Discriminative Training of Markov Logic Networks
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Transcript of Discriminative Training of Markov Logic Networks
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Discriminative Training of Markov Logic Networks
Parag Singla & Pedro Domingos
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Outline Motivation Review of MLNs Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future Work
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Outline MotivationMotivation Review of MLNs Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future Work
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Markov Logic Networks(MLNs) AI systems must be able to learn,
reason logically and handle uncertainty Markov Logic Networks [Richardson and
Domingos, 2004]- an effective way to combine first order logic and probability
Markov Networks are used as underlying representation
Features specfied using arbitrary formulas in finite first order logic
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Training of MLNs – Generative Approach Optimize the joint distribution of all the variables Parameters learnt independent of specific
inference task Maximum-likelihood (ML) training – computation
of the gradient involves inference – too slow! Use Psuedo-likelihood (PL) as an alternative –
easy to compute PL is suboptimal. Ignores any non-local
interactions between variables ML, PL – generative training approaches
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Training of MLNs -Discriminative Approach No need to optimize the joint
distribution of all the variables Optimize the conditional likelihood (CL)
of non-evidence variables given evidence variables
Parameters learnt for a specific inference task
Tends to do better than generative training in general
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Why is Discriminative Better?
Generative Parameters learnt are not optimized for the specific inference task.
Need to model all the dependencies in the data – learning might become complicated.
Example of generative models: MRFs
Discriminative Parameters learnt are optimized for the specific inference task.
Need not model dependencies between evidence variables – makes learning task easier.
Example of discriminative
models: CRFs [Lafferty, McCallum, Pereira 2001]
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Outline Motivation Review of MLNsReview of MLNs Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future Work
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Markov Logic Networks A Markov Logic Network (MLN) is a set of
pairs (F, w) where F is a formula in first-order logic w is a real number
Together with a finite set of constants,it defines a Markov network with One node for each grounding of each
predicate in the MLN One feature for each grounding of each
formula F in the MLN, with the corresponding weight w
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Likelihood
Fiii xnw
ZXP )(exp
1)(
Iterate over all MLN clauses
# true groundings of ith clause
Iterate over all ground clauses
1 if jth ground clause is true, 0 otherwise
Gjjj xgw
Z)(exp
1
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Gradient of Log-Likelihood
)()()(log xnExnxPw iwiwi
1st term: # true groundings of formula in DB2nd term: inference required (slow!)
Feature count according to data
Feature count according to model
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Pseudo-Likelihood [Besag, 1975]
Likelihood of each ground atom given its Markov blanket in the data
Does not require inference at each step
Optimized using L-BFGS [Liu & Nocedal, 1989]
( ) ( | ( ))x
PL X P x MB x
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Most terms not affected by changes in weights After initial setup, each iteration takes
O(# ground predicates x # first-order clauses)
( ) ( 0) ( 0) ( 1) ( 1)i i i ix
nsat x p x nsat x p x nsat x
Gradient ofPseudo-Log-Likelihood
where nsati(x=v) is the number of satisfied groundingsof clause i in the training data when x takes value v
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Outline Motivation Review of MLNs Discriminative Training Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future Work
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Conditional Likelihood (CL)
YFiii
x
yxnwZ
XYP ),(exp1
)|(
Normalize over all possible configurations of non-evidence variables
Iterate over all MLN clauses with at least one grounding containing query variables
Non-evidence variables
Evidence variables
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Derivative of log CL
),(),()|(log yxnEyxnxyPw iwiwi
1st term: # true groundings (involving query variables) of formula in DB2nd term: inference required, as before (slow!)
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Derivative of log CL
Approximate the expected count by MAP count
),(),()|(log *yxnyxnxyPw iiwi
MAP state
),(),()|(log yxnEyxnxyPw iwiwi
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Approximating the Expected Count Use Voted Perceptron Algorithm
[Collins, 2002] Approximate the expected count by
count for the most likely state (MAP) state
Used successfully for linear chain Markov networks
MAP state found using Viterbi
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Voted Perceptron Algorithm Initialize wi=0 For t=1 to T
Find the MAP configuration according to current set of weights.
wi,t= * (training count – MAP count)
wi= wi,t/T (Avoids over-fitting)
T
t 1
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Generalizing Voted Perceptron Finding the MAP configuration NP
hard for the general case. Can be reduced to a weighted
satisfiability (MaxSAT) problem. Given a SAT formula in clausal form
e.g. (x1 V x3 V x5) … (x5 V x7 Vx50) with clause i having weight of wi
Find the assignment maximizing the sum of weights of satisfied clauses.
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MaxWalkSAT [Kautz, Selman & Jiang 97] Assumes clauses with positive weights Mixes greedy search with random walks
Start with some configuration of variables. Randomly pick an unsatisfied clause. With probability p, flip the literal in the clause
which gives maximum gain. With probability 1-p flip a random literal in the clause.
Repeat for a pre-decided number of flips, storing the best seen configuration.
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Handling the Negative Weights MLN allows formulas with negative
weights. A formula with weight w can be
replaced by its negation with weight –w in the ground Markov network.
(x1 x3 x5) [w] => (x1 x3 x5) [-w]
=> (x1 x3 x5) [-w] (x1 x3 x5) [-w] => x1 , x3 , x5 [ -w/3]
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Weight Initialization and Learning Rate Weights initialized using log odds
of each clause being true in the data.
Determining the learning rate – use a validation set. Learning rate 1/#(ground predicates)
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Outline Motivation Review of MLNs Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future Work
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Outline Motivation Review of MLNs Discriminative Training Experiments
Link Prediction Link Prediction Object Identification
Conclusion and Future Work
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Link Prediction UW-CSE database
Used by Richardson & Domingos [2004] Database of people/courses/publications at UW-CSE 22 Predicates e.g. Student(P), Professor(P),
AdvisedBy(P1,P2) 1158 constants divided into 10 types 4,055,575 ground atoms 3212 true ground atoms 94 hand coded rules stating various regularities
Student(P) => !Professor(P) Predict AdvisedBy in the absence of information
about the predicates Professor and Student
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Systems Compared MLN(VP) MLN(ML) MLN(PL) KB CL NB BN
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Results on Link Prediction
0.033 0.063 0.034
0.530.693
1.237
0.0460
0.2
0.4
0.6
0.8
1
System
-CLL
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Results on Link Prediction
0.295
0.077
0.2320.114
0.006 0.065 0.020
0.2
0.4
0.6
0.8
1
System
AU
C
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Outline Motivation Review of MLNs Discriminative Training Experiments
Link Prediction Object IdentificationObject Identification
Conclusion and Future Work
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Object Identification Given a database of various records
referring to objects in the real world Each record represented by a set of
attribute values Want to find out which of the records
refer to the same object Example: A paper may have more than
one reference in a bibliography database
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Why is it Important? Data Cleaning and Integration – first
step in the KDD process Merging of data from multiple sources
results in duplicates Entity Resolution: Extremely important
for doing any sort of data-mining State of the art – far from what is
required. Citeseer has 30 different entries for the
AI textbook by Russell and Norvig
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Standard Approach [Fellegi & Sunter, 1969] Look at each pair of records independently Calculate the similarity score for each
attribute value pair based on some metric Find the overall similarity score Merge the records whose similarity is above
a threshold Take a transitive closure
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An Example
Subset of a Bibliography Relation
Record Title Author Venue
B1 Object Identification using MLNs
Linda Stewart KDD 2004
B2 Object Identification using MLNs
Linda Stewart SIGKDD 10
B3 Learning Boolean Formulas Bill Johnson KDD 2004
B4 Learning of Boolean Formulas William Johnson
SIGKDD 10
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Graphical Representation in Standard Model
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
Sim(KDD 2004, SIGKDD 10)
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
Venue
Record-pair node
Evidence node
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What’s Missing?
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
Sim(KDD 2004, SIGKDD 10)
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
Venue
If from b1=b2, you infer that “KDD 2004” is same as “SIGKDD 10”, how canyou use that to help figure out if b3=b4?
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Collective Model – Basic Idea Perform simultaneous inference for
all the candidate pairs Facilitate flow of information
through shared attribute values
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Representation in Standard Model
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
Sim(KDD 2004, SIGKDD 10)
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
Venue
No sharing of nodes
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Merging the Evidence Nodes
Author
Still does not solve the problem. Why?
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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Introducing Information Nodes
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Information node
Full representation in Collective Model
Sim(KDD 2004, SIGKDD 10)
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Flow of Information
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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Flow of Information
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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Flow of Information
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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Flow of Information
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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Flow of Information
b1=b2?
Sim(Linda Stewart, Linda Stewart)
b3=b4?
Author
Title
Venue
b1.T=b2.T?
b1.V=b2.V?b3.V=b4.V?
b3.A=b4.A?
b3.T=b4.T?
b1.A=b2.A?
Sim(Object Identification using MLNs, Object Identification using MLNs)
Sim(Bill Johnson, William Johnson)
Title
Author
Sim(Learning Boolean Formulas, Leraning of Boolean Formulas)
Sim(KDD 2004, SIGKDD 10)
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MLN Predicates for De-Duplicating Citation Databases If two bib entries are the same -
SameBib(b1,b2) If two field values are the same -
SameAuthor(a1,a2), SameTitle(t1,t2), SameVenue(v1,v2)
If cosine based TFIDF score of two field values lies in a particular range (0, 0 - .2, .2 - .4, etc.) – 6 predicates for each field. E.g. AuthorTFIDF.8(a1,a2) is true if TFIDF
similarity score of a1,a2 is in the range (.2, .4]
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MLN Rules for De-Duplicating Citation Databases Singleton Predicates
! SameBib(b1,b2) Two fields are same => corresponding bib entries are same.
Author(b1,a1) Author(b2,a2) SameAuthor(a1,a2)=> SameBib(b1,b2) Two papers are same => corresponding fields are same
Author(b1,a1) Author(b2,a2) SameBib(b1,b2)=> SameAuthor(a1,a2) High similarity score => two fields are same
AuthorTFIDF.8(a1,a2) =>SameAuthor(a1,a2) Transitive closure (currently not incorporated)
SameBib(b1,b2) SameBib(b2,b3) => SameBib(b1,b3) 25 first order predicates, 46 first order clauses.
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Cora Database Cleaned up version of McCallum’s Cora
database. 1295 citations to 132 difference Computer
Science research papers, each citation described by author, venue, title fields.
401,552 ground atoms. 82,026 tuples (true ground atoms) Predict SameBib, SameAuthor, SameVenue
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Systems Compared MLN(VP) MLN(ML) MLN(PL) KB CL NB BN
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Results on CoraPredicting the Citation Matches
0.069
13.261
0.699
8.629
0.461
0.082 0.067
0
0.2
0.4
0.6
0.8
1
System
-CLL
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Results on CoraPredicting the Citation Matches
0.973
0.111
0.722
0.149 0.187
0.945 0.951
0
0.2
0.4
0.6
0.8
1
System
AU
C
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Results on CoraPredicting the Author Matches
0.069
12.973 3.062 8.096 2.375
0.203 0.203
0
0.2
0.4
0.6
0.8
1
System
-CLL
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Results on CoraPredicting the Author Matches
0.969
0.162
0.3230.18
0.09
0.734 0.734
0
0.2
0.4
0.6
0.8
1
System
AU
C
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Results on CoraPredicting the Venue Matches
0.232
13.38
0.708
8.475 1.261
0.233 0.233
0
0.2
0.4
0.6
0.8
1
System
-CLL
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Results on CoraPredicting the Venue Matches
0.771
0.061
0.342
0.096 0.047
0.339 0.339
0
0.2
0.4
0.6
0.8
1
System
AU
C
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Outline Motivation Review of MLNs Discriminative Training Experiments
Link Prediction Object Identification
Conclusion and Future WorkConclusion and Future Work
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Conclusions Markov Logic Networks – a powerful
way of combining logic and probability. MLNs can be discriminatively trained
using a voted perceptron algorithm Discriminatively trained MLNs perform
better than purely logical approaches, purely probabilistic approaches as well as generatively trained MLNs.
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Future Work Discriminative learning of MLN
structure Max-margin type training of MLNs Extensions of MaxWalkSAT Further application to the link
prediction, object identification and possibly other application areas.