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Relational Learning from Ambiguous Examples Dominique Bouthinon Henry Soldano Laboratoire dInformatique de Paris Nord - UMR 7030 Universit Paris 13, Sorbonne Paris Cit ILP 2014 September 14-16, Nancy (France) Bouthinon, Soldano (LIPN)

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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

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Ambiguous examples

The ambiguity comes from a lack of information

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 2 / 27

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Ambiguous examples

The ambiguity comes from a lack of information

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 2 / 27

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Learning from ambiguous examples

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 3 / 27

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Extensional representation of ambiguous examples

An ambiguous example is a set of possibilities {x1, . . . , xn} containing a single unknown possibility that describes the actual example

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 4 / 27

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Extensional representation of ambiguous examples

An ambiguous example is a set of possibilities {x1, . . . , xn} 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

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Learning from ambiguous examples

Let H and e = {x1, . . . , . . . , xn}, x(e) is the actual example (one of the xis) hidden in e

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 5 / 27

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Learning from ambiguous examples

Let H and e = {x1, . . . , . . . , xn}, x(e) is the actual example (one of the xis) hidden in e

Credulous covering

H comp+e ⇔ ∃xi ∈ e, H covers xi H comp−e ⇔ ∃xi ∈ e, ¬(H covers xi)

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 5 / 27

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Learning from ambiguous examples

Let H and e = {x1, . . . , . . . , xn}, x(e) is the actual example (one of the xis) hidden in e

Credulous covering

H comp+e ⇔ ∃xi ∈ e, H covers xi H comp−e ⇔ ∃xi ∈ e, ¬(H covers xi)

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

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Sample complexity

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 6 / 27

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Sample complexity

LX (complete ex. ), Le = 2LX (ambiguous ex. ), LH (hypotheses)

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 7 / 27

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Sample complexity

LX (complete ex. ), Le = 2LX (ambiguous ex. ), LH (hypotheses) How many ambiguous examples of Le are needed to learn with a probability (1 − δ) a hypothesis H ∈ LH whose error on LX is less than ǫ ?

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 7 / 27

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Sample complexity

LX (complete ex. ), Le = 2LX (ambiguous ex. ), LH (hypotheses) How many ambiguous examples of Le are needed to learn with a probability (1 − δ) a hypothesis H ∈ LH whose error on LX is less than ǫ ?

Property

p(H is not compatible with e | H(x(e)) = c(x(e))) ≥ λ m ≥ 1 λ × ǫ × (ln(|LH|) + ln(1 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

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Learning relational rules from ambiguous clauses

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 8 / 27

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Learning relational rules from ambiguous clauses

(LH, Le, comp+, comp−) H ∈ LH is a set of clauses e ∈ Le 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

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Learning relational rules from ambiguous clauses

(LH, Le, comp+, comp−) H ∈ LH is a set of clauses e ∈ Le 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 ⇔ ∃xi ∈ e that is θ-subsumed by a clause of H H comp− e ⇔ ∃xi ∈ e that is θ-subsumed by no clause of H

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 9 / 27

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Minimal and maximal sets of clauses

Let e = {x1, . . . , xn} be an ambiguous example

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27

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Minimal and maximal sets of clauses

Let e = {x1, . . . , xn} be an ambiguous example min(e) = {xi ∈ e | ∀xj ∈ e, xj ⊂ xi} max(e) = {xi ∈ e | ∀xj ∈ e, xi ⊂ xj}

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 10 / 27

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Minimal and maximal sets of clauses

Let e = {x1, . . . , xn} be an ambiguous example min(e) = {xi ∈ e | ∀xj ∈ e, xj ⊂ xi} max(e) = {xi ∈ e | ∀xj ∈ e, xi ⊂ xj}

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

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Minimal and maximal sets of clauses

Let e = {x1, . . . , xn} be an ambiguous example min(e) = {xi ∈ e | ∀xj ∈ e, xj ⊂ xi} max(e) = {xi ∈ e | ∀xj ∈ e, xi ⊂ xj}

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

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Lear

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 11 / 27

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Lear

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27

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Lear

Greedy top-down algorithm using a separate and conquer strategy

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27

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Lear

Greedy top-down algorithm using a separate and conquer strategy

Principle

H is incrementally built H = { h1 , h2 , . . . , hm} E+ = E+

1

+ E+

2

+ . . . + E+

m

H comp+ E+ H comp− E−

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 12 / 27

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Lear

Greedy top-down algorithm using a separate and conquer strategy

Principle

H is incrementally built H = { h1 , h2 , . . . , hm} E+ = E+

1

+ E+

2

+ . . . + E+

m

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

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Experiments

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 13 / 27

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Objectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 14 / 27

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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

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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

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Bongard : accuracies on 1000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 bongard (artificial) 1000 ex Lear Nfoil Tilde ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 15 / 27

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Bongard : accuracies on 2000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 bongard (artificial) 2000 ex Lear Tilde Nfoil ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 16 / 27

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Bongard : accuracies on 4000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 bongard (artificial) 4 000 ex Lear Tilde Nfoil ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 17 / 27

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Bongard : accuracies on 16000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 bongard (artificial) 16 000 ex Lear Tilde Nfoil ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 18 / 27

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Loan : accuracies on 5000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 loan (UCI) 5000 ex Lear Tilde Nfoil ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 19 / 27

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Loan : accuracies 15000 examples

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 loan (UCI) 15 000 ex Lear Nfoil Tilde ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 20 / 27

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Why not completing the ambiguous examples ?

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 21 / 27

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Why not completing the ambiguous examples ?

Because we generally have no support to impute the truth values of the undetermined atoms

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 21 / 27

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Why not completing the ambiguous examples ?

Because we generally have no support to impute the truth values of the undetermined atoms

10 20 30 40 50 60 70 80 90 60 65 70 75 80 85 90 95 100 bongard (UCI) 1 500 ex. Lear comp. Lear ambiguity (%) accurracy (%) Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 21 / 27

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Perspectives

1

Learning from ambiguous examples

2

Sample complexity

3

Learning relational rules from ambiguous clauses

4

Lear

5

Experiments

6

Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 22 / 27

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Perspectives

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 23 / 27

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Perspectives

Extend the ambiguity to the heads of the clauses of the examples

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 23 / 27

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Perspectives

Extend the ambiguity to the heads of the clauses of the examples Apply Lear to learn action models and biological structures

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 23 / 27

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Perspectives

Extend the ambiguity to the heads of the clauses of the examples Apply Lear to learn action models and biological structures Adapt Lear to Probabilistic Logic Programming [De Raedt et al, 2007]

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 23 / 27

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Bongard (artificial)

5 relations

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 24 / 27

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Bongard (artificial)

5 relations Concept

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 24 / 27

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Bongard (artificial)

5 relations Concept B = { ← triangle(X) ∧ square(X), ← triangle(X) ∧ circle(X), triangle(X) ∨ circle(X) ∨ square(X), · · · }

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 24 / 27

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Loan (UCI)

8 relations

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 25 / 27

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Loan (UCI)

8 relations ec = no_paiement_due ← unemployed ∧ enlist(air_force) ∧ longest_absense_from_school(7) ∧ enrolled(occ, 9) ∧ enrolled(ucsd,5)

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 25 / 27

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Loan (UCI)

8 relations ec = no_paiement_due ← unemployed ∧ enlist(air_force) ∧ longest_absense_from_school(7) ∧ enrolled(occ, 9) ∧ enrolled(ucsd,5) B = ∅

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 25 / 27

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Multi-table representation

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 26 / 27

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Multi-table representation

the bodies of the clauses of each ambiguous example can be represented as a cross-product (may lead to an exponential gain in space)

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 26 / 27

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Multi-table representation

the bodies of the clauses of each ambiguous example can be represented as a cross-product (may lead to an exponential gain in space)

Exemple

e = { stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ red(b), stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ blue(b), stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ green(b), · · · stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ green(a) ∧ green(b)}

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 26 / 27

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Multi-table representation

the bodies of the clauses of each ambiguous example can be represented as a cross-product (may lead to an exponential gain in space)

Exemple

e = { stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ red(b), stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ blue(b), stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ red(a) ∧ green(b), · · · stable(a, b) ← on(a, b) ∧ on(b, floor) ∧ green(a) ∧ green(b)} T1 × T2 × T3

  • n(a, b) ∧ on(b, floor)

red(a) blue(a) green(a) red(b) blue(b) green(b)

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 26 / 27

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Methodology

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 27 / 27

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Methodology

C is a set of complete examples, B is the background knowledge Ci C = E =

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 27 / 27

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Methodology

C is a set of complete examples, B is the background knowledge Ci C = E = 10 × 10 cross-validation on C : For each fold Ci (test) one builds a set of ambiguous example E (train) from complete examples drawn in C − Ci each atom of a drawn complete example is made undetermined according to a probability p (blocking process [Schuurmans et Greiner, 1997])

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 27 / 27

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Methodology

C is a set of complete examples, B is the background knowledge Ci C = E = 10 × 10 cross-validation on C : For each fold Ci (test) one builds a set of ambiguous example E (train) from complete examples drawn in C − Ci each atom of a drawn complete example is made undetermined according to a probability p (blocking process [Schuurmans et Greiner, 1997]) Tilde and Nfoil assume each undetermined atom is false

Bouthinon, Soldano (LIPN) Relational Learning from Ambiguous Examples ILP 2014 (Nancy) 27 / 27