Introduction to Data-Driven Dependency Parsing

Transcription

1 Introduction to Data-Driven Dependency Parsing Introductory Course, ESSLLI 2007 Ryan McDonald 1 Joakim Nivre 2 1 Google Inc., New York, USA ryanmcd@google.com 2 Uppsala University and Växjö University, Sweden nivre@msi.vxu.se Introduction to Data-Driven Dependency Parsing 1(50)

2 Formal Conditions on Dependency Graphs From Last Lecture Last Lecture For a dependency graph G = (V,A) With label set L = {l 1,...,l L } G is (weakly) connected: If i, j V, i j. G is acyclic: If i j, then not j i. G obeys the single-head constraint: If i j, then not i j, for any i i. G is projective: If i j, then i i, for any i such that i <i <j or j <i <i. Introduction to Data-Driven Dependency Parsing 2(50)

3 From Last Lecture Dependency Graphs as Trees Consider a dependency graph G = (V, A) satisfying: G is (weakly) connected: If i, j V, i j. G obeys the single-head constraint: If i j, then not i j, for any i i. G obeys the single-root constraint: If i such that i j, then i such that i j, for any j j w 0 = root is always this node This dependency graph is by definition a tree For the rest of the course we assume that all dependency graphs are trees Introduction to Data-Driven Dependency Parsing 3(50)

4 From Last Lecture Dependency Graphs as Trees Satisfies: connected, single-head obj pc nmod sbj nmod nmod nmod Economic news had little effect on financial markets. Introduction to Data-Driven Dependency Parsing 4(50)

5 From Last Lecture Dependency Graphs as Trees Satisfies: connected, single-head, single-root p pred obj pc nmod sbj nmod nmod nmod root Economic news had little effect on financial markets. Introduction to Data-Driven Dependency Parsing 4(50)

6 Introduction Overview of the Course Dependency parsing (Joakim) Machine learning methods (Ryan) Transition-based models (Joakim) Graph-based models (Ryan) Loose ends (Joakim, Ryan): Other approaches Empirical results Available software Introduction to Data-Driven Dependency Parsing 5(50)

7 Introduction Data-Driven Parsing Data-Driven Machine Learning Parameterize a model Supervised: Learn parameters from annotated data Unsupervised: Induce parameters from a large corpora Data-Driven vs. Grammar-driven Can parse all sentences vs. generate specific language Data-driven = grammar of Σ Introduction to Data-Driven Dependency Parsing 6(50)

8 Introduction Lecture 2: Outline Feature Representations Linear Classifiers Perceptron Large-Margin Classifiers (SVMs, MIRA) Others Non-linear Classifiers K-NNs and Memory-based Learning Kernels Structured Learning Structured Perceptron Large-Margin Perceptron Others Introduction to Data-Driven Dependency Parsing 7(50)

9 Introduction Important Message This lecture contains a lot of details Not important if you do not follow all proofs and maths What is important Understand basic representation of data features Understand basic goal and structure of classifiers Understand important distinctions: linear vs. non-linear, binary vs. multiclass, multiclass vs. structured, etc. Interested in ML for NLP Check out afternoon course Machine learning methods for NLP Introduction to Data-Driven Dependency Parsing 8(50)

10 Feature Representations Feature Representations Input: x X e.g., document or sentence with some words x = w1... w n, or a series of previous actions Output: y Y e.g., dependency tree, document class, part-of-speech tags, next parsing action We assume a mapping from x to a high dimensional feature vector f(x) : X R m But sometimes it will be easier to think of a mapping from an input/output pair to a feature vector f(x, y) : X Y R m For any vector v R m, let v j be the j th value Introduction to Data-Driven Dependency Parsing 9(50)

11 Feature Representations Examples x is a document { 1 if x contains the word interest f j (x) = 0 otherwise f j (x) = The percentage of words than contain punctuation x is a word and y is a part-of-speech tag { 1 if x = bank and y = Verb f j (x,y) = 0 otherwise Introduction to Data-Driven Dependency Parsing 10(50)

12 Feature Representations Example 2 j 1 if x contains the word John f 0 (x) = 0 otherwise j 1 if x contains the word Mary f 1 (x) = 0 otherwise j 1 if x contains the word Harry f 2 (x) = 0 otherwise j 1 if x contains the word likes f 3 (x) = 0 otherwise x=john likes Mary f(x) = [ ] x=mary likes John f(x) = [ ] x=harry likes Mary f(x) = [ ] x=harry likes Harry f(x) = [ ] Introduction to Data-Driven Dependency Parsing 11(50)

13 Linear Classifiers Linear classifier: score (or probability) of a particular classification is based on a linear combination of features and their weights Let w R m be a high dimensional weight vector If we assume that w is known, then we can define two kinds of linear classifiers Reminder: v v = v j v j R Binary Classification: Y = { 1, 1} j y = sign(w f(x)) Multiclass Classification: Y = {0, 1,..., N} Linear Classifiers y = argmax y w f(x, y) Introduction to Data-Driven Dependency Parsing 12(50)

14 Linear Classifiers Binary Linear Classifier Divides all points: y = sign(w f(x)) Introduction to Data-Driven Dependency Parsing 13(50)

15 Linear Classifiers Multiclass Linear Classifier Defines regions of space: y = arg max y w f(x,y) Introduction to Data-Driven Dependency Parsing 14(50)

16 Linear Classifiers Separability A set of points is separable, if there exists a w such that classification is perfect Separable Not Separable Introduction to Data-Driven Dependency Parsing 15(50)

17 Linear Classifiers Supervised Learning Input: training examples T = {(x t,y t )} T t=1 Input: feature representation f Output: w that maximizes/minimizes some important function on the training set minimize error (Perceptron, SVMs, Boosting) maximize likelihood of data (Logistic Regression, CRFs) Assumption: The training data is separable Not necessary, just makes life easier There is a lot of good work in machine learning to tackle the non-separable case Introduction to Data-Driven Dependency Parsing 16(50)

18 Linear Classifiers Perceptron Minimize error Binary classification: Y = { 1, 1} w = arg min 1 1[y t = sign(w f(x t ))] w Multiclass classification: Y = {0, 1,..., N} w = arg min w t t 1[p] = 1 1[y t = arg maxw f(x t, y)] y { 1 p is true 0 otherwise Introduction to Data-Driven Dependency Parsing 17(50)

19 Perceptron Learning Algorithm (multiclass) Linear Classifiers Training data: T = {(x t,y t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. Let y = arg max y w (i) f(x t,y ) 5. if y y t 6. w (i+1) = w (i) + f(x t,y t ) f(x t,y ) 7. i = i return w i Introduction to Data-Driven Dependency Parsing 18(50)

20 Perceptron Learning Algorithm (multiclass) Linear Classifiers Given an training instance (x t,y t ), define: Ȳt = Y {y t } A training set T is separable with margin γ > 0 if there exists a vector u with u = 1 such that: u f(x t,y t ) u f(x t,y ) γ for all y Ȳ t and u = j u2 j Assumption: the training set is separable with margin γ Introduction to Data-Driven Dependency Parsing 19(50)

21 Perceptron Learning Algorithm (multiclass) Linear Classifiers Theorem: For any training set separable with a margin of γ, the following holds for the perceptron algorithm: Number of training errors R2 γ 2 where R f(x t,y t ) f(x t,y ) for all (x t,y t ) T and y Ȳ t Thus, after a finite number of training iterations, the error on the training set will converge to zero Let s prove it! (proof taken from Collins 02) Introduction to Data-Driven Dependency Parsing 20(50)

22 Perception Learning Algorithm (multiclass) Linear Classifiers Training data: T = {(x t, y t)} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. Let y = arg max y w (i) f(x t, y ) 5. if y y t 6. w (i+1) = w (i) + f(x t, y t) f(x t, y ) 7. i = i return w i w (k 1) are the weights before k th mistake Suppose k th mistake made at the t th example, (x t, y t) y = arg max y w (k 1) f(x t, y ) y y t w (k) = w (k 1) +f(x t, y t) f(x t, y ) Now: u w (k) = u w (k 1) + u (f(x t, y t) f(x t, y )) u w (k 1) + γ Now: w (0) = 0 and u w (0) = 0, by induction on k, u w (k) (k 1)γ Now: since u w (k) u w (k) and u = 1 then w (k) (k 1)γ Now: w (k) 2 = w (k 1) 2 + f(x t, y t) f(x t, y ) 2 + 2w (k 1) (f(x t, y t) f(x t, y )) w (k) 2 w (k 1) 2 + R 2 (since R f(x t, y t) f(x t, y ) and w (k 1) f(x t, y t) w (k 1) f(x t, y ) 0) Introduction to Data-Driven Dependency Parsing 21(50)

23 Perception Learning Algorithm (multiclass) We have just shown that w (k) (k 1)γ and w (k) 2 w (k 1) 2 + R 2 By induction on k and since w (0) = 0 and w (0) 2 = 0 Therefore, and solving for k w (k) 2 (k 1)R 2 (k 1) 2 γ 2 w (k) 2 (k 1)R 2 k 1 R2 γ 2 Therefore the number of errors is bounded! Linear Classifiers Introduction to Data-Driven Dependency Parsing 22(50)

24 Linear Classifiers Margin Training Testing Denote the value of the margin by γ Introduction to Data-Driven Dependency Parsing 23(50)

25 Linear Classifiers Margin Intuitively maximizing margin makes sense More importantly, generalization error to unseen test data is proportional to the inverse of the margin ǫ R 2 γ 2 T Perceptron: we have shown that: If a training set is separable by some margin, the perceptron will find a w that separates the data However, it does not pick a w to maximize the margin! Introduction to Data-Driven Dependency Parsing 24(50)

26 Linear Classifiers Max Margin = Min Norm Let γ > 0 Max Margin: max w 1 γ Min Norm: min 1 2 w 2 such that: y t (w f(x t )) γ (x t, y t ) T = such that: y t (w f(x t )) 1 (x t, y t ) T w is bound since scaling trivially produces larger margin y t ([βw] f(x t )) βγ, for some β 1 Instead of fixing w we fix the margin γ = 1 Introduction to Data-Driven Dependency Parsing 25(50)

27 Linear Classifiers Support Vector Machines Binary: Multiclass: min 1 2 w 2 min 1 2 w 2 such that: y t (w f(x t )) 1 (x t,y t ) T such that: w f(x t,y t ) w f(x t,y ) 1 (x t,y t ) T and y Ȳ t Both are quadratic programming problems a well known convex optimization problem Can be solved with out-of-the-box algorithms Introduction to Data-Driven Dependency Parsing 26(50)

28 Linear Classifiers Support Vector Machines Binary: min 1 2 w 2 such that: y t (w f(x t )) 1 Problem: Sometimes T is far too large Thus the number of constraints might make solving the quadratic programming problem very difficult Most common technique: Sequential Minimal Optimization (SMO) Sparse: solution only depends on support vectors (x t,y t ) T Introduction to Data-Driven Dependency Parsing 27(50)

29 Margin Infused Relaxed Algorithm (MIRA) Linear Classifiers Another option maximize margin using an online algorithm Batch vs. Online Batch update parameters based on entire training set (e.g., SVMs) Online update parameters based on a single training instance at a time (e.g., Perceptron) MIRA can be thought of as a max-margin perceptron or an online SVM Introduction to Data-Driven Dependency Parsing 28(50)

30 Linear Classifiers MIRA (multiclass) Batch (SVMs): min 1 2 w 2 such that: w f(x t, y t ) w f(x t, y ) 1 (x t, y t ) T and y Ȳ t Online (MIRA): Training data: T = {(x t, y t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. w (i+1) = arg min w* w* w (i) such that: w f(x t, y t ) w f(x t, y ) 1 y Ȳ t 5. i = i return w i MIRA has much smaller optimizations with only Ȳ t constraints Cost: sub-optimal optimization Introduction to Data-Driven Dependency Parsing 29(50)

31 Linear Classifiers Summary What we have covered Feature-based representations Linear Classifiers Perceptron Large-Margin SVMs (batch) and MIRA (online) What is next Non-linear classifiers Introduction to Data-Driven Dependency Parsing 30(50)

32 Non-Linear Classifiers Non-Linear Classifiers Some data sets require more than a linear classifier to be correctly modeled A lot of models out there K-Nearest Neighbours Decision Trees Kernels Neural Networks Will only discuss a couple due to time constraints Introduction to Data-Driven Dependency Parsing 31(50)

33 Non-Linear Classifiers K-Nearest Neighbours Simplest form: for a given test point x, find k-nearest neighbours in training set Neighbours vote for classification Distance is Euclidean distance d(x t,x r ) = (f j (x t ) f j (x r )) 2 j No linear classifier can correctly label data set. But 3-nearest neighbours does. Introduction to Data-Driven Dependency Parsing 32(50)

34 Non-Linear Classifiers K-Nearest Neighbours A training set T, distance function d, and value K define a non-linear classification boundary Introduction to Data-Driven Dependency Parsing 33(50)

35 Non-Linear Classifiers K-Nearest Neighbours K-NN is often called a lazy learning algorithm or memory based learning (MBL) K-NN generalized in the Tilburg Memory Based Learning Package Different distance functions Different voting schemes for classification Tie-breaking Memory representations Introduction to Data-Driven Dependency Parsing 34(50)

36 Non-Linear Classifiers Kernels A kernel is a similarity function between two points that is symmetric and positive semi-definite, which we denote by: φ(x t,x r ) R Mercer s Theorem: for any kernal φ, there exists an f, such that: φ(x t,x r ) = f(x t ) f(x r ) Introduction to Data-Driven Dependency Parsing 35(50)

37 Non-Linear Classifiers Kernel Trick Perceptron Algorithm Training data: T = {(x t, y t)} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. Let y = argmax y w (i) f(x t, y) 5. if y y t 6. w (i+1) = w (i) + f(x t, y t) f(x t, y) 7. i = i return w i Each feature function f(x t,y t ) is added and f(x t,y) is subtracted to w say α y,t times αy,t is the # of times during learning label y is predicted for example t Thus, w = t,y α y,t [f(x t,y t ) f(x t,y)] Introduction to Data-Driven Dependency Parsing 36(50)

38 Non-Linear Classifiers Kernel Trick Perceptron Algorithm We can re-write the argmax function as: y = arg maxw (i) f(x t, y ) y = arg max α y,t [f(x t, y t ) f(x t, y)] f(x t, y ) y t,y = arg max α y,t [f(x t, y t ) f(x t, y ) f(x t, y) f(x t, y )] y t,y = arg max α y,t [φ((x t, y t ), (x t, y )) φ((x t, y), (x t, y ))] y t,y We can then re-write the perceptron algorithm strictly with kernels Introduction to Data-Driven Dependency Parsing 37(50)

39 Non-Linear Classifiers Kernel Trick Perceptron Algorithm Training data: T = {(x t, y t)} T t=1 1. y,t set α y,t = 0 2. for n : 1..N 3. for t : 1..T 4. Let y = argmax y 5. if y y t 6. α y,t = α y,t + 1 P t,y αy,t[φ((xt, yt), (xt, y )) φ((x t, y), (x t, y ))] Given a new instance x y = arg max α y,t [φ((x t,y t ),(x,y )) φ((x t,y),(x,y ))] y t,y But it seems like we have just complicated things??? Introduction to Data-Driven Dependency Parsing 38(50)

40 Non-Linear Classifiers Kernels = Tractable Non-Linearity A linear classifier in a higher dimensional feature space is a non-linear classifier in the original space Computing a non-linear kernel is often better computationally than calculating the corresponding dot product in the high dimension feature space Thus, kernels allow us to efficiently learn non-linear classifiers Introduction to Data-Driven Dependency Parsing 39(50)

41 Non-Linear Classifiers Linear Classifiers in High Dimension Introduction to Data-Driven Dependency Parsing 40(50)

42 Non-Linear Classifiers Example: Polynomial Kernel f(x) R M, d 2 φ(x t,x s ) = (f(x t ) f(x s ) + 1) d O(M) to calculate for any d!! But in the original feature space (primal space) Consider d = 2, M = 2, and f(xt ) = [x t,1, x t,2 ] (f(x t) f(x s) + 1) 2 = ([x t,1,x t,2 ] [x s,1,x s,2 ] + 1) 2 = (x t,1 x s,1 + x t,2 x s,2 + 1) 2 = (x t,1 x s,1 ) 2 + (x t,2 x s,2 ) 2 + 2(x t,1 x s,1 ) + 2(x t,2 x s,2 ) +2(x t,1 x t,2 x s,1 x s,2 ) + (1) 2 which equals: [(x t,1 ) 2,(x t,2 ) 2, 2x t,1, 2x t,2, 2x t,1 x t,2,1] [(x s,1 ) 2,(x s,2 ) 2, 2x s,1, 2x s,2, 2x s,1 x s,2,1] Introduction to Data-Driven Dependency Parsing 41(50)

43 Non-Linear Classifiers Popular Kernels Polynomial kernel φ(x t,x s ) = (f(x t ) f(x s ) + 1) d Gaussian radial basis kernel (infinite feature space representation!) φ(x t,x s ) = exp( f(x t) f(x s ) 2 ) 2σ String kernels [Lodhi et al. 2002, Collins and Duffy 2002] Tree kernels [Collins and Duffy 2002] Introduction to Data-Driven Dependency Parsing 42(50)

44 Structured Learning Structured Learning Sometimes our output space Y is not simply a category Examples: Parsing: for a sentence x, Y is the set of possible parse trees Sequence tagging: for a sentence x, Y is the set of possible tag sequences, e.g., part-of-speech tags, named-entity tags Machine translation: for a source sentence x, Y is the set of possible target language sentences Can t we just use our multiclass learning algorithms? In all the cases, the size of the set Y is exponential in the length of the input x It is often non-trivial to solve our learning algorithms in such cases Introduction to Data-Driven Dependency Parsing 43(50)

45 Structured Learning Perceptron Training data: T = {(x t,y t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. Let y = arg max y w (i) f(x t,y ) (**) 5. if y y t 6. w (i+1) = w (i) + f(x t,y t ) f(x t,y ) 7. i = i return w i (**) Solving the argmax requires a search over an exponential space of outputs! Introduction to Data-Driven Dependency Parsing 44(50)

46 Structured Learning Large-Margin Classifiers Online (MIRA): Batch (SVMs): min 1 2 w 2 such that: w f(x t, y t ) w f(x t, y ) 1 (x t, y t ) T and y Ȳ t (**) Training data: T = {(x t, y t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. w (i+1) = arg min w* w* w (i) such that: w f(x t, y t ) w f(x t, y ) 1 y Ȳ t (**) 5. i = i return w i (**) There are exponential constraints in the size of each input!! Introduction to Data-Driven Dependency Parsing 45(50)

47 Structured Learning Factor the Feature Representations We can make an assumption that our feature representations factor relative to the output Example: Context Free Parsing: f(x, y) = A BC y f(x, A BC) Sequence Analysis Markov Assumptions: y f(x, y) = f(x, y i 1, y i ) These kinds of factorizations allow us to run algorithms like CKY and Viterbi to compute the argmax function i=1 Introduction to Data-Driven Dependency Parsing 46(50)

48 Structured Learning Structured Perceptron Exactly like original perceptron Except now the argmax function uses a factored feature representation All of the original analysis for the multiclass perceptron carries over!! Introduction to Data-Driven Dependency Parsing 47(50)

49 Structured Learning Structured SVMs min 1 2 w 2 such that: w f(x t,y t ) w f(x t,y ) L(y t,y ) (x t,y t ) T and y Ȳ t (**) Still have an exponential # of constraints Feature factorizations also allow for solutions Maximum Margin Markov Networks (Taskar et al. 03) Structured SVMs (Tsochantaridis et al. 04) Note: Old fixed margin of 1 is now a fixed loss L(y t,y ) between two structured outputs Introduction to Data-Driven Dependency Parsing 48(50)

50 Online Structured SVMs (or Online MIRA) Training data: T = {(x t,y t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. w (i+1) = arg min w* w* w (i) such that: w f(x t,y t ) w f(x t,y ) L(y t,y ) y Ȳ t and y k-best(x t,w (i) ) (**) 5. i = i return w i k-best(x t ) is set of outputs with highest scores using weight vector w (i) Structured Learning Simple Solution only consider outputs y Ȳ t that currently have highest score Introduction to Data-Driven Dependency Parsing 49(50)

51 Wrap Up Main Points of Lecture Feature representations Choose feature weights, w, to maximize some function (min error, max margin) Batch learning (SVMs) versus online learning (perceptron, MIRA) Linear versus Non-linear classifiers Structured Learning Introduction to Data-Driven Dependency Parsing 50(50)

52 References and Further Reading References and Further Reading A. L. Berger, S. A. Della Pietra, and V. J. Della Pietra A maximum entropy approach to natural language processing. Computational Linguistics, 22(1). P. M. Camerini, L. Fratta, and F. Maffioli The k best spanning arborescences of a network. Networks, 10(2): Y.J. Chu and T.H. Liu On the shortest arborescence of a directed graph. Science Sinica, 14: M. Collins and N. Duffy New ranking algorithms for parsing and tagging: Kernels over discrete structures, and the voted perceptron. In Proc. ACL. M. Collins Discriminative training methods for hidden Markov models: Theory and experiments with perceptron algorithms. In Proc. EMNLP. K. Crammer and Y. Singer On the algorithmic implementation of multiclass kernel based vector machines. JMLR. K. Crammer and Y. Singer Ultraconservative online algorithms for multiclass problems. JMLR. Introduction to Data-Driven Dependency Parsing 50(50)

53 References and Further Reading K. Crammer, O. Dekel, S. Shalev-Shwartz, and Y. Singer Online passive aggressive algorithms. In Proc. NIPS. K. Crammer, O. Dekel, J. Keshat, S. Shalev-Shwartz, and Y. Singer Online passive aggressive algorithms. JMLR. J. Edmonds Optimum branchings. Journal of Research of the National Bureau of Standards, 71B: J. Eisner Three new probabilistic models for dependency parsing: An exploration. In Proc. COLING. Y. Freund and R.E. Schapire Large margin classification using the perceptron algorithm. Machine Learning, 37(3): T. Joachims Learning to Classify Text using Support Vector Machines. Kluwer. D. Klein and C. Manning Corpus-based induction of syntactic structure: Models of dependency and constituency. In Proc. ACL. T. Koo, A. Globerson, X. Carreras, and M. Collins Structured prediction models via the matrix-tree theorem. In Proc. EMNLP. Introduction to Data-Driven Dependency Parsing 50(50)

54 J. Lafferty, A. McCallum, and F. Pereira Conditional random fields: Probabilistic models for segmenting and labeling sequence data. In Proc. ICML. H. Lodhi, C. Saunders, J. Shawe-Taylor, and N. Cristianini Classification with string kernels. Journal of Machine Learning Research. References and Further Reading A. McCallum, D. Freitag, and F. Pereira Maximum entropy Markov models for information extraction and segmentation. In Proc. ICML. R. McDonald and F. Pereira Online learning of approximate dependency parsing algorithms. In Proc EACL. R. McDonald and G. Satta On the complexity of non-projective data-driven dependency parsing. In Proc. IWPT. R. McDonald, K. Crammer, and F. Pereira Online large-margin training of dependency parsers. In Proc. ACL. K.R. Müller, S. Mika, G. Rätsch, K. Tsuda, and B. Schölkopf An introduction to kernel-based learning algorithms. IEEE Neural Networks, 12(2): M.A. Paskin Introduction to Data-Driven Dependency Parsing 50(50)

55 References and Further Reading Cubic-time parsing and learning algorithms for grammatical bigram models. Technical Report UCB/CSD , Computer Science Division, University of California Berkeley. K. Sagae and A. Lavie Parser combination by reparsing. In Proc. HLT/NAACL. F. Sha and F. Pereira Shallow parsing with conditional random fields. In Proc. HLT/NAACL, pages N. Smith and J. Eisner Guiding unsupervised grammar induction using contrastive estimation. In Working Notes of the International Joint Conference on Artificial Intelligence Workshop on Grammatical Inference Applications. D.A. Smith and N.A. Smith Probabilistic models of nonprojective dependency trees. In Proc. EMNLP. C. Sutton and A. McCallum An introduction to conditional random fields for relational learning. In L. Getoor and B. Taskar, editors, Introduction to Statistical Relational Learning. MIT Press. R.E. Tarjan Finding optimum branchings. Networks, 7: B. Taskar, C. Guestrin, and D. Koller Introduction to Data-Driven Dependency Parsing 50(50)

56 References and Further Reading Max-margin Markov networks. In Proc. NIPS. B. Taskar Learning Structured Prediction Models: A Large Margin Approach. Ph.D. thesis, Stanford. I. Tsochantaridis, T. Hofmann, T. Joachims, and Y. Altun Support vector learning for interdependent and structured output spaces. In Proc. ICML. W.T. Tutte Graph Theory. Cambridge University Press. D. Yuret Discovery of linguistic relations using lexical attraction. Ph.D. thesis, MIT. Introduction to Data-Driven Dependency Parsing 50(50)