Discriminative Training of Markov Logic Networks

Discriminative Training of Markov Logic Networks Parag Singla & Pedro Domingos

Outline • Motivation • Review of MLNs • Discriminative Training • Experiments • Link Prediction • Object Identification • Conclusion and Future Work

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

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

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

Why is Discriminative Better?

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

1 if jth ground clause is true, 0 otherwise Iterate over all ground clauses Likelihood # true groundings of ith clause Iterate over all MLN clauses

Gradient of Log-Likelihood Feature count according to data Feature count according to model 1st term: # true groundings of formula in DB 2nd term: inference required (slow!)

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]

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 • Most terms not affected by changes in weights • After initial setup, each iteration takesO(# ground predicates x # first-order clauses)

Conditional Likelihood (CL) Normalize over all possible configurations of non-evidence variables Non-evidence variables Iterate over all MLN clauses with at least one grounding containing query variables Evidence variables

Derivative of log CL 1st term: # true groundings (involving query variables) of formula in DB 2nd term: inference required, as before (slow!)

MAP state Derivative of log CL Approximate the expected count by MAP count

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

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)

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 Vx7 Vx50) with clause i having weight of wi • Find the assignment maximizing the sum of weights of satisfied clauses.

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.

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]

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)

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

Systems Compared • MLN(VP) • MLN(ML) • MLN(PL) • KB • CL • NB • BN

Results on Link Prediction

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

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

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

An Example Subset of a Bibliography Relation

Graphical Representation in Standard Model Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b1=b2 ? b3=b4 ? Sim(KDD 2004, SIGKDD 10) Sim(KDD 2004, SIGKDD 10) Venue Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author Record-pair node Evidence node

What’s Missing? Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b1=b2 ? b3=b4 ? Sim(KDD 2004, SIGKDD 10) Sim(KDD 2004, SIGKDD 10) Venue Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author If from b1=b2, you infer that “KDD 2004” is same as “SIGKDD 10”, how can you use that to help figure out if b3=b4?

Collective Model – Basic Idea • Perform simultaneous inference for all the candidate pairs • Facilitate flow of information through shared attribute values

Representation in Standard Model Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b3=b4 ? b1=b2 ? Sim(KDD 2004, SIGKDD 10) Sim(KDD 2004, SIGKDD 10) Venue Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author No sharing of nodes

Merging the Evidence Nodes Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b3=b4 ? b1=b2 ? Sim(KDD 2004, SIGKDD 10) Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author Author Still does not solve the problem. Why?

Introducing Information Nodes Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b1=b2 ? b3=b4 ? Information node b1.T=b2.T? b3.T=b4.T? b1.V=b2.V? b3.V=b4.V? b1.A=b2.A? b3.A=b4.A? Sim(KDD 2004, SIGKDD 10) Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author Full representation in Collective Model

Flow of Information Title Title Sim(Object Identification using MLNs, Object Identification using MLNs) Sim(Learning Boolean Formulas, Leraning of Boolean Formulas) b1=b2 ? b3=b4 ? b1.T=b2.T? b3.T=b4.T? b1.V=b2.V? b3.V=b4.V? b1.A=b2.A? b3.A=b4.A? Sim(KDD 2004, SIGKDD 10) Venue Sim(Linda Stewart, Linda Stewart) Sim(Bill Johnson, William Johnson) Author Author

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]

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.

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

Systems Compared • MLN(VP) • MLN(ML) • MLN(PL) • KB • CL • NB • BN

Results on Cora Predicting the Citation Matches

Discriminative Training of Markov Logic Networks