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**Markov Logic in Natural Language Processing**

Hoifung Poon Dept. of Computer Science & Eng. University of Washington

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**Overview Motivation Foundational areas Markov logic NLP applications**

Basics Supervised learning Unsupervised learning

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**Languages Are Structural**

governments lm$pxtm (according to their families)

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**Languages Are Structural**

govern-ment-s l-m$px-t-m (according to their families) NP VP V NP IL-4 induces CD11B Involvement of p70(S6)-kinase activation in IL-10 up-regulation in human monocytes by gp George Walker Bush was the 43rd President of the United States. …… Bush was the eldest son of President G. H. W. Bush and Babara Bush. ……. In November 1977, he met Laura Welch at a barbecue. involvement Theme Cause up-regulation activation Theme Cause Site Theme human monocyte IL-10 gp41 p70(S6)-kinase

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**Languages Are Structural**

govern-ment-s l-m$px-t-m (according to their families) NP VP V NP IL-4 induces CD11B Involvement of p70(S6)-kinase activation in IL-10 up-regulation in human monocytes by gp George Walker Bush was the 43rd President of the United States. …… Bush was the eldest son of President G. H. W. Bush and Babara Bush. ……. In November 1977, he met Laura Welch at a barbecue. involvement Theme Cause up-regulation activation Theme Cause Site Theme human monocyte IL-10 gp41 p70(S6)-kinase

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**Languages Are Structural**

Objects are not just feature vectors They have parts and subparts Which have relations with each other They can be trees, graphs, etc. Objects are seldom i.i.d. (independent and identically distributed) They exhibit local and global dependencies They form class hierarchies (with multiple inheritance) Objects’ properties depend on those of related objects Deeply interwoven with knowledge

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**First-Order Logic Main theoretical foundation of computer science**

General language for describing complex structures and knowledge Trees, graphs, dependencies, hierarchies, etc. easily expressed Inference algorithms (satisfiability testing, theorem proving, etc.)

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**Languages Are Statistical**

Microsoft buys Powerset Microsoft acquires Powerset Powerset is acquired by Microsoft Corporation The Redmond software giant buys Powerset Microsoft’s purchase of Powerset, … …… I saw the man with the telescope NP I saw the man with the telescope NP ADVP I saw the man with the telescope Here in London, Frances Deek is a retired teacher … In the Israeli town …, Karen London says … Now London says … G. W. Bush …… …… Laura Bush …… Mrs. Bush …… London PERSON or LOCATION? Which one?

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**Languages Are Statistical**

Languages are ambiguous Our information is always incomplete We need to model correlations Our predictions are uncertain Statistics provides the tools to handle this

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**Probabilistic Graphical Models**

Mixture models Hidden Markov models Bayesian networks Markov random fields Maximum entropy models Conditional random fields Etc. PCFG?

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**The Problem Logic is deterministic, requires manual coding**

Statistical models assume i.i.d. data, objects = feature vectors Historically, statistical and logical NLP have been pursued separately We need to unify the two! Burgeoning field in machine learning: Statistical relational learning

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**Costs and Benefits of Statistical Relational Learning**

Better predictive accuracy Better understanding of domains Enable learning with less or no labeled data Costs Learning is much harder Inference becomes a crucial issue Greater complexity for user

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**Progress to Date Probabilistic logic [Nilsson, 1986]**

Statistics and beliefs [Halpern, 1990] Knowledge-based model construction [Wellman et al., 1992] Stochastic logic programs [Muggleton, 1996] Probabilistic relational models [Friedman et al., 1999] Relational Markov networks [Taskar et al., 2002] Etc. This talk: Markov logic [Domingos & Lowd, 2009]

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**Markov Logic: A Unifying Framework**

Probabilistic graphical models and first-order logic are special cases Unified inference and learning algorithms Easy-to-use software: Alchemy Broad applicability Goal of this tutorial: Quickly learn how to use Markov logic and Alchemy for a broad spectrum of NLP applications 5k download Sep 2009

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**Overview Motivation Foundational areas Markov logic NLP applications**

Probabilistic inference Statistical learning Logical inference Inductive logic programming Markov logic NLP applications Basics Supervised learning Unsupervised learning

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**Markov Networks Smoking Cancer Asthma Cough**

Undirected graphical models Smoking Cancer Asthma Cough Potential functions defined over cliques Smoking Cancer Ф(S,C) False 4.5 True 2.7

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**Markov Networks Smoking Cancer Asthma Cough**

Undirected graphical models Smoking Cancer Asthma Cough Log-linear model: Weight of Feature i Feature i

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**Markov Nets vs. Bayes Nets**

Property Markov Nets Bayes Nets Form Prod. potentials Potentials Arbitrary Cond. probabilities Cycles Allowed Forbidden Partition func. Z = ? Z = 1 Indep. check Graph separation D-separation Indep. props. Some Inference MCMC, BP, etc. Convert to Markov

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**Inference in Markov Networks**

Goal: compute marginals & conditionals of Exact inference is #P-complete Conditioning on Markov blanket is easy: Gibbs sampling exploits this

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**MCMC: Gibbs Sampling state ← random truth assignment**

for i ← 1 to num-samples do for each variable x sample x according to P(x|neighbors(x)) state ← state with new value of x P(F) ← fraction of states in which F is true

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**Other Inference Methods**

Belief propagation (sum-product) Mean field / Variational approximations

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**MAP/MPE Inference Goal: Find most likely state of world given evidence**

Query Evidence

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**MAP Inference Algorithms**

Iterated conditional modes Simulated annealing Graph cuts Belief propagation (max-product) LP relaxation

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**Overview Motivation Foundational areas Markov logic NLP applications**

Probabilistic inference Statistical learning Logical inference Inductive logic programming Markov logic NLP applications Basics Supervised learning Unsupervised learning

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**Generative Weight Learning**

Maximize likelihood Use gradient ascent or L-BFGS No local maxima Requires inference at each step (slow!) No. of times feature i is true in data Expected no. times feature i is true according to model

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Pseudo-Likelihood Likelihood of each variable given its neighbors in the data Does not require inference at each step Widely used in vision, spatial statistics, etc. But PL parameters may not work well for long inference chains

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**Discriminative Weight Learning**

Maximize conditional likelihood of query (y) given evidence (x) Approximate expected counts by counts in MAP state of y given x No. of true groundings of clause i in data Expected no. true groundings according to model

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**Voted Perceptron wi ← 0 for t ← 1 to T do yMAP ← Viterbi(x)**

Originally proposed for training HMMs discriminatively Assumes network is linear chain Can be generalized to arbitrary networks wi ← 0 for t ← 1 to T do yMAP ← Viterbi(x) wi ← wi + η [counti(yData) – counti(yMAP)] return wi / T

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**Overview Motivation Foundational areas Markov logic NLP applications**

Probabilistic inference Statistical learning Logical inference Inductive logic programming Markov logic NLP applications Basics Supervised learning Unsupervised learning

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First-Order Logic Constants, variables, functions, predicates E.g.: Anna, x, MotherOf(x), Friends(x, y) Literal: Predicate or its negation Clause: Disjunction of literals Grounding: Replace all variables by constants E.g.: Friends (Anna, Bob) World (model, interpretation): Assignment of truth values to all ground predicates

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**Inference in First-Order Logic**

Traditionally done by theorem proving (e.g.: Prolog) Propositionalization followed by model checking turns out to be faster (often by a lot) Propositionalization: Create all ground atoms and clauses Model checking: Satisfiability testing Two main approaches: Backtracking (e.g.: DPLL) Stochastic local search (e.g.: WalkSAT)

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Satisfiability Input: Set of clauses (Convert KB to conjunctive normal form (CNF)) Output: Truth assignment that satisfies all clauses, or failure The paradigmatic NP-complete problem Solution: Search Key point: Most SAT problems are actually easy Hard region: Narrow range of #Clauses / #Variables

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**Stochastic Local Search**

Uses complete assignments instead of partial Start with random state Flip variables in unsatisfied clauses Hill-climbing: Minimize # unsatisfied clauses Avoid local minima: Random flips Multiple restarts

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**The WalkSAT Algorithm for i ← 1 to max-tries do**

solution = random truth assignment for j ← 1 to max-flips do if all clauses satisfied then return solution c ← random unsatisfied clause with probability p flip a random variable in c else flip variable in c that maximizes # satisfied clauses return failure

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**Overview Motivation Foundational areas Markov logic NLP applications**

Probabilistic inference Statistical learning Logical inference Inductive logic programming Markov logic NLP applications Basics Supervised learning Unsupervised learning

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Rule Induction Given: Set of positive and negative examples of some concept Example: (x1, x2, … , xn, y) y: concept (Boolean) x1, x2, … , xn: attributes (assume Boolean) Goal: Induce a set of rules that cover all positive examples and no negative ones Rule: xa ^ xb ^ … y (xa: Literal, i.e., xi or its negation) Same as Horn clause: Body Head Rule r covers example x iff x satisfies body of r Eval(r): Accuracy, info gain, coverage, support, etc.

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**Learning a Single Rule head ← y body ← Ø repeat for each literal x**

rx ← r with x added to body Eval(rx) body ← body ^ best x until no x improves Eval(r) return r

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**Learning a Set of Rules R ← Ø S ← examples repeat**

learn a single rule r R ← R U { r } S ← S − positive examples covered by r until S = Ø return R

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**First-Order Rule Induction**

y and xi are now predicates with arguments E.g.: y is Ancestor(x,y), xi is Parent(x,y) Literals to add are predicates or their negations Literal to add must include at least one variable already appearing in rule Adding a literal changes # groundings of rule E.g.: Ancestor(x,z) ^ Parent(z,y) Ancestor(x,y) Eval(r) must take this into account E.g.: Multiply by # positive groundings of rule still covered after adding literal

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**Overview Motivation Foundational areas Markov logic NLP applications**

Basics Supervised learning Unsupervised learning

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**Markov Logic Syntax: Weighted first-order formulas**

Semantics: Feature templates for Markov networks Intuition: Soften logical constraints Give each formula a weight (Higher weight Stronger constraint)

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**Example: Coreference Resolution**

Barack Obama, the 44th President of the United States, is the first African American to hold the office. ……

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**Example: Coreference Resolution**

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**Example: Coreference Resolution**

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**Example: Coreference Resolution**

Two mention constants: A and B Apposition(A,B) Head(A,“President”) Head(B,“President”) MentionOf(A,Obama) MentionOf(B,Obama) Head(A,“Obama”) Head(B,“Obama”) Apposition(B,A)

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**Markov Logic Networks MLN is template for ground Markov nets**

Probability of a world x: Typed variables and constants greatly reduce size of ground Markov net Functions, existential quantifiers, etc. Can handle infinite domains [Singla & Domingos, 2007] and continuous domains [Wang & Domingos, 2008] Weight of formula i No. of true groundings of formula i in x

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**Relation to Statistical Models**

Special cases: Markov networks Markov random fields Bayesian networks Log-linear models Exponential models Max. entropy models Gibbs distributions Boltzmann machines Logistic regression Hidden Markov models Conditional random fields Obtained by making all predicates zero-arity Markov logic allows objects to be interdependent (non-i.i.d.)

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**Relation to First-Order Logic**

Infinite weights First-order logic Satisfiable KB, positive weights Satisfying assignments = Modes of distribution Markov logic allows contradictions between formulas

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**MLN Algorithms: The First Three Generations**

Problem First generation Second generation Third generation MAP inference Weighted satisfiability Lazy inference Cutting planes Marginal inference Gibbs sampling MC-SAT Lifted inference Weight learning Pseudo-likelihood Voted perceptron Scaled conj. gradient Structure learning Inductive logic progr. ILP + PL (etc.) Clustering + pathfinding

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MAP/MPE Inference Problem: Find most likely state of world given evidence Query Evidence

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MAP/MPE Inference Problem: Find most likely state of world given evidence

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MAP/MPE Inference Problem: Find most likely state of world given evidence

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MAP/MPE Inference Problem: Find most likely state of world given evidence This is just the weighted MaxSAT problem Use weighted SAT solver (e.g., MaxWalkSAT [Kautz et al., 1997] )

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**The MaxWalkSAT Algorithm**

for i ← 1 to max-tries do solution = random truth assignment for j ← 1 to max-flips do if weights(sat. clauses) > threshold then return solution c ← random unsatisfied clause with probability p flip a random variable in c else flip variable in c that maximizes weights(sat. clauses) return failure, best solution found

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**Computing Probabilities**

P(Formula|MLN,C) = ? MCMC: Sample worlds, check formula holds P(Formula1|Formula2,MLN,C) = ? If Formula2 = Conjunction of ground atoms First construct min subset of network necessary to answer query (generalization of KBMC) Then apply MCMC Lifted

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**But … Insufficient for Logic**

Problem: Deterministic dependencies break MCMC Near-deterministic ones make it very slow Solution: Combine MCMC and WalkSAT → MC-SAT algorithm [Poon & Domingos, 2006]

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**Auxiliary-Variable Methods**

Main ideas: Use auxiliary variables to capture dependencies Turn difficult sampling into uniform sampling Given distribution P(x) Sample from f (x, u), then discard u An attractive solution is to use aux-var methods. The idea is to introduce auxiliary var to capture dependencies and turn diff sampling into uniform sampling. Given a dist p(x), we introduce a var u, and define joint dist over x, u, so that it is 1 iff u is btw 0 and p(x). Now if we marginalize over u, it becomes precisely p(x). So in order to sample p(x), we can instead sample from f(x,u), and discard the samples of u.

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**Slice Sampling [Damien et al. 1999]**

U P(x) Slice u(k) A representative of aux var methods is slice sampling. we are given a diff dist. Note the low-prob region between the two modes. This is exactly the type that cause trouble for Gibbs, because it is very slow to traverse thru this area. Now we see how slice sampling gets around this problem. It begins with a random start point. Now, suppose we have reached the kth-sample. We first sample the next U uniformly from the vertical bar from 0 to p(x). Then, we gather all x such that p(x) is above U, which comprise of the slice. We then sample the next x uniformly from this slice. And the process goes on. So w. slice sampling, we can jump from one mode to another mode in just one sampling, bypassing the low-prb region. X x(k) x(k+1)

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**Slice Sampling Identifying the slice may be difficult**

Introduce an auxiliary variable ui for each Фi The diff of slice sampling is in identifying the slice. E.g. p may be a product of a number of potentials, as in a GM. In this case, we can do is to introduce an aux var for each potential. In particular, as we shall see later, in MC-Sat, we introduce an aux var for the potential corresponding to each clause.

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**The MC-SAT Algorithm Select random subset M of satisfied clauses**

With probability 1 – exp ( – wi ) Larger wi Ci more likely to be selected Hard clause (wi ): Always selected Slice States that satisfy clauses in M Uses SAT solver to sample x | u. Orders of magnitude faster than Gibbs sampling, etc. In sum, the slice is decided by a random subset of sat clauses, M. For each sat clause, it is selected w. prb 1-e^-w_i. Larger weight means that the prb is higher, and so the clause is more likely to be selected. In the limit of infinite wt, i.e., for a hard clause, prb is 1 and it’s always selected. The slice comprises of all states that sat clauses in this set M. And we sample uniformly from these.

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**But … It Is Not Scalable 1000 researchers**

Coauthor(x,y): 1 million ground atoms Coauthor(x,y) Coauthor(y,z) Coauthor(x,z): 1 billion ground clauses Exponential in arity

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**Sparsity to the Rescue 1000 researchers**

Coauthor(x,y): 1 million ground atoms But … most atoms are false Coauthor(x,y) Coauthor(y,z) Coauthor(x,z): 1 billion ground clauses Most trivially satisfied if most atoms are false No need to explicitly compute most of them Rel inf is unscalable If we have to ground all… but we don’t need to Wide spread phenomena, e.g., sparse matrices

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**Lazy Inference LazySAT [Singla & Domingos, 2006a]**

Lazy version of WalkSAT [Selman et al., 1996] Grounds atoms/clauses as needed Greatly reduces memory usage The idea is much more general [Poon & Domingos, 2008a]

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**General Method for Lazy Inference**

If most variables assume the default value, wasteful to instantiate all variables / functions Main idea: Allocate memory for a small subset of “active” variables / functions Activate more if necessary as inference proceeds Applicable to a diverse set of algorithms: Satisfiability solvers (systematic, local-search), Markov chain Monte Carlo, MPE / MAP algorithms, Maximum expected utility algorithms, Belief propagation, MC-SAT, Etc. Reduce memory and time by orders of magnitude

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**Lifted Inference Consider belief propagation (BP)**

Often in large problems, many nodes are interchangeable: They send and receive the same messages throughout BP Basic idea: Group them into supernodes, forming lifted network Smaller network → Faster inference Akin to resolution in first-order logic

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Belief Propagation Features (f) Nodes (x) 66

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**Lifted Belief Propagation**

Features (f) Nodes (x) 67

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**Lifted Belief Propagation**

, : Functions of edge counts Features (f) Nodes (x) 68

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**Learning Data is a relational database**

Closed world assumption (if not: EM) Learning parameters (weights) Learning structure (formulas)

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**Parameter Learning Parameter tying: Groundings of same clause**

Generative learning: Pseudo-likelihood Discriminative learning: Conditional likelihood, use MC-SAT or MaxWalkSAT for inference No. of times clause i is true in data Expected no. times clause i is true according to MLN

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Parameter Learning Pseudo-likelihood + L-BFGS is fast and robust but can give poor inference results Voted perceptron: Gradient descent + MAP inference Scaled conjugate gradient

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**Voted Perceptron for MLNs**

HMMs are special case of MLNs Replace Viterbi by MaxWalkSAT Network can now be arbitrary graph wi ← 0 for t ← 1 to T do yMAP ← MaxWalkSAT(x) wi ← wi + η [counti(yData) – counti(yMAP)] return wi / T

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**Problem: Multiple Modes**

Not alleviated by contrastive divergence Alleviated by MC-SAT Warm start: Start each MC-SAT run at previous end state

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**Problem: Extreme Ill-Conditioning**

Solvable by quasi-Newton, conjugate gradient, etc. But line searches require exact inference Solution: Scaled conjugate gradient [Lowd & Domingos, 2008] Use Hessian to choose step size Compute quadratic form inside MC-SAT Use inverse diagonal Hessian as preconditioner

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Structure Learning Standard inductive logic programming optimizes the wrong thing But can be used to overgenerate for L1 pruning Our approach: ILP + Pseudo-likelihood + Structure priors For each candidate structure change: Start from current weights & relax convergence Use subsampling to compute sufficient statistics

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**Structure Learning Initial state: Unit clauses or prototype KB**

Operators: Add/remove literal, flip sign Evaluation function: Pseudo-likelihood + Structure prior Search: Beam search, shortest-first search

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**Alchemy Open-source software including: Full first-order logic syntax**

Generative & discriminative weight learning Structure learning Weighted satisfiability, MCMC, lifted BP Programming language features alchemy.cs.washington.edu

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Alchemy Prolog BUGS Represent-ation F.O. Logic + Markov nets Horn clauses Bayes nets Inference Model check- ing, MCMC, lifted BP Theorem proving MCMC Learning Parameters & structure No Params. Uncertainty Yes Relational

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**Constrained Conditional Model**

Representation: Integer linear programs Local classifiers + Global constraints Inference: LP solver Parameter learning: None for constraints Weights of soft constraints set heuristically Local weights typically learned independently Structure learning: None to date But see latest development in NAACL-10

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**Running Alchemy Programs Options MLN file Database files Infer**

Learnwts Learnstruct Options MLN file Types (optional) Predicates Formulas Database files

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**Overview Motivation Foundational areas Markov logic NLP applications**

Basics Supervised learning Unsupervised learning

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**Uniform Distribn.: Empty MLN**

Example: Unbiased coin flips Type: flip = { 1, … , 20 } Predicate: Heads(flip)

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**Binomial Distribn.: Unit Clause**

Example: Biased coin flips Type: flip = { 1, … , 20 } Predicate: Heads(flip) Formula: Heads(f) Weight: Log odds of heads: By default, MLN includes unit clauses for all predicates (captures marginal distributions, etc.)

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**Multinomial Distribution**

Example: Throwing die Types: throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face) Formulas: Outcome(t,f) ^ f != f’ => !Outcome(t,f’). Exist f Outcome(t,f). Too cumbersome!

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**Multinomial Distrib.: ! Notation**

Example: Throwing die Types: throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face!) Formulas: Semantics: Arguments without “!” determine arguments with “!”. Also makes inference more efficient (triggers blocking).

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**Multinomial Distrib.: + Notation**

Example: Throwing biased die Types: throw = { 1, … , 20 } face = { 1, … , 6 } Predicate: Outcome(throw,face!) Formulas: Outcome(t,+f) Semantics: Learn weight for each grounding of args with “+”.

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**Logistic Regression (MaxEnt)**

Type: obj = { 1, ... , n } Query predicate: C(obj) Evidence predicates: Fi(obj) Formulas: a C(x) bi Fi(x) ^ C(x) Resulting distribution: Therefore: Alternative form: Fi(x) => C(x)

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**Hidden Markov Models obs = { Red, Green, Yellow }**

state = { Stop, Drive, Slow } time = { 0, ..., 100 } State(state!,time) Obs(obs!,time) State(+s,0) State(+s,t) ^ State(+s',t+1) Obs(+o,t) ^ State(+s,t) Sparse HMM: State(s,t) => State(s1,t+1) v State(s2, t+1) v ... .

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Bayesian Networks Use all binary predicates with same first argument (the object x). One predicate for each variable A: A(x,v!) One clause for each line in the CPT and value of the variable Context-specific independence: One clause for each path in the decision tree Logistic regression: As before Noisy OR: Deterministic OR + Pairwise clauses

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**Relational Models Knowledge-based model construction**

Allow only Horn clauses Same as Bayes nets, except arbitrary relations Combin. function: Logistic regression, noisy-OR or external Stochastic logic programs Weight of clause = log(p) Add formulas: Head holds Exactly one body holds Probabilistic relational models Allow only binary relations Same as Bayes nets, except first argument can vary

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**Relational Models Relational Markov networks Bayesian logic**

SQL → Datalog → First-order logic One clause for each state of a clique + syntax in Alchemy facilitates this Bayesian logic Object = Cluster of similar/related observations Observation constants + Object constants Predicate InstanceOf(Obs,Obj) and clauses using it Unknown relations: Second-order Markov logic S. Kok & P. Domingos, “Statistical Predicate Invention”, in Proc. ICML-2007.

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**Overview Motivation Foundational areas Markov logic NLP applications**

Basics Supervised learning Unsupervised learning

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**Text Classification ……**

The 56th quadrennial United States presidential election was held on November 4, Outgoing Republican President George W. Bush's policies and actions and the American public's desire for change were key issues throughout the campaign. …… Topic = politics The Chicago Bulls are an American professional basketball team based in Chicago, Illinois, playing in the Central Division of the Eastern Conference in the National Basketball Association (NBA). …… Topic = sports ……

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**Text Classification page = {1, ..., max} word = { ... }**

topic = { ... } Topic(page,topic) HasWord(page,word) Topic(p,t) HasWord(p,+w) => Topic(p,+t) If topics mutually exclusive: Topic(page,topic!)

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**Text Classification page = {1, ..., max} word = { ... }**

topic = { ... } Topic(page,topic) HasWord(page,word) Links(page,page) Topic(p,t) HasWord(p,+w) => Topic(p,+t) Topic(p,t) ^ Links(p,p') => Topic(p',t) Cf. S. Chakrabarti, B. Dom & P. Indyk, “Hypertext Classification Using Hyperlinks,” in Proc. SIGMOD-1998.

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**Entity Resolution AUTHOR: H. POON & P. DOMINGOS**

TITLE: UNSUPERVISED SEMANTIC PARSING VENUE: EMNLP-09 SAME? AUTHOR: Hoifung Poon and Pedro Domings TITLE: Unsupervised semantic parsing VENUE: Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing AUTHOR: Poon, Hoifung and Domings, Pedro TITLE: Unsupervised ontology induction from text VENUE: Proceedings of the Forty-Eighth Annual Meeting of the Association for Computational Linguistics SAME? AUTHOR: H. Poon, P. Domings TITLE: Unsupervised ontology induction VENUE: ACL-10

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**Entity Resolution Problem: Given database, find duplicate records**

HasToken(token,field,record) SameField(field,record,record) SameRecord(record,record) HasToken(+t,+f,r) ^ HasToken(+t,+f,r’) => SameField(f,r,r’) SameField(f,r,r’) => SameRecord(r,r’)

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**Entity Resolution Problem: Given database, find duplicate records**

HasToken(token,field,record) SameField(field,record,record) SameRecord(record,record) HasToken(+t,+f,r) ^ HasToken(+t,+f,r’) => SameField(f,r,r’) SameField(f,r,r’) => SameRecord(r,r’) SameRecord(r,r’) ^ SameRecord(r’,r”) => SameRecord(r,r”) Cf. A. McCallum & B. Wellner, “Conditional Models of Identity Uncertainty with Application to Noun Coreference,” in Adv. NIPS 17, 2005.

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**Entity Resolution Can also resolve fields:**

HasToken(token,field,record) SameField(field,record,record) SameRecord(record,record) HasToken(+t,+f,r) ^ HasToken(+t,+f,r’) => SameField(f,r,r’) SameField(f,r,r’) <=> SameRecord(r,r’) SameRecord(r,r’) ^ SameRecord(r’,r”) => SameRecord(r,r”) SameField(f,r,r’) ^ SameField(f,r’,r”) => SameField(f,r,r”) More: P. Singla & P. Domingos, “Entity Resolution with Markov Logic”, in Proc. ICDM-2006.

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**Information Extraction**

Unsupervised Semantic Parsing, Hoifung Poon and Pedro Domingos. Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing. Singapore: ACL. UNSUPERVISED SEMANTIC PARSING. H. POON & P. DOMINGOS. EMNLP-2009.

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**Information Extraction**

Author Title Venue Unsupervised Semantic Parsing, Hoifung Poon and Pedro Domingos. Proceedings of the 2009 Conference on Empirical Methods in Natural Language Processing. Singapore: ACL. SAME? UNSUPERVISED SEMANTIC PARSING. H. POON & P. DOMINGOS. EMNLP-2009.

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**Information Extraction**

Problem: Extract database from text or semi-structured sources Example: Extract database of publications from citation list(s) (the “CiteSeer problem”) Two steps: Segmentation: Use HMM to assign tokens to fields Entity resolution: Use logistic regression and transitivity

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**Information Extraction**

Token(token, position, citation) InField(position, field!, citation) SameField(field, citation, citation) SameCit(citation, citation) Token(+t,i,c) => InField(i,+f,c) InField(i,+f,c) ^ InField(i+1,+f,c) Token(+t,i,c) ^ InField(i,+f,c) ^ Token(+t,i’,c’) ^ InField(i’,+f,c’) => SameField(+f,c,c’) SameField(+f,c,c’) <=> SameCit(c,c’) SameField(f,c,c’) ^ SameField(f,c’,c”) => SameField(f,c,c”) SameCit(c,c’) ^ SameCit(c’,c”) => SameCit(c,c”)

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**Information Extraction**

Token(token, position, citation) InField(position, field!, citation) SameField(field, citation, citation) SameCit(citation, citation) Token(+t,i,c) => InField(i,+f,c) InField(i,+f,c) ^ !Token(“.”,i,c) ^ InField(i+1,+f,c) Token(+t,i,c) ^ InField(i,+f,c) ^ Token(+t,i’,c’) ^ InField(i’,+f,c’) => SameField(+f,c,c’) SameField(+f,c,c’) <=> SameCit(c,c’) SameField(f,c,c’) ^ SameField(f,c’,c”) => SameField(f,c,c”) SameCit(c,c’) ^ SameCit(c’,c”) => SameCit(c,c”) More: H. Poon & P. Domingos, “Joint Inference in Information Extraction”, in Proc. AAAI-2007.

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**Biomedical Text Mining**

Traditionally, name entity recognition or information extraction E.g., protein recognition, protein-protein identification BioNLP-09 shared task: Nested bio-events Much harder than traditional IE Top F1 around 50% Naturally calls for joint inference

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Bio-Event Extraction Involvement of p70(S6)-kinase activation in IL-10 up-regulation in human monocytes by gp41 envelope protein of human immunodeficiency virus type 1 ... involvement Theme Cause up-regulation activation Theme Cause Site Theme human monocyte p70(S6)-kinase IL-10 gp41

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**Bio-Event Extraction Logistic regression Token(position, token)**

DepEdge(position, position, dependency) IsProtein(position) EvtType(position, evtType) InArgPath(position, position, argType!) Token(i,+w) => EvtType(i,+t) Token(j,w) ^ DepEdge(i,j,+d) => EvtType(i,+t) DepEdge(i,j,+d) => InArgPath(i,j,+a) Token(i,+w) ^ DepEdge(i,j,+d) => InArgPath(i,j,+a) … Logistic regression

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**Adding a few joint inference rules doubles the F1**

Bio-Event Extraction Token(position, token) DepEdge(position, position, dependency) IsProtein(position) EvtType(position, evtType) InArgPath(position, position, argType!) Token(i,+w) => EvtType(i,+t) Token(j,w) ^ DepEdge(i,j,+d) => EvtType(i,+t) DepEdge(i,j,+d) => InArgPath(i,j,+a) Token(i,+w) ^ DepEdge(i,j,+d) => InArgPath(i,j,+a) … InArgPath(i,j,Theme) => IsProtein(j) v (Exist k k!=i ^ InArgPath(j, k, Theme)). More: H. Poon and L. Vanderwende, “Joint Inference for Knowledge Extraction from Biomedical Literature”, 10:40 am, June 4, Gold Room. Adding a few joint inference rules doubles the F1

109
**Temporal Information Extraction**

Identify event times and temporal relations (BEFORE, AFTER, OVERLAP) E.g., who is the President of U.S.A.? Obama: 1/20/2009 present G. W. Bush: 1/20/2001 1/19/2009 Etc.

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**Temporal Information Extraction**

DepEdge(position, position, dependency) Event(position, event) After(event, event) DepEdge(i,j,+d) ^ Event(i,p) ^ Event(j,q) => After(p,q) After(p,q) ^ After(q,r) => After(p,r)

111
**Temporal Information Extraction**

DepEdge(position, position, dependency) Event(position, event) After(event, event) Role(position, position, role) DepEdge(I,j,+d) ^ Event(i,p) ^ Event(j,q) => After(p,q) Role(i,j,ROLE-AFTER) ^ Event(i,p) ^ Event(j,q) => After(p,q) After(p,q) ^ After(q,r) => After(p,r) More: K. Yoshikawa, S. Riedel, M. Asahara and Y. Matsumoto, “Jointly Identifying Temporal Relations with Markov Logic”, in Proc. ACL-2009. X. Ling & D. Weld, “Temporal Information Extraction”, in Proc. AAAI-2010.

112
**Semantic Role Labeling**

Problem: Identify arguments for a predicate Two steps: Argument identification: Determine whether a phrase is an argument Role classification: Determine the type of an argument (agent, theme, temporal, adjunct, etc.)

113
**Semantic Role Labeling**

Token(position, token) DepPath(position, position, path) IsPredicate(position) Role(position, position, role!) HasRole(position, position) Token(i,+t) => IsPredicate(i) DepPath(i,j,+p) => Role(i,j,+r) HasRole(i,j) => IsPredicate(i) IsPredicate(i) => Exist j HasRole(i,j) HasRole(i,j) => Exist r Role(i,j,r) Role(i,j,r) => HasRole(i,j) Cf. K. Toutanova, A. Haghighi, C. Manning, “A global joint model for semantic role labeling”, in Computational Linguistics 2008.

114
**Joint Semantic Role Labeling and Word Sense Disambiguation**

Token(position, token) DepPath(position, position, path) IsPredicate(position) Role(position, position, role!) HasRole(position, position) Sense(position, sense!) Token(i,+t) => IsPredicate(i) DepPath(i,j,+p) => Role(i,j,+r) Sense(I,s) => IsPredicate(i) HasRole(i,j) => IsPredicate(i) IsPredicate(i) => Exist j HasRole(i,j) HasRole(i,j) => Exist r Role(i,j,r) Role(i,j,r) => HasRole(i,j) Token(i,+t) ^ Role(i,j,+r) => Sense(i,+s) More: I. Meza-Ruiz & S. Riedel, “Jointly Identifying Predicates, Arguments and Senses using Markov Logic”, in Proc. NAACL-2009.

115
**Practical Tips: Modeling**

Add all unit clauses (the default) How to handle uncertain data: R(x,y) ^ R’(x,y) (the “HMM trick”) Implications vs. conjunctions For soft correlation, conjunctions often better Implication: A => B is equivalent to !(A ^ !B) Share cases with others like A => C Make learning unnecessarily harder

116
**Practical Tips: Efficiency**

Open/closed world assumptions Low clause arities Low numbers of constants Short inference chains

117
**Practical Tips: Development**

Start with easy components Gradually expand to full task Use the simplest MLN that works Cycle: Add/delete formulas, learn and test

118
**Overview Motivation Foundational areas Markov logic NLP applications**

Basics Supervised learning Unsupervised learning

119
**Unsupervised Learning: Why?**

Virtually unlimited supply of unlabeled text Labeling is expensive (Cf. Penn-Treebank) Often difficult to label with consistency and high quality (e.g., semantic parses) Emerging field: Machine reading Extract knowledge from unstructured text with high precision/recall and minimal human effort Check out LBR-Workshop (WS9) on Sunday

120
**Unsupervised Learning: How?**

I.i.d. learning: Sophisticated model requires more labeled data Statistical relational learning: Sophisticated model may require less labeled data Relational dependencies constrain problem space One formula is worth a thousand labels Small amount of domain knowledge large-scale joint inference

121
**Unsupervised Learning: How?**

Ambiguities vary among objects Joint inference Propagate information from unambiguous objects to ambiguous ones E.g.: G. W. Bush … He … … Mrs. Bush … Are they coreferent?

122
**Unsupervised Learning: How**

Ambiguities vary among objects Joint inference Propagate information from unambiguous objects to ambiguous ones E.g.: G. W. Bush … He … … Mrs. Bush … Should be coreferent

123
**Unsupervised Learning: How**

Ambiguities vary among objects Joint inference Propagate information from unambiguous objects to ambiguous ones E.g.: G. W. Bush … He … … Mrs. Bush … So must be singular male!

124
**Unsupervised Learning: How**

Ambiguities vary among objects Joint inference Propagate information from unambiguous objects to ambiguous ones E.g.: G. W. Bush … He … … Mrs. Bush … Must be singular female!

125
**Unsupervised Learning: How**

Ambiguities vary among objects Joint inference Propagate information from unambiguous objects to ambiguous ones E.g.: G. W. Bush … He … … Mrs. Bush … Verdict: Not coreferent!

126
**Parameter Learning Marginalize out hidden variables**

Use MC-SAT to approximate both expectations May also combine with contrastive estimation [Poon & Cherry & Toutanova, NAACL-2009] Sum over z, conditioned on observed x Summed over both x and z

127
**Unsupervised Coreference Resolution**

Head(mention, string) Type(mention, type) MentionOf(mention, entity) MentionOf(+m,+e) Type(+m,+t) Head(+m,+h) ^ MentionOf(+m,+e) MentionOf(a,e) ^ MentionOf(b,e) => (Type(a,t) <=> Type(b,t)) … (similarly for Number, Gender etc.) Mixture model Joint inference formulas: Enforce agreement

128
**Unsupervised Coreference Resolution**

Head(mention, string) Type(mention, type) MentionOf(mention, entity) Apposition(mention, mention) MentionOf(+m,+e) Type(+m,+t) Head(+m,+h) ^ MentionOf(+m,+e) MentionOf(a,e) ^ MentionOf(b,e) => (Type(a,t) <=> Type(b,t)) … (similarly for Number, Gender etc.) Apposition(a,b) => (MentionOf(a,e) <=> MentionOf(b,e)) More: H. Poon and P. Domingos, “Joint Unsupervised Coreference Resolution with Markov Logic”, in Proc. EMNLP-2008. Joint inference formulas: Leverage apposition

129
**Relational Clustering: Discover Unknown Predicates**

Cluster relations along with objects Use second-order Markov logic [Kok & Domingos, 2007, 2008] Key idea: Cluster combination determines likelihood of relations InClust(r,+c) ^ InClust(x,+a) ^ InClust(y,+b) => r(x,y) Input: Relational tuples extracted by TextRunner [Banko et al., 2007] Output: Semantic network

130
**Recursive Relational Clustering**

Unsupervised semantic parsing [Poon & Domingos, EMNLP-2009] Text Knowledge Start directly from text Identify meaning units + Resolve variations Use high-order Markov logic (variables over arbitrary lambda forms and their clusters) End-to-end machine reading: Read text, then answer questions

131
**IL-4 protein induces CD11b**

Semantic Parsing INDUCE(e1) IL-4 protein induces CD11b INDUCER(e1,e2) INDUCED(e1,e3) IL-4(e2) CD11B(e3) Structured prediction: Partition + Assignment induces induces INDUCE nsubj dobj INDUCER nsubj dobj INDUCED protein protein CD11b CD11b nn nn CD11B IL-4 IL-4 IL-4

132
**Challenge: Same Meaning, Many Variations**

IL-4 up-regulates CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is induced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … ……

133
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, …

134
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … Cluster same forms at the atom level

135
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … Cluster forms in composition with same forms

136
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … Cluster forms in composition with same forms

137
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … Cluster forms in composition with same forms

138
**Unsupervised Semantic Parsing**

USP Recursively cluster arbitrary expressions composed with / by similar expressions IL-4 induces CD11b Protein IL-4 enhances the expression of CD11b CD11b expression is enhanced by IL-4 protein The cytokin interleukin-4 induces CD11b expression IL-4’s up-regulation of CD11b, … Cluster forms in composition with same forms

139
**Unsupervised Semantic Parsing**

Exponential prior on number of parameters Event/object/property cluster mixtures: InClust(e,+c) ^ HasValue(e,+v) Object/Event Cluster: INDUCE Property Cluster: INDUCER induces 0.1 nsubj 0.5 IL-4 0.2 None 0.1 enhances 0.4 … agent 0.4 IL-8 0.1 One 0.8 … … … …

140
**But … State Space Too Large**

Coreference: #-clusters #-mentions USP: #-clusters exp(#-tokens) Also, meaning units often small and many singleton clusters Use combinatorial search

141
**Inference: Hill-Climb Probability**

? induces ? nsubj dobj ? Initialize ? protein CD11B ? nn ? IL-4 ? Lambda reduction protein protein ? Search Operator nn nn ? ? IL-4 IL-4 ?

142
**Learning: Hill-Climb Likelihood**

… induces 1 enhances 1 IL-4 1 protein Initialize 1 MERGE COMPOSE induces 1 enhances 1 IL-4 1 protein 1 Search Operator induces 0.2 IL-4 protein 1 enhances 0.8

143
**Unsupervised Ontology Induction**

Limitations of USP: No ISA hierarchy among clusters Little smoothing Limited capability to generalize OntoUSP [Poon & Domingos, ACL-2010] Extends USP to also induce ISA hierarchy Joint approach for ontology induction, population, and knowledge extraction To appear in ACL (see you in Uppsala :-)

144
**OntoUSP Modify the cluster mixture formula**

InClust(e,c) ^ ISA(c,+d) ^ HasValue(e,+v) Hierarchical smoothing + clustering New operator in learning: MERGE with REGULATE? ABSTRACTION induces 0.3 enhances 0.1 induces 0.6 inhibits 0.2 INDUCE up-regulates 0.2 suppresses 0.1 … … ISA ISA INDUCE INHIBIT inhibits 0.4 inhibits 0.4 INHIBIT suppresses induces 0.6 0.2 up-regulates suppresses 0.2 0.2 … … …

145
**End of The Beginning … Growth area for machine learning and NLP**

Not merely a user guide of MLN and Alchemy Statistical relational learning: Growth area for machine learning and NLP

146
**Future Work: Inference**

Scale up inference Cutting-planes methods (e.g., [Riedel, 2008]) Unify lifted inference with sampling Coarse-to-fine inference Alternative technology E.g., linear programming, lagrangian relaxation

147
**Future Work: Supervised Learning**

Alternative optimization objectives E.g., max-margin learning [Huynh & Mooney, 2009] Learning for efficient inference E.g., learning arithmetic circuits [Lowd & Domingos, 2008] Structure learning: Improve accuracy and scalability E.g., [Kok & Domingos, 2009]

148
**Future Work: Unsupervised Learning**

Model: Learning objective, formalism, etc. Learning: Local optima, intractability, etc. Hyperparameter tuning Leverage available resources Semi-supervised learning Multi-task learning Transfer learning (e.g., domain adaptation) Human in the loop E.g., interative ML, active learning, crowdsourcing

149
**Future Work: NLP Applications**

Existing application areas: More joint inference opportunities Additional domain knowledge Combine multiple pipeline stages A “killer app”: Machine reading Many, many more awaiting YOU to discover

150
**Summary We need to unify logical and statistical NLP**

Markov logic provides a language for this Syntax: Weighted first-order formulas Semantics: Feature templates of Markov nets Inference: Satisfiability, MCMC, lifted BP, etc. Learning: Pseudo-likelihood, VP, PSCG, ILP, etc. Growing set of NLP applications Open-source software: Alchemy Book: Domingos & Lowd, Markov Logic, Morgan & Claypool, 2009. alchemy.cs.washington.edu

151
References [Banko et al., 2007] Michele Banko, Michael J. Cafarella, Stephen Soderland, Matt Broadhead, Oren Etzioni, "Open Information Extraction From the Web", In Proc. IJCAI-2007. [Chakrabarti et al., 1998] Soumen Chakrabarti, Byron Dom, Piotr Indyk, "Hypertext Classification Using Hyperlinks", in Proc. SIGMOD-1998. [Damien et al., 1999] Paul Damien, Jon Wakefield, Stephen Walker, "Gibbs sampling for Bayesian non-conjugate and hierarchical models by auxiliary variables", Journal of the Royal Statistical Society B, 61:2. [Domingos & Lowd, 2009] Pedro Domingos and Daniel Lowd, Markov Logic, Morgan & Claypool. [Friedman et al., 1999] Nir Friedman, Lise Getoor, Daphne Koller, Avi Pfeffer, "Learning probabilistic relational models", in Proc. IJCAI-1999.

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References [Halpern, 1990] Joe Halpern, "An analysis of first-order logics of probability", Artificial Intelligence 46. [Huynh & Mooney, 2009] Tuyen Huynh and Raymond Mooney, "Max-Margin Weight Learning for Markov Logic Networks", In Proc. ECML-2009. [Kautz et al., 1997] Henry Kautz, Bart Selman, Yuejun Jiang, "A general stochastic approach to solving problems with hard and soft constraints", In The Satisfiability Problem: Theory and Applications. AMS. [Kok & Domingos, 2007] Stanley Kok and Pedro Domingos, "Statistical Predicate Invention", In Proc. ICML-2007. [Kok & Domingos, 2008] Stanley Kok and Pedro Domingos, "Extracting Semantic Networks from Text via Relational Clustering", In Proc. ECML-2008.

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References [Kok & Domingos, 2009] Stanley Kok and Pedro Domingos, "Learning Markov Logic Network Structure via Hypergraph Lifting", In Proc. ICML-2009. [Ling & Weld, 2010] Xiao Ling and Daniel S. Weld, "Temporal Information Extraction", In Proc. AAAI-2010. [Lowd & Domingos, 2007] Daniel Lowd and Pedro Domingos, "Efficient Weight Learning for Markov Logic Networks", In Proc. PKDD-2007. [Lowd & Domingos, 2008] Daniel Lowd and Pedro Domingos, "Learning Arithmetic Circuits", In Proc. UAI-2008. [Meza-Ruiz & Riedel, 2009] Ivan Meza-Ruiz and Sebastian Riedel, "Jointly Identifying Predicates, Arguments and Senses using Markov Logic", In Proc. NAACL-2009.

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References [Muggleton, 1996] Stephen Muggleton, "Stochastic logic programs", in Proc. ILP-1996. [Nilsson, 1986] Nil Nilsson, "Probabilistic logic", Artificial Intelligence 28. [Page et al., 1998] Lawrence Page, Sergey Brin, Rajeev Motwani, Terry Winograd, "The PageRank Citation Ranking: Bringing Order to the Web", Tech. Rept., Stanford University, 1998. [Poon & Domingos, 2006] Hoifung Poon and Pedro Domingos, "Sound and Efficient Inference with Probabilistic and Deterministic Dependencies", In Proc. AAAI-06. [Poon & Domingos, 2007] Hoifung Poon and Pedro Domingo, "Joint Inference in Information Extraction", In Proc. AAAI-07.

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References [Poon & Domingos, 2008a] Hoifung Poon, Pedro Domingos, Marc Sumner, "A General Method for Reducing the Complexity of Relational Inference and its Application to MCMC", In Proc. AAAI-08. [Poon & Domingos, 2008b] Hoifung Poon and Pedro Domingos, "Joint Unsupervised Coreference Resolution with Markov Logic", In Proc. EMNLP-08. [Poon & Domingos, 2009] Hoifung and Pedro Domingos, "Unsupervised Semantic Parsing", In Proc. EMNLP-09. [Poon & Cherry & Toutanova, 2009] Hoifung Poon, Colin Cherry, Kristina Toutanova, "Unsupervised Morphological Segmentation with Log-Linear Models", In Proc. NAACL-2009.

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References [Poon & Vanderwende, 2010] Hoifung Poon and Lucy Vanderwende, "Joint Inference for Knowledge Extraction from Biomedical Literature", In Proc. NAACL-10. [Poon & Domingos, 2010] Hoifung and Pedro Domingos, "Unsupervised Ontology Induction From Text", In Proc. ACL-10. [Riedel 2008] Sebatian Riedel, "Improving the Accuracy and Efficiency of MAP Inference for Markov Logic", In Proc. UAI-2008. [Riedel et al., 2009] Sebastian Riedel, Hong-Woo Chun, Toshihisa Takagi and Jun'ichi Tsujii, "A Markov Logic Approach to Bio-Molecular Event Extraction", In Proc. BioNLP 2009 Shared Task. [Selman et al., 1996] Bart Selman, Henry Kautz, Bram Cohen, "Local search strategies for satisfiability testing", In Cliques, Coloring, and Satisfiability: Second DIMACS Implementation Challenge. AMS.

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References [Singla & Domingos, 2006a] Parag Singla and Pedro Domingos, "Memory-Efficient Inference in Relational Domains", In Proc. AAAI-2006. [Singla & Domingos, 2006b] Parag Singla and Pedro Domingos, "Entity Resolution with Markov Logic", In Proc. ICDM-2006. [Singla & Domingos, 2007] Parag Singla and Pedro Domingos, "Markov Logic in Infinite Domains", In Proc. UAI-2007. [Singla & Domingos, 2008] Parag Singla and Pedro Domingos, "Lifted First-Order Belief Propagation", In Proc. AAAI-2008. [Taskar et al., 2002] Ben Taskar, Pieter Abbeel, Daphne Koller, "Discriminative probabilistic models for relational data", in Proc. UAI-2002.

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References [Toutanova & Haghighi & Manning, 2008] Kristina Toutanova, Aria Haghighi, Chris Manning, "A global joint model for semantic role labeling", Computational Linguistics. [Wang & Domingos, 2008] Jue Wang and Pedro Domingos, "Hybrid Markov Logic Networks", In Proc. AAAI-2008. [Wellman et al., 1992] Michael Wellman, John S. Breese, Robert P. Goldman, "From knowledge bases to decision models", Knowledge Engineering Review 7. [Yoshikawa et al., 2009] Katsumasa Yoshikawa, Sebastian Riedel, Masayuki Asahara and Yuji Matsumoto, "Jointly Identifying Temporal Relations with Markov Logic", In Proc. ACL-2009.

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Markov Logic: A Simple and Powerful Unification Of Logic and Probability Pedro Domingos Dept. of Computer Science & Eng. University of Washington Joint.

Markov Logic: A Simple and Powerful Unification Of Logic and Probability Pedro Domingos Dept. of Computer Science & Eng. University of Washington Joint.

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