nips nips2006 nips2006-172 knowledge-graph by maker-knowledge-mining

172 nips-2006-Scalable Discriminative Learning for Natural Language Parsing and Translation

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Author: Joseph Turian, Benjamin Wellington, I. D. Melamed

Abstract: Parsing and translating natural languages can be viewed as problems of predicting tree structures. For machine learning approaches to these predictions, the diversity and high dimensionality of the structures involved mandate very large training sets. This paper presents a purely discriminative learning method that scales up well to problems of this size. Its accuracy was at least as good as other comparable methods on a standard parsing task. To our knowledge, it is the ﬁrst purely discriminative learning algorithm for translation with treestructured models. Unlike other popular methods, this method does not require a great deal of feature engineering a priori, because it performs feature selection over a compound feature space as it learns. Experiments demonstrate the method’s versatility, accuracy, and eﬃciency. Relevant software is freely available at http://nlp.cs.nyu.edu/parser and http://nlp.cs.nyu.edu/GenPar. 1

Reference: text

Summary: the most important sentenses genereted by tfidf model

sentIndex sentText sentNum sentScore

1 edu Computer Science Department New York University New York, New York 10003 Abstract Parsing and translating natural languages can be viewed as problems of predicting tree structures. [sent-4, score-0.267]

2 Its accuracy was at least as good as other comparable methods on a standard parsing task. [sent-7, score-0.272]

3 To our knowledge, it is the ﬁrst purely discriminative learning algorithm for translation with treestructured models. [sent-8, score-0.363]

4 Unlike other popular methods, this method does not require a great deal of feature engineering a priori, because it performs feature selection over a compound feature space as it learns. [sent-9, score-0.424]

5 State of the art methods for both parsing and translation use discriminative methods, but they are still limited by their reliance on generative models that can be estimated relatively cheaply. [sent-20, score-0.632]

6 For example, some parsers and translators use a generative model to generate a list of candidates, and then rerank them using a discriminative reranker (e. [sent-21, score-0.303]

7 Others use a generative model as a feature in a discriminative framework, because otherwise training is impractically slow (Collins & Roark, 2004; Taskar et al. [sent-25, score-0.461]

8 Similarly, the best machine translation (MT) systems use discriminative methods only to calibrate the weights of a handful of diﬀerent knowledge sources, which are either enumerated by hand or learned automatically but not discriminatively (e. [sent-27, score-0.375]

9 A state is correct if it is possible to infer zero or more items to obtain the ﬁnal state that corresponds to the training data tree. [sent-43, score-0.34]

10 The ﬁrst, which we use for our parsing experiments, is to severely restrict the order in which items can be inferred. [sent-50, score-0.37]

11 However, in contrast to traditional context-free parsing algorithms, that computation can involve context-sensitive features. [sent-53, score-0.235]

12 An important design decision in learning the inference cost function cΘ is the choice of feature set. [sent-54, score-0.254]

13 1 The Training Set The training data used for both parsing and translation initially comes in the form of trees. [sent-68, score-0.523]

14 2 These gold-standard trees are used to generate training examples, each of which is a candidate inference: Starting at the initial state, we randomly choose a sequence of correct inferences that lead to the (gold-standard) ﬁnal state. [sent-69, score-0.307]

15 All the candidate inferences that can possibly follow each state in this sequence become part of the training set. [sent-70, score-0.253]

16 We will use X f (i) to refer to the element of 1 2 What counts as a complete tree is problem-speciﬁc. [sent-78, score-0.267]

17 , in parsing, a complete tree is one that covers the input and has a root labeled TOP. [sent-81, score-0.301]

18 The learner then induces compound features, each of which is a conjunction of possibly negated atomic features. [sent-88, score-0.378]

19 Each atomic feature can have one of three values (yes/no/don’t care), so the size of the compound feature space is 3|A| , exponential in the number of atomic features. [sent-89, score-0.672]

20 The tree cost CΘ (Equation 1) is obtained by computing the objective function with y(i) = +1 and b(i) = 1 for every inference in the tree, and treating the penalty term ΩΘ as constant. [sent-98, score-0.42]

21 3 Boosting 1 -Regularized Decision Trees We use an ensemble of conﬁdence-rated decision trees (Schapire & Singer, 1999) to represent hΘ . [sent-105, score-0.242]

22 3 Each internal node is split on an atomic feature. [sent-106, score-0.363]

23 The path from the root to each node n in a decision tree corresponds to a compound feature f, and we write ϕ(n) = f. [sent-107, score-0.777]

24 Each leaf node n keeps track of the parameter value Θϕ(n) . [sent-109, score-0.35]

25 To score an inference i using a decision tree, we percolate the inference down to a leaf n and return conﬁdence Θϕ(n) . [sent-110, score-0.536]

26 The score hΘ (i) given to an inference i by the whole ensemble is the sum of the conﬁdences returned by all trees in the ensemble. [sent-111, score-0.24]

27 We grow the ensemble until the objective cannot be further reduced for the current 3 Turian and Melamed (2005) built more accurate parsers more quickly using decision trees rather than decision stumps, so we build full decision trees. [sent-115, score-0.519]

28 First, we choose some compound features that will allow us to decrease the objective function. [sent-120, score-0.319]

29 We do this by building a decision tree, whose leaf node paths represent the chosen compound features. [sent-121, score-0.645]

30 Second, we conﬁdence-rate each leaf to minimize the objective over the examples that percolate down to that leaf. [sent-122, score-0.357]

31 Finally, we append the decision tree to the ensemble and update parameter vector Θ accordingly. [sent-123, score-0.407]

32 In this manner, compound feature selection is performed incrementally during training, as opposed to a priori. [sent-124, score-0.288]

33 , 2003), extended to work over the compound feature space. [sent-126, score-0.288]

34 The construction of each decision tree begins with a root node, which corresponds to a dummy “always true” feature. [sent-127, score-0.376]

35 Speciﬁcally, we consider splitting each leaf node n using atomic feature a, where ˆ a = arg max GΘ (I; f ∧ a) + GΘ (I; f ∧ ¬a) ˆ (7) a∈A Splitting using a would create children nodes n1 and n2 , with ϕ(n1 ) = f ∧ a and ϕ(n2 ) = f ∧ ¬ˆ . [sent-132, score-0.707]

36 We ˆ ˆ a split node n using a only if the total gain of these two children exceeds the gain of the unsplit node, ˆ i. [sent-133, score-0.311]

37 if: GΘ (I; f ∧ a) + GΘ (I; f ∧ ¬ˆ ) > GΘ (I; f ) ˆ a (8) Otherwise, n remains a leaf node of the decision tree, and Θϕ(n) becomes one of the values to be optimized during the parameter update step. [sent-135, score-0.425]

38 Parameter update is done sequentially on only the most recently added compound features, which correspond to the leaves of the new decision tree. [sent-136, score-0.342]

39 After the entire tree is built, we percolate each example down to its appropriate leaf node. [sent-137, score-0.582]

40 A convenient property of decision trees is that the leaves’ compound features are mutually exclusive, so their parameters can be directly optimized independently of each other. [sent-138, score-0.454]

41 We use a line search to choose for each leaf node n the parameter Θϕ(n) that minimizes the objective over the examples in n. [sent-139, score-0.392]

42 3 Parsing The parsing algorithm starts from an initial state that contains one terminal item per input word, labeled with a part-of-speech (POS) tag by the method of Ratnaparkhi (1996). [sent-140, score-0.402]

43 However, in our experiments, the inference evaluation function was learned accurately enough to guide the parser to the optimal parse reasonably quickly without pruning, and thus without search errors. [sent-143, score-0.355]

44 (2004), we trained and tested a parser using the algorithm in Section 2 on ≤ 15 word sentences from the English Penn Treebank (Taylor et al. [sent-145, score-0.457]

45 Our atomic feature set A contained features of the form “is there an item in group J whose label/headword/headtag/headtagclass is X? [sent-165, score-0.402]

46 (2004) and Turian and Melamed (2005) for their discriminative parsers, which were also trained and tested on ≤ 15 word sentences. [sent-173, score-0.283]

47 4 We also compared our parser to a representative non-discriminative parser (Bikel, 2004)5 , the only one that we were able to train and test under exactly the same experimental conditions, including the use of POS tags from Ratnaparkhi (1996). [sent-174, score-0.417]

48 The accuracy of our parser is at least as high as that of comparable parsers in the literature. [sent-178, score-0.309]

49 By comparison, it took several CPU-months to train the parser of Taskar et al. [sent-182, score-0.253]

50 4 Translation The experiments in this section employed the tree transduction approach to translation, which is used by today’s best MT systems (Marcu et al. [sent-186, score-0.452]

51 To translate by tree transduction, we assume that the input sentence has already been parsed by a parser like the one described in Section 3. [sent-188, score-0.643]

52 The transduction algorithm performs a sequence of inferences to transform this input parse tree into an output parse tree, which has words of the target language in its leaves, often in a diﬀerent order than the corresponding words in the source tree. [sent-189, score-0.867]

53 The words are then read oﬀ the target tree and outputted; the rest of the tree is discarded. [sent-190, score-0.571]

54 Inferences are ordered by their cost, just like in ordinary parsing, and tree transduction stops when each source node has been transduced. [sent-191, score-0.561]

55 From this corpus, we extracted sentence pairs where both sentences had between 5 and 40 words, and where the ratio of their lengths was no more than 2:1. [sent-193, score-0.252]

56 Typical MT systems in the literature are trained on hundreds of thousands of sentence pairs, so our main experiment used 100K sentence pairs of training data. [sent-196, score-0.464]

57 Where noted, preliminary experiments were performed using 10K sentence pairs of training data. [sent-197, score-0.309]

58 We computed parse trees for all the English sentences in all data sets. [sent-198, score-0.249]

59 For each of our two training sets, we induced word alignments using the default conﬁguration of GIZA++ (Och & Ney, 2003). [sent-199, score-0.302]

60 Their parser beat the generative model of Bikel (2004) only after using the output from a generative model as a feature. [sent-202, score-0.361]

61 word alignments and English parse trees were fed into the default French-English hierarchical alignment algorithm distributed with the GenPar system (Burbank et al. [sent-204, score-0.456]

62 Tree alignments are the ideal form of training data for tree transducers, because they fully specify the relation between nodes in the source tree and nodes in the target tree. [sent-206, score-0.822]

63 We experimented with a simplistic tree transducer that involves only two types of inferences. [sent-207, score-0.403]

64 The ﬁrst type transduces words at the leaves of the source tree; the second type transduces internal nodes. [sent-208, score-0.277]

65 To transduce a word w at the leaf, the transducer replaces it with a single word v that is a translation of w. [sent-209, score-0.619]

66 This transducer is grossly inadequate for modeling real translations (Galley et al. [sent-216, score-0.241]

67 One could apply the same learning methods to more sophisticated tree transducers. [sent-220, score-0.267]

68 When inducing leaf transducers using 10K training sentence pairs, there were 819K training inferences and 80. [sent-221, score-0.862]

69 And for inducing internal node transducers using 100K training sentence pairs, there were 1. [sent-225, score-0.586]

70 We parallelized training of the word transducers according to the source and target word pair (w, v). [sent-229, score-0.592]

71 Prior to training, we ﬁltered out word translation examples that were likely to be noise. [sent-230, score-0.331]

72 6K diﬀerent word transducers over 10K training sentence pairs, and 41. [sent-232, score-0.533]

73 We used several kinds of features to evaluate leaf transductions. [sent-234, score-0.328]

74 “Window” features included the source words and part-of-speech (POS) tags within a 2-word window around the word in the leaf (the “focus” word), along with their relative positions (from -2 to +2). [sent-235, score-0.588]

75 The literature on monolingual parsing gives a standard procedure for annotating each node in an English parse tree with its “lexical head word. [sent-238, score-0.753]

76 The features used to evaluate transductions of internal nodes included all those listed for leaf transduction above, where the focus words were the head words of the children of the internal node. [sent-240, score-0.733]

77 Using these features, we applied the method of Section 2 to induce conﬁdence-rating binary classiﬁers for each word pair in the lexicon, and additional binary classiﬁers for predicting the permutations of the children of internal tree nodes. [sent-241, score-0.531]

78 (2005), which learned word transduction classiﬁers using logistic regression with 2 regularization. [sent-243, score-0.271]

79 We induced word transduction classiﬁers over the 10K training data using this e model and our own, and tested them on the development set. [sent-245, score-0.418]

80 7K non-zero compound features over an even smaller number of atomic features. [sent-249, score-0.435]

81 , the highest node that has the focus word as its lexical head; if it is a leaf, then that label is a POS tag. [sent-258, score-0.319]

82 Table 2 Accuracy of tree transducers using 100K sentence pairs of training data. [sent-259, score-0.693]

83 The generative model was a top-down tree transducer (Comon et al. [sent-276, score-0.556]

84 , 1997), which stochastically generates the target tree top-down given the source tree. [sent-277, score-0.329]

85 Our hypothesis was that the discriminative approach would be more accurate than the generative model, because its evaluation of each inference could take into account a greater variety of information in the tree, including its entire yield (string), not just the information in nearby nodes. [sent-281, score-0.291]

86 For eﬃciency, we used a chart to keep track of item costs, and pruned items whose cost was more than 103 times the cost of the least expensive item in the same chart cell. [sent-283, score-0.517]

87 We also pruned items whenever the number of items in the same cell exceeded 40. [sent-284, score-0.27]

88 Our entire tree transduction algorithm was equivalent to bottom-up synchronous parsing (Melamed, 2004) where the source side of the output bi-tree is constrained by the input (source) tree. [sent-285, score-0.683]

89 We compared the generative and discriminative models by reading out the string encoded in their predicted trees, and computing the F-measure between that string and the reference target sentence in the test corpus. [sent-286, score-0.451]

90 The generative transducer achieved its highest F-measure when the input parse trees were computed by the generative parser of Bikel (2004). [sent-294, score-0.694]

91 The discriminatively trained transducer was most accurate when the source trees were computed by the parser in Section 3. [sent-295, score-0.552]

92 Conclusion We have demonstrated how to predict tree structures using binary classiﬁers. [sent-298, score-0.267]

93 These classiﬁers are discriminatively induced by boosting conﬁdence-rated decision trees to minimize the 1 -regularized log-loss. [sent-299, score-0.277]

94 For large problems in tree-structured prediction, such as natural language parsing and translation, this learning algorithm has several attractive properties. [sent-300, score-0.294]

95 It learned a purely discriminative machine over 40 million training examples and 1. [sent-301, score-0.345]

96 1 million atomic features, using no generative model of any kind. [sent-302, score-0.297]

97 The method did not require a great deal of feature engineering a priori, because it performed feature selection over a compound feature space as it learned. [sent-303, score-0.424]

98 In future work, we plan to integrate the parsing and translation methods described in our experiments, to reduce compounded error. [sent-305, score-0.414]

99 Final report on statistical machine translation by parsing (Tech. [sent-331, score-0.414]

100 SPMT: Statistical machine translation with syntactiﬁed target language phrases. [sent-414, score-0.238]

similar papers computed by tfidf model

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Figure 1(a) is a 2-stream MDBN expanded representation of the three models, in this case ℓ = 4 and m = 3. As shown, it appears that the fan-in to node fi will be ℓ and thus will grow without bound. However, a switching mechanism whereby P (fi |e, ai ) = P (fi |eai ) limits the number of parameters regardless of L. This means that the alignment variable ai indicates the English word eai that should be aligned to French word fi . The variable e0 is a null word that connects to French words not explained by any of e1 , . . . , eℓ . The graph expresses all three models — the difference is that, in Models 1 and 2, there are no edges between aj and aj+1 . In Model 1, p(aj = ℓ) is uniform on the set {1, . . . , L}; in Model 2, the distribution over aj is a function only of its position j, and on the English and French lengths ℓ and m respectively. In the M-HMM model, the ai variables form a ﬁrst order Markov chain. l e0 ℓ e1 e3 e2 e1 e4 e2 e3 φ1 φ2 φ3 m’ φ0 τ01 a1 f2 a2 f3 a3 m (a) Models 1,2 and M-HMM τ12 τ13 τ21 π02 π11 π12 π13 π21 f2 f3 f4 f5 f6 a1 u v τ11 f1 f1 τ02 a2 a3 a4 a5 a6 π01 w y x m (b) Expanded M3 graph Figure 1: Expanded 2-stream MDBN description of IBM Models 1 and 2, and the M-HMM model for MT; and the expanded MDBN description of IBM Model 3 with fertility assignment φ0 = 2, φ1 = 3, φ2 = 1, φ3 = 0. From the above, we see that it would be difﬁcult to express this model graphically using a standard DBN since L and M are unequal random variables. Indeed, there are two DBNs in operation, one consisting of the English string, and the other consisting of the French string and its alignment. Moreover, the fully connected structure of the graph in the ﬁgure can represent the appropriate family of model, but it also represents models whose parameter space grows without bound — the switching function allows the model template to stay ﬁnite regardless of L and M . With our MDBN descriptive abilities complete, it is now possible to describe the more complex IBM models 3, and 4[3] (an MDBN for Model3 is depicted in ﬁg. 1(b)). The top most random variable, ℓ, is a hidden switching existence variable corresponding to the length of the English string. The box abutting ℓ includes all the nodes whose existence depends on the value of ℓ. In the ﬁgure, ℓ = 3, thus resulting in three English words e1 , e2 , and e3 connected using a second-order Markov chain. To each English word ei corresponds a conditionally dependent fertility eb-node φi , which indicates how many times ei is used by words in the French string. Each φi in turn controls the existence of a set of variables under it. Given the fertilities (the ﬁgure depicts the case φ1 = 3, φ2 = 1, φ3 = 0), for each word ei , φi French word variables are granted existence and are denoted by τi1 , τi2 , . . . , τiφi , what is called the tablet [3] of ei . The values taken by the τ variables need to match the actual observed French sequence f1 , . . . , fm . This is represented as a shared constraint between all the f , π, and τ variables which have incoming edges into the observed variable v. v’s conditional probability table is such that it is one only when the associated constraint is satisﬁed2 . The variable 2 This type of encoding of constraints corresponds to the standard mechanism used by Pearl [14]. A naive implementation, however, would enumerate a number of conﬁgurations exponential in the number of constrained variables, while typically only a small fraction of the conﬁgurations would have positive probability. πi,k ∈ {1, . . . , m} is a switching dependency parent with respect to the constraint variable v and determines which fj participates in an equality constraint with τi,k . The bottom variable m is a switching existence node (observed to be 6 in the ﬁgure) with corresponding French word sequence and alignment variables. The French sequence participates in the v constraint described above, while the alignment variables aj ∈ {1, . . . , ℓ}, j ∈ 1, . . . , m constrain the fertilities to take their unique allowable values (for the given alignment). Alignments also restrict the domain of permutation variables, π, using the constraint variable x. Finally, the domain size of each aj has to lie in the interval [0, ℓ] and that is enforced by the variable u. The dashed edges connecting the alignment a variables represent an extension to implement an M3/M-HMM hybrid. ℓ The null submodel involving the deterministic node m′ (= i=1 φi ) and eb-node φ0 accounts for French words that are not explained by any of the English words e1 , . . . , eℓ . In this submodel, successive permutation variables are ordered and this constraint is implemented using the observed child w of π0i and π0(i+1) . Model 4 [3] is similar to Model 3 except that the former is based on a more elaborate distortion model that uses relative instead of absolute positions both within and between tablets. 3 Inference, Parameter Estimation and MPE Multi-dynamic Bayesian Networks are amenable to any type of inference that is applicable to regular Bayesian networks as long as switching existence relationships are respected and all the constraints (aggregation for example) are satisﬁed. Unfortunately DBN inference procedures that take advantage of the repeatable template and can preprocess it ofﬂine, are not easy to apply to MDBNs. A case in point is the Junction Tree algorithm [11]. Triangulation algorithms exist that create an ofﬂine triangulated version of the input graph and do not re-triangulate it for each different instance of the input data [12, 2]. In MDBNs, due to the ﬂexibility to unroll templates in several dimensions and to specify dependencies and constraints spanning the entire unrolled graph, it is not obvious how we can exploit any repetitive patterns in a Junction Tree-style ofﬂine triangulation of the graph template. In section 4, we discuss sampling inference methods we have used. Here we discuss our extension to a backtracking search algorithm with the same performance guarantees as the JT algorithm, but with the advantage of easily handling determinism, existence uncertainty, and constraints, both learned and explicitly stated. Value Elimination (VE) ([1]), is a backtracking Bayesian network inference technique that caches factors associated with portions of the search tree and uses them to avoid iterating again over the same subtrees. We follow the notation introduced in [1] and refer the reader to that paper for details about VE inference. We have extended the VE inference approach to handle explicitly encoded constraints, existence uncertainty, and to perform approximate local domain pruning (see section 4). We omit these details as well as others in the original paper and brieﬂy describe the main data structure required by VE and sketch the algorithm we refer to as FirstPass (ﬁg. 1) since it constitutes the ﬁrst step of the learning procedure, our main contribution in this section. A VE factor, F , is such that we can write the following marginal of the joint distribution P (X = x, Y = y, Z) = F.val × f (Z) X=x such that (X∪Y)∩Z = ∅, F.val is a constant, and f (Z) a function of Z only. Y is a set of variables previously instantiated in the current branch of search tree to the value vector y. The pair (Y, y) is referred to as a dependency set (F.Dset). X is referred to as a subsumed set (F.Sset). By caching the tuple (F.Dset, F.Sset, F.val), we avoid recomputing the marginal again whenever (1) F.Dset is active, meaning all nodes stored in F.Dset are assigned their cached values in the current branch of the search tree; and (2) none of the variables in F.Sset are assigned yet. FirstPass (alg. 1) visits nodes in the graph in Depth First fashion. In line 7, we get the values of all Newly Single-valued (NSV) CPTs i.e., CPTs that involve the current node, V , and in which all We use a general directed domain pruning constraint. Deterministic relationships then become a special case of our constraint whereby the domain of the child variable is constrained to a single value with probability one. Variable traversal order: A, B, C, and D. Factors are numbered by order of creation. *Fi denotes the activation of factor i. Tau values propagated recursively F7: Dset={} Sset={A,B,C,D} val=P(E=e) F7.tau = 1.0 = P(Evidence)/F7.val A F5: Dset={A=0} Sset={B,C,D} F2 D *F1 *F2 Factor values needed for c(A=0) and c(C=0,B=0) computation: F5.val=P(B=0|A=0)*F3.val+P(B=1|A=0)*F4.val F3.val=P(C=0|B=0)*F1.val+P(C=1|B=0)*F2.val F4.val=P(C=0|B=1)*F1.val+P(C=1|B=1)*F2.val F1.val=P(D=0|C=0)P(E=e|D=0)+P(D=1|C=0)P(E=e|D=1) F2.val=P(D=0|C=1)P(E=e|D=0)+P(D=1|C=1)P(E=e|D=1) First pass C *F3 *F4 Second pass D B F4 C F6.tau = F7.tau * P(A=1) 1 B F3: Dset={B=0} Sset={C,D} F1 F5.tau = F7.tau * P(A=0) F6 0 F3.tau = F5.tau * P(B=0|A=0) + F6.tau * P(B=0|A=1) = P(B=0) F4.tau = F5.tau * P(B=1|A=0) + F6.tau * P(B=1|A=1) = P(B=1) F1.tau = F3.tau * P(C=0|B=0) + F4.tau * P(C=0|B=1) = P(C=0) F2.tau = F3.tau * P(C=1|B=0) + F4.tau * P(C=1|B=1) = P(C=1) c(A=0)=(1/P(e))*(F7.tau*P(A=0)*F5.val)=(1/P(e))(P(A=0)*P(E=e|A=0))=P(A=0|E=e) c(C=0,B=0)=(1/P(e))*F3.tau*P(C=0|B=0)*F1.val =(1/P(e) * (P(A=0,B=0)+P(A=1,B=0)) * P(C=0|B=0) * F1.val =(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val =(1/P(e)) * P(B=0) * P(C=0|B=0) * F1.val =(1/P(e)) * P(C=0,B=0) * F1.val =P(C=0,B=0,E=e)/P(e)=P(C=0,B=0|E=e) Figure 2: Learning example using the Markov chain A → B → C → D → E, where E is observed. In the ﬁrst pass, factors (Dset, Sset and val) are learned in a bottom up fashion. Also, the normalization constant P (E = e) (probability of evidence) is obtained. In the second pass, tau values are updated in a top-down fashion and used to calculate expected counts c(F.head, pa(F.head)) corresponding to each F.head (the ﬁgure shows the derivations for (A=0) and (C=0,B=0), but all counts are updated in the same pass). other variables are already assigned (these variables and their values are accumulated into Dset). We also check for factors that are active, multiply their values in, and accumulate subsumed vars in Sset (to avoid branching on them). In line 10, we add V to the Sset. In line 11, we cache a new factor F with value F.val = sum. We store V into F.head, a pointer to the last variable to be inserted into F.Sset, and needed for parameter estimation described below. F.Dset consists of all the variables, except V , that appeared in any NSV CPT or the Dset of an activated factor at line 6. Regular Value Elimination is query-based, similar to variable elimination and recursive conditioning—what this means is that to answer a query of the type P (Q|E = e), where Q is query variable and E a set of evidence nodes, we force Q to be at the top of the search tree, run the backtracking algorithm and then read the answers to the queries P (Q = q|E = e), q ∈ Dom[Q], along each of the outgoing edges of Q. Parameter estimation would require running a number of queries on the order of the number of parameters to estimate. We extend VE into an algorithm that allows us to obtain Expectation Maximization sufﬁcient statistics in a single run of Value Elimination plus a second pass, which can never take longer than the ﬁrst one (and in practice is much faster). This two-pass procedure is analogous to the collect-distribute evidence procedure in the Junction Tree algorithm, but here we do this via a search tree. Let θX=x|pa(X)=y be a parameter associated with variable X with value x and parents Y = pa(X) when they have value y. Assuming a maximum likelihood learning scenario3 , to estimate θX=x|pa(X)=y , we need to compute f (X = x, pa(X) = y, E = e) = P (W, X = x, pa(X) = y, E = e) W\{X,pa(X)} which is a sum of joint probabilities of all conﬁgurations that are consistent with the assignment {X = x, pa(X) = y}. If we were to turn off factor caching, we would enumerate all such variable conﬁgurations and could compute the sum. When standard VE factors are used, however, this is no longer possible whenever X or any of its parents becomes subsumed. Fig. 2 illustrates an example of a VE tree and the factors that are learned in the case of a Markov chain with an evidence node at the end. We can readily estimate the parameters associated with variables A and B as they are not subsumed along any branch. C and D become subsumed, however, and we cannot obtain the correct counts along all the branches that would lead to C and D in the full enumeration case. To address this issue, we store a special value, F.tau, in each factor. F.tau holds the sum over all path probabilities from the ﬁrst level of the search tree to the level at which the factor F was 3 For Bayesian networks the likelihood function decomposes such that maximizing the expectation of the complete likelihood is equivalent to maximizing the “local likelihood” of each variable in the network. either created or activated. For example, F 6.tau in ﬁg. 2 is simply P (A = 1). Although we can compute F 3.tau directly, we can also compute it recursively using F 5.tau and F 6.tau as shown in the ﬁgure. This is because both F 5 and F 6 subsume F 3: in the context {F 5.Dset}, there exists a (unique) value dsub of F 5.head4 s.t. F 3 becomes activable. Likewise for F 6. We cannot compute F 1.tau directly, but we can, recursively, from F 3.tau and F 4.tau by taking advantage of a similar subsumption relationship. In general, we can show that the following recursive relationship holds: F pa .tau × N SVF pa .head=dsub × F.tau ← F pa ∈F pa Fact .val F.val Fact ∈Fact (1) where F pa is the set of factors that subsume F , Fact is the set of all factors (including F ) that become active in the context of {F pa .Dset, F pa .head = dsub } and N SVF pa .head=dsub is the product of all newly single valued CPTs under the same context. For top-level factors (not subsumed by any factor), F.tau = Pevidence /F.val, which is 1.0 when there is a unique top-level factor. Alg. 2 is a simple recursive computation of eq. 1 for each factor. We visit learned factors in the reverse order in which they were learned to ensure that, for any factor F ′ , F ′ .tau is incremented (line 13) by any F that might have activated F ′ (line 12). For example, in ﬁg. 2, F 4 uses F 1 and F 2, so F 4.tau needs to be updated before F 1.tau and F 2.tau. In line 11, we can increment the counts for any NSV CPT entries since F.tau will account for the possible ways of reaching the conﬁguration {F.Dset, F.head = d} in an equivalent full enumeration tree. Algorithm 1: FirstPass(level) 1 2 3 4 5 6 7 8 9 10 Input: Graph G Output: A list of learned factors and Pevidence Select var V to branch on if V ==NONE then return Sset={}, Dset={} for d ∈ Dom[V ] do V ←d prod = productOfAllNSVsAndActiveFactors(Dset, Sset) if prod != 0 then FirstPass(level+1) sum += prod Sset = Sset ∪ {V } cacheNewFactor(F.head ← V ,F.val ← sum, F.Sset ← Sset, F.Dset ← Dset); Algorithm 2: SecondPass() 1 2 3 4 5 6 7 8 9 10 11 12 13 Input: F : List of factors in the reverse order learned in the ﬁrst pass and Pevidence . Result: Updated counts foreach F ∈ F do if F.Dset = {} then F.tau ← Pevidence /F.val else F.tau ← 0.0 Assign vars in F.Dset to their values V ← F.head (last node to have been subsumed in this factor) foreach d ∈ Dom[V ] do prod = productOfAllNSVsAndActiveFactors() prod∗ = F.tau foreach newly single-valued CPT C do count(C.child,C.parents)+=prod/Pevidence F ′ =getListOfActiveFactors() for F ′ ∈ F ′ do F ′ .tau+ = prod/F ′ .val Most Probable Explanation We compute MPE using a very similar two-pass algorithm. In the ﬁrst pass, factors are used to store a maximum instead of a summation over variables in the Sset. We also keep track of the value of F.head at which the maximum is achieved. In the second pass, we recursively ﬁnd the optimal variable conﬁguration by following the trail of factors that are activated when we assign each F.head variable to its maximum value starting from the last learned factor. 4 Recall, F.head is the last variable to be added to a newly created factor in line 10 of alg. 1 4 MACHINE TRANSLATION WORD ALIGNMENT EXPERIMENTS A major motivation for pursuing the type of representation and inference described above is to make it possible to solve computationally-intensive real-world problems using large amounts of data, while retaining the full generality and expressiveness afforded by the MDBN modeling language. In the experiments below we compare running times of MDBNs to GIZA++ on IBM Models 1 through 4 and the M-HMM model. GIZA++ is a special-purpose optimized MT word alignment C++ tool that is widely used in current state-of-the-art phrase-based MT systems [10] and at the time of this writing is the only publicly available software that implements all of the IBM Models. We test on French-English 107 hand-aligned sentences5 from a corpus of the European parliament proceedings (Europarl [9]) and train on 10000 sentence pairs from the same corpus and of maximum number of words 40. The Alignment Error Rate (AER) [13] evaluation metric quantiﬁes how well the MPE assignment to the hidden alignment variables matches human-generated alignments. Several pruning and smoothing techniques are used by GIZA and MDBNs. GIZA prunes low lexical (P (f |e)) probability values and uses a default small value for unseen (or pruned) probability table entries. For models 3 and 4, for which there is no known polynomial time algorithm to perform the full E-step or compute MPE, GIZA generates a set of high probability alignments using an MHMM and hill-climbing and collects EM counts over these alignments using M3 or M4. For MDBN models we use the following pruning strategy: at each level of the search tree we prune values which, together, account for the lowest speciﬁed percentage of the total probability mass of the product of all newly active CPTs in line 6 of alg. 1. This is a more effective pruning than simply removing low-probability values of each CPD because it factors in the joint contribution of multiple active variables. Table 1 shows a comparison of timing numbers obtained GIZA++ and MDBNs. The runtime numbers shown are for the combined tasks of training and decoding; however, training time dominates given the difference in size between train and test sets. For models 1 and 2 neither GIZA nor MDBNs perform any pruning. For the M-HMM, we prune 60% of probability mass at each level and use a Dirichlet prior over the alignment variables such that long-range transitions are exponentially less likely than shorter ones.6 This model achieves similar times and AER to GIZA’s. Interestingly, without any pruning, the MDBN M-HMM takes 160 minutes to complete while only marginally improving upon the pruned model. Experimenting with several pruning thresholds, we found that AER would worsen much more slowly than runtime decreases. Models 3 and 4 have treewidth equal to the number of alignment variables (because of the global constraints tying them) and therefore require approximate inference. Using Model 3, and a drastic pruning threshold that only keeps the value with the top probability at each level, we were able to achieve an AER not much higher than GIZA’s. For M4, it achieves a best AER of 31.7% while we do not improve upon Model3, most likely because a too restrictive pruning. Nevertheless, a simple variation on Model3 in the MDBN framework achieves a lower AER than our regular M3 (with pruning still the same). The M3-HMM hybrid model combines the Markov alignment dependencies from the M-HMM model with the fertility model of M3. MCMC Inference Sampling is widely used for inference in high-treewidth models. Although MDBNs support Likelihood Weighing, it is very inefﬁcient when the probability of evidence is very small, as is the case in our MT models. Besides being slow, Markov chain Monte Carlo can be problematic when the joint distribution is not positive everywhere, in particular in the presence of determinism and hard constraints. Techniques such as blocking Gibbs sampling [8] try to address the problem. Often, however, one has to carefully choose a problem-dependent proposal distribution. We used MCMC to improve training of the M3-HMM model. We were able to achieve an AER of 32.8% (down from 39.1%) but using 400 minutes of uniprocessor time. 5 CONCLUSION The existing classes of graphical models are not ideally suited for representing SMT models because “natural” semantics for specifying the latter combine ﬂavors of different GM types on top of standard directed Bayesian network semantics: switching parents found in Bayesian Multinets [6], aggregation relationships such as in Probabilistic Relational Models [5], and existence uncertainty [7]. We 5 Available at http://www.cs.washington.edu/homes/karim French and English have similar word orders. On a different language pair, a different prior might be more appropriate. With a uniform prior, the MDBN M-HMM has 36.0% AER. 6 Model Init M1 M2 M-HMM M3 M4 M3-HMM GIZA++ M1 M-HMM 1m45s (47.7%) N/A 2m02s (41.3%) N/A 4m05s (35.0%) N/A 2m50 (45%) 5m20s (38.5%) 5m20s (34.8%) 7m45s (31.7%) N/A MDBN M1 3m20s (48.0%) 5m30s (41.0%) 4m15s (33.0%) 12m (43.6%) 25m (43.6%) 9m30 (41.0%) M-HMM N/A N/A N/A 9m (42.5%) 23m (42.6%) 9m15s (39.1%) MCMC 400m (32.8%) Table 1: MDBN VE-based learning versus GIZA++ timings and %AER using 5 EM iterations. The columns M1 and M-HMM correspond to the model that is used to initialize the model in the corresponding row. The last row is a hybrid Model3-HMM model that we implemented using MDBNs and is not expressible using GIZA. have introduced a generalization of dynamic Bayesian networks to easily and concisely build models consisting of varying-length parallel asynchronous and interacting data streams. We have shown that our framework is useful for expressing various statistical machine translation models. We have also introduced new parameter estimation and decoding algorithms using exact and approximate searchbased probability computation. While our timing results are not yet as fast as a hand-optimized C++ program on the equivalent model, we have shown that even in this general-purpose framework of MDBNs, our timing numbers are competitive and usable. Our framework can of course do much more than the IBM and HMM models. One of our goals is to use this framework to rapidly prototype novel MT systems and develop methods to statistically induce an interlingua. We also intend to use MDBNs in other domains such as multi-party social interaction analysis. References [1] F. Bacchus, S. Dalmao, and T. Pitassi. Value elimination: Bayesian inference via backtracking search. In UAI-03, pages 20–28, San Francisco, CA, 2003. Morgan Kaufmann. [2] J. Bilmes and C. Bartels. On triangulating dynamic graphical models. In Uncertainty in Artiﬁcial Intelligence: Proceedings of the 19th Conference, pages 47–56. Morgan Kaufmann, 2003. [3] P. F. Brown, J. Cocke, S. A. Della Piettra, V. J. Della Piettra, F. Jelinek, J. D. Lafferty, R. L. Mercer, and P. S. Roossin. A statistical approach to machine translation. Computational Linguistics, 16(2):79–85, June 1990. [4] T. Dean and K. Kanazawa. Probabilistic temporal reasoning. AAAI, pages 524–528, 1988. [5] N. Friedman, L. Getoor, D. Koller, and A. Pfeffer. Learning probabilistic relational models. In IJCAI, pages 1300–1309, 1999. [6] D. Geiger and D. Heckerman. Knowledge representation and inference in similarity networks and Bayesian multinets. Artif. Intell., 82(1-2):45–74, 1996. [7] L. Getoor, N. Friedman, D. Koller, and B. Taskar. Learning probabilistic models of link structure. Journal of Machine Learning Research, 3(4-5):697–707, May 2003. [8] C. Jensen, A. Kong, and U. Kjaerulff. Blocking Gibbs sampling in very large probabilistic expert systems. In International Journal of Human Computer Studies. Special Issue on Real-World Applications of Uncertain Reasoning., 1995. [9] P. Koehn. Europarl: A multilingual corpus for evaluation of machine http://www.isi.edu/koehn/publications/europarl, 2002. translation. [10] P. Koehn, F. Och, and D. Marcu. Statistical phrase-based translation. In NAACL/HLT 2003, 2003. [11] S. Lauritzen. Graphical Models. Oxford Science Publications, 1996. [12] K. Murphy. Dynamic Bayesian Networks: Representation, Inference and Learning. PhD thesis, U.C. Berkeley, Dept. of EECS, CS Division, 2002. [13] F. J. Och and H. Ney. Improved statistical alignment models. In ACL, pages 440–447, Oct 2000. [14] J. Pearl. Probabilistic Reasoning in Intelligent Systems: Networks of Plausible Inference. Morgan Kaufmann, 2nd printing edition, 1988. [15] S. Vogel, H. Ney, and C. Tillmann. HMM-based word alignment in statistical translation. 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