Inference and Learning via Integer Linear Programming

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Inference and Learning via Integer Linear
Programming

• Vasin,Dan,Scott,Dav

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Outline

• Problem Definition
• Integer Linear Programming (ILP)
• Its generality
• Learning and Inference via ILP
• Experiments
• Extension to hierarchical learning
• Future Direction
• Hidden Variables

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Problem Definition

• X (X1,...,Xk) ? X k X
• Y (Y1,...,Yl) ? Y l Y
• Given X x, find Y y
• Notation agreements
• Capital letters mean variables
• Non-capital letters mean values
• Bold indicates vectors or matrixes
• X,Y is a set

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Example (Text Chunking)
y
NP
ADJP
VP
ADVP
VP
The
guy
presenting
now
is
so
tired
x
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Classifiers

• A classifier
• h X ?Y (l-1)?Y ? 1,..,l ?R
• Example
• score(x,y-3,NP,3) 0.3
• score(x,y-3,VP,3) 0.5
• score(x,y-3,ADVP,3) 0.2
• score(x,y-3,ADJP,3) 1.2
• score(x,y-3,NULL,3) 0.1

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Inference

• Goal x ? y
• Given
• x input
• score(x,y-t,y,t) for all (y-t,y) ? Y l, t ?
  1,..,l
• C A set of constraints over Y
• Find y
• maximizes global function
• score(x,y) ?t score(x,y-t,yt,t)
• satisfies constraints C

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Integer Linear Programming

• Boolean variables U (U1,...,Ud) ? 0,1d
• Cost vector p (p1,,pd) ? Rd
• Cost Function p?U
• Constraint Matrix c ? Re?Rd
• Maximize p?U
• Subject to cU ? 0 (cU0, cU?3, possible)

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ILP (Example)

• U (U1,U2,U3)
• p (0.3, 0.5, 0.8)
• c 1 2 3
• -1 -2 2
• 0 -3 2
• Maximize p?U
• Subject to cU ? 0

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Boolean Functions as Linear Constraints

• Conjunction
• U1?U2?U3 ? U11, U21, U31
• Disjunction
• U1?U2?U3 ? U1 U2 U3 ? 1
• CNF
• (U1?U2)?(U3?U4) ? U1U2 ? 1, U3U4 ? 1

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Text Chunking

• Indicator Variables
• U1,NP,U1,NULL,U2,VP...? y1NP, y1NULL, y2VP,..
• U1,NP indicates that phrase 1 is labeled NP
• Cost Vector
• p1,NP score(x,NP,1)
• p1,NULL score(x,NULL,1)
• p2,VP score(x,VP,2)
• ...
• p?U score(x,y) ?t score(x,yt,t), subject to
  constraints

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Structural Constraints

• Coherency
• yt can take only one value
• ?y?NP,..NULL Ut,y 1
• Non-Overlapping
• y1 and y2 overlap
• U1,NULL U2,NULL 1

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Linguistic Constraints

• Every sentence must have at least one VP
• ?t Ut,VP ? 1
• Every sentence must have at least one NP
• ?t Ut,NP ? 1
• ...

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Interacting Classifiers

• Classifier for an output yt uses other outputs
  y-t as inputs
• score(x,y-t,y,t)
• Need to ensure that the final output from ILP
  computed from a consistent y
• Introduce additional variables
• Introduce additional coherency constraints

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Interacting Classifiers

• Additional variables
• Yy ? UY,y for all possible y-t,y
• Additional coherency constraints
• UY,y 1 iff Ut,yt 1 for all yt in y
• ?yt in y Ut,yt - UY,y ? l 1
• ?yt in y Ut,yt - lUY,y ? 0

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Learning Classifiers

• score(x,y-t,y,t) ?y??y(x,y-t,t)
• Learn ?y, for all y ? Y
• Multi-class learning
• Example (x,y) ? ?y(x,y-t,t),ytt1..l
• Learn each classifier independently

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Learn with Inference Feedback

• Learn by observing global behavior
• For each example (x,y)
• Make prediction with the current classifiers and
  ILP
• y argmaxy ?t score(x,y-t,y,t)
• For each t, update
• If yt ? yt
• Promote score(x,y-t,yt,t)
• Demote score(x,y-t,yt,t)

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Experiments

• Semantics Role Labeling
• Assume correct boundaries are given
• Only sentences with more than 5 arguments are
  included

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Experimental Results
Winnow
Perceptron

• For difficult task
• Inference feedback during training improves
  performance
• For easy task
• Learning without inference feedback is better

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Conservative Updating

• Update only if necessary
• Example
• U1 U2 1
• Predict (U1, U2) (1,0)
• Correct (U1, U2) (0,1)
• Feedback ?Demote class 1, promote class 2
• So, U10 ? U21, so only demote class 1

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Conservative Updating

• S minset(Constraints)
• Set of functions that, if changed, would make
  global prediction correct.
• Promote (Demote) only those functions in the
  minset S

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Hierarchical Learning

• Given x
• Compute hierarchically
• z1 h1(x)
• z2 h2(x,z1)
• y hs1(x,z1,,zs)
• Assume all z are known in training

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Hierarchical Learning

• Assume each hj can be computed via ILP
• pj, Uj, cj
• y argmaxymaxz1,zs ?j?jpjUj
• Subject to
• c1U1 ? 0, c2U2 ? 0, , cs1Us1 ? 0
• where ?j is a large enough constant to preserve
  hierarchy

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Hidden Variables

• Given x
• y h(x,z)
• z is not known in training
• y argmaxymaxz score(x,z,y-t,y,t)
• Subject to some constraints

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Learning with Hidden Variables

• Truncated EM styled learning
• For each example (x,y)
• Compute z with the current classifiers and ILP
• z argmaxz score(x,z,y-t,y,t)
• Make prediction with the current classifiers and
  ILP
• (y,z) argmaxy,z ?t score(x,z,y-t,y,t)
• For each t, update
• If yt ? yt
• Promote score(x,z,y-t,yt,t)
• Demote score(x,z,y-t,yt,t)

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Conclusion

• ILP is
• powerful
• general
• learnable
• useful
• fast (or at least not too slow)
• extendable

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Boolean Functions as Linear Constraints

• Conjunction
• a?b?c ? Ua Ub Uc ? 3
• Disjunction
• a?b?c ? Ua Ub Uc ? 1
• DNF
• ab cd ? Iab Icd ? 1
• Introduce new variables Iab, Icd

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Helper Variables

• We must link Ia, Ib, and Iab
• Iab ?ab
• IaIb ? Iab
• Ia Ib lt Iab
• Iab ? IaIb
• 2Iab lt Ia Ib

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Semantic Role Labeling

• a,b,c...? ph1A0, ph1A1,ph2A0,..
• Cost Vector
• pa score(ph1A0)
• pb score(ph1A1)
• ...
• Indicator Variables
• Ia indicates that phrase 1 is labeled A0
• paIa 0.3 if Ia and 0 ow

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Learning

• X (X1,...,Xk) ? X1,,Xk X
• Y-t (Y1,...,Yt-1,Yt1,Yl)
• ? Y1,,Yt-1,Yt1,,Yl Y -t
• Yt ? Yt
• Given X x, and Y-t y-t, find Yt yt or score
  of each possible yt
• X ?Y t ? Yt or X ?Y t?Yt ?R

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SRL via Generalized Inference
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Outline

• Find potential argument candidates
• Classify arguments to types
• Inference for Argument Structure
• Integer linear programming (ILP)
• Cost Function
• Constraints
• Features

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Find Potential Arguments

• Every chunk can be an argument
• Restrict potential arguments
• BEGIN(word)
• BEGIN(word) 1 ? word begins argument
• END(word)
• END(word) 1 ? word ends argument
• Argument
• (wi,...,wj) is a potential argument iff
• BEGIN(wi) 1 and END(wj) 1
• Reduce set of potential argments

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Details...

• BEGIN(word)
• Learn a function
• ?B(word,context,structure) ? 0,1
• END(word)
• Learn a function
• ?E(word,context,structure) ? 0,1
• POTARG arg BEGIN(first(arg)) and
  END(last(arg))

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Arguments Type Likelihood

• Assign type-likelihood
• How likely is it that arg a is type t?
• For all a ? POTARG , t ? T
• P (argument a type t )

0.3 0.2 0.2 0.3
0.6 0.0 0.0 0.4
A0
CA1
A1
Ø
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Details...

• Learn a classifier
• ARGTYPE(arg)
• ?P(arg) ? A0,A1,...,CA0,...,LOC,...
• argmaxt?A0,A1,...,CA0,...,LOC,... wt ?P(arg)
• Estimate Probabilites
• P(a t) wt ?P(a) / Z

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What is a Good Assignment?

• Likelihood of being correct
• P(Arg a Type t)
• if t is the correct type for argument a
• For a set of arguments a1, a2, ..., an
• Expected number of arguments correct
• ? i P( ai ti )
• We search for the assignment with maximum
  expected correct

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Inference

• Maximize expected number correct
• T argmaxT ? i P( ai ti )
• Subject to some constraints
• Structural and Linguistic

I left my nice pearls to her
I left my nice pearls to her
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Everything is Linear

• Cost function
• ?a ? POTARG P(at) ?a ? POTARG , t ? T
  P(at)Iat
• Constraints
• Non-Overlapping
• a and a overlap ? IaØ IaØ 0
• Linguistic
• ? CA0 ? ? A0 ? ?a IaCA0 ?a IaA0 ? 1
• Integer Linear Programming

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Features are Important

• Here, a discussion of the features should go.
• Which are most important?
• Comparison to other people.

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