Relational Learning from Ambiguous Examples Dominique Bouthinon Henry Soldano Laboratoire d’Informatique de Paris Nord - UMR 7030 Université Paris 13, Sorbonne Paris Cité ILP 2014 September 14-16, Nancy (France) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 1 / 27
Ambiguous examples The ambiguity comes from a lack of information Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 2 / 27
Ambiguous examples The ambiguity comes from a lack of information Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 2 / 27
Learning from ambiguous examples Learning from ambiguous examples 1 Sample complexity 2 Learning relational rules from ambiguous clauses 3 Lear 4 Experiments 5 Perspectives 6 Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 3 / 27
Extensional representation of ambiguous examples An ambiguous example is a set of possibilities { x 1 , . . . , x n } containing a single unknown possibility that describes the actual example Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 4 / 27
Extensional representation of ambiguous examples An ambiguous example is a set of possibilities { x 1 , . . . , x n } containing a single unknown possibility that describes the actual example Background knowledge, if available, reduces the number of valid possibilities. Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 4 / 27
Learning from ambiguous examples Let H and e = { x 1 , . . . , . . . , x n } , x ( e ) is the actual example (one of the x i s) hidden in e Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 5 / 27
Learning from ambiguous examples Let H and e = { x 1 , . . . , . . . , x n } , x ( e ) is the actual example (one of the x i s) hidden in e Credulous covering H comp + e ⇔ ∃ x i ∈ e , H covers x i H comp − e ⇔ ∃ x i ∈ e , ¬ ( H covers x i ) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 5 / 27
Learning from ambiguous examples Let H and e = { x 1 , . . . , . . . , x n } , x ( e ) is the actual example (one of the x i s) hidden in e Credulous covering H comp + e ⇔ ∃ x i ∈ e , H covers x i H comp − e ⇔ ∃ x i ∈ e , ¬ ( H covers x i ) Coherence of an hypothesis Let E = E + ∪ E − , then H is coherent iff : ∀ e ∈ E + , H comp + e ∀ e ∈ E − , H comp − e Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 5 / 27
Sample complexity Learning from ambiguous examples 1 Sample complexity 2 Learning relational rules from ambiguous clauses 3 Lear 4 Experiments 5 Perspectives 6 Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 6 / 27
Sample complexity L X (complete ex. ), L e = 2 L X (ambiguous ex. ), L H (hypotheses) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 7 / 27
Sample complexity L X (complete ex. ), L e = 2 L X (ambiguous ex. ), L H (hypotheses) How many ambiguous examples of L e are needed to learn with a probability ( 1 − δ ) a hypothesis H ∈ L H whose error on L X is less than ǫ ? Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 7 / 27
Sample complexity L X (complete ex. ), L e = 2 L X (ambiguous ex. ), L H (hypotheses) How many ambiguous examples of L e are needed to learn with a probability ( 1 − δ ) a hypothesis H ∈ L H whose error on L X is less than ǫ ? Property p ( H is not compatible with e | H ( x ( e )) � = c ( x ( e ))) ≥ λ λ × ǫ × ( ln ( | L H | ) + ln ( 1 1 m ≥ δ )) H ( x ( e )) � = c ( x ( e )) : classification error of the actual example x ( e ) ∈ e Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 7 / 27
Learning relational rules from ambiguous clauses Learning from ambiguous examples 1 Sample complexity 2 Learning relational rules from ambiguous clauses 3 Lear 4 Experiments 5 Perspectives 6 Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 8 / 27
Learning relational rules from ambiguous clauses ( L H , L e , comp + , comp − ) H ∈ L H is a set of clauses e ∈ L e is a set of grounded clauses Example H = { stable(A,B) ← on(A,B) ∧ on(B,floor), stable(A,B) ← on(A,floor) ∧ on(B,floor) } e = { stable(a,b) ← on(a, b) ∧ red(a), stable(a,b) ← on(a,floor) ∧ on(b,floor) ∧ blue(b) } Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 9 / 27
Learning relational rules from ambiguous clauses ( L H , L e , comp + , comp − ) H ∈ L H is a set of clauses e ∈ L e is a set of grounded clauses Example H = { stable(A,B) ← on(A,B) ∧ on(B,floor), stable(A,B) ← on(A,floor) ∧ on(B,floor) } e = { stable(a,b) ← on(a, b) ∧ red(a), stable(a,b) ← on(a,floor) ∧ on(b,floor) ∧ blue(b) } Compatibility relations H comp + e ⇔ ∃ x i ∈ e that is θ -subsumed by a clause of H H comp − e ⇔ ∃ x i ∈ e that is θ -subsumed by no clause of H Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 9 / 27
Minimal and maximal sets of clauses Let e = { x 1 , . . . , x n } be an ambiguous example Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27
Minimal and maximal sets of clauses Let e = { x 1 , . . . , x n } be an ambiguous example min ( e ) = { x i ∈ e | ∀ x j ∈ e , x j �⊂ x i } max ( e ) = { x i ∈ e | ∀ x j ∈ e , x i �⊂ x j } Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27
Minimal and maximal sets of clauses Let e = { x 1 , . . . , x n } be an ambiguous example min ( e ) = { x i ∈ e | ∀ x j ∈ e , x j �⊂ x i } max ( e ) = { x i ∈ e | ∀ x j ∈ e , x i �⊂ x j } Property H comp + e ⇔ H comp + max ( e ) H comp − e ⇔ H comp − min ( e ) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27
Minimal and maximal sets of clauses Let e = { x 1 , . . . , x n } be an ambiguous example min ( e ) = { x i ∈ e | ∀ x j ∈ e , x j �⊂ x i } max ( e ) = { x i ∈ e | ∀ x j ∈ e , x i �⊂ x j } Property H comp + e ⇔ H comp + max ( e ) H comp − e ⇔ H comp − min ( e ) One represents each positive ambiguous example e by max ( e ) , and each negative ambiguous example e by min ( e ) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27
Lear Learning from ambiguous examples 1 Sample complexity 2 Learning relational rules from ambiguous clauses 3 Lear 4 Experiments 5 Perspectives 6 Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 11 / 27
Lear Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27
Lear Greedy top-down algorithm using a separate and conquer strategy Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27
Lear Greedy top-down algorithm using a separate and conquer strategy Principle H is incrementally built = { h 1 , . . . , h m } H , h 2 E + = E + + E + + . . . + E + m 1 2 H comp + E + H comp − E − Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27
Lear Greedy top-down algorithm using a separate and conquer strategy Principle H is incrementally built = { h 1 , . . . , h m } H , h 2 E + = E + + E + + . . . + E + m 1 2 H comp + E + H comp − E − Lear uses an ambiguous seed to reduce the hypothesis search space Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27
Experiments Learning from ambiguous examples 1 Sample complexity 2 Learning relational rules from ambiguous clauses 3 Lear 4 Experiments 5 Perspectives 6 Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 13 / 27
Objectives Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 14 / 27
Objectives Study the accuracy (computed on complete examples) when learning from ambiguous examples Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 14 / 27
Objectives Study the accuracy (computed on complete examples) when learning from ambiguous examples Compare LEar , Tilde ( [Blockeel & De Raedt, 1998] ) and Nfoil ( [Landwehr et al., 2007] ) when learning from incomplete data Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 14 / 27
Bongard : accuracies on 1000 examples bongard (artificial) 1000 ex 100 95 90 Lear accurracy (%) 85 80 Nfoil 75 Tilde 70 65 60 0 10 20 30 40 50 60 70 80 90 ambiguity (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 15 / 27
Bongard : accuracies on 2000 examples bongard (artificial) 2000 ex 100 95 90 Lear accurracy (%) 85 80 Tilde 75 Nfoil 70 65 60 0 10 20 30 40 50 60 70 80 90 ambiguity (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 16 / 27
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