gpu accelerated end to end differentiable planning and
play

GPU-accelerated End-to-end Differentiable Planning and Reasoning - PowerPoint PPT Presentation

Jul 14, 2023 •141 likes •2.05k views

GPU-accelerated End-to-end Differentiable Planning and Reasoning Tim Rockt aschel Whiteson Research Lab, University of Oxford http://rockt.github.com Twitter: @ rockt tim.rocktaschel@cs.ox.ac.uk Talk at GTC Europe, ID 23372 12th of

  1. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  2. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Rule : head :– body . head : atom body : (possibly empty) list of atoms representing conjunction grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  3. Notation Constant : homer , bart , lisa etc. (lowercase) Variable : X , Y etc. (uppercase, universally quantified) Term : constant or variable Predicate : fatherOf , parentOf etc. function from terms to a Boolean Atom : predicate and terms, e.g., parentOf ( X , bart ) Rule : head :– body . head : atom body : (possibly empty) list of atoms representing conjunction grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Fact : ground rule (no free variables) with empty body, e.g., parentOf ( homer , bart ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 9/39

  4. Example Knowledge Base 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , lisa ) . 3. parentOf ( homer , bart ) . 4. grandpaOf ( abe , lisa ) . 5. grandfatherOf ( abe , maggie ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 10/39

  5. Example Knowledge Base 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , lisa ) . 3. parentOf ( homer , bart ) . 4. grandpaOf ( abe , lisa ) . 5. grandfatherOf ( abe , maggie ) . 6. grandfatherOf ( X 1 , Y 1 ) :– fatherOf ( X 1 , Z 1 ) , parentOf ( Z 1 , Y 1 ) . 7. grandparentOf ( X 2 , Y 2 ) :– grandfatherOf ( X 2 , Y 2 ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 10/39

  6. Prolog Backward Chaining Example Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  7. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  8. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  9. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  10. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? 1 2 3 failure success failure { X / abe , Y / bart , Z / homer } Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  11. Prolog Backward Chaining Example grandfatherOf ( abe , bart )? Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1 2 3 2. parentOf ( homer , bart ) . success failure failure 3. grandfatherOf ( X , Y ) :– { X / abe , Y / bart } fatherOf ( X , Z ) , 3.1 fatherOf ( abe , Z )? parentOf ( Z , Y ) . 1 2 3 success failure failure { X / abe , Y / bart , Z / homer } 3.2 parentOf ( homer , bart )? 1 2 3 failure success failure { X / abe , Y / bart , Z / homer } What about grandpaOf ( abe , bart )? Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 11/39

  12. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  13. Symbolic Representations Symbols (constants and predicates) do not share any information: grandpaOf � = grandfatherOf No notion of similarity: apple ∼ orange , professorAt ∼ lecturerAt Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  14. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  15. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  16. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  17. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 12/39

  18. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  19. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  20. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  21. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  22. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) But... need large amount of training data Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  23. Neural Representations Lower-dimensional fixed-length vector representations of symbols (predicates and constants): v apple , v orange , v fatherOf , . . . ∈ R k Can capture similarity and even semantic hierarchy of symbols: v grandpaOf = v grandfatherOf , v apple ∼ v orange , v apple < v fruit Can be trained from raw task data (e.g. facts in a knowledge base) Can be compositional v ‘‘is the father of’’ = RNN θ ( v is , v the , v father , v of ) But... need large amount of training data No direct way of incorporating prior knowledge v grandfatherOf ( X , Y ) :– v fatherOf ( X , Z ) , v parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 13/39

  24. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  25. Machine Learning & Logic Fuzzy Logic (Zadeh, 1965) Probabilistic Logic Programming, e.g., Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  26. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  27. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  28. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  29. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  30. 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  31. 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 (Garcez and Zaverucha, 1999) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  32. 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 (Garcez and Zaverucha, 1999) First-order inference (no training of symbol representations): Unification Neural Networks (Holld¨ obler, 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 GPU-accelerated End-to-end Differentiable Planning and Reasoning 14/39

  33. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  34. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) v s , v i , v j ∈ R k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  35. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) v s , v i , v j ∈ R k f ( v s , v i , v j ) = v ⊤ s ( v i ⊙ v j ) � = v sk v ik v jk k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  36. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ s ( v i ⊙ v j ) � = v sk v ik v jk k Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  37. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ f ( v s , v i , v j ) = s ( v i ⊙ v j ) � real( v s ) ⊤ (real( v i ) ⊙ real( v j )) = v sk v ik v jk + real( v s ) ⊤ (imag( v i ) ⊙ imag( v j )) k + imag( v s ) ⊤ (real( v i ) ⊙ imag( v j )) − imag( v s ) ⊤ (imag( v i ) ⊙ real( v j )) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  38. State-of-the-art Neural Link Prediction livesIn ( melinda , seattle )? = f ( v livesIn , v melinda , v seattle ) DistMult (Yang et al., 2015) ComplEx (Trouillon et al., 2016) v s , v i , v j ∈ R k v s , v i , v j ∈ C k f ( v s , v i , v j ) = v ⊤ f ( v s , v i , v j ) = s ( v i ⊙ v j ) � real( v s ) ⊤ (real( v i ) ⊙ real( v j )) = v sk v ik v jk + real( v s ) ⊤ (imag( v i ) ⊙ imag( v j )) k + imag( v s ) ⊤ (real( v i ) ⊙ imag( v j )) − imag( v s ) ⊤ (imag( v i ) ⊙ real( v j )) Training Loss � L = − y log ( σ ( f ( v s , v i , v j ))) − (1 − y ) log (1 − σ ( f ( v s , v i , v j ))) r s ( e i , e j ) , y ∈ T Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 15/39

  39. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  40. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  41. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , ∅ parentOf ( Z , Y ) . Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  42. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  43. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  44. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  45. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  46. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) Y / bart Z / homer · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  47. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  48. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · parentOf homer bart Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  49. Differentiable Proving in a Nutshell grandfatherOf abe bart Example Knowledge Base: 1. fatherOf ( abe , homer ) . 1. fatherOf ( abe , homer ) 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) . 3. grandfatherOf ( X , Y ) :– 2. parentOf ( homer , bart ) fatherOf ( X , Z ) , 3.1 fatherOf ( X , Z ) ∅ X / abe · · · parentOf ( Z , Y ) . Y / bart 3.2 parentOf ( Z , Y ) fatherOf abe Z 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 1. fatherOf ( abe , homer ) X / abe 2. parentOf ( homer , bart ) 3.2 parentOf ( Z , Y ) Y / bart Z / homer · · · · · · parentOf homer bart 3. grandfatherOf ( X , Y ) :– fatherOf ( X , Z ) , parentOf ( Z , Y ) 2. parentOf ( homer , bart ) · · · 1. fatherOf ( abe , homer ) X / abe Y / bart X / abe Z / homer Y / bart Z / homer Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 16/39

  50. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  51. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  52. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  53. Proof States S = (Ψ , ρ ) Substitution set Ψ constructed in the proof so far Neural network ρ that outputs a real-valued proof success score X / Q Y / bart S Ψ S ρ Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 17/39

  54. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

  55. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Discrete objects (rules, atoms etc.) are used to instantiate proof modules Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

  56. Proof Modules unify θ , or K θ , and K θ Modular construction of differentiable prover Discrete objects (rules, atoms etc.) are used to instantiate proof modules Modules transform proof states into new proof states Tim Rockt¨ aschel GPU-accelerated End-to-end Differentiable Planning and Reasoning 18/39

Recommend

Differentiable Tree Planning for Deep RL Greg Farquhar 1 In Collaboration With Tim

Differentiable Tree Planning for Deep RL Greg Farquhar 1 In Collaboration With Tim

Differentiable Tree Planning for Deep RL Greg Farquhar 1 In Collaboration With Tim Rocktaschel, Maximilian Igl, &amp; Shimon Whiteson Greg Farquhar 2 / 35 Overview Reinforcement learning Model-based RL and online planning

385 views • 35 slides
Accelerated Reader What is Accelerated Reader? Accelerated Reader is the number one software

Accelerated Reader What is Accelerated Reader? Accelerated Reader is the number one software

Camerado Springs Middle School A closer look at Accelerated Reader What is Accelerated Reader? Accelerated Reader is the number one software program (or tool) that helps manage reading practice, monitor daily progress, and plan instruction. The

484 views • 13 slides
Accelerated first-order methods Geoff Gordon &amp; Ryan Tibshirani Optimization 10-725 / 36-725

Accelerated first-order methods Geoff Gordon &amp; Ryan Tibshirani Optimization 10-725 / 36-725

Accelerated first-order methods Geoff Gordon &amp; Ryan Tibshirani Optimization 10-725 / 36-725 1 Remember generalized gradient descent We want to solve x R n g ( x ) + h ( x ) , min for g convex and differentiable, h convex Generalized

746 views • 27 slides
Learning to map between ferns with differentiable binary embedding networks Maximilian Blendowski

Learning to map between ferns with differentiable binary embedding networks Maximilian Blendowski

DIFFERENTIABLE BINARY EMBEDDING USING FERNS Learning to map between ferns with differentiable binary embedding networks Maximilian Blendowski &amp; Mattias P. Heinrich Institute of Medical Informatics University of Lbeck Short paper @ MIDL

95 views • 5 slides
NVGRAPH,FIREHOSE,PAGERANK GPU ACCELERATED ANALYTICS NOV 2016 Joe Eaton Ph.D. Accelerated

NVGRAPH,FIREHOSE,PAGERANK GPU ACCELERATED ANALYTICS NOV 2016 Joe Eaton Ph.D. Accelerated

NVGRAPH,FIREHOSE,PAGERANK GPU ACCELERATED ANALYTICS NOV 2016 Joe Eaton Ph.D. Accelerated Computing nvGRAPH New Features Coming Soon Agenda Dynamic Graphs GraphBLAS 2 ACCELERATED COMPUTING 10x Performance &amp; 5x Energy Efficiency GPU

855 views • 23 slides
Differentiable Rendering for Mesh and Implicit Surface Weikai Chen Tencent America GAMES

Differentiable Rendering for Mesh and Implicit Surface Weikai Chen Tencent America GAMES

Differentiable Rendering for Mesh and Implicit Surface Weikai Chen Tencent America GAMES Graphics And Mixed Environment Seminar Outline Motivation SoftRas: A Differentiable Renderer for Triangular Mesh (ICCV19 ) Learning to

1.17k views • 34 slides
Accelerated Photodynamic Cancer Therapy Planning with FullMonte Jeffrey Cassidy, PhD Candidate

Accelerated Photodynamic Cancer Therapy Planning with FullMonte Jeffrey Cassidy, PhD Candidate

Accelerated Photodynamic Cancer Therapy Planning with FullMonte Jeffrey Cassidy, PhD Candidate University of Toronto (Canada) #OpenPOWERSummit Photodynamic Therapy (PDT) for Cancer Photosensitizer Light Exposure (drug) (fluence J/cm 2 )

1.24k views • 15 slides
Learning with Differentiable Perturbed Optimizers Quentin Berthet Optimization for ML - CIRM -

Learning with Differentiable Perturbed Optimizers Quentin Berthet Optimization for ML - CIRM -

Learning with Differentiable Perturbed Optimizers Quentin Berthet Optimization for ML - CIRM - 2020 Q. Berthet M.Blondel O.Teboul M. Cuturi J-P. Vert F.Bach Learning with Differentiable Perturbed Optimizers Preprint: arXiv:2002.08676

857 views • 27 slides
Differentiable Cloth Simulation for Inverse Problems Junbang Liang 1 Content Motivation

Differentiable Cloth Simulation for Inverse Problems Junbang Liang 1 Content Motivation

Differentiable Cloth Simulation for Inverse Problems Junbang Liang 1 Content Motivation Related Work Our Method Simulation pipeline Gradient Computation Results 2 Motivation Differentiable Physics

520 views • 40 slides
Relay : a high level differentiable IR Jared Roesch TVMConf December 12th, 2018 1 This

Relay : a high level differentiable IR Jared Roesch TVMConf December 12th, 2018 1 This

Relay : a high level differentiable IR Jared Roesch TVMConf December 12th, 2018 1 This represents months of joint work with lots of great folks: 2 TVM Stack Optimization High-Level Differentiable IR Relay AutoTVM Tensor Expression

634 views • 36 slides
The Differentiable Curry Martin Abadi, Dan Belov, Gordon Plotkin, Richard Wei, Dimitrios Vytiniotis

The Differentiable Curry Martin Abadi, Dan Belov, Gordon Plotkin, Richard Wei, Dimitrios Vytiniotis

The Differentiable Curry Martin Abadi, Dan Belov, Gordon Plotkin, Richard Wei, Dimitrios Vytiniotis DeepMind and Google Brain thanks to the many from the Swift For Tensorflow and JAX teams r o f * s l a The Differentiable Curry i t n e

345 views • 19 slides
An Enriched Perspective on Differentiable Stacks Benjamin MacAdam Joint work with Jonathan

An Enriched Perspective on Differentiable Stacks Benjamin MacAdam Joint work with Jonathan

An Enriched Perspective on Differentiable Stacks Motivation An Enriched Perspective on Differentiable Stacks Benjamin MacAdam Joint work with Jonathan Gallagher July 9, 2019 An Enriched Perspective on Differentiable Stacks Motivation

323 views • 20 slides
Learning with Differentiable Perturbed Optimizers Quentin Berthet Youth in High-dimensions -

Learning with Differentiable Perturbed Optimizers Quentin Berthet Youth in High-dimensions -

Learning with Differentiable Perturbed Optimizers Quentin Berthet Youth in High-dimensions - ICTP - 2020 Q. Berthet M.Blondel O.Teboul M. Cuturi J-P. Vert F.Bach Learning with Differentiable Perturbed Optimizers Preprint:

517 views • 17 slides
What is Accelerated Reader? Accelerated Reader is a computer program that helps teachers manage

What is Accelerated Reader? Accelerated Reader is a computer program that helps teachers manage

What is Accelerated Reader? Accelerated Reader is a computer program that helps teachers manage and monitor childrens independent reading practice. Your child picks a book at his/her own level and reads it at his/her own pace. When finished,

453 views • 21 slides
Accelerated Learning - for Breakthrough Results Whole brain, person, systems approach Debbie

Accelerated Learning - for Breakthrough Results Whole brain, person, systems approach Debbie

Accelerated Learning - for Breakthrough Results Whole brain, person, systems approach Debbie Craig, Kerryn Kohl, Darryn Van Den Berg, Natalie Cunningham The Book PART 1: Paradigm shift for Accelerated Learning The need for accelerated

930 views • 41 slides
ACCELERATED COMPUTING FOR AI Bryan Catanzaro, 7 December 2018 ACCELERATED COMPUTING: REDUCE

ACCELERATED COMPUTING FOR AI Bryan Catanzaro, 7 December 2018 ACCELERATED COMPUTING: REDUCE

ACCELERATED COMPUTING FOR AI Bryan Catanzaro, 7 December 2018 ACCELERATED COMPUTING: REDUCE LATENCY OF IDEA GENERATION Research as a sequential, cyclic process Idea Limit: Limit: Programmability Ingenuity Hack Invent Test Code Train

543 views • 30 slides
SEQ 3 : Differentiable Sequence-to-Sequence-to-Sequence Autoencoder for Unsupervised Abstractive

SEQ 3 : Differentiable Sequence-to-Sequence-to-Sequence Autoencoder for Unsupervised Abstractive

SEQ 3 : Differentiable Sequence-to-Sequence-to-Sequence Autoencoder for Unsupervised Abstractive Sentence Compression Christos Baziotis, Ion Androutsopoulos, Ioannis Konstas, Alexandros Potamianos Ed nburgh NLP University of Edinburgh Natural

772 views • 51 slides
Singularities and Characteristic classes for Differentiable Maps

Singularities and Characteristic classes for Differentiable Maps

Singularities and Characteristic classes for Differentiable Maps Toru Ohmoto, Hokkaido Univ. July 20, Nihon Univ. Contents 1. Thom polynomials 2. Fukudas formula for Morin maps 3.

584 views • 37 slides
A Unified Distributed Algorithm for Non- Games Non-cooperative, Non-convex, and Non-differentiable

A Unified Distributed Algorithm for Non- Games Non-cooperative, Non-convex, and Non-differentiable

A Unified Distributed Algorithm for Non- Games Non-cooperative, Non-convex, and Non-differentiable Jong-Shi Pang and Meisam Razaviyayn presented at Workshop on Optimization for Modern Computation Peking University, Beijing, China 10:10

658 views • 20 slides
USE OF 3D MODELLING IN IOWA DOT ACCELERATED BRIDGE CONSTRUCTION (ABC) PROJECTS Jim Nelson ABC

USE OF 3D MODELLING IN IOWA DOT ACCELERATED BRIDGE CONSTRUCTION (ABC) PROJECTS Jim Nelson ABC

USE OF 3D MODELLING IN IOWA DOT ACCELERATED BRIDGE CONSTRUCTION (ABC) PROJECTS Jim Nelson ABC Definition ABC is bridge construction that uses innovative planning, design, materials and construction methods in a safe and cost- effective

343 views • 11 slides
Programming With A Differentiable Forth Interpreter Varun Gangal, CMU Based on the work of

Programming With A Differentiable Forth Interpreter Varun Gangal, CMU Based on the work of

Programming With A Differentiable Forth Interpreter Varun Gangal, CMU Based on the work of Matko Bosnjak et al 1 Whats Forth? Kind of like a cross between Python and Assembly High-level imperative programming language BUT Can

924 views • 32 slides
Reparameterization Gradient for Non-differentiable Models Wonyeol Lee Hangyeol Yu Hongseok

Reparameterization Gradient for Non-differentiable Models Wonyeol Lee Hangyeol Yu Hongseok

Reparameterization Gradient for Non-differentiable Models Wonyeol Lee Hangyeol Yu Hongseok Yang KAIST Published at NeurIPS 2018 Reparameterization Gradient for Non-differentiable Models Wonyeol Lee Hangyeol Yu Hongseok Yang

1.19k views • 95 slides
Automatic Differentiation (or Differentiable Programming) Atlm Gne Baydin National

Automatic Differentiation (or Differentiable Programming) Atlm Gne Baydin National

Automatic Differentiation (or Differentiable Programming) Atlm Gne Baydin National University of Ireland Maynooth Joint work with Barak Pearlmutter Alan Turing Institute, February 5, 2016 A brief introduction to AD My ongoing work

593 views • 30 slides
Alexandru Suciu Northeastern University Session on the Geometry and Topology of Differentiable

Alexandru Suciu Northeastern University Session on the Geometry and Topology of Differentiable

T OPOLOGY OF COMPLEX ARRANGEMENTS Alexandru Suciu Northeastern University Session on the Geometry and Topology of Differentiable Manifolds and Algebraic Varieties The Eighth Congress of Romanian Mathematicians Ia si, Romania, June 30, 2015

510 views • 23 slides

More recommend