PSDDs for Tractable Learning in Structured and Unstructured Spaces

Transcription

1 PSDDs for Tractable Learning in Structured and Unstructured Spaces Guy Van den Broeck DeLBP Aug 18, 2017

2 References Probabilistic Sentential Decision Diagrams Doga Kisa, Guy Van den Broeck, Arthur Choi and Adnan Darwiche KR, 2014 Learning with Massive Logical Constraints Doga Kisa, Guy Van den Broeck, Arthur Choi and Adnan Darwiche ICML LTPM workshop, 2014 Tractable Learning for Structured Probability Spaces Arthur Choi, Guy Van den Broeck and Adnan Darwiche IJCAI, 2015 Tractable Learning for Complex Probability Queries Jessa Bekker, Jesse Davis, Arthur Choi, Adnan Darwiche, Guy Van den Broeck. NIPS, 2015 Learning the Structure of PSDDs Yitao Liang, Jessa Bekker and Guy Van den Broeck UAI, 2017 Towards Compact Interpretable Models: Learning and Shrinking PSDDs Yitao Liang and Guy Van den Broeck IJCAI XAI workshop, 2017

3 (P)SDDs in Melbourne Sunday: Logical Foundations for Uncertainty and Machine Learning Workshop Adnan Darwiche: On the Role of Logic in Probabilistic Inference and Machine Learning YooJung Choi: Optimal Feature Selection for Decision Robustness in Bayesian Networks Sunday: Explainable AI Workshop Yitao Liang: Towards Compact Interpretable Models: Learning and Shrinking PSDDs Tuesday: IJCAI YooJung Choi (again)

4 Structured vs. unstructured probability spaces?

5 Running Example Courses: Logic (L) Knowledge Representation (K) Probability (P) Artificial Intelligence (A) Data

6 Running Example Courses: Logic (L) Knowledge Representation (K) Probability (P) Artificial Intelligence (A) Data Constraints Must take at least one of Probability or Logic. Probability is a prerequisite for AI. The prerequisites for KR is either AI or Logic.

7 Probability Space unstructured L K P A

8 Structured Probability Space unstructured L K P A Must take at least one of Probability or Logic. Probability is a prerequisite for AI. The prerequisites for KR is either AI or Logic. 7 out of 16 instantiations are impossible structured L K P A

9 Learning with Constraints Data Constraints (Background Knowledge) (Physics) Learn Statistical Model (Distribution)

10 Learning with Constraints Data Constraints (Background Knowledge) (Physics) Learn Statistical Model (Distribution) Learn a statistical model that assigns zero probability to instantiations that violate the constraints.

11 Example: Video [Lu, W. L., Ting, J. A., Little, J. J., & Murphy, K. P. (2013). Learning to track and identify players from broadcast sports videos.]

12 Example: Video [Lu, W. L., Ting, J. A., Little, J. J., & Murphy, K. P. (2013). Learning to track and identify players from broadcast sports videos.]

13 Example: Robotics [Wong, L. L., Kaelbling, L. P., & Lozano-Perez, T., Collision-free state estimation. ICRA 2012]

14 Example: Robotics [Wong, L. L., Kaelbling, L. P., & Lozano-Perez, T., Collision-free state estimation. ICRA 2012]

15 Example: Language Non-local dependencies: At least one verb in each sentence [Chang, M., Ratinov, L., & Roth, D. (2008). Constraints as prior knowledge],, [Chang, M. W., Ratinov, L., & Roth, D. (2012). Structured learning with constrained conditional models.], [

16 Example: Language Non-local dependencies: At least one verb in each sentence Sentence compression If a modifier is kept, its subject is also kept [Chang, M., Ratinov, L., & Roth, D. (2008). Constraints as prior knowledge],, [Chang, M. W., Ratinov, L., & Roth, D. (2012). Structured learning with constrained conditional models.], [

17 Example: Language Non-local dependencies: At least one verb in each sentence Sentence compression If a modifier is kept, its subject is also kept Information extraction [Chang, M., Ratinov, L., & Roth, D. (2008). Constraints as prior knowledge],, [Chang, M. W., Ratinov, L., & Roth, D. (2012). Structured learning with constrained conditional models.], [

18 Example: Language Non-local dependencies: At least one verb in each sentence Sentence compression If a modifier is kept, its subject is also kept Information extraction Semantic role labeling and many more! [Chang, M., Ratinov, L., & Roth, D. (2008). Constraints as prior knowledge],, [Chang, M. W., Ratinov, L., & Roth, D. (2012). Structured learning with constrained conditional models.], [

19 Example: Deep Learning [Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., Grabska-Barwińska, A., et al.. (2016). Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626), ]

20 Example: Deep Learning [Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., Grabska-Barwińska, A., et al.. (2016). Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626), ]

21 Example: Deep Learning [Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., Grabska-Barwińska, A., et al.. (2016). Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626), ]

22 Example: Deep Learning [Graves, A., Wayne, G., Reynolds, M., Harley, T., Danihelka, I., Grabska-Barwińska, A., et al.. (2016). Hybrid computing using a neural network with dynamic external memory. Nature, 538(7626), ]

23 What are people doing now? Ignore constraints Handcraft into models Use specialized distributions Find non-structured encoding Try to learn constraints Hack your way around

24 What are people doing now? Ignore constraints Handcraft into models Use specialized distributions Find non-structured encoding Try to learn constraints Hack your way around + Accuracy? Specialized skill? Intractable inference? Intractable learning? Waste parameters? Risk predicting out of space? you are on your own

25 Structured Probability Spaces Everywhere in ML! Configuration problems, inventory, video, text, deep learning Planning and diagnosis (physics) Causal models: cooking scenarios (interpreting videos) Combinatorial objects: parse trees, rankings, directed acyclic graphs, trees, simple paths, game traces, etc.

26 Structured Probability Spaces Everywhere in ML! Configuration problems, inventory, video, text, deep learning Planning and diagnosis (physics) Causal models: cooking scenarios (interpreting videos) Combinatorial objects: parse trees, rankings, directed acyclic graphs, trees, simple paths, game traces, etc. Some representations: constrained conditional models, mixed networks, probabilistic logics.

27 Structured Probability Spaces Everywhere in ML! Configuration problems, inventory, video, text, deep learning Planning and diagnosis (physics) Causal models: cooking scenarios (interpreting videos) Combinatorial objects: parse trees, rankings, directed acyclic graphs, trees, simple paths, game traces, etc. Some representations: constrained conditional models, mixed networks, probabilistic logics. No statistical ML boxes out there that take constraints as input!

28 Structured Probability Spaces Everywhere in ML! Configuration problems, inventory, video, text, deep learning Planning and diagnosis (physics) Causal models: cooking scenarios (interpreting videos) Combinatorial objects: parse trees, rankings, directed acyclic graphs, trees, simple paths, game traces, etc. Some representations: constrained conditional models, mixed networks, probabilistic logics. No statistical ML boxes out there that take constraints as input! Goal: Constraints as important as data! General purpose!

29 Specification Language: Logic

30 Structured Probability Space unstructured L K P A Must take at least one of Probability or Logic. Probability is a prerequisite for AI. The prerequisites for KR is either AI or Logic. 7 out of 16 instantiations are impossible structured L K P A

31 Boolean Constraints unstructured L K P A out of 16 instantiations are impossible structured L K P A

32 Combinatorial Objects: Rankings rank sushi rank sushi 1 fatty tuna 2 sea urchin 3 salmon roe 4 shrimp 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll 1 shrimp 2 sea urchin 3 salmon roe 4 fatty tuna 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll 10 items: 3,628,800 rankings 20 items: 2,432,902,008,176,640,000 rankings

33 Combinatorial Objects: Rankings rank sushi 1 fatty tuna 2 sea urchin 3 salmon roe 4 shrimp 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll rank sushi 1 shrimp 2 sea urchin 3 salmon roe 4 fatty tuna 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll A ij item i at position j (n items require n 2 Boolean variables)

34 Combinatorial Objects: Rankings rank sushi 1 fatty tuna 2 sea urchin 3 salmon roe 4 shrimp 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll rank sushi 1 shrimp 2 sea urchin 3 salmon roe 4 fatty tuna 5 tuna 6 squid 7 tuna roll 8 see eel 9 egg 10 cucumber roll A ij item i at position j (n items require n 2 Boolean variables) An item may be assigned to more than one position A position may contain more than one item

35 Encoding Rankings in Logic A ij : item i at position j pos 1 pos 2 pos 3 pos 4 item 1 A 11 A 12 A 13 A 14 item 2 A 21 A 22 A 23 A 24 item 3 A 31 A 32 A 33 A 34 item 4 A 41 A 42 A 43 A 44

36 Encoding Rankings in Logic A ij : item i at position j pos 1 pos 2 pos 3 pos 4 constraint: each item i assigned to a unique position (n constraints) item 1 A 11 A 12 A 13 A 14 item 2 A 21 A 22 A 23 A 24 item 3 A 31 A 32 A 33 A 34 item 4 A 41 A 42 A 43 A 44

37 Encoding Rankings in Logic A ij : item i at position j pos 1 pos 2 pos 3 pos 4 constraint: each item i assigned to a unique position (n constraints) item 1 A 11 A 12 A 13 A 14 item 2 A 21 A 22 A 23 A 24 item 3 A 31 A 32 A 33 A 34 item 4 A 41 A 42 A 43 A 44 constraint: each position j assigned a unique item (n constraints)

38 Encoding Rankings in Logic A ij : item i at position j pos 1 pos 2 pos 3 pos 4 constraint: each item i assigned to a unique position (n constraints) item 1 A 11 A 12 A 13 A 14 item 2 A 21 A 22 A 23 A 24 item 3 A 31 A 32 A 33 A 34 item 4 A 41 A 42 A 43 A 44 constraint: each position j assigned a unique item (n constraints)

39 Structured Space for Paths cf. Nature paper

40 Structured Space for Paths cf. Nature paper Good variable assignment (represents route) 184

41 Structured Space for Paths cf. Nature paper Good variable assignment (represents route) 184 Bad variable assignment (does not represent route) 16,777,032

42 Structured Space for Paths cf. Nature paper Good variable assignment (represents route) 184 Bad variable assignment (does not represent route) 16,777,032 Space easily encoded in logical constraints See [Choi, Tavabi, Darwiche, AAAI 2016]

43 Structured Space for Paths cf. Nature paper Good variable assignment (represents route) 184 Bad variable assignment (does not represent route) 16,777,032 Space easily encoded in logical constraints See [Choi, Tavabi, Darwiche, AAAI 2016] Unstructured probability space: ,777,032 = 2 24

44 Deep Architecture Logic + Probability

45 Logical Circuits L K L P A P L L P A P L K L P P K K A A A A

46 Property: Decomposability L K L P A P L L P A P L K L P P K K A A A A

47 Property: Decomposability L K L P A P L L P A P L K L P P K K A A A A

48 Property: Determinism L K L P A P L L P A P L K L P P Input: L, K, P, A K K A A A A

49 Sentential Decision Diagram (SDD) L K L P A P L L P A P L K L P P Input: L, K, P, A K K A A A A

50 Sentential Decision Diagram (SDD) L K L P A P L L P A P L K L P P Input: L, K, P, A K K A A A A

51 Sentential Decision Diagram (SDD) L K L P A P L L P A P L K L P P Input: L, K, P, A K K A A A A

52 Tractable for Logical Inference Is structured space empty? (SAT) Count size of structured space (#SAT) Check equivalence of spaces

53 Tractable for Logical Inference Is structured space empty? (SAT) Count size of structured space (#SAT) Check equivalence of spaces Algorithms linear in circuit size (pass up, pass down, similar to backprop)

54 Tractable for Logical Inference Is structured space empty? (SAT) Count size of structured space (#SAT) Check equivalence of spaces Algorithms linear in circuit size (pass up, pass down, similar to backprop)

55 PSDD: Probabilistic SDD L K L P A P L L P A P L K L P P K K A A A A

56 PSDD: Probabilistic SDD L K L P A P L L P A P L K L P P K K A A A A Input: L, K, P, A

57 PSDD: Probabilistic SDD L K L P A P L L P A P L K L P P K K A A A A Input: L, K, P, A

58 PSDD: Probabilistic SDD L K L P A P L L P A P L K L P P K K A A A A Input: L, K, P, A Pr(L,K,P,A) = 0.3 x 1.0 x 0.8 x 0.4 x 0.25 = 0.024

59 PSDD nodes induce a normalized distribution! L K L P A P L L P A P L K L P P A A A A A A

60 PSDD nodes induce a normalized distribution! L K L P A P L L P A P L K L P P A A A A A A

61 PSDD nodes induce a normalized distribution! L K L P A P L L P A P L K L P P A A A A A A Can read probabilistic independences off the circuit structure

62 Tractable for Probabilistic Inference MAP inference: Find most-likely assignment (otherwise NP-complete) Computing conditional probabilities Pr(x y) (otherwise PP-complete) Sample from Pr(x y)

63 Tractable for Probabilistic Inference MAP inference: Find most-likely assignment (otherwise NP-complete) Computing conditional probabilities Pr(x y) (otherwise PP-complete) Sample from Pr(x y) Algorithms linear in circuit size (pass up, pass down, similar to backprop)

64 Learning PSDDs Logic + Probability + ML

65 Parameters are Interpretable L K L P A P L L P A P L K L P P K K A A A A Explainable AI DARPA Program

66 Parameters are Interpretable L K L P A P L L P A P L K L P P K K A A A A Student takes course L Explainable AI DARPA Program

67 Parameters are Interpretable L K L P A P L L P A P L K L P P Student takes course L K K A A Student takes course P A A Explainable AI DARPA Program

68 Parameters are Interpretable Probability of P given L L K L P A P L L P A P L K L P P Student takes course L K K A A Student takes course P A A Explainable AI DARPA Program

69 Learning Algorithms Parameter learning: Closed form max likelihood from complete data One pass over data to estimate Pr(x y)

70 Learning Algorithms Parameter learning: Closed form max likelihood from complete data One pass over data to estimate Pr(x y) Not a lot to say: very easy!

71 Learning Algorithms Parameter learning: Closed form max likelihood from complete data One pass over data to estimate Pr(x y) Not a lot to say: very easy! Circuit learning (naïve): Compile constraints to SDD circuit Use SAT solver technology Circuit does not depend on data

72 Learning Preference Distributions Special-purpose distribution: Mixture-of-Mallows # of components from 1 to 20 EM with 10 random seeds implementation of Lu & Boutilier PSDD

73 Learning Preference Distributions Special-purpose distribution: Mixture-of-Mallows # of components from 1 to 20 EM with 10 random seeds implementation of Lu & Boutilier PSDD This is the naive approach, circuit does not depend on data!

74 Learn Circuit from Data Even in unstructured spaces

75 Tractable Learning Bayesian networks Markov networks

76 Tractable Learning Bayesian networks Markov networks Do not support linear-time exact inference

77 Tractable Learning Historically: Polytrees, Chow-Liu trees, etc. SPNs Cutset Networks Both are Arithmetic Circuits (ACs) [Darwiche, JACM 2003]

78 PSDDs are Arithmetic Circuits n * * * * 1 * 2 * n p 1 s 1 p 2 s 2 PSDD p n s n p 1 s 1 p 2 s 2 p n s n AC

79 Tractable Learning Strong Properties Representational Freedom

80 Tractable Learning DNN Strong Properties Representational Freedom

81 Tractable Learning DNN Strong Properties Representational Freedom

82 Tractable Learning DNN Cutset SPN Strong Properties Representational Freedom

83 Tractable Learning DNN Cutset SPN Strong Properties Representational Freedom Perhaps the most powerful circuit proposed to date

84 PSDDs for the Logic-Phobic

85 PSDDs for the Logic-Phobic Bottom-up each node is a distribution

86 PSDDs for the Logic-Phobic Bottom-up each node is a distribution

87 PSDDs for the Logic-Phobic

88 PSDDs for the Logic-Phobic Multiply independent distributions

89 PSDDs for the Logic-Phobic

90 PSDDs for the Logic-Phobic Weighted mixture of lower level distributions

91 PSDDs for the Logic-Phobic

92 PSDDs for the Logic-Phobic

93 Variable Trees (vtrees) PSDD Vtree Correspondence

94 Learning Variable Trees How much do vars depend on each other? Learn vtree by hierarchical clustering

95 Learning Variable Trees How much do vars depend on each other? Learn vtree by hierarchical clustering

96 Learning Primitives

97 Learning Primitives

98 Learning Primitives Primitives maintain PSDD properties and structured space!

99 LearnPSDD 1 Vtree learning 2 Construct the most naïve PSDD 3 LearnPSDD (search for better structure)

100 LearnPSDD 1 Vtree learning 2 Construct the most naïve PSDD Generate candidate operations Simulate operations 3 LearnPSDD (search for better structure) Execute the best

101 Experiments on 20 datasets

102 Experiments on 20 datasets Compare with O-SPN: smaller size in 14, better LL in 11, win on both in 6 Compare with L-SPN: smaller size in 14, better LL in 6, win on both in 2

103 Experiments on 20 datasets Compare with O-SPN: smaller size in 14, better LL in 11, win on both in 6 Compare with L-SPN: smaller size in 14, better LL in 6, win on both in 2 Comparable in performance & Smaller in size

104 Ensembles of PSDDs

105 Ensembles of PSDDs EM/Bagging

106 Ensembles of PSDDs EM/Bagging

107 State-of-the-Art Performance

108 State-of-the-Art Performance State of the art in 6 datasets

109 What happens if you ignore constraints?

110 What happens if you ignore constraints? Roadmap Compile logic into a SDD Convert to a PSDD: Parameter estimation LearnPSDD

111 What happens if you ignore constraints? Roadmap Compile logic into a SDD Convert to a PSDD: Parameter estimation LearnPSDD Generate candidate operations Execute the best Simulate operations

112 What happens if you ignore constraints? Discrete multi-valued data A: a 1, a 2, a 3 a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3

113 What happens if you ignore constraints? Discrete multi-valued data A: a 1, a 2, a 3 a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3

114 What happens if you ignore constraints? Discrete multi-valued data A: a 1, a 2, a 3 a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3

115 What happens if you ignore constraints? Discrete multi-valued data A: a 1, a 2, a 3 a 1 a 2 a 3 a 1 a 2 a 3 a 1 a 2 a 3 Never omit domain constraints

116 Complex queries and Learning from constraints

117 Incomplete Data a classical complete dataset id X Y Z 1 x 1 y 2 z 1 2 x 2 y 1 z 2 3 x 2 y 1 z 2 4 x 1 y 1 z 1 5 x 1 y 2 z 2 closed-form (maximum-likelihood estimates are unique)

118 Incomplete Data a classical complete dataset id X Y Z 1 x 1 y 2 z 1 2 x 2 y 1 z 2 3 x 2 y 1 z 2 4 x 1 y 1 z 1 5 x 1 y 2 z 2 closed-form (maximum-likelihood estimates are unique) a classical incomplete dataset id X Y Z 1 x 1 y 2? 2 x 2 y 1? 3?? z 2 4? y 1 z 1 5 x 1 y 2 z 2 EM algorithm (on PSDDs)

119 Incomplete Data a classical complete dataset id X Y Z 1 x 1 y 2 z 1 2 x 2 y 1 z 2 3 x 2 y 1 z 2 4 x 1 y 1 z 1 a classical incomplete dataset id X Y Z 1 x 1 y 2? 2 x 2 y 1? 3?? z 2 4? y 1 z 1 a new type of incomplete dataset id X Y Z 1 X Z 2 x 2 and (y 2 or z 2 ) 3 x 2 y 1 4 X Y Z 1 5 x 1 y 2 z 2 closed-form (maximum-likelihood estimates are unique) 5 x 1 y 2 z 2 EM algorithm (on PSDDs) 5 x 1 and y 2 and z 2 Missed in the ML literature

120 Structured Datasets a classical complete dataset (e.g., total rankings) a classical incomplete dataset (e.g., top-k rankings) id st sushi fatty tuna fatty tuna 3 tuna 4 fatty tuna 2 nd sushi sea urchin 3 rd sushi salmon roe tuna shrimp tuna roll salmon roe sea eel tuna id st sushi fatty tuna 3 tuna 4 2 nd sushi 3 rd sushi sea urchin? fatty tuna?? fatty tuna tuna roll? salmon roe? 5 egg squid shrimp 5 egg??

121 Structured Datasets a classical complete dataset (e.g., total rankings) a new type of incomplete dataset (e.g., partial rankings) id st sushi fatty tuna fatty tuna 3 tuna 4 fatty tuna 2 nd sushi sea urchin 3 rd sushi salmon roe tuna shrimp tuna roll salmon roe sea eel tuna 5 egg squid shrimp id st sushi 2 nd sushi 3 rd sushi (fatty tuna > sea urchin) and (tuna > sea eel) (fatty tuna is 1 st ) and (salmon roe > egg) 3 tuna > squid 4 egg is last 5 egg > squid > shrimp (represents constraints on possible total rankings)

122 Learning from Incomplete Data Movielens Dataset: 3,900 movies, 6,040 users, 1m ratings take ratings from 64 most rated movies ratings 1-5 converted to pairwise prefs. PSDD for partial rankings 4 tiers 18,711 parameters movies by expected tier rank movie 1 The Godfather 2 The Usual Suspects 3 Casablanca 4 The Shawshank Redemption 5 Schindler s List 6 One Flew Over the Cuckoo s Nest 7 The Godfather: Part II 8 Monty Python and the Holy Grail 9 Raiders of the Lost Ark 10 Star Wars IV: A New Hope

123 PSDD Sizes

124 Structured Queries rank movie 1 Star Wars V: The Empire Strikes Back 2 Star Wars IV: A New Hope 3 The Godfather 4 The Shawshank Redemption 5 The Usual Suspects

125 Structured Queries rank movie 1 Star Wars V: The Empire Strikes Back 2 Star Wars IV: A New Hope 3 The Godfather 4 The Shawshank Redemption 5 The Usual Suspects no other Star Wars movie in top-5 at least one comedy in top-5

126 Structured Queries no other Star Wars movie in top-5 at least one comedy in top-5 rank movie 1 Star Wars V: The Empire Strikes Back 2 Star Wars IV: A New Hope 3 The Godfather 4 The Shawshank Redemption 5 The Usual Suspects rank movie 1 Star Wars V: The Empire Strikes Back 2 American Beauty 3 The Godfather 4 The Usual Suspects 5 The Shawshank Redemption

127 Structured Queries no other Star Wars movie in top-5 at least one comedy in top-5 rank movie 1 Star Wars V: The Empire Strikes Back 2 Star Wars IV: A New Hope 3 The Godfather 4 The Shawshank Redemption 5 The Usual Suspects rank movie 1 Star Wars V: The Empire Strikes Back 2 American Beauty 3 The Godfather 4 The Usual Suspects 5 The Shawshank Redemption diversified recommendations via logical constraints

128 Conclusions Structured spaces are everywhere PSDDs build on logical circuits 1. Tractability 2. Semantics 3. Natural encoding of structured spaces Learning is effective 1. From constraints encoding structured space State of the art learning preference distributions 2. From standard unstructured datasets using search State of the art on standard tractable learning datasets Novel settings for inference and learning Structured spaces / learning from constraints / complex queries

129 References Probabilistic Sentential Decision Diagrams Doga Kisa, Guy Van den Broeck, Arthur Choi and Adnan Darwiche KR, 2014 Learning with Massive Logical Constraints Doga Kisa, Guy Van den Broeck, Arthur Choi and Adnan Darwiche ICML LTPM workshop, 2014 Tractable Learning for Structured Probability Spaces Arthur Choi, Guy Van den Broeck and Adnan Darwiche IJCAI, 2015 Tractable Learning for Complex Probability Queries Jessa Bekker, Jesse Davis, Arthur Choi, Adnan Darwiche, Guy Van den Broeck. NIPS, 2015 Learning the Structure of PSDDs Yitao Liang, Jessa Bekker and Guy Van den Broeck UAI, 2017 Towards Compact Interpretable Models: Learning and Shrinking PSDDs Yitao Liang and Guy Van den Broeck IJCAI XAI workshop, 2017

130 (P)SDDs in Melbourne Sunday: Logical Foundations for Uncertainty and Machine Learning Workshop Adnan Darwiche: On the Role of Logic in Probabilistic Inference and Machine Learning YooJung Choi: Optimal Feature Selection for Decision Robustness in Bayesian Networks Sunday: Explainable AI Workshop Yitao Liang: Towards Compact Interpretable Models: Learning and Shrinking PSDDs Tuesday: IJCAI YooJung Choi (again)

131 Conclusions Statistical ML Probability Symbolic AI Logic PSDD Connectionism Deep

132 Questions? PSDD with 15,000 nodes LearnPSDD code: Other PSDD code: SDD code: