Tekniker för storskalig parsning: Dependensparsning 2

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1 Tekniker för storskalig parsning: Dependensparsning 2 Joakim Nivre Uppsala Universitet Institutionen för lingvistik och filologi joakim.nivre@lingfil.uu.se Dependensparsning 2 1(45)

2 Data-Driven Dependency Parsing Transition-based [Yamada and Matsumoto 2003, Nivre et al. 2004]: Parameterize parsing model by state transitions Learn to predict transitions given the input and a history Predict new trees using greedy inference Graph-based [Eisner 1996, McDonald et al. 2005]: Parameterize parsing model by dependency arcs Learn to predict entire trees given the input Predict new trees using spanning tree algorithms CoNLL shared task 2006: 10 transition-based systems; best score: [Nivre et al. 2006] 6 graph-based systems; best score: [McDonald et al. 2006] Dependensparsning 2 2(45)

3 Some Graph Theory Reminders A graph G = (V, A) is a set of vertices V and arcs (i, j) A, where i, j V Undirected graphs: (i, j) A (j, i) A Directed graphs (digraphs): (i, j) A (j, i) A Dependensparsning 2 3(45)

4 Multi-Digraphs A multi-digraph is a digraph where there can be multiple arcs between vertices G = (V, A) (i, j, k) A represents the k th arc from vertex i to vertex j Dependensparsning 2 4(45)

5 Directed Spanning Trees (a.k.a. Arborescence) A directed spanning tree of a (multi-)digraph G = (V, A), is a subgraph G = (V, A ) such that: V = V A A, and A = V 1 G is a tree (acyclic) A spanning tree of the following (multi-)digraphs Dependensparsning 2 5(45)

6 Weighted Directed Spanning Trees Assume we have a weight function for each arc in a multi-digraph G = (V, A) Define w k ij 0 to be the weight of (i, j, k) A for a multi-digraph Define the weight of directed spanning tree G of graph G as w(g ) = (i,j,k) G w k ij Notation: (i, j, k) G = (V, A) the arc (i, j, k) A Dependensparsning 2 6(45)

7 Maximum Spanning Trees (MST) of (Multi-)Digraphs Let T (G) be the set of all spanning trees for graph G The MST Problem: Find the spanning tree G of the graph G that has highest weight G = arg max w(g ) = arg max G T (G) G T (G) Solutions... to come. (i,j,k) G w k ij Dependensparsning 2 7(45)

8 Arc-Factored Dependency Models Remember: Data-driven parsing parameterizes model and then learns parameters from data Arc-factored model Assumes that the score / weight of a dependency graph factors by its arcs w(g) = (i,j,k) G w k ij look familiar? w k ij is the weight of creating a dependency from word w i to w j with label l k Thus there is an assumption that each dependency decision is independent Strong assumption! Will address this later. Dependensparsning 2 8(45)

9 Arc-Factored Dependency Models Example Weight of dependency graph is = 70 Dependensparsning 2 9(45)

10 Important Concept G x For input sentence x = w 0,..., w n, define G x = (V x, A x ) as: Vx = {0, 1,..., n} A x = {(i, j, k) i, j V x and l k L} Thus, G x is complete multi-digraph over vertex set representing words Theorem Every valid dependency graph for sentence x is equivalent to a directed spanning tree for G x that originates out of vertex 0 Falls out of definitions of tree constrained dependency graphs and spanning trees Both are spanning/connected (contain all words) Both are trees Dependensparsning 2 10(45)

11 Three Important Problems Theorem Every valid dependency graph for sentence x is equivalent to a directed spanning tree for G x that originates out of vertex 0 1. Inference finding the MST of G x G = arg max G T (G x ) w(g) = arg max G T (G x ) (i,j,k) G w k ij 2. Defining w k ij and its feature space 3. Learning w k ij Dependensparsning 2 11(45)

12 Inference - Getting Rid of Arc Labels G = arg max G T (G x ) w(g) = arg max G T (G x ) (i,j,k) G w k ij Consider all the arcs between vertexes i and j Now, consider the arc (i, j, k) such that, (i, j, k) = arg max wij k k Theorem If the highest weighted dependency tree for sentence x contains an arc (i, j, k ) then k = k (assuming no ties) Easy proof: if not, sub in (i, j, k) and get higher weighted tree Dependensparsning 2 12(45)

13 Inference - Getting Rid of Arc Labels G = arg max G T (G x ) w(g) = arg max G T (G x ) (i,j,k) G w k ij Thus, we can reduce G x from a multi-digraph to a simple digraph Just remove all arcs that do not satisfy (i, j, k) = arg max wij k k Problem is now equal to the MST problem for digraphs We will use the Chu-Liu-Edmonds Algorithm [Chu and Liu 1965, Edmonds 1967] Dependensparsning 2 13(45)

14 Chu-Liu-Edmonds Algorithm Finds the MST originating out of a vertex of choice Assumes weight of tree is sum of arc weights x = root John saw Mary Remove all arcs into the root node Dependensparsning 2 14(45)

15 Chu-Liu-Edmonds Find highest scoring incoming arc for each vertex If this is a tree, then we have found MST!! Dependensparsning 2 15(45)

16 Chu-Liu-Edmonds If not a tree, identify cycle and contract Recalculate arc weights into and out-of cycle Dependensparsning 2 16(45)

17 Chu-Liu-Edmonds Outgoing arc weights Equal to the max of outgoing arc over all vertexes in cycle e.g., John Mary is 3 and saw Mary is 30 Dependensparsning 2 17(45)

18 Chu-Liu-Edmonds Incoming arc weights Equal to the weight of best spanning tree that includes head of incoming arc, and all nodes in cycle root saw John is 40 (**) root John saw is 29 Dependensparsning 2 18(45)

19 Chu-Liu-Edmonds Theorem The weight of the MST of this contracted graph is equal to the weight of the MST for the original graph Therefore, recursively call algorithm on new graph Dependensparsning 2 19(45)

20 Chu-Liu-Edmonds This is a tree and the MST for the contracted graph!! Go back up recursive call and reconstruct final graph Dependensparsning 2 20(45)

21 Chu-Liu-Edmonds This is the MST!! Dependensparsning 2 21(45)

22 Chu-Liu-Edmonds Code Chu-Liu-Edmonds(G x, w) 1. Let M = {(i, j) : j V x, i = arg max i w ij } 2. Let G M = (V x, M) 3. If G M has no cycles, then it is an MST: return G M 4. Otherwise, find a cycle C in G M 5. Let < G C, c, ma >= contract(g, C, w) 6. Let G = Chu-Liu-Edmonds(G C, w) 7. Find vertex i C such that (i, c) G and ma(i, c) = i 8. Find arc (i, i) C 9. Find all arc (c, i ) G 10. G = G {(ma(c, i ), i )} (c,i ) G C {(i, i)} {(i, i)} 11. Remove all vertices and arcs in G containing c 12. return G Reminder: w ij = arg max k w k ij Dependensparsning 2 22(45)

23 Chu-Liu-Edmonds Code (II) contract(g = (V, A), C, w) 1. Let G C be the subgraph of G excluding nodes in C 2. Add a node c to G C representing cycle C 3. For i V C : i C (i, i) A Add arc (c, i) to G C with ma(c, i) = arg max i C score(i, i) i = ma(c, i) score(c, i) = score(i, i) 4. For i V C : i C (i, i ) A Add edge (i, c) to G C with ma(i, c) = arg max i C [score(i, i ) score(a(i ), i )] i = ma(i, c) score(i, c) = [score(i, i ) score(a(i ), i ) + score(c)] where a(v) is the predecessor of v in C and score(c) = P v C score(a(v), v) 5. return < G C, c, ma > Dependensparsning 2 23(45)

24 Chu-Liu-Edmonds Naive implementation O(n 3 + L n 2 ) Converting Gx to a digraph O( L n 2 ) Finding best arc O(n 2 ) Contracting cycles O(n 2 ) At most n recursive calls Better algorithms run in O( L n 2 ) [Tarjan 1977] Chu-Liu-Edmonds searches all dependency graphs Both projective and non-projective Thus, it is an exact non-projective search algorithm!!! What about the projective case? Dependensparsning 2 24(45)

25 Arc-factored Projective Parsing Projective dependency structures are nested Can use CFG like parsing algorithms chart parsing Each chart item (triangle) represents the weight of the best tree rooted at word h spanning all the words from i to j Analog in CFG parsing: items represent best tree rooted at non-terminal NT spanning words i to j Goal: Find chart item rooted at 0 spanning 0 to n Base case Length 1, h = i = j, has weight 0 Dependensparsning 2 25(45)

26 Arc-factored Projective Parsing All projective graphs can be written as the combination of two smaller adjacent graphs Inductive hypothesis algorithm has calculated score of smaller items correctly (just like CKY) Dependensparsning 2 26(45)

27 Arc-factored Projective Parsing Chart item filled in a bottom-up manner First do all strings of length 1, then 2, etc. just like CKY Weight of new item: max l,j,k w(a) + w(b) + w k hh Algorithm runs in O( L n 5 ) Use back-pointers to extract best parse (like CKY) Dependensparsning 2 27(45)

28 Arc-factored Projective Parsing O( L n 5 ) is not that good [Eisner 1996] showed how this can be reduced to O( L n 3 ) Key: split items so that sub-roots are always on periphery Dependensparsning 2 28(45)

29 Inference in Arc-Factored Models Non-projective case O( L n 2 ) with the Chu-Liu-Edmonds MST algorithm Projective case O( L n 3 ) with the Eisner algorithm But we still haven t defined the form of w k ij Or how to learn these parameters Dependensparsning 2 29(45)

30 Arc weights as linear classifiers w k ij = w f(i, j, k) Arc weights are a linear combination of features of the arc, f, and a corresponding weight vector w What arc features? [McDonald et al. 2005] discuss a number of binary features Dependensparsning 2 30(45)

31 Arc Features: f(i, j, k) Features from [McDonald et al. 2005]: Identities of the words wi and w j and the label l k head=saw & dependent=with Dependensparsning 2 31(45)

32 Arc Features: f(i, j, k) Features from [McDonald et al. 2005]: Part-of-speech tags of the words wi and w j and the label l k head-pos=verb & dependent-pos=preposition Dependensparsning 2 32(45)

33 Arc Features: f(i, j, k) Features from [McDonald et al. 2005]: Part-of-speech of words surrounding and between wi and w j inbetween-pos=noun inbetween-pos=adverb dependent-pos-right=pronoun head-pos-left=noun... Dependensparsning 2 33(45)

34 Arc Features: f(i, j, k) Features from [McDonald et al. 2005]: Number of words between w i and w j, and their orientation arc-distance=3 arc-direction=right Dependensparsning 2 34(45)

35 Arc Features: f(i, j, k) Label features arc-label=pp Dependensparsning 2 35(45)

36 Arc Features: f(i, j, k) Combos of the above head-pos=verb & dependent-pos=preposition & arc-label=pp head-pos=verb & dependent=with & arc-distance=3... No limit: any feature over arc (i, j, k) or input x Dependensparsning 2 36(45)

37 Inference-based Learning e.g., The Perceptron Training data: T = {(x t, G t )} T t=1 1. w (0) = 0; i = 0 2. for n : 1..N 3. for t : 1..T 4. Let G = arg max G w (i) f(g ) (**) 5. if G G t 6. w (i+1) = w (i) + f(g t ) f(g ) 7. i = i return w i Dependensparsning 2 37(45)

38 Beyond Arc-factored Models Arc-factored models make strong independence assumptions Can we do better? Rest of lecture NP-hardness of Markovization for non-projective parsing But... projective case has polynomial solutions!! Approximate non-projective algorithms Dependensparsning 2 38(45)

39 Vertical and Horizontal Markovization Dependency graphs weight factors over neighbouring arcs Vertical versus Horizontal neighbourhoods Dependensparsning 2 39(45)

40 N th Order Horizontal Markov Factorization Assume the unlabeled parsing case (adding labels is easy) Weights factor over neighbourhoods of size N Normal (arc-factored = first-order) m k=1 w i0 i k Second-order weights over pairs of adjacent (same side) arcs j 1 w i0 i k i k+1 + w i0 i j + w i0 i j+1 + k=1 m 1 k=j+1 w i0 i k i k+1 Dependensparsning 2 40(45)

41 Projective Horizontal Markovization Can simply augment chart parsing algorithm Same for the Eisner algorithm runtime still O( L n 3 ) Dependensparsning 2 41(45)

42 Approx Non-proj Horizontal Markovization Non-projective second-order parsing is NP-hard Two properties: Projective parsing is polynomial w/ horizontal Markovization Most non-projective graphs are still primarily projective Use these facts to get an approximate non-projective algorithm Find a high scoring projective parse Iteratively modify to create a higher scoring non-projective parse Post-process non-projectivity, which is related to pseudo-projective parsing Dependensparsning 2 42(45)

43 Approx Non-proj Horizontal Markovization Algorithm 1. Let G be the highest weighted projective graph 2. Find the arc (i, j, k) G, a node i and label l k such that G = G {(i, j, k )} {(i, j, k)} is a valid graph (tree) G has highest weight of all possibly changes 3. if w(g ) > w(g) then G = G and return to step 2 4. Otherwise return G Intuition: Start with a high weighted graph and make local changes that increase the graphs weight until convergence Works well in practice [McDonald and Pereira 2006] Dependensparsning 2 43(45)

44 Beyond Arc-Factorization For the non-projective case, increasing scope of weights (and as a result features) makes parsing intractable However, chart parsing nature of projective algorithms allows for simple augmentations Can approximate the non-projective case using the exact projective algorithms plus a post-process optimization Further reading: [McDonald and Pereira 2006, McDonald and Satta 2007] Dependensparsning 2 44(45)

45 Summary Graph-based Methods Arc-factored models Maximum spanning tree formulation Projective and non-projective inference algorithms Partition function and arc expectation algorithms Matrix Tree Theorem Beyond Arc-factored Models Vertical and horizontal markovization Approximations Dependensparsning 2 45(45)

46 References and Further Reading Y. J. Chu and T. H. Liu On the shortest arborescence of a directed graph. Science Sinica, 14: Jack Edmonds Optimum branchings. Journal of Research of the National Bureau of Standards, 71B: Jason M. Eisner Three new probabilistic models for dependency parsing: An exploration. In Proceedings of the 16th International Conference on Computational Linguistics (COLING), pages Ryan McDonald and Fernando Pereira Online learning of approximate dependency parsing algorithms. In Proceedings of the 11th Conference of the European Chapter of the Association for Computational Linguistics (EACL), pages Ryan McDonald and Giorgio Satta On the complexity of non-projective data-driven dependency parsing. In Proceedings of the 10th International Conference on Parsing Technologies (IWPT), pages Ryan McDonald, Koby Crammer, and Fernando Pereira Dependensparsning 2 45(45)

47 Online large-margin training of dependency parsers. In Proceedings of the 43rd Annual Meeting of the Association for Computational Linguistics (ACL), pages Ryan McDonald, Kevin Lerman, and Fernando Pereira Multilingual dependency analysis with a two-stage discriminative parser. In Proceedings of the 10th Conference on Computational Natural Language Learning (CoNLL), pages Joakim Nivre, Johan Hall, and Jens Nilsson Memory-based dependency parsing. In Proceedings of the 8th Conference on Computational Natural Language Learning, pages Joakim Nivre, Johan Hall, Jens Nilsson, Gülsen Eryiugit, and Svetoslav Marinov Labeled pseudo-projective dependency parsing with support vector machines. In Proceedings of the 10th Conference on Computational Natural Language Learning (CoNLL), pages R. E. Tarjan Finding optimum branchings. Networks, 7: Hiroyasu Yamada and Yuji Matsumoto Statistical dependency analysis with support vector machines. In Proceedings of the 8th International Workshop on Parsing Technologies (IWPT), pages Dependensparsning 2 45(45)

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