Learning MR-Sort rules with coalitional veto Olivier Sobrie 1,2 Vincent Mousseau 1 Marc Pirlot 2 1 Université Paris-Saclay - CentraleSupélec 2 Université de Mons - Faculté polytechnique November 7, 2016 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 1 / 24
1 Sorting problem 2 MR-Sort 3 Learning a MR-Sort model 4 MR-Sort with coalitional veto 5 Learning a MR-SortCV model 6 Experimental results 7 Conclusion Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 2 / 24
1. Sorting problem 1 Sorting problem 2 MR-Sort 3 Learning a MR-Sort model 4 MR-Sort with coalitional veto 5 Learning a MR-SortCV model 6 Experimental results 7 Conclusion Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 3 / 24
1. Sorting problem Sorting problem Settings ◮ Assignment of alternatives in categories ◮ Categories are ordered ◮ Alternatives are evaluated on monotone criteria Example of sorting problem ◮ Assignment of hotels in two categories : “Bad” and “Good” . . . distance to the beach 600m 300m 50m 200m . . . distance to the center 500m 100m 600m 300m . . . price 150 e 130 e 90 e 80 e . . . 45m 2 35m 2 30m 2 25m 2 size . . . rating . . . Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 4 / 24
1. Sorting problem Sorting problem Settings ◮ Assignment of alternatives in categories ◮ Categories are ordered ◮ Alternatives are evaluated on monotone criteria Example of sorting problem ◮ Assignment of hotels in two categories : “Bad” and “Good” Good Bad Front Maritim Travelhodge Majestic Plaza Rambla Hotel W ≻ Hilton Miramar Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 4 / 24
2. MR-Sort 1 Sorting problem 2 MR-Sort 3 Learning a MR-Sort model 4 MR-Sort with coalitional veto 5 Learning a MR-SortCV model 6 Experimental results 7 Conclusion Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 5 / 24
2. MR-Sort Majority rule sorting model ◮ Sorting model ( p ordered categories , i.e. C p ≻ C p − 1 ≻ . . . ≻ C 1 ) ◮ Axiomatized by Bouyssou and Marchant (2007a,b) ◮ n weights ( w 1 , . . . , w n ) ◮ 1 majority threshold ( λ ) C 3 ◮ p − 1 profiles ( b 1 , . . . , b p − 1 ) b 2 C 2 Assignment rule b 1 a ∈ C h C 1 ⇔ � � w j ≥ λ and w j < λ j : a j ≥ b h − 1 j : a j ≥ b h j j crit. 1 crit. 2 crit. 3 crit. 4 crit. 5 w 1 w 2 w 3 w 4 w 5 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 6 / 24
2. MR-Sort MR-Sort applied to the introductory example ◮ Sorting accommodations in two categories : Good and Bad 45m 2 Assignment rule 0m 0m 0 e 5 � hotel ∈ Good ⇔ w j ≥ λ j : a j ≥ b 1 Good j 25m 2 200m 400m 100 e 3 b 1 Bad 5m 2 600m 800m 200 e 1 crit. beach center price size rating 0 . 2 w j 0 . 2 0 . 2 0 . 2 0 . 2 λ = 0 . 6 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 7 / 24
2. MR-Sort MR-Sort applied to the introductory example ◮ Sorting accommodations in two categories : Good and Bad 45m 2 Assignment rule 0m 0m 0 e 5 � hotel ∈ Good ⇔ 50m w j ≥ λ j : a j ≥ b 1 Good j 30m 2 90 e 25m 2 300m 400m 100 e 3 b 1 Hilton Bad 600m ∈ Good 5m 2 600m 800m 200 e 1 � w j = 0 . 8 crit. beach center price size rating 0 . 2 w j 0 . 2 0 . 2 0 . 2 0 . 2 j : a j ≥ b 1 j λ = 0 . 6 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 7 / 24
2. MR-Sort MR-Sort applied to the introductory example ◮ Sorting accommodations in two categories : Good and Bad 45m 2 Assignment rule 0m 0m 0 e 5 � hotel ∈ Good ⇔ w j ≥ λ 35m 2 j : a j ≥ b 1 Good 4 j 25m 2 300m 400m 100 e 3 b 1 Plaza 130 e 300m 500m Bad ∈ Bad 5m 2 600m 800m 200 e 1 � w j = 0 . 4 crit. beach center price size rating 0 . 2 w j 0 . 2 0 . 2 0 . 2 0 . 2 j : a j ≥ b 1 j λ = 0 . 6 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 7 / 24
3. Learning a MR-Sort model 1 Sorting problem 2 MR-Sort 3 Learning a MR-Sort model 4 MR-Sort with coalitional veto 5 Learning a MR-SortCV model 6 Experimental results 7 Conclusion Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 8 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Learning set N mod MR-Sort models LP learning the weights and the majority threshold Reinitialize Heuristic adjus- � � N mod 2 ting the profiles worst models Stopping criterion met ? MR-Sort model Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Profiles initialized with a heu- Learning set N mod MR-Sort ristic with some randomness models LP learning the weights and the majority threshold Reinitialize Heuristic adjus- � � N mod 2 ting the profiles worst models Stopping criterion met ? MR-Sort model Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Profiles initialized with a heuristic Learning set N mod MR-Sort with some randomness models LP learning the Fixed profiles weights and the Maximization of the CA majority threshold Reinitialize Heuristic adjus- � � N mod 2 ting the profiles worst models Stopping criterion met ? MR-Sort model Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Profiles initialized with a heuristic Learning set N mod MR-Sort with some randomness models LP learning the Fixed profiles weights and the Maximization of the CA majority threshold Fixed weights and majority Reinitialize Heuristic adjus- � � N mod threshold 2 ting the profiles Maximization of the CA worst models Stopping criterion met ? MR-Sort model Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Profiles initialized with a heuristic Learning set N mod MR-Sort with some randomness models LP learning the Fixed profiles weights and the Maximization of the CA majority threshold Fixed weights and majority thre- Reinitialize Heuristic adjus- � � N mod shold 2 ting the profiles Maximization of the CA worst models Once a model restores all the Stopping assignment examples criterion met ? or after N it iterations MR-Sort model Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
3. Learning a MR-Sort model Heuristic algorithm for learning a MR-Sort model Initialization of Profiles initialized with a heuristic Learning set N mod MR-Sort with some randomness models LP learning the Fixed profiles weights and the Maximization of the CA majority threshold Fixed weights and majority thre- Reinitialize Heuristic adjus- � � N mod shold 2 ting the profiles Maximization of the CA worst models Once a model restores all the Stopping assignment examples criterion met ? or after N it iterations MR-Sort The best model regarding CA model or AUC is returned Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 9 / 24
4. MR-Sort with coalitional veto 1 Sorting problem 2 MR-Sort 3 Learning a MR-Sort model 4 MR-Sort with coalitional veto 5 Learning a MR-SortCV model 6 Experimental results 7 Conclusion Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 10 / 24
4. MR-Sort with coalitional veto MR-Sort with binary veto rule ◮ Sorting model ( p ordered categories , i.e. C p ≻ C p − 1 ≻ . . . ≻ C 1 ) ◮ Veto if alternative worse than the veto profile on any criterion ◮ n weights ( w 1 , . . . , w n ) ◮ 1 majority threshold ( λ ) C 3 ◮ p − 1 profiles ( b 1 , . . . , b p − 1 ) b 2 ◮ p − 1 veto profiles ( v 1 , . . . , v p − 1 ) v 2 C 2 b 1 Assignment rule a ∈ C h C 1 v 1 ⇔ � w j ≥ λ and ∄ j : a j ≤ v h − 1 j j : a j ≥ b h − 1 j crit. 1 crit. 2 crit. 3 crit. 4 crit. 5 AND w 1 w 2 w 3 w 4 w 5 � w j < λ or ∃ j : a j ≤ v h j j : a j ≥ b h j Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 11 / 24
4. MR-Sort with coalitional veto MR-Sort with binary veto rule ◮ Veto if alternative worse than the veto profile on any criterion Assignment rule 45m 2 0 e 0m 0m 5 hotel ∈ Good ⇔ 50m � w j ≥ λ and ∄ j : a j ≤ v 1 j 200m Good j : a j ≥ b 1 j 30m 2 25m 2 300m 400m 100 e 3 Rambla b 1 125 e ∈ Bad Bad 150 e 2 550m v 1 700m 5m 2 600m 800m 200 e 1 center crit. beach price size rating 0 . 2 w j 0 . 2 0 . 2 0 . 2 0 . 2 λ = 0 . 6 Learning MR-Sort rules with coalitional veto O. Sobrie - November 7, 2016 12 / 24
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