End-to-End Differentiable Proving Tim Rockt aschel Whiteson - - PowerPoint PPT Presentation

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End-to-End Differentiable Proving Tim Rockt aschel Whiteson - - PowerPoint PPT Presentation

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End-to-End Differentiable Proving Tim Rockt aschel Whiteson Research Lab, University of Oxford Twitter: @ rockt tim.rocktaschel@cs.ox.ac.uk http://rockt.github.com Logic and Learning Workshop at The Alan Turing Institute January 12, 2018

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End-to-End Differentiable Proving

Tim Rockt¨ aschel

Whiteson Research Lab, University of Oxford http://rockt.github.com Twitter: @ rockt tim.rocktaschel@cs.ox.ac.uk

Logic and Learning Workshop at The Alan Turing Institute January 12, 2018

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Joint Work With

Sebastian Riedel Pasquale Minervini

University College London University College London

Thomas Demeester Sameer Singh

Ghent University University of California, Irvine

Tim Rockt¨ aschel End-to-End Differentiable Proving 1/30

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XKCD, 17th May 2017

Tim Rockt¨ aschel End-to-End Differentiable Proving 3/30

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XKCD, 17th May 2017

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Tim Rockt¨ aschel End-to-End Differentiable Proving 3/30

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SLIDE 12

XKCD, 17th May 2017

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Answers & Explanations

  • Rules
  • Programs
  • Natural Language
  • Plans

Tim Rockt¨ aschel End-to-End Differentiable Proving 3/30

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XKCD, 17th May 2017

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Answers & Explanations

  • Rules
  • Programs
  • Natural Language
  • Plans

Answers & Explanations

  • Rules
  • Programs
  • Natural Language
  • Plans

Tim Rockt¨ aschel End-to-End Differentiable Proving 3/30

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XKCD, 17th May 2017

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Data & Explanations

  • Rules
  • (Partial) Programs
  • Natural Language

Answers & Explanations

  • Rules
  • Programs
  • Natural Language
  • Plans

Answers & Explanations

  • Rules
  • Programs
  • Natural Language
  • Plans

Data Efficiency & Model Interpretability

Tim Rockt¨ aschel End-to-End Differentiable Proving 3/30

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Expert Systems

  • No/little training data
  • Interpretable
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Expert Systems

  • No/little training data
  • Interpretable
  • Rules manually defined
  • No generalization
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Expert Systems

  • No/little training data
  • Interpretable
  • Rules manually defined
  • No generalization
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Expert Systems

  • No/little training data
  • Interpretable
  • Rules manually defined
  • No generalization

Neural Networks

  • Trained end-to-end
  • Strong generalization
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Expert Systems

  • No/little training data
  • Interpretable
  • Rules manually defined
  • No generalization

Neural Networks

  • Trained end-to-end
  • Strong generalization
  • Need a lot of training data
  • Not interpretable
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Expert Systems

  • No/little training data
  • Interpretable

Neural Networks

  • Trained end-to-end
  • Strong generalization
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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965)

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . . Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007) Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007)

Neural-symbolic Connectionism

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007)

Neural-symbolic Connectionism

Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (d’Avila Garcez and Zaverucha, 1999) Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007)

Neural-symbolic Connectionism

Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (d’Avila Garcez and Zaverucha, 1999) First-order inference (no training of symbol representations): Unification Neural Networks (H¨

  • lldobler, 1990; Komendantskaya, 2011), SHRUTI

(Shastri, 1992), Neural Prolog (Ding, 1995), CLIP++ (Franca et al., 2014), Lifted Relational Networks (Sourek et al., 2015) Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007)

Neural-symbolic Connectionism

Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (d’Avila Garcez and Zaverucha, 1999) First-order inference (no training of symbol representations): Unification Neural Networks (H¨

  • lldobler, 1990; Komendantskaya, 2011), SHRUTI

(Shastri, 1992), Neural Prolog (Ding, 1995), CLIP++ (Franca et al., 2014), Lifted Relational Networks (Sourek et al., 2015)

Recent: Logic Tensor Networks (Serafini and d’Avila Garcez, 2016), TensorLog (Cohen, 2016), Differentiable Inductive Logic (Evans and Grefenstette, 2017)

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Machine Learning & Logic

Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g.,

IBAL (Pfeffer, 2001), BLOG (Milch et al., 2005), Markov Logic Networks (Richardson and Domingos, 2006), ProbLog (De Raedt et al., 2007) . . .

Inductive Logic Programming, e.g.,

Plotkin (1970), Shapiro (1991), Muggleton (1991), De Raedt (1999) . . . Statistical Predicate Invention (Kok and Domingos, 2007)

Neural-symbolic Connectionism

Propositional rules: EBL-ANN (Shavlik and Towell, 1989), KBANN (Towell and Shavlik, 1994), C-LIP (d’Avila Garcez and Zaverucha, 1999) First-order inference (no training of symbol representations): Unification Neural Networks (H¨

  • lldobler, 1990; Komendantskaya, 2011), SHRUTI

(Shastri, 1992), Neural Prolog (Ding, 1995), CLIP++ (Franca et al., 2014), Lifted Relational Networks (Sourek et al., 2015)

Recent: Logic Tensor Networks (Serafini and d’Avila Garcez, 2016), TensorLog (Cohen, 2016), Differentiable Inductive Logic (Evans and Grefenstette, 2017) For overviews see Besold et al. (2017) and d’Avila Garcez et al. (2012)

Tim Rockt¨ aschel End-to-End Differentiable Proving 5/30

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Outline

1 Link prediction & symbolic vs. neural representations Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015)

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016)

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016) Model-agnostic and fast (Minervini et al., 2017)

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016) Model-agnostic and fast (Minervini et al., 2017)

3 End-to-end differentiable proving (Rockt¨

aschel and Riedel, 2017)

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016) Model-agnostic and fast (Minervini et al., 2017)

3 End-to-end differentiable proving (Rockt¨

aschel and Riedel, 2017)

Explicit multi-hop reasoning using neural networks

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016) Model-agnostic and fast (Minervini et al., 2017)

3 End-to-end differentiable proving (Rockt¨

aschel and Riedel, 2017)

Explicit multi-hop reasoning using neural networks Inducing rules using gradient descent

Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Outline

1 Link prediction & symbolic vs. neural representations 2 Regularize neural representations using logical rules

Model-agnostic but slow (Rockt¨ aschel et al., 2015) Fast but restricted (Demeester et al., 2016) Model-agnostic and fast (Minervini et al., 2017)

3 End-to-end differentiable proving (Rockt¨

aschel and Riedel, 2017)

Explicit multi-hop reasoning using neural networks Inducing rules using gradient descent

4 Outlook & Summary Tim Rockt¨ aschel End-to-End Differentiable Proving 6/30

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Notation

Constant: homer, bart, lisa etc. (lowercase)

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified)

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk Predicate: fatherOf, parentOf etc. function from terms to a Boolean

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk Predicate: fatherOf, parentOf etc. function from terms to a Boolean Atom: predicate and terms, e.g., parentOf(X, bart)

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk Predicate: fatherOf, parentOf etc. function from terms to a Boolean Atom: predicate and terms, e.g., parentOf(X, bart) Literal: atom or negated or atom, e.g., not parentOf(bart, lisa)

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk Predicate: fatherOf, parentOf etc. function from terms to a Boolean Atom: predicate and terms, e.g., parentOf(X, bart) Literal: atom or negated or atom, e.g., not parentOf(bart, lisa) Rule: head :– body. head: atom body: (possibly empty) list of literals representing conjunction Restricted to Horn clauses in this talk

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Notation

Constant: homer, bart, lisa etc. (lowercase) Variable: X, Y etc. (uppercase, universally quantified) Term: constant or variable Restricted to function-free terms in this talk Predicate: fatherOf, parentOf etc. function from terms to a Boolean Atom: predicate and terms, e.g., parentOf(X, bart) Literal: atom or negated or atom, e.g., not parentOf(bart, lisa) Rule: head :– body. head: atom body: (possibly empty) list of literals representing conjunction Restricted to Horn clauses in this talk Fact: ground rule (no free variables) with empty body, e.g., parentOf(homer, bart).

Tim Rockt¨ aschel End-to-End Differentiable Proving 7/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete!

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people!

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly Weak logical relationships that can be used for inferring facts

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly Weak logical relationships that can be used for inferring facts melinda seattle livesIn?

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly Weak logical relationships that can be used for inferring facts melinda bill spouseOf seattle livesIn?

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly Weak logical relationships that can be used for inferring facts melinda bill spouseOf microsoft chairmanOf seattle livesIn?

Das et al. (2017) 8/30

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Link Prediction

Real world knowledge bases (like Freebase, DBPedia, YAGO, etc.) are incomplete! placeOfBirth attribute is missing for 71% of people! Commonsense knowledge often not stated explicitly Weak logical relationships that can be used for inferring facts melinda bill spouseOf microsoft chairmanOf seattle headquarteredIn livesIn?

Das et al. (2017) 8/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf No notion of similarity: apple ∼ orange, professorAt ∼ lecturerAt

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf No notion of similarity: apple ∼ orange, professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit(apple), apple ∼ organge, isFruit(orange)?

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf No notion of similarity: apple ∼ orange, professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit(apple), apple ∼ organge, isFruit(orange)? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’(X, Y)

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf No notion of similarity: apple ∼ orange, professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit(apple), apple ∼ organge, isFruit(orange)? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’(X, Y) But... leads to powerful inference mechanisms and proofs for predictions: fatherOf(abe, homer). parentOf(homer, lisa). parentOf(homer, bart). grandfatherOf(X, Y) :– fatherOf(X, Z), parentOf(Z, Y). grandfatherOf(abe, Q)? {Q/lisa}, {Q/bart}

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Symbolic Representations

Symbols (constants and predicates) do not share any information: grandpaOf = grandfatherOf No notion of similarity: apple ∼ orange, professorAt ∼ lecturerAt No generalization beyond what can be symbolically inferred: isFruit(apple), apple ∼ organge, isFruit(orange)? Hard to work with language, vision and other modalities ‘‘is a film based on the novel of the same name by’’(X, Y) But... leads to powerful inference mechanisms and proofs for predictions: fatherOf(abe, homer). parentOf(homer, lisa). parentOf(homer, bart). grandfatherOf(X, Y) :– fatherOf(X, Z), parentOf(Z, Y). grandfatherOf(abe, Q)? {Q/lisa}, {Q/bart} Fairly easy to debug and trivial to incorporate domain knowledge: Show to domain expert and let her change/add rules and facts

Tim Rockt¨ aschel End-to-End Differentiable Proving 9/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk Can capture similarity and even semantic hierarchy of symbols: vgrandpaOf = vgrandfatherOf, vapple ∼ vorange, vapple < vfruit

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk Can capture similarity and even semantic hierarchy of symbols: vgrandpaOf = vgrandfatherOf, vapple ∼ vorange, vapple < vfruit Can be trained from raw task data (e.g. facts in a knowledge base)

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk Can capture similarity and even semantic hierarchy of symbols: vgrandpaOf = vgrandfatherOf, vapple ∼ vorange, vapple < vfruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v‘‘is the father of’’ = RNNθ(vis, vthe, vfather, vof)

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk Can capture similarity and even semantic hierarchy of symbols: vgrandpaOf = vgrandfatherOf, vapple ∼ vorange, vapple < vfruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v‘‘is the father of’’ = RNNθ(vis, vthe, vfather, vof) But... need large amount of training data

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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Neural Representations

Lower-dimensional fixed-length vector representations of symbols (predicates and constants): vapple, vorange, vfatherOf, . . . ∈ Rk Can capture similarity and even semantic hierarchy of symbols: vgrandpaOf = vgrandfatherOf, vapple ∼ vorange, vapple < vfruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v‘‘is the father of’’ = RNNθ(vis, vthe, vfather, vof) But... need large amount of training data No direct way of incorporating prior knowledge vgrandfatherOf(X, Y) :– vfatherOf(X, Z), vparentOf(Z, Y).

Tim Rockt¨ aschel End-to-End Differentiable Proving 10/30

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State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle)

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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SLIDE 73

State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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SLIDE 74

State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

ComplEx (Trouillon et al., 2016) vs, vi, vj ∈ Ck

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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SLIDE 75

State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

ComplEx (Trouillon et al., 2016) vs, vi, vj ∈ Ck

fθ(vs, vi, vj) = real(vs)⊤(real(vi) ⊙ real(vj)) + real(vs)⊤(imag(vi) ⊙ imag(vj)) + imag(vs)⊤(real(vi) ⊙ imag(vj)) − imag(vs)⊤(imag(vi) ⊙ real(vj))

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

ComplEx (Trouillon et al., 2016) vs, vi, vj ∈ Ck

fθ(vs, vi, vj) = real(vs)⊤(real(vi) ⊙ real(vj)) + real(vs)⊤(imag(vi) ⊙ imag(vj)) + imag(vs)⊤(real(vi) ⊙ imag(vj)) − imag(vs)⊤(imag(vi) ⊙ real(vj))

Training Loss

L =

  • rs(ei,ej),y ∈ T

−y log (σ(fθ(vs, vi, vj))) − (1 − y) log (1 − σ(fθ(vs, vi, vj)))

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

slide-77
SLIDE 77

State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

ComplEx (Trouillon et al., 2016) vs, vi, vj ∈ Ck

fθ(vs, vi, vj) = real(vs)⊤(real(vi) ⊙ real(vj)) + real(vs)⊤(imag(vi) ⊙ imag(vj)) + imag(vs)⊤(real(vi) ⊙ imag(vj)) − imag(vs)⊤(imag(vi) ⊙ real(vj))

Training Loss

L =

  • rs(ei,ej),y ∈ T

−y log (σ(fθ(vs, vi, vj))) − (1 − y) log (1 − σ(fθ(vs, vi, vj))) Learn vs, vi, vj from data

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

slide-78
SLIDE 78

State-of-the-art Neural Link Prediction

livesIn(melinda, seattle)? = fθ(vlivesIn, vmelinda, vseattle) DistMult (Yang et al., 2015) vs, vi, vj ∈ Rk

fθ(vs, vi, vj) = v⊤

s (vi ⊙ vj)

=

  • k

vskvikvjk

ComplEx (Trouillon et al., 2016) vs, vi, vj ∈ Ck

fθ(vs, vi, vj) = real(vs)⊤(real(vi) ⊙ real(vj)) + real(vs)⊤(imag(vi) ⊙ imag(vj)) + imag(vs)⊤(real(vi) ⊙ imag(vj)) − imag(vs)⊤(imag(vi) ⊙ real(vj))

Training Loss

L =

  • rs(ei,ej),y ∈ T

−y log (σ(fθ(vs, vi, vj))) − (1 − y) log (1 − σ(fθ(vs, vi, vj))) Learn vs, vi, vj from data Obtain gradients ∇vsL, ∇viL, ∇vjL by backprop

Tim Rockt¨ aschel End-to-End Differentiable Proving 11/30

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SLIDE 79

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

Rockt¨ aschel et al. (2015), NAACL 12/30

slide-80
SLIDE 80

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

fatherOf(X, Y) :– parentOf(X, Y), ¬ motherOf(X, Y)

Rockt¨ aschel et al. (2015), NAACL 12/30

slide-81
SLIDE 81

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

fatherOf(X, Y) :– parentOf(X, Y), ¬ motherOf(X, Y) p(F) = F =

            

fθ(s, i, j) if F = s(i, j) 1 − A if F = ¬ A A B if F = A ∧ B A + B − A B if F = A ∨ B B (A − 1) + 1 if F = A :– B

Rockt¨ aschel et al. (2015), NAACL 12/30

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SLIDE 82

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

u7 1 − • u9 ∗ u8

  • − 1

u10 ∗ u11

  • + 1

Differentiable Rule

fatherOf(X, Y) :– parentOf(X, Y), ¬ motherOf(X, Y) p(F) = F =

            

fθ(s, i, j) if F = s(i, j) 1 − A if F = ¬ A A B if F = A ∧ B A + B − A B if F = A ∨ B B (A − 1) + 1 if F = A :– B

Rockt¨ aschel et al. (2015), NAACL 12/30

slide-83
SLIDE 83

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

u7 1 − • u9 ∗ u8

  • − 1

u10 ∗ u11

  • + 1

Differentiable Rule

loss −log

Loss

fatherOf(X, Y) :– parentOf(X, Y), ¬ motherOf(X, Y) p(F) = F =

            

fθ(s, i, j) if F = s(i, j) 1 − A if F = ¬ A A B if F = A ∧ B A + B − A B if F = A ∨ B B (A − 1) + 1 if F = A :– B L fatherOf(homer, bart) :– parentOf(homer, bart) ∧ ¬ motherOf(homer, bart)

Rockt¨ aschel et al. (2015), NAACL 12/30

slide-84
SLIDE 84

Regularization by Propositional Logic

parentOf homer, bart motherOf fatherOf u1 u3 u2 dot dot dot u4 u5 u6 sigm sigm sigm

Link Predictor

u7 1 − • u9 ∗ u8

  • − 1

u10 ∗ u11

  • + 1

Differentiable Rule

loss −log

Loss

fatherOf(X, Y) :– parentOf(X, Y), ¬ motherOf(X, Y) p(F) = F =

            

fθ(s, i, j) if F = s(i, j) 1 − A if F = ¬ A A B if F = A ∧ B A + B − A B if F = A ∨ B B (A − 1) + 1 if F = A :– B L fatherOf(homer, bart) :– parentOf(homer, bart) ∧ ¬ motherOf(homer, bart)

L(f) = − log (∀X, Y : f(X, Y)) = −

(ei,ej)∈C2 log f(ei, ej)

Rockt¨ aschel et al. (2015), NAACL 12/30

slide-85
SLIDE 85

Zero-shot Learning Results

Neural Link Prediction (LP)

20 40 3 10 21 33 38 weighted Mean Average Precision

Tim Rockt¨ aschel End-to-End Differentiable Proving 13/30

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SLIDE 86

Zero-shot Learning Results

Neural Link Prediction (LP) Deduction

20 40 3 10 21 33 38 weighted Mean Average Precision

Tim Rockt¨ aschel End-to-End Differentiable Proving 13/30

slide-87
SLIDE 87

Zero-shot Learning Results

Neural Link Prediction (LP) Deduction Deduction after LP

20 40 3 10 21 33 38 weighted Mean Average Precision

Tim Rockt¨ aschel End-to-End Differentiable Proving 13/30

slide-88
SLIDE 88

Zero-shot Learning Results

Neural Link Prediction (LP) Deduction Deduction after LP Deduction before LP

20 40 3 10 21 33 38 weighted Mean Average Precision

Tim Rockt¨ aschel End-to-End Differentiable Proving 13/30

slide-89
SLIDE 89

Zero-shot Learning Results

Neural Link Prediction (LP) Deduction Deduction after LP Deduction before LP Regularization

20 40 3 10 21 33 38 weighted Mean Average Precision

Tim Rockt¨ aschel End-to-End Differentiable Proving 13/30

slide-90
SLIDE 90

Lifted Regularization by Implications

Every father is a parent Every mother is a parent

mother of father of parent of Demeester et al. (2016), EMNLP 14/30

slide-91
SLIDE 91

Lifted Regularization by Implications

Every father is a parent Every mother is a parent

mother of father of parent of implied by father of Demeester et al. (2016), EMNLP 14/30

slide-92
SLIDE 92

Lifted Regularization by Implications

Every father is a parent Every mother is a parent

Before mother of father of parent of implied by father of After father of parent of Demeester et al. (2016), EMNLP 14/30

slide-93
SLIDE 93

Lifted Regularization by Implications

Every father is a parent Every mother is a parent

Before mother of father of parent of implied by father of After father of mother of parent of Demeester et al. (2016), EMNLP 14/30

slide-94
SLIDE 94

Lifted Regularization by Implications

Every father is a parent Generalises to similar relations (e.g. dad) Every mother is a parent Generalises to similar relations (e.g. mum)

Before mother of father of parent of implied by father of After father of mother of parent of mum of dad of Demeester et al. (2016), EMNLP 14/30

slide-95
SLIDE 95

Lifted Regularization by Implications

Every father is a parent Generalises to similar relations (e.g. dad) Every mother is a parent Generalises to similar relations (e.g. mum) Every parent is a relative No training facts needed!

Before mother of father of parent of implied by father of After father of mother of parent of mum of dad of relative of Demeester et al. (2016), EMNLP 14/30

slide-96
SLIDE 96

Lifted Regularization by Implications

Every father is a parent Generalises to similar relations (e.g. dad) Every mother is a parent Generalises to similar relations (e.g. mum) Every parent is a relative No training facts needed!

Before mother of father of parent of implied by father of After father of mother of parent of mum of dad of relative of

∀X, Y : h(X, Y) :– b(X, Y) ∀(ei, ej) ∈ C2 : h⊤ ei, ej ≥ b⊤ ei, ej h ≥ b , ∀(ei, ej) ∈ C2 : ei, ej ∈ Rk

+

Demeester et al. (2016), EMNLP 14/30

slide-97
SLIDE 97

Adversarial Regularization

x y x z z y Clause A: h(X, Y) :– b1(X, Z) ∧ b2(Z, Y) Link Predictor Link Predictor Link Predictor φh(x, y) φb1(x, z) φb2(z, y) JI [φh(x, y) :– φb1(x, z) ∧ φb2(z, y)] Inconsistency Loss

Regularization by propositional rules needs grounding – does not scale to large domains!

Minervini et al. (2017), UAI 14/30

slide-98
SLIDE 98

Adversarial Regularization

x y x z z y Clause A: h(X, Y) :– b1(X, Z) ∧ b2(Z, Y) Link Predictor Link Predictor Link Predictor φh(x, y) φb1(x, z) φb2(z, y) JI [φh(x, y) :– φb1(x, z) ∧ φb2(z, y)] Inconsistency Loss

Regularization by propositional rules needs grounding – does not scale to large domains! Lifted regularization only supports direct implications

Minervini et al. (2017), UAI 14/30

slide-99
SLIDE 99

Adversarial Regularization

x y x z z y y x z Adversary Clause A: h(X, Y) :– b1(X, Z) ∧ b2(Z, Y) Link Predictor Link Predictor Link Predictor φh(x, y) φb1(x, z) φb2(z, y) JI [φh(x, y) :– φb1(x, z) ∧ φb2(z, y)] Inconsistency Loss Adversarial Set S

Regularization by propositional rules needs grounding – does not scale to large domains! Lifted regularization only supports direct implications Idea: let grounding be generated by an adversary and optimize minimax game...

Minervini et al. (2017), UAI 14/30

slide-100
SLIDE 100

Adversarial Regularization

x y x z z y y x z Adversary Clause A: h(X, Y) :– b1(X, Z) ∧ b2(Z, Y) Link Predictor Link Predictor Link Predictor φh(x, y) φb1(x, z) φb2(z, y) JI [φh(x, y) :– φb1(x, z) ∧ φb2(z, y)] Inconsistency Loss Adversarial Set S

Regularization by propositional rules needs grounding – does not scale to large domains! Lifted regularization only supports direct implications Idea: let grounding be generated by an adversary and optimize minimax game... Adversary finds maximally violating grounding for a given rule

Minervini et al. (2017), UAI 14/30

slide-101
SLIDE 101

Adversarial Regularization

x y x z z y y x z Adversary Clause A: h(X, Y) :– b1(X, Z) ∧ b2(Z, Y) Link Predictor Link Predictor Link Predictor φh(x, y) φb1(x, z) φb2(z, y) JI [φh(x, y) :– φb1(x, z) ∧ φb2(z, y)] Inconsistency Loss Adversarial Set S

Regularization by propositional rules needs grounding – does not scale to large domains! Lifted regularization only supports direct implications Idea: let grounding be generated by an adversary and optimize minimax game... Adversary finds maximally violating grounding for a given rule Neural link predictor attempts to minimize rule violation for given generated groundings

Minervini et al. (2017), UAI 14/30

slide-102
SLIDE 102

End-to-End Differentiable Prover

Neural network for proving queries to a knowledge base

Rockt¨ aschel and Riedel (2017), NIPS 15/30

slide-103
SLIDE 103

End-to-End Differentiable Prover

Neural network for proving queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols

Rockt¨ aschel and Riedel (2017), NIPS 15/30

slide-104
SLIDE 104

End-to-End Differentiable Prover

Neural network for proving queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Learn vector representations of symbols end-to-end from proof success

Rockt¨ aschel and Riedel (2017), NIPS 15/30

slide-105
SLIDE 105

End-to-End Differentiable Prover

Neural network for proving queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Learn vector representations of symbols end-to-end from proof success Make use of provided rules in soft proofs

Rockt¨ aschel and Riedel (2017), NIPS 15/30

slide-106
SLIDE 106

End-to-End Differentiable Prover

Neural network for proving queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Learn vector representations of symbols end-to-end from proof success Make use of provided rules in soft proofs Induce interpretable rules end-to-end from proof success

Rockt¨ aschel and Riedel (2017), NIPS 15/30

slide-107
SLIDE 107

Approach

Tim Rockt¨ aschel End-to-End Differentiable Proving 16/30

slide-108
SLIDE 108

Approach

Tim Rockt¨ aschel End-to-End Differentiable Proving 16/30

slide-109
SLIDE 109

Approach

Let’s neuralize Prolog’s Backward Chaining using a Radial Basis Function kernel for unifying vector representations of symbols!

Tim Rockt¨ aschel End-to-End Differentiable Proving 16/30

slide-110
SLIDE 110

Prolog’s Backward Chaining

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

Tim Rockt¨ aschel End-to-End Differentiable Proving 17/30

slide-111
SLIDE 111

Prolog’s Backward Chaining

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

Intuition: Backward chaining translates a query into subqueries via rules, e.g.,

grandfatherOf(abe, bart) 3. fatherOf(abe, Z), parentOf(Z, bart)

Tim Rockt¨ aschel End-to-End Differentiable Proving 17/30

slide-112
SLIDE 112

Prolog’s Backward Chaining

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

Intuition: Backward chaining translates a query into subqueries via rules, e.g.,

grandfatherOf(abe, bart) 3. fatherOf(abe, Z), parentOf(Z, bart)

It attempts this for all rules in the knowledge base and thus specifies a depth-first search

Tim Rockt¨ aschel End-to-End Differentiable Proving 17/30

slide-113
SLIDE 113

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-114
SLIDE 114

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 1. fatherOf abe homer Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-115
SLIDE 115

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 1. fatherOf abe homer

?

=

?

=

?

=

Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-116
SLIDE 116

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 1. fatherOf abe homer FAIL SUCCESS FAIL

?

=

?

=

?

=

Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-117
SLIDE 117

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 1. fatherOf abe homer FAIL SUCCESS FAIL

?

=

?

=

?

=

State t

∅

SUCCESS Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-118
SLIDE 118

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 1. fatherOf abe homer FAIL SUCCESS FAIL

?

=

?

=

?

=

State t

∅

SUCCESS

State t + 1

∅

FAIL Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-119
SLIDE 119

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 2. parentOf homer bart FAIL FAIL SUCCESS

?

=

?

=

?

=

State t

∅

SUCCESS

State t + 1

∅

FAIL Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-120
SLIDE 120

Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandfatherOf abe bart 3. grandfatherOf X Y SUCCESS X/abe Y/bart

?

=

?

=

?

=

State t

∅

SUCCESS

State t + 1

X/abe Y/bart

SUCCESS Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-121
SLIDE 121

Unification Failure

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandpaOf abe bart 3. grandfatherOf X Y FAIL X/abe Y/bart

?

=

?

=

?

=

State t

∅

SUCCESS

State t + 1

X/abe Y/bart

FAIL Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

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SLIDE 122

Neural Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandpaOf abe bart 3. X Y X/abe Y/bart

?

=

?

=

?

=

State t

∅

1.0

State t + 1

X/abe Y/bart

Tim Rockt¨ aschel End-to-End Differentiable Proving 18/30

slide-123
SLIDE 123

Neural Unification

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y). Query grandpaOf abe bart 3. X Y X/abe Y/bart

?

=

?

=

?

=

State t

∅

1.0

State t + 1

X/abe Y/bart

min

  • 1.0, exp

−vgrandpaOf−vgrandfatherOf2 2µ2

  • Tim Rockt¨

aschel End-to-End Differentiable Proving 18/30

slide-124
SLIDE 124

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-125
SLIDE 125

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-126
SLIDE 126

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-127
SLIDE 127

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y)

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-128
SLIDE 128

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-129
SLIDE 129

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z FAIL 3.

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-130
SLIDE 130

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1. FAIL 3.

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-131
SLIDE 131

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1. FAIL 3. parentOf homer bart

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

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SLIDE 132

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1. FAIL 3. parentOf homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2.

Tim Rockt¨ aschel End-to-End Differentiable Proving 19/30

slide-133
SLIDE 133

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1.

X/abe Y/bart Z/bart

3.2 parentOf(Z, Y) 2. FAIL 3. parentOf homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2.

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SLIDE 134

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1.

X/abe Y/bart Z/bart

3.2 parentOf(Z, Y) 2. FAIL 3. parentOf homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2. parentOf bart bart

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SLIDE 135

Differentiable Prover

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1.

X/abe Y/bart Z/bart

3.2 parentOf(Z, Y) 2. FAIL 3. parentOf homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2. parentOf bart bart

X/abe Y/bart Z/bart X/abe Y/bart Z/bart

FAIL 1. 3. 2.

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SLIDE 136

Neural Program Induction

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. grandfatherOf(X, Y) :–

fatherOf(X, Z), parentOf(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 fatherOf(X, Z) 3.2 parentOf(Z, Y) fatherOf abe Z

X/abe Y/bart Z/homer

3.2 parentOf(Z, Y) 1.

X/abe Y/bart Z/bart

3.2 parentOf(Z, Y) 2. FAIL 3. parentOf homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2. parentOf bart bart

X/abe Y/bart Z/bart X/abe Y/bart Z/bart

FAIL 1. 3. 2.

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SLIDE 137

Neural Program Induction

Example Knowledge Base:

  • 1. fatherOf(abe, homer).
  • 2. parentOf(homer, bart).
  • 3. θ1(X, Y) :–

θ2(X, Z), θ3(Z, Y).

∅

1.0

grandpaOf abe bart

∅

1.

∅

2.

X/abe Y/bart

3. 3.1 θ2(X, Z) 3.2 θ3(Z, Y) θ2 abe Z

X/abe Y/bart Z/homer

3.2 θ3(Z, Y) 1.

X/abe Y/bart Z/bart

3.2 θ3(Z, Y) 2. FAIL 3. θ3 homer bart

X/abe Y/bart Z/homer X/abe Y/bart Z/homer

FAIL 1. 3. 2. θ3 bart bart

X/abe Y/bart Z/bart X/abe Y/bart Z/bart

FAIL 1. 3. 2.

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SLIDE 138

Training Objective

grandpaOf abe bart

∅ ∅

X/abe Y/bart Z/homer X/abe Y/bart Z/homer X/abe Y/bart Z/bart X/abe Y/bart Z/bart

1. 1. 3. 1. 1. 3. 1. 2. 3. 2. 1. 3. 2. 2.

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SLIDE 139

Training Objective

grandpaOf abe bart

∅ ∅

X/abe Y/bart Z/homer X/abe Y/bart Z/homer X/abe Y/bart Z/bart X/abe Y/bart Z/bart

1. 1. 3. 1. 1. 3. 1. 2. 3. 2. 1. 3. 2. 2.

fθ(grandpaOf(abe, bart))

max pooling

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SLIDE 140

Training Objective

grandpaOf abe bart

∅ ∅

X/abe Y/bart Z/homer X/abe Y/bart Z/homer X/abe Y/bart Z/bart X/abe Y/bart Z/bart

1. 1. 3. 1. 1. 3. 1. 2. 3. 2. 1. 3. 2. 2.

fθ(grandpaOf(abe, bart))

max pooling

Loss: negative log-likelihood w.r.t. target proof success

Tim Rockt¨ aschel End-to-End Differentiable Proving 20/30

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SLIDE 141

Training Objective

grandpaOf abe bart

∅ ∅

X/abe Y/bart Z/homer X/abe Y/bart Z/homer X/abe Y/bart Z/bart X/abe Y/bart Z/bart

1. 1. 3. 1. 1. 3. 1. 2. 3. 2. 1. 3. 2. 2.

fθ(grandpaOf(abe, bart))

max pooling

Loss: negative log-likelihood w.r.t. target proof success Trained end-to-end using stochastic gradient descent

Tim Rockt¨ aschel End-to-End Differentiable Proving 20/30

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SLIDE 142

Training Objective

grandpaOf abe bart

∅ ∅

X/abe Y/bart Z/homer X/abe Y/bart Z/homer X/abe Y/bart Z/bart X/abe Y/bart Z/bart

1. 1. 3. 1. 1. 3. 1. 2. 3. 2. 1. 3. 2. 2.

fθ(grandpaOf(abe, bart))

max pooling

Loss: negative log-likelihood w.r.t. target proof success Trained end-to-end using stochastic gradient descent Vectors are learned such that proof success is high for known facts and low for sampled negative facts

Tim Rockt¨ aschel End-to-End Differentiable Proving 20/30

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SLIDE 143

Calculation on GPU

Q

parentOf dadOf homer abe

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SLIDE 144

Calculation on GPU

Q

parentOf dadOf homer abe fatherOf parentOf grandmaOf abe homer mona homer bart lisa

Tim Rockt¨ aschel End-to-End Differentiable Proving 21/30

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SLIDE 145

Calculation on GPU

Q

parentOf dadOf homer abe fatherOf parentOf grandmaOf abe homer mona homer bart lisa unify unify

Tim Rockt¨ aschel End-to-End Differentiable Proving 21/30

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SLIDE 146

Calculation on GPU

Q Q /

parentOf dadOf homer abe fatherOf parentOf grandmaOf abe homer mona homer bart lisa homer bart lisa homer bart lisa unify unify unify (symbolic)

Tim Rockt¨ aschel End-to-End Differentiable Proving 21/30

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SLIDE 147

Calculation on GPU

Q Q /

parentOf dadOf homer abe fatherOf parentOf grandmaOf abe homer mona homer bart lisa homer bart lisa homer bart lisa unify unify unify (symbolic)

Tim Rockt¨ aschel End-to-End Differentiable Proving 21/30

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SLIDE 148

Experiments

Benchmark Knowledge Bases: Kinship, Nations, UMLS (Kok and Domingos, 2007), and Countries (Bouchard et al., 2015) Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn

Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn

Tim Rockt¨ aschel End-to-End Differentiable Proving 22/30

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SLIDE 149

Experiments

Benchmark Knowledge Bases: Kinship, Nations, UMLS (Kok and Domingos, 2007), and Countries (Bouchard et al., 2015) Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn

Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn

Tim Rockt¨ aschel End-to-End Differentiable Proving 22/30

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SLIDE 150

Experiments

Benchmark Knowledge Bases: Kinship, Nations, UMLS (Kok and Domingos, 2007), and Countries (Bouchard et al., 2015) Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn locatedIn locatedIn

Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn

Tim Rockt¨ aschel End-to-End Differentiable Proving 22/30

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SLIDE 151

Experiments

Benchmark Knowledge Bases: Kinship, Nations, UMLS (Kok and Domingos, 2007), and Countries (Bouchard et al., 2015) Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn locatedIn

Test Country Train Country Region Subregion

neighborOf locatedIn locatedIn locatedIn locatedIn

Tim Rockt¨ aschel End-to-End Differentiable Proving 22/30

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SLIDE 152

Details

Models implemented in TensorFlow

Tim Rockt¨ aschel End-to-End Differentiable Proving 23/30

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SLIDE 153

Details

Models implemented in TensorFlow

ComplEx Neural link prediction model by Trouillon et al. (2016)

Tim Rockt¨ aschel End-to-End Differentiable Proving 23/30

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SLIDE 154

Details

Models implemented in TensorFlow

ComplEx Neural link prediction model by Trouillon et al. (2016) Prover End-to-end differentiable prover

Tim Rockt¨ aschel End-to-End Differentiable Proving 23/30

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SLIDE 155

Details

Models implemented in TensorFlow

ComplEx Neural link prediction model by Trouillon et al. (2016) Prover End-to-end differentiable prover Proverλ Same, but representations trained with ComplEx as auxiliary task

Tim Rockt¨ aschel End-to-End Differentiable Proving 23/30

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SLIDE 156

Details

Models implemented in TensorFlow

ComplEx Neural link prediction model by Trouillon et al. (2016) Prover End-to-end differentiable prover Proverλ Same, but representations trained with ComplEx as auxiliary task

Rule Templates:

Kinship, Nations & UMLS 20 #1(X, Y) :– #2(X, Y). 20 #1(X, Y) :– #2(Y, X). 20 #1(X, Y) :– #2(X, Z), #3(Z, Y). Countries S1 3 #1(X, Y) :– #1(Y, X). 3 #1(X, Y) :– #2(X, Z), #2(Z, Y). Countries S2 3 #1(X, Y) :– #2(X, Z), #3(Z, Y). Countries S3 3 #1(X, Y) :– #2(X, Z), #3(Z, W), #4(W, Y). Tim Rockt¨ aschel End-to-End Differentiable Proving 23/30

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SLIDE 157

Tim Rockt¨ aschel End-to-End Differentiable Proving 24/30

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SLIDE 158

Results

ComplEx Countries S3 Kinship Nations UMLS 20 40 60 80 100 48 70 62 82 57 48 62 82 77 76 59 87 Accuracy / HITS@1

Tim Rockt¨ aschel End-to-End Differentiable Proving 25/30

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SLIDE 159

Results

ComplEx Prover Countries S3 Kinship Nations UMLS 20 40 60 80 100 48 70 62 82 57 48 62 82 77 76 59 87 Accuracy / HITS@1

Tim Rockt¨ aschel End-to-End Differentiable Proving 25/30

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SLIDE 160

Results

ComplEx Prover Proverλ Countries S3 Kinship Nations UMLS 20 40 60 80 100 48 70 62 82 57 48 62 82 77 76 59 87 Accuracy / HITS@1

Tim Rockt¨ aschel End-to-End Differentiable Proving 25/30

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SLIDE 161

Examples of Induced Rules

Corpus Induced rules and their confidence Countries S1 0.90 locatedIn(X,Y) :– locatedIn(X,Z), locatedIn(Z,Y). S2 0.63 locatedIn(X,Y) :– neighborOf(X,Z), locatedIn(Z,Y). S3 0.32 locatedIn(X,Y) :– neighborOf(X,Z), neighborOf(Z,W), locatedIn(W,Y). Nations 0.68 blockpositionindex(X,Y) :– blockpositionindex(Y,X). 0.46 expeldiplomats(X,Y) :– negativebehavior(X,Y). 0.38 negativecomm(X,Y) :– commonbloc0(X,Y). 0.38 intergovorgs3(X,Y) :– intergovorgs(Y,X). UMLS 0.88 interacts with(X,Y) :– interacts with(X,Z), interacts with(Z,Y). 0.77 isa(X,Y) :– isa(X,Z), isa(Z,Y). 0.71 derivative of(X,Y) :– derivative of(X,Z), derivative of(Z,Y). Tim Rockt¨ aschel End-to-End Differentiable Proving 26/30

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SLIDE 162

Outlook

Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 163

Outlook

User

Question My patient is not responding after three days of codeine treatment. What could have happened? Question My patient is not responding after three days of codeine treatment. What could have happened? Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 164

Outlook

Structured Data

Databases User

Question My patient is not responding after three days of codeine treatment. What could have happened? Structured Data Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 165

Outlook

Structured Data

Databases

Explanations

Teacher User

Question My patient is not responding after three days of codeine treatment. What could have happened? Structured Data Explanations Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 166

Outlook

Structured Data

Databases

Explanations

Teacher

Text Text

Publications User

Question My patient is not responding after three days of codeine treatment. What could have happened? Structured Data Explanations Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 167

Outlook

Structured Data

Databases

Explanations

Teacher

Text Text

Publications User

Question My patient is not responding after three days of codeine treatment. What could have happened? Structured Data Explanations Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 168

Outlook

Structured Data

Databases

Explanations

Teacher

Text

Publications User

Question My patient is not responding after three days of codeine treatment. What could have happened? Answer

Morphine intoxication

Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 169

Outlook

Structured Data

Databases

Explanations

Teacher

Text

Publications User

Question My patient is not responding after three days of codeine treatment. What could have happened? Answer

Morphine intoxication

Proof

  • Codeine is metabolized to morphine
  • Mutation in CYP2D6 can cause ultrarapid metabolization
  • Ultrarapid metabolization can lead to morphine overdose
  • Morphine overdose is an intoxication

Tim Rockt¨ aschel End-to-End Differentiable Proving 27/30

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SLIDE 170

Summary

We proposed various ways of regularizing vector representations of symbols using rules

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 171

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 172

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 173

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 174

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 175

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent Induces interpretable rules from data via gradient descent

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 176

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent Induces interpretable rules from data via gradient descent Various computational optimizations: batch proving, tree pruning etc.

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 177

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent Induces interpretable rules from data via gradient descent Various computational optimizations: batch proving, tree pruning etc. Future research:

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 178

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent Induces interpretable rules from data via gradient descent Various computational optimizations: batch proving, tree pruning etc. Future research:

Scaling up to larger knowledge bases

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 179

Summary

We proposed various ways of regularizing vector representations of symbols using rules We used Prolog’s backward chaining as recipe for recursively constructing a neural network to prove queries to a knowledge base Proof success differentiable w.r.t. vector representations of symbols Symbolic rule application but neural unification Learns vector representations of symbols from data via gradient descent Induces interpretable rules from data via gradient descent Various computational optimizations: batch proving, tree pruning etc. Future research:

Scaling up to larger knowledge bases Connecting to RNNs for proving with natural language statements

Tim Rockt¨ aschel End-to-End Differentiable Proving 28/30

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SLIDE 180

Thank you!

http://rockt.github.com tim.rocktaschel@cs.ox.ac.uk Twitter: @ rockt

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SLIDE 181

References I

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uhnberger, L. C. Lamb, D. Lowd, P. M. V. Lima, L. de Penning, G. Pinkas, H. Poon, and G. Zaverucha. Neural-symbolic learning and reasoning: A survey and interpretation. CoRR, abs/1711.03902, 2017. URL http://arxiv.org/abs/1711.03902.

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Business Media, 2012.

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aschel, and S. Riedel. Lifted rule injection for relation embeddings. In Proceedings of the 2016 Conference on Empirical Methods in Natural Language Processing, EMNLP 2016, Austin, Texas, USA, November 1-4, 2016, pages 1389–1399, 2016. URL http://aclweb.org/anthology/D/D16/D16-1146.pdf.

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21st Century., IEEE International Conference on, volume 4, pages 3603–3608. IEEE, 1995.

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Conference (ICML 2007), Corvallis, Oregon, USA, June 20-24, 2007, pages 433–440, 2007. doi: 10.1145/1273496.1273551. URL http://doi.acm.org/10.1145/1273496.1273551.

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doi: 10.1093/jigpal/jzq012. URL http://dx.doi.org/10.1093/jigpal/jzq012.

  • P. Minervini, T. Demeester, T. Rockt¨

aschel, and S. Riedel. Adversarial sets for regularised neural link predictors. In Proceedings of the 33rd Conference on Uncertainty in Artificial Intelligence (UAI), 2017.

  • T. Rockt¨

aschel and S. Riedel. End-to-end differentiable proving. In Advances in Neural Information Processing Systems 30: Annual Conference on Neural Information Processing Systems 2017, 4-9 December 2017, Long Beach, CA, USA, pages 3791–3803, 2017. URL http://papers.nips.cc/paper/6969-end-to-end-differentiable-proving.

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aschel, S. Singh, and S. Riedel. Injecting logical background knowledge into embeddings for relation extraction. In NAACL HLT 2015, The 2015 Conference of the North American Chapter of the Association for Computational Linguistics: Human Language Technologies, Denver, Colorado, USA, May 31 - June 5, 2015, pages 1119–1129, 2015. URL http://aclweb.org/anthology/N/N15/N15-1118.pdf.

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connectionist model based on temporal synchrony. In Proceedings of the Fourteenth Annual Conference of the Cognitive Science Society: July 29 to August 1, 1992, Cognitive Science Program, Indiana University, Bloomington, volume 14, page 159. Psychology Press, 1992.

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231–253, 1989.

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y, and O. Kuzelka. Lifted relational neural networks. In Proceedings of the NIPS Workshop on Cognitive Computation: Integrating Neural and Symbolic Approaches co-located with the 29th Annual Conference on Neural Information Processing Systems (NIPS 2015), Montreal, Canada, December 11-12, 2015., 2015. URL http://ceur-ws.org/Vol-1583/CoCoNIPS_2015_paper_7.pdf.

  • G. G. Towell and J. W. Shavlik. Knowledge-based artificial neural networks. Artif. Intell., 70(1-2):119–165, 1994. doi:

10.1016/0004-3702(94)90105-8. URL http://dx.doi.org/10.1016/0004-3702(94)90105-8.

  • T. Trouillon, J. Welbl, S. Riedel, ´
  • E. Gaussier, and G. Bouchard. Complex embeddings for simple link prediction. In Proceedings of the

33nd International Conference on Machine Learning, ICML 2016, New York City, NY, USA, June 19-24, 2016, pages 2071–2080, 2016. URL http://jmlr.org/proceedings/papers/v48/trouillon16.html.

  • B. Yang, W. Yih, X. He, J. Gao, and L. Deng. Embedding entities and relations for learning and inference in knowledge bases. In

International Conference on Learning Representations (ICLR), 2015. URL http://arxiv.org/abs/1412.6575.