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A Pareto-Compliant Surrogate Approach
    for Multiobjective Optimization


     Ilya Loshchilov, Marc Schoenauer and Michéle Sebag
              TAO, INRIA, Univesité Paris-Sud XI
Main Idea → Pareto Surrogate


        WE HAVE:
          • Current Pareto and dominated points
        WE WANT TO FIND F: decision space →
          • F < 0 for dominated points

                • F ≈ 0 for current Pareto points
        WE EXPECT:
          • F > 0 for new Pareto points



A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 2
Outline


          Support Vector Machine (SVM)
          A Pareto-Compliant Approach for MOO
          Experiments
          Discussion and on-going work




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 3
Support Vector Machine (1): Classification




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 4
Support Vector Machine (2): Regression




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 5
Support Vector Machine (3): One-Class SVM




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 6
Outline


          Support Vector Machine (SVM)
          A Pareto-Compliant Approach for MOO
          Experiments
          Discussion and on-going work




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 7
Pareto Support Vector Machine (Pareto-SVM)




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 8
SVM-informed EMOA

     How we can use F?
         For optimization
                 • If we are sure that the model F is accurate enough far
                   from the current Pareto points
         For filtering (select the most promising children according to F):
                 • F(children) > F(parent) → dangerous because of
                   approximation noise (epsilon)

                 • F(children) > F(nearest point from population) is more
                   robust criterion




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.        Page 9
SVM-informed EMOA: Filtering
     Filtering procedure:
         Generate            pre-children
         For each pre-children and the nearest parent
         calculate
         New children is the point with the maximum value of




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 10
Outline


          Support Vector Machine (SVM)
          A Pareto-Compliant Approach for MOO
          Experiments
          Discussion and on-going work




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 11
Pareto-SVM model validation

           Generate 20.000 points at a given distance from a (known)
           nearly-optimal Pareto front

           Apply Non-dominated sorting to rank non-dominated fronts
                     , where

           Choose             and        as the training data,
           where             – current Pareto points,       – dominated points

           Update Pareto-SVM model

           Compare                  on different           (use          as the Test data)




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.                       Page 12
Pareto-SVM model for ZDT1 and IHR1 problems


           A, b,c,d




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 13
SVM-informed S-NSGA-2 and MO-CMA-ES on ZDT's and IHR's




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 14
Strong and Weak Points


       + Speedup of 2-2.5 on ZDT and IHR for first 15 000 evaluations
       + CPU cost of one learning ~ 0.5-1.0 sec. (complete run<15mn)


       - High selection pressure may lead to premature convergence;
       - The formulation of the SVM optimization problem still requires the
          user decision*.




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.        Page 15
Issue: A more robust Pareto-SVM formulation

       Original Formulation:
       In feature space:
       - Pareto set → cluster around single line
       - Dominated points → one side of this line
       Issues:
         - Which side?
            - Preserve symmetry (            and                         )
       Solutions(?):
            - Manually choose
       - New Formulation:
              D: additional parameter → harder learning problem




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.       Page 16
Perspective (1): Quality indicator-based Mono surrogate




           Regression of indicator values (e.g. hypervolume)
           Issues: - value for dominated points?
                   - value for extremal points?


A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 17
Perspective (2): Rank-SVM (Ordinal Regression SVM)
             Ordinal Regression SVM: surrogate constrained by ordered pairs
             MO-case: derive constraints from Pareto dominance
                      If X dominates Y, add constraint (F(X)<F(Y))
             Learning time
                     - 'quadratic' with number of constraints
                     - quasi-linear with dimension
             Which constraints?


                                                  Results for single objective problems(*):
                                                  Cost of learning/testing with n-1 constraints
                                                  increases quasi-linearly with dimension.


                                                  (*) - "Comparison-Based Optimizers Need Comparison-Based
                                                   Surrogates” I.Loshchilov, M. Schoenauer, M. Sebag.
                                                   In PPSN-XI (2010).



A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.                                Page 18
Summary

       Contributions
            Single-valued surrogate to capture Pareto
            dominance
            Several alternative formulations
            Using surrogate as filter limits possible speedup
       Further work
            Comparison with multi-surrogate
            Rank-based SVM

A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 19
Thank you




A Pareto-Compliant Surrogate Approach for Multiobjective Optimization.   Page 20

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A Pareto-Compliant Surrogate Approach for Multiobjective Optimization

  • 1. A Pareto-Compliant Surrogate Approach for Multiobjective Optimization Ilya Loshchilov, Marc Schoenauer and Michéle Sebag TAO, INRIA, Univesité Paris-Sud XI
  • 2. Main Idea → Pareto Surrogate WE HAVE: • Current Pareto and dominated points WE WANT TO FIND F: decision space → • F < 0 for dominated points • F ≈ 0 for current Pareto points WE EXPECT: • F > 0 for new Pareto points A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 2
  • 3. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going work A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 3
  • 4. Support Vector Machine (1): Classification A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 4
  • 5. Support Vector Machine (2): Regression A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 5
  • 6. Support Vector Machine (3): One-Class SVM A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 6
  • 7. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going work A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 7
  • 8. Pareto Support Vector Machine (Pareto-SVM) A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 8
  • 9. SVM-informed EMOA How we can use F? For optimization • If we are sure that the model F is accurate enough far from the current Pareto points For filtering (select the most promising children according to F): • F(children) > F(parent) → dangerous because of approximation noise (epsilon) • F(children) > F(nearest point from population) is more robust criterion A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 9
  • 10. SVM-informed EMOA: Filtering Filtering procedure: Generate pre-children For each pre-children and the nearest parent calculate New children is the point with the maximum value of A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 10
  • 11. Outline Support Vector Machine (SVM) A Pareto-Compliant Approach for MOO Experiments Discussion and on-going work A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 11
  • 12. Pareto-SVM model validation Generate 20.000 points at a given distance from a (known) nearly-optimal Pareto front Apply Non-dominated sorting to rank non-dominated fronts , where Choose and as the training data, where – current Pareto points, – dominated points Update Pareto-SVM model Compare on different (use as the Test data) A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 12
  • 13. Pareto-SVM model for ZDT1 and IHR1 problems A, b,c,d A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 13
  • 14. SVM-informed S-NSGA-2 and MO-CMA-ES on ZDT's and IHR's A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 14
  • 15. Strong and Weak Points + Speedup of 2-2.5 on ZDT and IHR for first 15 000 evaluations + CPU cost of one learning ~ 0.5-1.0 sec. (complete run<15mn) - High selection pressure may lead to premature convergence; - The formulation of the SVM optimization problem still requires the user decision*. A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 15
  • 16. Issue: A more robust Pareto-SVM formulation Original Formulation: In feature space: - Pareto set → cluster around single line - Dominated points → one side of this line Issues: - Which side? - Preserve symmetry ( and ) Solutions(?): - Manually choose - New Formulation: D: additional parameter → harder learning problem A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 16
  • 17. Perspective (1): Quality indicator-based Mono surrogate Regression of indicator values (e.g. hypervolume) Issues: - value for dominated points? - value for extremal points? A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 17
  • 18. Perspective (2): Rank-SVM (Ordinal Regression SVM) Ordinal Regression SVM: surrogate constrained by ordered pairs MO-case: derive constraints from Pareto dominance If X dominates Y, add constraint (F(X)<F(Y)) Learning time - 'quadratic' with number of constraints - quasi-linear with dimension Which constraints? Results for single objective problems(*): Cost of learning/testing with n-1 constraints increases quasi-linearly with dimension. (*) - "Comparison-Based Optimizers Need Comparison-Based Surrogates” I.Loshchilov, M. Schoenauer, M. Sebag. In PPSN-XI (2010). A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 18
  • 19. Summary Contributions Single-valued surrogate to capture Pareto dominance Several alternative formulations Using surrogate as filter limits possible speedup Further work Comparison with multi-surrogate Rank-based SVM A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 19
  • 20. Thank you A Pareto-Compliant Surrogate Approach for Multiobjective Optimization. Page 20