                       2. Implementation
                                                                                                                       3. Test Problems
     Surrogate Model Layer                                                                                             3. Results
                                                                                                                       4. Discussion
                                                                                                                       5. Conclusions




           Map input examples to schema-index matrix
           – Consider all the schemas:



                                   {0*0, 0*1, 1*0, 1*1, *0*, *1*}
                                    [0 2]          [1]
           – In general, the number of schemas to be considered is:


                                                               Number of linkage
                                                                   groups


                                                                                               Cardinality of the variable jth
                                                                                                 of the ith linkage group




                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                               Slide 7
IWLCS’07
1. Methodology
                                                                                                                     2. Implementation
                                                                                                                     3. Test Problems
     Surrogate Model Layer                                                                                           3. Results
                                                                                                                     4. Discussion
                                                                                                                     5. Conclusions




           Map each example to a binary vector of size msch
           – For each building block, the entry corresponding to the schemata contained
             in the example is one. All the other are zero.

                                                                [0 2]                                 [1]

                                                 0*0           0*1        1*0        1*1         *0*        *1*

                            010                    1            0           0          0          0              1

                            100                    0            0           1          0          1              0

                            111                    0            0           0          1          0              1




                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                           Slide 8
IWLCS’07
1. Methodology
                                                                                                              2. Implementation
                                                                                                              3. Test Problems
     Surrogate Model Layer                                                                                    3. Results
                                                                                                              4. Discussion
                                                                                                              5. Conclusions




           This results in the following matrix



                                                                                                  n input
                                                                                                 examples




           Each class is mapped to an integer




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 9
IWLCS’07
1. Methodology
                                                                                                                     2. Implementation
                                                                                                                     3. Test Problems
     Surrogate Model Layer                                                                                           3. Results
                                                                                                                     4. Discussion
                                                                                                                     5. Conclusions




           Solve the following linear system to estimate the
           coefficients of the surrogate model

                Input instances mapped                                           Class or label of each
                 to the schemas matrix                                              input instance

                                                  Coefficients of the
                                                   surrogate model



           To solve it, we use multi-dimensional least squares fitting
           approach:


           – Minimize:




                               Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 10
IWLCS’07
1. Methodology
                                                                                                                    2. Implementation
                                                                                                                    3. Test Problems
     Overview of the Methodology                                                                                    3. Results
                                                                                                                    4. Discussion
                                                                                                                    5. Conclusions




             Structural                                Surrogate                                             Classification
            Model Layer                               Model Layer                                            Model Layer




                                                •Infer structural form                                   •Use the surrogate
           • Extract dependencies
                                                 and the coefficients                                     function to predict the
            between attributes
                                                 based on the                                             class of new
           • Expressed as:
                                                 structural model                                         examples
              • Linkage Groups
              • Matrices                        (Sastry, Lima & Goldberg, 2006)
              • Bayesian Networks
           • Example: x-or



                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                          Slide 11
IWLCS’07
1. Methodology
                                                                                                                       2. Implementation
                                                                                                                       3. Test Problems
     Classification Model Layer                                                                                        3. Results
                                                                                                                       4. Discussion
                                                                                                                       5. Conclusions




           Use the surrogate to predict the class of a new input instance



                                              Map the example
              New input                  to a vector e of length msch                                     Output = round (x·e)
               example                   according to the schemas
                                                                                                                         Coefficients of the
                                                                                                                          surrogate model




                                   0*0        0*1       1*0      1*1       *0*      *1*
                                                                                                                Output = x·(001010)’
                001                  0          0        1         0        1        0




                          Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                              Slide 12
IWLCS’07
1. Methodology
                                                                                                                    2. Implementation
                                                                                                                    3. Test Problems
     Overview of the Methodology                                                                                    3. Results
                                                                                                                    4. Discussion
                                                                                                                    5. Conclusions




             Structural                                Surrogate                                             Classification
            Model Layer                               Model Layer                                            Model Layer




                                                •Infer structural form                                   •Use the surrogate
           • Extract dependencies
                                                 and the coefficients                                     function to predict the
            between attributes
                                                 based on the                                             class of new
           • Expressed as:
                                                 structural model                                         examples
              • Linkage Groups
              • Matrices                        (Sastry, Lima & Goldberg, 2006)
              • Bayesian Networks
           • Example: x-or



                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                          Slide 13
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 14
IWLCS’07
1. Methodology

     How do we obtain the structural                                                                             2. Implementation
                                                                                                                 3. Test Problems
                                                                                                                 3. Results

     model?                                                                                                      4. Discussion
                                                                                                                 5. Conclusions




           Greedy extraction of the structural model for classification
           (gESMC): Greedy search à la eCGA (Harik, 98)
           1. Start considering all variables independent. sModel = [1] [2] … [n]
           2. Divide the dataset in ten-fold cross-validation sets: train and test
           3. Build the surrogate with the train set and evaluate with the test set
           4. While there is improvement
               1. Create all combinations of two subset merges
                    Eg. { [1,2] .. [n]; [1,n] .. [2]; [1]..[2,n]}
               2. Build the surrogate for all candidates. Evaluate the quality with the test set
               3. Select the candidate with minimum error
               4. If the quality of the candidate is better than the parent, accept the model
                  (update smodel) and go to 4.
               5. Otherwise, stop. sModel is optimal.


                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 15
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 16
IWLCS’07
1. Methodology
                                                                                                                 2. Implementation
                                                                                                                 3. Test Problems
     Test Problems                                                                                               3. Results
                                                                                                                 4. Discussion
                                                                                                                 5. Conclusions




           Two-level hierarchical problems (Butz, Pelikan, Llorà & Goldberg, 2006)




                           Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 17
IWLCS’07
1. Methodology
                                                                                                                2. Implementation
                                                                                                                3. Test Problems
     Test Problems                                                                                              3. Results
                                                                                                                4. Discussion
                                                                                                                5. Conclusions




           Problems in the lower level

                                                  Condition
                                                   length (l)
                                                                                               Position of the left-most
           – Position Problem:                  000110 :3                                           one-valued bit



                                                  Condition
                                                  length (l)

           – Parity Problem:                                                                      Number of 1 mod 2
                                              01001010 :1




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                         Slide 18
IWLCS’07
1. Methodology
                                                                                                                2. Implementation
                                                                                                                3. Test Problems
     Test Problems                                                                                              3. Results
                                                                                                                4. Discussion
                                                                                                                5. Conclusions




           Problems in the higher level

                                                    Condition
                                                    length (l)
                                                                                              Integer value of the input
           – Decoder Problem:                      000110 :5



                                                       Condition
                                                       length (l)
           – Count-ones Problem:                                                               Number of ones of the input
                                                      000110 :2

           Combinations of both levels:
           – Parity + decoder (hParDec)                              - Parity + count-ones (hParCount)
           – Position + decoder (hPosDec)                             - Position + count-ones (hPosCount)



                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                         Slide 19
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 20
IWLCS’07
1. Methodology
                                                                                                                    2. Implementation
                                                                                                                    3. Test Problems
     Results with 2-bit lower level blocks                                                                          3. Results
                                                                                                                    4. Discussion
                                                                                                                    5. Conclusions




           First experiment:
           –   hParDec, hPosDec, hParCount, hPosCount
           –   Binary inputs of length 17.
           –   The first 6 bits are relevant. The others are irrelevant
           –   3 blocks of two-bits low order problems
           –   Eg: hPosDec


                                                                                                                Interacting variables:
                                     10 01 00 00101010101                                                       [x0 x1] [x2 x3] [x4 x5]

                                                                                       irrelevant bits
                                       210
                                                                   Output = 21


                          Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                           Slide 21
IWLCS’07
1. Methodology
                                                                                                           2. Implementation
                                                                                                           3. Test Problems
     Results with 2-bit lower level blocks                                                                 3. Results
                                                                                                           4. Discussion
                                                                                                           5. Conclusions




           Performance of gESMC compared to SMO and C4.5:




                     Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 22
IWLCS’07
1. Methodology
                                                                                                               2. Implementation
                                                                                                               3. Test Problems
     Results with 2-bit lower level blocks                                                                     3. Results
                                                                                                               4. Discussion
                                                                                                               5. Conclusions




           Form and coefficients of the surrogate models
           – Interacting variables [x0 x1] [x2 x3] [x4 x5]




           – Linkages between variables are detected
           – Easy visualization of the salient variable interactions
           – All models only consist of the six relevant variables

                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 23
IWLCS’07
1. Methodology
                                                                                                               2. Implementation
                                                                                                               3. Test Problems
     Results with 2-bit lower level blocks                                                                     3. Results
                                                                                                               4. Discussion
                                                                                                               5. Conclusions




           C4.5 created trees:
           – HPosDec and HPosCount
               • Trees of 53 nodes and 27 leaves on average
               • Only the six relevant variables in the decision nodes
           – HParDec
               • Trees of 270 nodes and 136 leaves on average
               • Irrelevant attributes in the decision nodes
           – HParCount
               • Trees of 283 nodes and 142 leaves on average
               • Irrelevant attributes in the decision nodes


           SMO built support vector machines:
           – 351, 6, 6, and 28 machines with 17 weights each one were
             created for HPosDec, HPosCount, HParDec, and HParCount


                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 24
IWLCS’07
1. Methodology
                                                                                                                    2. Implementation
                                                                                                                    3. Test Problems
     Results with 3-bit lower level blocks                                                                          3. Results
                                                                                                                    4. Discussion
                                                                                                                    5. Conclusions




           Second experiment:
           –   hParDec, hPosDec, hParCount, hPosCount
           –   Binary inputs of length 17.
           –   The first 6 bits are relevant. The others are irrelevant
           –   2 blocks of three-bits low order problems
           –   Eg: hPosDec


                                                                                                                Interacting variables:
                                     000 100 00101010101                                                         [x0 x1 x2] [x3 x4 x5]

                                                                                       irrelevant bits
                                        0             3
                                                                   Output = 3


                          Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                           Slide 25
IWLCS’07
1. Methodology
                                                                                                                  2. Implementation
                                                                                                                  3. Test Problems
     Results with 3-bit lower level blocks                                                                        3. Results
                                                                                                                  4. Discussion
                                                                                                                  5. Conclusions




           Second experiment:
           – 2 blocks of three-bits low order problems




                                                           Interesting observations:
                                                           •As expected, the method stops the search when no
   M1: [0][1][2][3][4][5]                                  improvement is found
   M2: [0 1][2][3][4][5]                                   •In HParDec, half of the times the system gets a
                                                           model that represents the salient interacting
   M3: [0 1 2][3][4][5]                                    variables due to the stochasticity of the 10-fold CV
                                                           estimation. Eg. [0 1 2 8] [3 4 9 5 6]

                            Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 26
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 27
IWLCS’07
1. Methodology
                                                                                                              2. Implementation
                                                                                                              3. Test Problems
     Discussion                                                                                               3. Results
                                                                                                              4. Discussion
                                                                                                              5. Conclusions




           Lack of guidance from low-order substructures
           – Increase the order of substructural merges




           – Select randomly one of the new structural models
               • Follow schemes such as simulated annealing (Korst & Aarts, 1997)




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 28
IWLCS’07
1. Methodology
                                                                                                               2. Implementation
                                                                                                               3. Test Problems
     Discussion                                                                                                3. Results
                                                                                                               4. Discussion
                                                                                                               5. Conclusions




           Creating substructural models with overlapping
           structures
           – Problems with overlapping linkages
           – Eg: multiplexer problems
           – gESMC on the 6-bit mux results in: [0 1 2 3 4 5 ]
           – What we would like to obtain is: [0 1 2] [0 1 3] [0 1 4] [0 1 5]
           – Enhance the methodology permitting to represent overlapping
             linkages, using methods such as the design structure matrix
             genetic algorithm (DMSGA) (Yu, 2006)




                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 29
IWLCS’07
Outline


           1. Methodology

           2. Implementation

           3. Test Problems

           4. Results

           5. Discussion

           6. Conclusions




                        Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems   Slide 30
IWLCS’07
1. Methodology
                                                                                                               2. Implementation
                                                                                                               3. Test Problems
     Conclusions                                                                                               3. Results
                                                                                                               4. Discussion
                                                                                                               5. Conclusions




           Methodology for learning from decomposable problems
           – Structural model
           – Surrogate model
           – Classification model
           First implementation approach
           – greedy search for the best structural model
           Main advantage of gESMC over other learners
           – It provides the structural model jointly with the classification
             model.
           Some limitations of gESMC:
           – It depends on the guidance on the low-order structure
           – Several approaches outlined as further work.

                         Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems                       Slide 31
IWLCS’07
Substructural Surrogates for Learning
Decomposable Classification Problems:
  Implementation and First Results


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

               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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Substructrual surrogates for learning decomposable classification problems: implementation and first results

  • 1. Substructural Surrogates for Learning Decomposable Classification Problems: Implementation and First Results Albert Orriols-Puig1,2 Kumara Sastry2 David E. Goldberg2 Ester Bernadó-Mansilla1 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. Motivation Nearly decomposable functions in engineering systems • (Simon,69; Gibson,79; Goldberg,02) Design decomposition principle in: – Genetic Algorithms (GAs) • (Goldberg,02; Pelikan,05; Pelikan, Sastry & Cantú-Paz, 06) – Genetic Programming (GPs) • (Sastry & Goldberg, 03) – Learning Classifier Systems (LCS) • (Butz, Pelikan, Llorà & Goldberg, 06) • (Llorà, Sastry, Goldberg & de la Ossa, 06) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 2 IWLCS’07
  • 3. Aim Our approach: build surrogates for classification. – Probabilistic models Induce the form of a surrogate – Regression Estimate the coefficients of the surrogate – Use the surrogate to classify new examples Surrogates used for efficiency enhancement of: – GAs: • (Sastry, Pelikan & Goldberg, 2004) • (Pelikan & Sastry, 2004) • (Sastry, Lima & Goldberg, 2006) – LCSs: • (Llorà, Sastry, Yu & Goldberg, 2007) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 3 IWLCS’07
  • 4. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 4 IWLCS’07
  • 5. 1. Methodology 2. Implementation 3. Test Problems Overview of the Methodology 3. Results 4. Discussion 5. Conclusions Structural Surrogate Classification Model Layer Model Layer Model Layer •Infer structural form •Use the surrogate • Extract dependencies and the coefficients function to predict the between attributes based on the class of new • Expressed as: structural model examples • Linkage Groups • Matrices (Sastry, Lima & Goldberg, 2006) • Bayesian Networks • Example: x-or Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 5 IWLCS’07
  • 6. 1. Methodology 2. Implementation 3. Test Problems Surrogate Model Layer 3. Results 4. Discussion 5. Conclusions Input: Database of labeled examples with nominal attributes Substructure identified by the structural model layer – Example: [0 2] [1] Surrogate form induced from the structural model: f(x0, x1, x2) = cI + c0x0 + c1x1 + c2x2 + c0,2x0x2 Coefficients of surrogates: linear regression methods Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 6 IWLCS’07
  • 7. 1. Methodology 2. Implementation 3. Test Problems Surrogate Model Layer 3. Results 4. Discussion 5. Conclusions Map input examples to schema-index matrix – Consider all the schemas: {0*0, 0*1, 1*0, 1*1, *0*, *1*} [0 2] [1] – In general, the number of schemas to be considered is: Number of linkage groups Cardinality of the variable jth of the ith linkage group Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 7 IWLCS’07
  • 8. 1. Methodology 2. Implementation 3. Test Problems Surrogate Model Layer 3. Results 4. Discussion 5. Conclusions Map each example to a binary vector of size msch – For each building block, the entry corresponding to the schemata contained in the example is one. All the other are zero. [0 2] [1] 0*0 0*1 1*0 1*1 *0* *1* 010 1 0 0 0 0 1 100 0 0 1 0 1 0 111 0 0 0 1 0 1 Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 8 IWLCS’07
  • 9. 1. Methodology 2. Implementation 3. Test Problems Surrogate Model Layer 3. Results 4. Discussion 5. Conclusions This results in the following matrix n input examples Each class is mapped to an integer Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 9 IWLCS’07
  • 10. 1. Methodology 2. Implementation 3. Test Problems Surrogate Model Layer 3. Results 4. Discussion 5. Conclusions Solve the following linear system to estimate the coefficients of the surrogate model Input instances mapped Class or label of each to the schemas matrix input instance Coefficients of the surrogate model To solve it, we use multi-dimensional least squares fitting approach: – Minimize: Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 10 IWLCS’07
  • 11. 1. Methodology 2. Implementation 3. Test Problems Overview of the Methodology 3. Results 4. Discussion 5. Conclusions Structural Surrogate Classification Model Layer Model Layer Model Layer •Infer structural form •Use the surrogate • Extract dependencies and the coefficients function to predict the between attributes based on the class of new • Expressed as: structural model examples • Linkage Groups • Matrices (Sastry, Lima & Goldberg, 2006) • Bayesian Networks • Example: x-or Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 11 IWLCS’07
  • 12. 1. Methodology 2. Implementation 3. Test Problems Classification Model Layer 3. Results 4. Discussion 5. Conclusions Use the surrogate to predict the class of a new input instance Map the example New input to a vector e of length msch Output = round (x·e) example according to the schemas Coefficients of the surrogate model 0*0 0*1 1*0 1*1 *0* *1* Output = x·(001010)’ 001 0 0 1 0 1 0 Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 12 IWLCS’07
  • 13. 1. Methodology 2. Implementation 3. Test Problems Overview of the Methodology 3. Results 4. Discussion 5. Conclusions Structural Surrogate Classification Model Layer Model Layer Model Layer •Infer structural form •Use the surrogate • Extract dependencies and the coefficients function to predict the between attributes based on the class of new • Expressed as: structural model examples • Linkage Groups • Matrices (Sastry, Lima & Goldberg, 2006) • Bayesian Networks • Example: x-or Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 13 IWLCS’07
  • 14. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 14 IWLCS’07
  • 15. 1. Methodology How do we obtain the structural 2. Implementation 3. Test Problems 3. Results model? 4. Discussion 5. Conclusions Greedy extraction of the structural model for classification (gESMC): Greedy search à la eCGA (Harik, 98) 1. Start considering all variables independent. sModel = [1] [2] … [n] 2. Divide the dataset in ten-fold cross-validation sets: train and test 3. Build the surrogate with the train set and evaluate with the test set 4. While there is improvement 1. Create all combinations of two subset merges Eg. { [1,2] .. [n]; [1,n] .. [2]; [1]..[2,n]} 2. Build the surrogate for all candidates. Evaluate the quality with the test set 3. Select the candidate with minimum error 4. If the quality of the candidate is better than the parent, accept the model (update smodel) and go to 4. 5. Otherwise, stop. sModel is optimal. Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 15 IWLCS’07
  • 16. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 16 IWLCS’07
  • 17. 1. Methodology 2. Implementation 3. Test Problems Test Problems 3. Results 4. Discussion 5. Conclusions Two-level hierarchical problems (Butz, Pelikan, Llorà & Goldberg, 2006) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 17 IWLCS’07
  • 18. 1. Methodology 2. Implementation 3. Test Problems Test Problems 3. Results 4. Discussion 5. Conclusions Problems in the lower level Condition length (l) Position of the left-most – Position Problem: 000110 :3 one-valued bit Condition length (l) – Parity Problem: Number of 1 mod 2 01001010 :1 Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 18 IWLCS’07
  • 19. 1. Methodology 2. Implementation 3. Test Problems Test Problems 3. Results 4. Discussion 5. Conclusions Problems in the higher level Condition length (l) Integer value of the input – Decoder Problem: 000110 :5 Condition length (l) – Count-ones Problem: Number of ones of the input 000110 :2 Combinations of both levels: – Parity + decoder (hParDec) - Parity + count-ones (hParCount) – Position + decoder (hPosDec) - Position + count-ones (hPosCount) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 19 IWLCS’07
  • 20. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 20 IWLCS’07
  • 21. 1. Methodology 2. Implementation 3. Test Problems Results with 2-bit lower level blocks 3. Results 4. Discussion 5. Conclusions First experiment: – hParDec, hPosDec, hParCount, hPosCount – Binary inputs of length 17. – The first 6 bits are relevant. The others are irrelevant – 3 blocks of two-bits low order problems – Eg: hPosDec Interacting variables: 10 01 00 00101010101 [x0 x1] [x2 x3] [x4 x5] irrelevant bits 210 Output = 21 Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 21 IWLCS’07
  • 22. 1. Methodology 2. Implementation 3. Test Problems Results with 2-bit lower level blocks 3. Results 4. Discussion 5. Conclusions Performance of gESMC compared to SMO and C4.5: Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 22 IWLCS’07
  • 23. 1. Methodology 2. Implementation 3. Test Problems Results with 2-bit lower level blocks 3. Results 4. Discussion 5. Conclusions Form and coefficients of the surrogate models – Interacting variables [x0 x1] [x2 x3] [x4 x5] – Linkages between variables are detected – Easy visualization of the salient variable interactions – All models only consist of the six relevant variables Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 23 IWLCS’07
  • 24. 1. Methodology 2. Implementation 3. Test Problems Results with 2-bit lower level blocks 3. Results 4. Discussion 5. Conclusions C4.5 created trees: – HPosDec and HPosCount • Trees of 53 nodes and 27 leaves on average • Only the six relevant variables in the decision nodes – HParDec • Trees of 270 nodes and 136 leaves on average • Irrelevant attributes in the decision nodes – HParCount • Trees of 283 nodes and 142 leaves on average • Irrelevant attributes in the decision nodes SMO built support vector machines: – 351, 6, 6, and 28 machines with 17 weights each one were created for HPosDec, HPosCount, HParDec, and HParCount Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 24 IWLCS’07
  • 25. 1. Methodology 2. Implementation 3. Test Problems Results with 3-bit lower level blocks 3. Results 4. Discussion 5. Conclusions Second experiment: – hParDec, hPosDec, hParCount, hPosCount – Binary inputs of length 17. – The first 6 bits are relevant. The others are irrelevant – 2 blocks of three-bits low order problems – Eg: hPosDec Interacting variables: 000 100 00101010101 [x0 x1 x2] [x3 x4 x5] irrelevant bits 0 3 Output = 3 Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 25 IWLCS’07
  • 26. 1. Methodology 2. Implementation 3. Test Problems Results with 3-bit lower level blocks 3. Results 4. Discussion 5. Conclusions Second experiment: – 2 blocks of three-bits low order problems Interesting observations: •As expected, the method stops the search when no M1: [0][1][2][3][4][5] improvement is found M2: [0 1][2][3][4][5] •In HParDec, half of the times the system gets a model that represents the salient interacting M3: [0 1 2][3][4][5] variables due to the stochasticity of the 10-fold CV estimation. Eg. [0 1 2 8] [3 4 9 5 6] Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 26 IWLCS’07
  • 27. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 27 IWLCS’07
  • 28. 1. Methodology 2. Implementation 3. Test Problems Discussion 3. Results 4. Discussion 5. Conclusions Lack of guidance from low-order substructures – Increase the order of substructural merges – Select randomly one of the new structural models • Follow schemes such as simulated annealing (Korst & Aarts, 1997) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 28 IWLCS’07
  • 29. 1. Methodology 2. Implementation 3. Test Problems Discussion 3. Results 4. Discussion 5. Conclusions Creating substructural models with overlapping structures – Problems with overlapping linkages – Eg: multiplexer problems – gESMC on the 6-bit mux results in: [0 1 2 3 4 5 ] – What we would like to obtain is: [0 1 2] [0 1 3] [0 1 4] [0 1 5] – Enhance the methodology permitting to represent overlapping linkages, using methods such as the design structure matrix genetic algorithm (DMSGA) (Yu, 2006) Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 29 IWLCS’07
  • 30. Outline 1. Methodology 2. Implementation 3. Test Problems 4. Results 5. Discussion 6. Conclusions Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 30 IWLCS’07
  • 31. 1. Methodology 2. Implementation 3. Test Problems Conclusions 3. Results 4. Discussion 5. Conclusions Methodology for learning from decomposable problems – Structural model – Surrogate model – Classification model First implementation approach – greedy search for the best structural model Main advantage of gESMC over other learners – It provides the structural model jointly with the classification model. Some limitations of gESMC: – It depends on the guidance on the low-order structure – Several approaches outlined as further work. Illinois Genetic Algorithms Laboratory and Group of Research in Intelligent Systems Slide 31 IWLCS’07
  • 32. Substructural Surrogates for Learning Decomposable Classification Problems: Implementation and First Results Albert Orriols-Puig1,2 Kumara Sastry2 David E. Goldberg2 Ester Bernadó-Mansilla1 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