Scalable Statistical - PowerPoint Presentation

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Scalable Statistical Relational Learning for NLP

William Y. WangWilliam W. CohenMachine Learning Dept and Language Technologies Inst.joint work with:Kathryn Rivard Mazaitis

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OutlineMotivationBackgroundLogicProbabilityCombining logic and probabilities: MLNsProPPRKey ideasLearning methodResults for parameter learning

Structure learning for ProPPR for KB completionJoint IE and KB completionComparison to neural KBC modelsBeyond ProPPR….

Slide3

Motivation

Slide4

Slide5

KR & ReasoningInference Methods, Inference Rules

AnswersQueries…Challenges for KR

:

Robustness

: noise, incompleteness, ambiguity (“

Sunnybrook

”), statistical information (“

foundInRoom

(bathtub, bathroom)

”), …

Complex queries:

“which Canadian hockey teams have won the Stanley Cup?”

Learning

: how to

acquire and maintain

knowledge and inference rules as well as how to use it

“Expressive, probabilistic, efficient:

pick any two”

Current state of the art

What if the DB/KB or inference rules are imperfect?

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Large ML-based software system

Machine Learning(for complex tasks)

Relational, joint learning and inference

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Background

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

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Background: Logic ProgramsA program with one definite clause (Horn clauses):grandparent(X,Y) :- parent(X,Z),parent(Z,Y)Logical variables: X,Y,ZConstant symbols: bob, alice, …

We’ll consider two types of clauses:Horn clauses A:-B1,…,Bk with no constantsUnit clauses A:- with no variables (facts):parent(alice,bob):- or parent(alice,bob)

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

head

body

“neck”

Intensional definition,

rules

Extensional definition, database

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Background: Logic ProgramsA program with one definite clause:grandparent(X,Y) :- parent(X,Z),parent(Z,Y)Logical variables: X,Y,Z

Constant symbols: bob, alice, …Predicates: grandparent/2, parent/2Alphabet: set of possible predicates and constantsAtomic formulae: parent(X,Y), parent(alice,bob)Ground atomic formulae: parent(alice,bob), …

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

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Background: Logic ProgramsThe set of all ground atomic formulae (consistent with a fixed alphabet) is the Herbrand base of a program: {parent(alice,alice),parent(alice,bob),…,parent(zeke,zeke),grandparent(alice,alice),…}The

interpretation of a program is a subset of the Herbrand base. An interpretation M is a model of a program if For any A:-B1,…,Bk in the program and any mapping Theta from the variables in A,B1,..,Bk to constants:If Theta(B1) in M and … and Theta(Bk) in M then Theta(A) in MA program defines a unique least Herbrand model

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

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Background: Logic ProgramsA program defines a unique least Herbrand modelExample program:grandparent(X,Y):-parent(X,Z),parent(Z,Y).parent(alice,bob). parent(bob,chip). parent(bob,dana).

The least Herbrand model also includes grandparent(alice,dana) and grandparent(alice,chip).Finding the least Herbrand model: theorem proving…Usually we case about answering queries: What are values of W: grandparent(alice,W) ?

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

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MotivationInference Methods, Inference Rules

AnswersQueriesChallenges for KR:Robustness: noise, incompleteness, ambiguity (“Sunnybrook”), statistical information (“foundInRoom(bathtub, bathroom)”), …Complex queries: “which Canadian hockey teams have won the Stanley Cup?”Learning: how to acquire and maintain knowledge and inference rules as well as how to use itquery(T):- play(T,hockey), hometown(T,C), country(C,canada)

{T : query(T) } ?

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BackgroundRandom variable: burglary, earthquake, …Usually denote with upper-case letters: B,E,A,J,MJoint distribution: Pr(B,E,A,J,M)

H/T: “Probabilistic Logic Programming, De Raedt and KerstingBE

A

J

M

prob

T

T

T

T

T

0.00001

F

T

T

T

T

0.03723

…

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Background: Bayes networksRandom variable: B,E,A,J,MJoint distribution: Pr(B,E,A,J,M)Directed graphical models

give one way of defining a compact model of the joint distribution:Queries: Pr(A=t|J=t,M=f) ?

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

A

J

Prob

(J|A)

F

F

0.95

F

T

0.05

T

F

0.25

T

T

0.75

A

M

Prob

(J|A)

F

F

0.80

…

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BackgroundRandom variable: B,E,A,J,MJoint distribution: Pr(B,E,A,J,M)Directed graphical models

give one way of defining a compact model of the joint distribution:Queries: Pr(A=t|J=t,M=f) ?

H/T: “Probabilistic Logic Programming, De Raedt and Kersting

A

J

Prob

(J|A)

F

F

0.95

F

T

0.05

T

F

0.25

T

T

0.75

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Background: Markov networksRandom variable: B,E,A,J,MJoint distribution: Pr(B,E,A,J,M)Undirected graphical models

give another way of defining a compact model of the joint distribution…via potential functions.ϕ(A=a,J=j) is a scalar measuring the “compatibility” of A=a J=j

x

x

x

x

A

J

ϕ

(

a,j

)

F

F

20

F

T

1

T

F

0.1

T

T

0.4

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Backgroundϕ(A=a,J=j) is a scalar measuring the “compatibility” of A=a J=j

x

x

x

x

A

J

ϕ

(

a,j

)

F

F

20

F

T

1

T

F

0.1

T

T

0.4

clique potential

…

…

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MotivationInference Methods, Inference Rules

AnswersQueriesChallenges for KR:Robustness: noise, incompleteness, ambiguity (“Sunnybrook”), statistical information (“foundInRoom(bathtub, bathroom)”), …Complex queries: “which Canadian hockey teams have won the Stanley Cup?”Learning: how to acquire and maintain knowledge and inference rules as well as how to use it

In space of “flat” propositions corresponding to single random variables

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Background

H/T: “Probabilistic Logic Programming, De Raedt and Kersting???

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OutlineMotivationBackgroundLogicProbabilityCombining logic and probabilities: MLNsProPPRKey ideasLearning methodResults for parameter learning

Structure learning for ProPPR for KB completionJoint IE and KB completionComparison to neural KBC modelsBeyond ProPPR….

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Markov Networks: [Review]Undirected graphical modelsCancer

CoughAsthmaSmokingSmoking

Cancer

Ф

(S,C)

False

False

4.5

False

True

4.5

True

False

2.7

True

True

4.5

H/T: Pedro Domingos

x

= vector

x

c

= short vector

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Markov Logic: IntuitionA logical KB is a set of hard constraintson the set of possible worldsLet’s make them soft constraints

:When a world violates a formula,It becomes less probable, not impossibleGive each formula a weight(Higher weight  Stronger constraint)

H/T: Pedro Domingos

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Markov Logic: DefinitionA Markov Logic Network (MLN) is a set of pairs (F, w) whereF is a formula in first-order logicw

is a real numberTogether with a set of constants,it defines a Markov network withOne node for each grounding of each predicate in the MLNOne feature for each grounding of each formula F in the MLN, with the corresponding weight w

H/T: Pedro Domingos

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Example: Friends & Smokers

H/T: Pedro Domingos

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Example: Friends & Smokers

H/T: Pedro Domingos

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Example: Friends & Smokers

H/T: Pedro Domingos

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Example: Friends & Smokers

Two constants:

Anna

(A) and

Bob

(B)

H/T: Pedro Domingos

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Example: Friends & Smokers

Cancer(A)Smokes(A)

Smokes(B)

Cancer(B)

Two constants:

Anna

(A) and

Bob

(B)

H/T: Pedro Domingos

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Example: Friends & Smokers

Cancer(A)Smokes(A)

Friends(A,A)

Friends(B,A)

Smokes(B)

Friends(A,B)

Cancer(B)

Friends(B,B)

Two constants:

Anna

(A) and

Bob

(B)

H/T: Pedro Domingos

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Example: Friends & Smokers

Cancer(A)Smokes(A)

Friends(A,A)

Friends(B,A)

Smokes(B)

Friends(A,B)

Cancer(B)

Friends(B,B)

Two constants:

Anna

(A) and

Bob

(B)

H/T: Pedro Domingos

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Example: Friends & Smokers

Cancer(A)Smokes(A)

Friends(A,A)

Friends(B,A)

Smokes(B)

Friends(A,B)

Cancer(B)

Friends(B,B)

Two constants:

Anna

(A) and

Bob

(B)

H/T: Pedro Domingos

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Markov Logic NetworksMLN is template for ground Markov netsProbability of a world x:

Weight of formula

i

No. of true groundings of formula

i

in

x

H/T: Pedro Domingos

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MLNs generalize many statistical models Special cases:Markov networksMarkov random fieldsBayesian networksLog-linear models

Exponential modelsMax. entropy modelsGibbs distributionsBoltzmann machinesLogistic regressionHidden Markov modelsConditional random fieldsObtained by making all predicates zero-arityMarkov logic allows objects to be interdependent (non-i.i.d.)

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MLNs generalize logic programs Subsets of Herbrand base  domain of joint distributionInterpretation  element of the joint

Consistency with all clauses A:-B1,…,Bk == “model of program”  compatibility with program as determined by clique potentialsReaches logic in the limit when potentials are infinite.

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MLNs are expensive Inference done by explicitly building a ground MLNHerbrand base is huge for reasonable programs: It grows faster than the size of the DB of facts

You’d like to able to use a huge DB—NELL is O(10M)Inference on an arbitrary MLN is expensive: #P-completeIt’s not obvious how to restrict the template so the MLNs will be tractable

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What’s the alternative?There are many probabilistic LPs:Compile to other 0th-order formats: (Bayesian LPs, ProbLog, ….), Impose a distribution over proofs, not interpretations (Probabilistic Constraint LPs, Stochastic LPs, …): requires generating all proofs to answer queries, also a large spaceSample from the space of proofs (PRISM, Blog)

Limited relational extensions to 0th-order models (PRMs, RDTs, MEBNs, …)Probabilistic programming languages (Church, …)Imperative languages for defining complex probabilistic models (Related LP work: PRISM)

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OutlineMotivationBackgroundLogicProbabilityCombining logic and probabilities: MLNsProPPRKey ideasLearning methodResults for parameter learning

Structure learning for ProPPR for KB completionJoint IE and KB completionComparison to neural KBC modelsBeyond ProPPR….