emnlp emnlp2012 emnlp2012-119 knowledge-graph by maker-knowledge-mining
Source: pdf
Author: Paramveer Dhillon ; Jordan Rodu ; Michael Collins ; Dean Foster ; Lyle Ungar
Abstract: Recently there has been substantial interest in using spectral methods to learn generative sequence models like HMMs. Spectral methods are attractive as they provide globally consistent estimates of the model parameters and are very fast and scalable, unlike EM methods, which can get stuck in local minima. In this paper, we present a novel extension of this class of spectral methods to learn dependency tree structures. We propose a simple yet powerful latent variable generative model for dependency parsing, and a spectral learning method to efficiently estimate it. As a pilot experimental evaluation, we use the spectral tree probabilities estimated by our model to re-rank the outputs of a near state-of-theart parser. Our approach gives us a moderate reduction in error of up to 4.6% over the baseline re-ranker. .
Reference: text
sentIndex sentText sentNum sentScore
1 iend dsu u@} }c, ,s Abstract Recently there has been substantial interest in using spectral methods to learn generative sequence models like HMMs. [sent-14, score-0.709]
2 Spectral methods are attractive as they provide globally consistent estimates of the model parameters and are very fast and scalable, unlike EM methods, which can get stuck in local minima. [sent-15, score-0.151]
3 In this paper, we present a novel extension of this class of spectral methods to learn dependency tree structures. [sent-16, score-0.822]
4 We propose a simple yet powerful latent variable generative model for dependency parsing, and a spectral learning method to efficiently estimate it. [sent-17, score-0.966]
5 As a pilot experimental evaluation, we use the spectral tree probabilities estimated by our model to re-rank the outputs of a near state-of-theart parser. [sent-18, score-0.754]
6 Adding latent variables to these models gives us additional modeling power and have shown success in applications like POS tagging (Merialdo, 1994), speech recognition (Rabiner, 1989) and object recognition (Quattoni et al. [sent-23, score-0.11]
7 (2008) has shown that globally consistent estimates of the parameters of HMMs can be found by using spectral methods, particularly by singular value decomposition (SVD) of appropriately defined linear systems. [sent-29, score-0.733]
8 Besides ducking the NP hard problem, the spectral methods are very fast and scalable to train compared to EM methods. [sent-34, score-0.631]
9 (2008) to learn dependency tree structures with latent variables. [sent-36, score-0.264]
10 (2006) and Musillo and Merlo (2008) have shown that learning PCFGs and dependency grammars respectively with latent variables can produce parsers with very good generalization performance. [sent-38, score-0.285]
11 However, both these approaches rely on EM for parameter estimation and can benefit from using spectral methods. [sent-39, score-0.688]
12 We propose a simple yet powerful latent variable generative model for use with dependency pars1Actually, instead of using the model by Hsu et al. [sent-40, score-0.335]
13 lc L2a0n1g2ua Agseso Pcrioactieosnsi fnogr a Cnodm Cpoumtaptiuotna tilo Lnianlg Nuaist uircasl ing which has one hidden node for each word in the sentence, like the one shown in Figure 1 and work out the details for the parameter estimation of the corresponding spectral learning model. [sent-45, score-0.806]
14 At a very high level, the parameter estimation of our model involves collecting unigram, bigram and trigram counts sensitive to the underlying dependency structure of the given sentence. [sent-46, score-0.198]
15 (2012) have also proposed a spectral method for dependency parsing, however they deal with horizontal markovization and use hidden states to model sequential dependencies within a word’s sequence of children. [sent-48, score-0.939]
16 In contrast with that, in this paper, we propose a spectral learning algorithm where latent states are not restricted to HMM-like distributions of modifier sequences for a particular head, but instead allow information to be propagated through the entire tree. [sent-49, score-0.752]
17 (2012) have proposed a spectral method for learning PCFGs. [sent-51, score-0.631]
18 (2008) to latent variable dependency trees like us but under the restrictive conditions that model parameters are trained for a specified, albeit arbitrary, tree topology. [sent-54, score-0.415]
19 2 In other words, all training sentences and test sentences must have identical tree topologies. [sent-55, score-0.084]
20 By doing this they allow for node-specific model parameters, but must retrain the model entirely when a different tree topology is encountered. [sent-56, score-0.084]
21 Our model on the other hand allows the flexibility and efficiency of processing sentences with a variety of tree topologies from a single training run. [sent-57, score-0.126]
22 Most of the current state-of-the-art dependency parsers are discriminative parsers (Koo et al. [sent-58, score-0.175]
23 Also, as is common in statistical parsing, re-ranking the outputs of a parser leads to significant reductions in error (Collins and Koo, 2005). [sent-61, score-0.085]
24 Since our spectral learning algorithm uses a gen- 2This can be useful in modeling phylogeny trees for instance, but precludes most NLP applications, since there is a need to model the full set of different tree topologies possible in parsing. [sent-62, score-0.797]
25 206 Figure 1: Sample dependency parsing tree for “Kilroy was here” erative model of words given a tree structure, it can score a tree structure i. [sent-63, score-0.407]
26 In the next section we introduce the notation and give a brief overview of the spectral algorithm for learning HMMs (Hsu et al. [sent-68, score-0.665]
27 In Section 3 we describe our proposed model for dependency parsing in detail and work out the theory behind it. [sent-71, score-0.155]
28 2 Spectral Algorithm For Learning HMMs In this section we describe the spectral algorithm for learning HMMs. [sent-74, score-0.631]
29 1 Notation The HMM that we consider in this section is a sequence of hidden states h ∈ {1, . [sent-76, score-0.201]
30 (2008), but does further dimensionality reduction and thus has lower sample complexity. [sent-93, score-0.071]
31 The parameters of this HMM are: • • • A vector π of length k where πi = p(h1 = i) : TAh vee probability onfgtthhe k kst warht estreate π in the sequence being i. [sent-95, score-0.131]
32 A matrix T of size k k where Ti,j = p(ht+1 = ie |ht = j): Terhee probability of transitioning to= =st ia|the i, given that the previous state was j. [sent-96, score-0.125]
33 A matrix O of size n k where Oi,j = p(x = i|h = j): hTehree probability of state h= emitting o ib|hser =va tjio):n x. [sent-97, score-0.125]
34 Define δj to be the vector of length n with a 1in the jth entry and 0 everywhere else, and diag(v) to be the matrix with the entries of v on the diagonal and 0 everywhere else. [sent-98, score-0.22]
35 , hm) mY−1 = πh1 Ym Y Thj,hj−1 YOxj,hj Yj=2 Yj=1 Now, we can write the marginal probability of a sequence of observations as p(x1, . [sent-111, score-0.183]
36 ,hm which can be expressed in matrix form4 as: p(x1, . [sent-123, score-0.078]
37 Since Axm depends on the hidden state, it is not observable, and hence cannot be directly estimated. [sent-128, score-0.118]
38 4This is essentially the matrix form of the standard dynamic program (forward algorithm) used to estimate HMMs. [sent-129, score-0.078]
39 (2012) showed that under certain conditions there exists a fully observable representation of the observable operator model. [sent-132, score-0.196]
40 2 Fully observable representation Before presenting the model, we need to address a few more points. [sent-134, score-0.098]
41 (2012) discuss U in more detail, but U can, for example, be obtained by the SVD of the bigram probability matrix (where µ Pij = p(xt+1 = i = j)) or by doing CCA on |xt neighboring n-grams (Dhillon et al. [sent-138, score-0.192]
42 ˆp 3 Spectral Algorithm For Learning Dependency Trees In this section, we first describe a simple latent variable generative model for dependency parsing. [sent-151, score-0.285]
43 We then define some extra notation and finally present the details of the corresponding spectral learning algorithm for dependency parsing, and prove that our learning algorithm provides a consistent estimation of the marginal probabilities. [sent-152, score-0.93]
44 It is worth mentioning that an alternate way of approaching the spectral estimation of latent states for dependency parsing is by converting the dependency trees into linear sequences from root-to-leaf and doing a spectral estimation of latent states using Hsu et al. [sent-153, score-1.92]
45 However, this approach would not give us the correct probability distribution over trees as the probability calculations for different paths through the trees are not independent. [sent-155, score-0.174]
46 Thus, although one could calculate the probability of a path from the root to a leaf, one cannot generalize from this probability to say anything about the neighboring nodes or words. [sent-156, score-0.127]
47 Put another way, when a parent has more than the one descendant, one has to be careful to take into account that the hidden variables at each child node are all conditioned on the hidden variable of the parent. [sent-157, score-0.456]
48 1 A latent variable generative model for dependency parsing In the standard setting, we are given training examples where each training example consists of a sequence of words x1, . [sent-159, score-0.368]
49 , xm together with a dependency structure over those words, and we want to estimate the probability of the observed structure. [sent-162, score-0.315]
50 This marginal probability estimates can then be used to build an actual generative dependency parser or, since the marginal probability is conditioned on the tree structure, it can be used re-rank the outputs of a parser. [sent-163, score-0.668]
51 As in the conventional HMM described in the previous section, we can define a simple latent variable first order dependency parsing model by introducing a hidden variable hi for each word xi. [sent-164, score-0.47]
52 The joint probability of a sequence of observed nodes x1, . [sent-165, score-0.082]
53 , hm) Ym Ym = πh1 Ytd(j)(hj|hpa(j))Yo(xj|hj) Yj=2 Yj=1 (2) 208 Figure 2: Dependency parsing tree with observed variables y1, y2, and y3. [sent-177, score-0.169]
54 where pa(j) is the parent of node j and d(j) ∈ {L, R} pina(djic)at ises whehe pthareern hj ifs a oldefet or a right jn)od ∈e {ofL hpa(j) . [sent-178, score-0.115]
55 dFicora simplicity, t hhe number of hidden and observed nodes in our tree are the same, however they are not required to be so. [sent-179, score-0.202]
56 As is the case with the conventional HMM, the parameters used to calculate this joint probability are unobservable, but it turns out that under suitable conditions a fully observable model is also possible for the dependency tree case with the parameterization as described below. [sent-180, score-0.385]
57 2 Model parameters We will define both the theoretical representations of our observable parameters, and the sampling versions of these parameters. [sent-182, score-0.147]
58 Define Td and Tdu where d ∈ {L, R} to be the hidden state transitionw mhearteric des ∈ ∈fr {omL, parent btoe l tehfet or right child, and from left or right child to parent (hence the u for ‘up’), respectively. [sent-184, score-0.338]
59 Further, recall the notation diag(v), which is a matrix with elements of v on its diagonal, then: Define the k-dimensional vector counts): • (unigram = Gπ = Xn [ µˆ]i X ¯c(u)Uu(i) Xu=1 cN(u1), where c(u) = c(u) is the count of observation u in the training sample, and N1 = Pu∈nc(u). [sent-186, score-0.112]
60 To see this, let hch(i) be the set of hidden children of hidden node i (in Figure 2 for instance, hch(1) = {2, 3}) and let och(i) be the set of obshecrhve(1d) )c h=ild {re2,n3 o}f) h aidndde lne tn oodche ii (in eth teh same figure och(i) = {1}). [sent-195, score-0.236]
61 , xm) nfr comom Equation m2 as ri(h) = Y αj(h) j∈Yoch(i) o(xj|h) Y j∈Yhch(i) (3) where αi (h) is defined by summing over all the hidden random variables i. [sent-199, score-0.155]
62 This can ) bre written in a compact matrix form as →ri> 1> Y = diag(Td>jr −→j) j∈Yhch(i) · Y diag(O>δxj) (4) j∈Yoch(i) 5Note than ΩR = ΩLT, which is not immediately obvious from the matrix representations. [sent-202, score-0.156]
63 6The details of the derivation follow directly from the matrix versions of the variables. [sent-203, score-0.078]
64 where →ri is a vector of size k (the dimensionality of the hidden space) of values ri(h). [sent-204, score-0.151]
65 Note that since in Equation 2 we condition on whether xj is the left or right child of its parent, we have separate transition matrices for left and right transitions from a given hidden node dj ∈ {L, R}. [sent-205, score-0.398]
66 The recursive computation can be written in terms of observables as: →ri>= c>∞j∈hYch(i)D(Ed>jr −→j) Y D((U>U)−1U>δxj) · j∈Yoch(i) The final calculation for the marginal probability of a given sequence is pˆ(x1, . [sent-206, score-0.231]
67 ,xm) = r→1>c1 (5) The spectral estimation procedure is described below in Algorithm 1. [sent-209, score-0.688]
68 h, sequence x(i) = 2: Compute the spectral parameters µˆ, ΩˆL, and Kˆ #Now, for a given sentence, we can recursively compute the following: 3: for for j ∈ {mi, . [sent-217, score-0.715]
69 ,xmi) = r→1>c1 #The marginal probability of an entire tree. [sent-227, score-0.148]
70 4 Sample complexity Our main theoretical result states that the above scheme for spectral estimation of marginal probabilities provides a guaranteed consistent estimation scheme for the marginal probabilities: 210 Theorem 3. [sent-229, score-0.995]
71 The proof with directional transition parameters is almost identical. [sent-261, score-0.192]
72 4 Experimental Evaluation Since our algorithm can score any given tree structure by computing its marginal probability, a natural way to benchmark our parser is to generate nbest dependency trees using some standard parser and then use our algorithm to re-rank the candidate dependency trees, e. [sent-262, score-0.531]
73 using the log spectral probability as described in Algorithm 1 as a feature in a discriminative re-ranker. [sent-264, score-0.678]
74 1 Experimental Setup Our base parser was the discriminatively trained MSTParser (McDonald, 2006), which implements both first and second order parsers and is trained using MIRA (Crammer et al. [sent-266, score-0.08]
75 We used the PennConverter7 tool to convert Penn Treebank from constituent to dependency format. [sent-273, score-0.107]
76 2 Details of spectral learning For the spectral learning phase, we need to just collect word counts from the training data as described above, so there are no tunable parameters as such. [sent-282, score-1.311]
77 Using k = 10 we were able to estimate our spectral learning parameters ΣL,R, ΩL,R, K from the entire training data in under 2 minutes on a 64 bit Intel 2. [sent-291, score-0.68]
78 3 Re-ranking the outputs of MST parser We could not find any previous work which describes features for discriminative re-ranking for dependency parsing, which is due to the fact that unlike constituency parsing, the base parsers for depen- dency parsing are discriminative (e. [sent-294, score-0.274]
79 However, parse re-ranking is a good testbed for our spectral dependency parser which can score a given tree. [sent-297, score-0.784]
80 Accuracy is the number of words which correctly identified their parent and Complete is the number of sentences for which the entire dependency tree was correct. [sent-310, score-0.25]
81 5 Discussion and Future Work Spectral learning of structured latent variable models in general is a promising direction as has been shown by the recent interest in this area. [sent-328, score-0.135]
82 It allows us to circumvent the ubiquitous problem of getting stuck in local minima when estimating the latent variable models via EM. [sent-329, score-0.184]
83 In this paper we ex8One might be able to come up with better features for dependency parse re-ranking. [sent-330, score-0.107]
84 tended the spectral learning ideas to learn a simple yet powerful dependency parser. [sent-332, score-0.788]
85 As future work, we are working on building an end-to-end parser which would involve coming up with a spectral version of the inside-outside algorithm for our setting. [sent-333, score-0.677]
86 We are also working on extending it to learn more powerful grammars e. [sent-334, score-0.084]
87 6 Conclusion In this paper we proposed a novel spectral method for dependency parsing. [sent-337, score-0.738]
88 Unlike EM trained generative latent variable models, our method does not get stuck in local optima, it gives consistent parameter estimates, and it is extremely fast to train. [sent-338, score-0.227]
89 We worked out the theory of a simple yet powerful generative model and showed how it can be learned us- ing a spectral method. [sent-339, score-0.724]
90 As a pilot experimental evaluation we showed the efficacy of our approach by using the spectral probabilities output by our model for re-ranking the outputs of MST parser. [sent-340, score-0.67]
91 7 Appendix This appendix offers a sketch of the proof of Theorem 1. [sent-343, score-0.164]
92 The proof uses the following definitions, which are slightly modified from those of Foster et al. [sent-344, score-0.108]
93 The proof relies on the fact that a row vector mul- tiplied by a series of matrices, and finally multiplied by a column vector amounts to a sum over all possible products of individual entries in the vectors and matrices. [sent-351, score-0.148]
94 With this in mind, if we bound the largest relative error of any particular entry in the matrix by, say, ω, and there are, say, s parameters (vectors and 212 matrices) being multiplied together, then by simple algebra the total relative error of the sum over the products is bounded by ωs. [sent-352, score-0.213]
95 Then, to calculate the exponent s one simply counts the number of parameters multiplied together when calculating the probability of a particular sequence of observations. [sent-355, score-0.171]
96 Since each hidden node is associated with exactly one observed node, it follows that s = 12m + 2L, where L is the number of levels (for instance in our example “Kilroy was here” there are two levels). [sent-356, score-0.118]
97 s can be easily computed for arbitrary tree topologies. [sent-357, score-0.084]
98 (2012) shows how to incorporate the accuracy of the estimates into the sample complexity. [sent-366, score-0.091]
99 Efficient parsing for bilexical context-free grammars and headautomaton grammars. [sent-403, score-0.082]
100 A tutorial on hidden markov models and selected applications in speech recognition. [sent-465, score-0.118]
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Abstract: Independence between sentences is an assumption deeply entrenched in the models and algorithms used for statistical machine translation (SMT), particularly in the popular dynamic programming beam search decoding algorithm. This restriction is an obstacle to research on more sophisticated discourse-level models for SMT. We propose a stochastic local search decoding method for phrase-based SMT, which permits free document-wide dependencies in the models. We explore the stability and the search parameters ofthis method and demonstrate that it can be successfully used to optimise a document-level semantic language model. 1 Motivation In the field oftranslation studies, it is undisputed that discourse-wide context must be considered care- fully for good translation results (Hatim and Mason, 1990). By contrast, the state of the art in statistical machine translation (SMT), despite significant advances in the last twenty years, still assumes that texts can be translated sentence by sentence under strict independence assumptions, even though it is well known that certain linguistic phenomena such as pronominal anaphora cannot be translated correctly without referring to extra-sentential context. This is true both for the phrase-based and the syntaxbased approach to SMT. In the rest of this paper, we shall concentrate on phrase-based SMT. One reason why it is difficult to experiment with document-wide models for phrase-based SMT is that the dynamic programming (DP) algorithm 1179 which has been used almost exclusively for decoding SMT models in the recent literature has very strong assumptions of locality built into it. DP beam search for phrase-based SMT was described by Koehn et al. (2003), extending earlier work on word-based SMT (Tillmann et al., 1997; Och et al., 2001 ; Tillmann and Ney, 2003). This algorithm con- structs output sentences by starting with an empty hypothesis and adding output words at the end until translations for all source words have been generated. The core models of phrase-based SMT, in particular the n-gram language model (LM), only depend on a constant number of output words to the left of the word being generated. This fact is exploited by the search algorithm with a DP technique called hypothesis recombination (Och et al., 2001), which permits the elimination of hypotheses from the search space if they coincide in a certain number of final words with a better hypothesis and no future expansion can possibly invert the relative ranking of the two hypotheses under the given models. Hypothesis recombination achieves a substantial reduction of the search space without affecting search optimality and makes it possible to use aggressive pruning techniques for fast search while still obtaining good results. The downside of this otherwise excellent approach is that it only works well with models that have a local dependency structure similar to that of an n-gram language model, so they only depend on a small context window for each target word. Sentence-local models with longer dependencies can be added, but doing so greatly increases the risk for search errors by inhibiting hypothesis recombination. Cross-sentence dependencies cannot be directly integrated into DP SMT decoding in LParnogcue agdein Lgesa ornf tihneg, 2 p0a1g2e Jso 1in17t C9–o1n1f9e0re,n Jce ju on Is Elanmdp,ir Kicoarlea M,e 1t2h–o1d4s J iunly N 2a0tu1r2a.l ? Lc a2n0g1u2ag Aes Psorcoicaetsiosin fgo arn Cdo Cmopmutpauti oantiaoln Lailn Ngautiustriacls any obvious way, especially if joint optimisation of a number of interdependent decisions over an entire document is required. Research into models with a more varied, non-local dependency structure is to some extent stifled by the difficulty of decoding such models effectively, as can be seen by the problems some researchers encountered when they attempted to solve discourse-level problems. Consider, for instance, the work on cache-based language models by Tiedemann (2010) and Gong et al. (201 1), where error propagation was a serious issue, or the works on pronominal anaphora by Le Nagard and Koehn (2010), who implemented cross-sentence dependencies with an ad-hoc two-pass decoding strategy, and Hardmeier and Federico (2010) with the use of an external decoder driver to manage backward-only dependencies between sentences. In this paper, we present a method for decoding complete documents in phrase-based SMT. Our decoder uses a local search approach whose state consists of a complete translation of an entire document at any time. The initial state is improved by the application of a series of operations using a hill climbing strategy to find a (local) maximum of the score function. This setup gives us complete freedom to define scoring functions over the entire document. Moreover, by optionally initialising the state with the output of a traditional DP decoder, we can ensure that the final hypothesis is no worse than what would have been found by DP search alone. We start by describing the decoding algorithm and the state operations used by our decoder, then we present empirical results demonstrating the effectiveness of our approach and its usability with a document-level semantic language model, and finally we discuss some related work. 2 SMT Decoding by Hill Climbing In this section, we formally describe the phrasebased SMT model implemented by our decoder as well as the decoding algorithm we use. 2.1 SMT Model Our decoder is based on local search, so its state at any time is a representation of a complete translation of the entire document. Even though the decoder operates at the document level, it is important to keep 1180 track of sentence boundaries, and the individual operations that are applied to the state are still confined to sentence scope, so it is useful to decompose the state of a document into the state of its sentences, and we define the overall state S as a sequence of sentence states: S = S1S2 . . .SN, (1) where N is the number of sentences. This implies that we constrain the decoder to emit exactly one output sentence per input sentence. Let ibe the number of a sentence and mi the number of input tokens of this sentence, p and q (with 1 ≤ p ≤ q ≤ mi) be positions in the input sentence a1n ≤d [p; q] qde ≤no mte the set ofpositions from p up to and including q. We say that [p; q] precedes [p0; q0], or [p; q] ≺ [p0; q0], if q < p0. Let Φi([p; q]) be the set of t[pra;nqs]l ≺atio [pns for the source phrase covering positions [p; q] in the input sentence ias given by the phrase table. We call A = h[p; q] ,φi an anchored phrase pair w.it Wh coverage C(A) = [p; q] nif a φ ∈ Φi([p; q]) sise a target phrase translating =th [ep source w∈o Φrds at positions [p; q] . Then a sequence of ni anchored phrase pairs Si = A1A2 . . .Ani (2) is a valid sentence state for sentence iif the following two conditions hold: 1. The coverage sets C(Aj) for j in 1, . . . , ni are mutually disjoint, and 2. the anchored phrase pairs jointly cover the complete input sentence, or [niC(Aj) = [1;mi]. (3) [j=1 Let f(S) be a scoring function mapping a state S to a real number. As usual in SMT, it is assumed that the scoring function can be decomposed into a linear combination of K feature functions hk(S), each with a constant weight λk, so f(S) =k∑K=1λkhk(S). (4) The problem addressed by the decoder is the search for the state with maximal score, such that Sˆ Sˆ = argSmaxf(S). (5) The feature functions implemented in our baseline system are identical to the ones found in the popular Moses SMT system (Koehn et al., 2007). In particular, our decoder has the following feature functions: 1. phrase translation scores provided by the phrase table including forward and backward conditional probabilities, lexical weights and a phrase penalty (Koehn et al., 2003), 2. n-gram language model scores implemented with the KenLM toolkit (Heafield, 2011), 3. a word penalty score, 4. a distortion model with geometric (Koehn et al., 2003), and decay 5. a feature indicating the number of times a given distortion limit is exceeded in the current state. In our experiments, the last feature is used with a fixed weight of negative infinity in order to limit the gaps between the coverage sets of adjacent anchored phrase pairs to a maximum value. In DP search, the distortion limit is usually enforced directly by the search algorithm and is not added as a feature. In our decoder, however, this restriction is not required to limit complexity, so we decided to add it among the scoring models. 2.2 Decoding Algorithm The decoding algorithm we use (algorithm 1) is very simple. It starts with a given initial document state. In the main loop, which extends from line 3 to line 12, it generates a successor state S0 for the current state S by calling the function Neighbour, which non-deterministically applies one of the operations described in section 3 of this paper to S. The score of the new state is compared to that of the previous one. If it meets a given acceptance criterion, S0 becomes the current state, else search continues from the previous state S. For the experiments in this paper, we use the hill climbing acceptance criterion, which simply accepts a new state if its score is higher than that of the current state. Other acceptance criteria are possible and could be used to endow the search algorithm with stochastic behaviour. 1181 The main loop is repeated until a maximum number of steps (step limit) is reached or until a maximum number of moves are rejected in a row (rejection limit). Algorithm 1 Decoding algorithm Input: an initial document state S; search parameters maxsteps and maxrejected Output: a modified document state 1: nsteps ← 0 2: nrejected ← 0 3: nwrhejileec nsteps < maxsteps and nrejected < maxrejected do 4: S0 ← Neighbour (S) 5: if Accept (f(S0) , f(S)) then 6: S ← S0 7: nrejected ← 0 8: elsner 9: nrejected ← nrejected + 1 10: enndr eifj 11: nsteps ← nsteps + 1 12: ennds wtephsile ← 13: return S A notable difference between this algorithm and other hill climbing algorithms that have been used for SMT decoding (Germann et al., 2004; Langlais et al., 2007) is its non-determinism. Previous work for sentence-level decoding employed a steepest ascent strategy which amounts to enumerating the complete neighbourhood of the current state as defined by the state operations and selecting the next state to be the best state found in the neighbourhood of the current one. Enumerating all neighbours of a given state, costly as it is, has the advantage that it makes it easy to prove local optimality of a state by recognising that all possible successor states have lower scores. It can be rather inefficient, since at every step only one modification will be adopted; many of the modifications that are discarded will very likely be generated anew in the next iteration. As we extend the decoder to the document level, the size of the neighbourhood that would have to be explored in this way increases considerably. Moreover, the inefficiency of the steepest ascent approach potentially increases as well. Very likely, a promising move in one sentence will remain promising after a modification has been applied to another sentence, even though this is not guaranteed to be true in the presence of cross-sentence models. We therefore adopt a first-choice hill climbing strategy that non-deterministically generates successor states and accepts the first one that meets the acceptance criterion. This frees us from the necessity of generating the full set of successors for each state. On the downside, if the full successor set is not known, it is no longer possible to prove local optimality of a state, so we are forced to use a different condition for halting the search. We use a combination of two limits: The step limit is a hard limit on the resources the user is willing to expend on the search problem. The value of the rejection limit determines how much of the neighbourhood is searched for better successors before a state is accepted as a solution; it is related to the probability that a state returned as a solution is in fact locally optimal. To simplify notations in the description of the individual state operations, we write Si −→ Si0 (6) to signify that a state operation, when presented with a document state as in equation 1 and acting on sentence i, returns a new document state of S0 = S1 . . .Si−1 Si0 Si+1 . . .SN. (7) Similarly, Si : Aj . . .Aj+h−1 −→ A01 . . .A0h0 (8) is equivalent to Si −→ A1 . . .Aj−1 A01 . . .A0h0 Aj+h . . .Ani (9) and indicates that the operation returns a state in which a sequence of h consecutive anchored phrase pairs has been replaced by another sequence of h0 anchored phrase pairs. 2.3 Efficiency Considerations When implementing the feature functions for the decoder, we have to exercise some care to avoid recomputing scores for the whole document at every iteration. To achieve this, the scores are computed completely only once, at the beginning of the decoding run. In subsequent iterations, scoring functions are presented with the scores of the previous 1182 iteration and a list of modifications produced by the state operation, a set of tuples hi, r, s,A01 . . .A0h0i, each indicating tthioant ,t ahe s edto ocfu tmupelnets s hhio,ru,sld, Abe modifii,e eda as described by Si :Ar . . .As −→ A01 . . .A0h0 . (10) If a feature function is decomposable in some way, as all the standard features developed under the constraints of DP search are, it can then update the state simply by subtracting and adding score components pertaining to the modified parts of the document. Feature functions have the possibility to store their own state information along with the document state to make sure the required information is available. Thus, the framework makes it possible to exploit decomposability for efficient scoring without impos- ing any particular decomposition on the features as beam search does. To make scoring even more efficient, scores are computed in two passes: First, every feature function is asked to provide an upper bound on the score that will be obtained for the new state. In some cases, it is possible to calculate reasonable upper bounds much more efficiently than computing the exact feature value. If the upper bound fails to meet the acceptance criterion, the new state is discarded right away; if not, the full score is computed and the acceptance criterion is tested again. Among the basic SMT models, this two-pass strategy is only used for the n-gram LM, which requires fairly expensive parameter lookups for scoring. The scores of all the other baseline models are fully computed during the first scoring pass. The n-gram model is more complex. In its state information, it keeps track of the LM score and LM library state for each word. The first scoring pass then identifies the words whose LM scores are affected by the current search step. This includes the words changed by the search operation as well as the words whose LM history is modified. The range of the history de- pendencies can be determined precisely by considering the “valid state length” information provided by the KenLM library. In the first pass, the LM scores of the affected words are subtracted from the total score. The model only looks up the new LM scores for the affected words and updates the total score if the new search state passes the first acceptance check. This two-pass scoring approach allows us to avoid LM lookups altogether for states that will be rejected anyhow because of low scores from the other models, e. g. because the distortion limit is violated. Model score updates become more complex and slower as the number of dependencies of a model increases. While our decoding algorithm does not impose any formal restrictions on the number or type of dependencies that can be handled, there will be practical limits beyond which decoding becomes unacceptably slow or the scoring code becomes very difficult to maintain. These limits are however fairly independent of the types of dependencies handled by a model, which permits the exploration of more varied model types than those handled by DP search. 2.4 State Initialisation Before the hill climbing decoding algorithm can be run, an initial state must be generated. The closer the initial state is to an optimum, the less work remains to be done for the algorithm. If the algorithm is to be self-contained, initialisation must be relatively uninformed and can only rely on some general prior assumptions about what might be a good initial guess. On the other hand, if optimal results are sought after, it pays off to invest some effort into a good starting point. One way to do this is to run DP search first. For uninformed initialisation, we chose to implement a very simple procedure based only on the observation that, at least for language pairs involving the major European languages, it is usually a good guess to keep the word order of the output very similar to that of the input. We therefore create the initial state by selecting, for each sentence in the document, a sequence of anchored phrase pairs covering the input sentence in monotonic order, that is, such that for all pairs of adjacent anchored phrase pairs Aj and Aj+1, we have that C(Aj) ≺ C(Aj+1 ). For initialisation with DP search, we first run the Moses decoder (Koehn et al., 2007) with default search parameters and the same models as those used by our decoder. Then we extract the best output hypothesis from the search graph of the decoder and map it into a sequence of anchored phrase pairs in the obvious way. When the document-level decoder is used with models that are incompatible with beam search, Moses can be run with a subset of the models in order to find an approximation of the solution 1183 which is then refined with the complete feature set. 3 State Operations Given a document state S, the decoder uses a neighbourhood function Neighbour to simulate a move in the state space. The neighbourhood function nondeterministically selects a type of state operation and a location in the document to apply it to and returns the resulting new state. We use a set of three operations that has the property that every possible document state can be reached from every other state in a sequence of moves. Designing operations for state transitions in local search for phrase-based SMT is a problem that has been addressed in the literature (Langlais et al., 2007; Arun et al., 2010). Our decoder’s first- choice hill climbing strategy never enumerates the full neighbourhood of a state. We therefore place less emphasis than previous work on defining a compact neighbourhood, but allow the decoder to make quite extensive changes to a state in a single step with a certain probability. Otherwise our operations are similar to those used by Arun et al. (2010). All of the operations described in this paper make changes to a single sentence only. Each time it is called, the Neighbour function selects a sentence in the document with a probability proportional to the number of input tokens in each sentence to ensure a fair distribution ofthe decoder’s attention over the words in the document regardless of varying sentence lengths. 3.1 Changing Phrase Translations The change-phrase-translation operation replaces the translation of a single phrase with a random translation with the same coverage taken from the phrase table. Formally, the operation selects an anchored phrase pair Aj by drawing uniformly from the elements of Si and then draws a new translation φ0 uniformly from the set Φi(C(Aj)). The new state is given by Si : Aj −→ hC(Aj), φ0i. (11) 3.2 Changing Word Order The swap-phrases operation affects the output word order without changing the phrase translations. It exchanges two anchored phrase pairs Aj and Aj+h, resulting in an output state of Si : Aj . . .Aj+h −→ Aj+h Aj+1 . . .Aj+h−1 Aj. (12) The start location j is drawn uniformly from the eligible sentence positions; the swap range h comes from a geometric distribution with configurable decay. Other word-order changes such as a one-way move operation that does not require another movement in exchange or more advanced permutations can easily be defined. 3.3 Resegmentation The most complex operation is resegment, which allows the decoder to modify the segmentation ofthe source phrase. It takes a number of anchored phrase pairs that form a contiguous block both in the input and in the output and replaces them with a new set of phrase pairs covering the same span of the input sentence. Formally, Si : Aj . . .Aj+h−1 −→ A01 . . .A0h0 (13) such that j+[h−1 [h0 [ C(Aj0) = [ C(A0j0) = [p;q] j[0=j (14) j[0=1 for some p and q, where, for j0 = 1, . . . ,h0, we have that A0j0 = h[pj0; qj0] , φj0i, all [pj0; qj0] are mutually disjoint =an hd[ peach φj0 isi randomly drawn from Φi([pj0;qj0]). Regardless of the ordering of Aj . . .Aj+h−1 , the resegment operation always generates a sequence of anchored phrase pairs in linear order, such that C(A0j0) ≺ C(A0j0+1 ) for j0 = 1, . . . ,h0 −1 . As )f o≺r Cth(eA other operations, j is− generated uniformly and h is drawn from a geometric distribution with a decay parameter. The new segmentation is generated by extending the sequence of anchored phrase pairs with random elements starting at the next free position, proceeding from left to right until the whole range [p; q] is covered. 4 Experimental Results In this section, we present the results of a series of experiments with our document decoder. The 1184 goal of our experiments is to demonstrate the behaviour of the decoder and characterise its response to changes in the fundamental search parameters. The SMT models for our experiments were created with a subset of the training data for the English-French shared task at the WMT 2011workshop (Callison-Burch et al., 2011). The phrase table was trained on Europarl, news-commentary and UN data. To reduce the training data to a manageable size, singleton phrase pairs were removed before the phrase scoring step. Significance-based filtering (Johnson et al., 2007) was applied to the resulting phrase table. The language model was a 5gram model with Kneser-Ney smoothing trained on the monolingual News corpus with IRSTLM (Federico et al., 2008). Feature weights were trained with Minimum Error-Rate Training (MERT) (Och, 2003) on the news-test2008 development set using the DP beam search decoder and the MERT implementation of the Moses toolkit (Koehn et al., 2007). Experimental results are reported for the newstest2009 test set, a corpus of 111 newswire documents totalling 2,525 sentences or 65,595 English input tokens. 4.1 Stability An important difference between our decoder and the classical DP decoder as well as previous work in SMT decoding with local search is that our decoder is inherently non-deterministic. This implies that repeated runs of the decoder with the same search parameters, input and models will not, in general, find the same local maximum of the score space. The first empirical question we ask is therefore how different the results are under repeated runs. The results in this and the next section were obtained with random state initialisation, i. e. without running the DP beam search decoder. Figure 1 shows the results of 7 decoder runs with the models described above, translating the newstest2009 test set, with a step limit of 227 and a rejection limit of 100,000. The x-axis of both plots shows the number of decoding steps on a logarithmic scale, so the number of steps is doubled between two adjacent points on the same curve. In the left plot, the y-axis indicates the model score optimised by the decoder summed over all 2525 sentences of the document. In the right plot, the case-sensitive BLEU score (Papineni et al., 2002) of the current decoder Figure 1: Score stability in repeated decoder runs state against a reference translation is displayed. We note, as expected, that the decoder achieves a considerable improvement of the initial state with diminishing returns as decoding continues. Between 28 and 214 steps, the score increases at a roughly logarithmic pace, then the curve flattens out, which is partly due to the fact that decoding for some documents effectively stopped when the maximum number of rejections was reached. The BLEU score curve shows a similar increase, from an initial score below 5 % to a maximum of around 21.5 %. This is below the score of 22.45 % achieved by the beam search decoder with the same models, which is not surprising considering that our decoder approximates a more difficult search problem, from which a number of strong independence assumptions have been lifted, without, at the moment, having any stronger models at its disposal to exploit this additional freedom for better translation. In terms of stability, there are no dramatic differences between the decoder runs. Indeed, the small differences that exist are hardly discernible in the plots. The model scores at the end of the decoding run range between −158767.9 and −158716.9, a g re rlautniv rea ndgieffe breetnwceee nof − only a6b7.o9ut a n0d.0 −3 %15.8 F1i6n.a9l, BLEU scores range from 21.41 % to 21.63 %, an interval that is not negligible, but comparable to the variance observed when, e. g., feature weights from repeated MERT runs are used with one and the same SMT system. Note that these results were obtained with random state initialisation. With DP initialisation, score differences between repeated runs rarely 1185 exceed 0.02 absolute BLEU percentage points. Overall, we conclude that the decoding results of our algorithm are reasonably stable despite the nondeterminism inherent in the procedure. In our subsequent experiments, the evaluation scores reported are calculated as the mean of three runs for each experiment. 4.2 Search Algorithm Parameters The hill climbing algorithm we use has two parameters which govern the trade-off between decoding time and the accuracy with which a local maximum is identified: The step limit stops the search process after a certain number of steps regardless of the search progress made or lack thereof. The rejection limit stops the search after a certain number of unsuccessful attempts to make a step, when continued search does not seem to be promising. In most of our experiments, we used a step limit of 227 ≈ 1.3 · 108 and a rejection limit of 105. In practice, decoding terminates by reaching the rejection limit for the vast majority of documents. We therefore examined the effect of different rejection limits on the learning curves. The results are shown in figure 2. The results show that continued search does pay off to a certain extent. Indeed, the curve for rejection limit 107 seems to indicate that the model score increases roughly logarithmically, albeit to a higher base, even after the curve has started to flatten out at 214 steps. At a certain point, however, the probability of finding a good successor state drops rather sharply by about two orders of magnitude, as Figure 2: Search performance at different rejection limits evidenced by the fact that a rejection limit of 106 does not give a large improvement over one of 105, while one of 107 does. The continued model score improvement also results in an increase in BLEU scores, and with a BLEU score of 22. 1% the system with rejection limit 107 is fairly close to the score of 22.45 % obtained by DP beam search. Obviously, more exact search comes at a cost, and in this case, it comes at a considerable cost, which is an explosion of the time required to decode the test set from 4 minutes at rejection limit 103 to 224 minutes at rejection limit 105 and 38 hours 45 minutes at limit 107. The DP decoder takes 3 1 minutes for the same task. We conclude that the rejection limit of 105 selected for our experiments, while technically suboptimal, realises a good trade-off between decoding time and accuracy. 4.3 A Semantic Document Language Model In this section, we present the results of the application of our decoder to an actual SMT model with cross-sentence features. Our model addresses the problem of lexical cohesion. In particular, it rewards the use of semantically related words in the translation output by the decoder, where semantic distance is measured with a word space model based on Latent Semantic Analysis (LSA). LSA has been applied to semantic language modelling in previous research with some success (Coccaro and Jurafsky, 1998; Bellegarda, 2000; Wandmacher and Antoine, 2007). In SMT, it has mostly been used for domain adaptation (Kim and Khudanpur, 2004; Tam et al., 1186 2007), or to measure sentence similarities (Banchs and Costa-juss a`, 2011). The model we use is inspired by Bellegarda (2000). It is a Markov model, similar to a standard n-gram model, and assigns to each content word a score given a history of n preceding content words, where n = 30 below. Scoring relies on a 30dimensional LSA word vector space trained with the S-Space software (Jurgens and Stevens, 2010). The score is defined based on the cosine similarity between the word vector of the predicted word and the mean word vector of the words in the history, which is converted to a probability by histogram lookup as suggested by Bellegarda (2000). The model is structurally different from a regular n-gram model in that word vector n-grams are defined over content words occurring in the word vector model only and can cross sentence boundaries. Stop words, identified by an extensive stop word list and amounting to around 60 % of the tokens, are scored by a different mechanism based on their relative frequency (undiscounted unigram probability) in the training corpus. In sum, the score produced by the semantic document LM has the following form: wh(er|h)α=is tεpαheuncipgors(wp)o|hrtinof w fci os nakutneskotn wpon w ,onerldse,in(ls1teh5) training corpus and ε is a small fixed probability. It is integrated into the decoder as an extra feature function. Since we lack an automatic method for training the feature weights of document-wide features, its weight was selected by grid search over a number of values, comparing translation performance for the newstest2009 test set. In these experiments, we used DP beam search to initialise the state of our local search decoder. Three results are presented (table 1): The first table row shows the baseline performance using DP beam search with standard sentence-local features only. The scores in the second row were obtained by running the hill climbing decoder with DP initialisation, but without adding any models. A marginal increase in scores for all three test sets demonstrates that the hill climbing decoder manages to fix some of the search errors made by the DP search. The last row contains the scores obtained by adding in the semantic language model. Scores are presented for three publicly available test sets from recent WMT Machine Translation shared tasks, of which one (newstest2009) was used to monitor progress during development and select the final model. Adding the semantic language model results in a small increase in NIST scores (Doddington, 2002) for all three test sets as well as a small BLEU score gain (Papineni et al., 2002) for two out of three corpora. We note that the NIST score turned out to react more sensitively to improvements due to the semantic LM in all our experiments, which is reasonable because the model specifically targets content words, which benefit from the information weighting done by the NIST score. While the results we present do not constitute compelling evidence in favour of our semantic LM in its current form, they do suggest that this model could be improved to realise higher gains from cross-sentence semantic information. They support our claim that cross- sentence models should be examined more closely and that existing methods should be adapted to deal with them, a problem addressed by our main contribution, the local search document decoder. 5 Related Work Even though DP beam search (Koehn et al., 2003) has been the dominant approach to SMT decoding in recent years, methods based on local search have been explored at various times. For word-based SMT, greedy hill-climbing techniques were advo1187 cated as a faster replacement for beam search (Germann et al., 2001 ; Germann, 2003; Germann et al., 2004), and a problem formulation specifically targeting word reordering with an efficient word reordering algorithm has been proposed (Eisner and Tromble, 2006). A local search decoder has been advanced as a faster alternative to beam search also for phrasebased SMT (Langlais et al., 2007; Langlais et al., 2008). That work anticipates many of the features found in our decoder, including the use of local search to refine an initial hypothesis produced by DP beam search. The possibility of using models that do not fit well into the beam search paradigm is mentioned and illustrated with the example of a reversed n-gram language model, which the authors claim would be difficult to implement in a beam search decoder. Similarly to the work by Germann et al. (2001), their decoder is deterministic and explores the entire neighbourhood of a state in order to identify the most promising step. Our main contribution with respect to the work by Langlais et al. (2007) is the introduction of the possibility of handling document-level models by lifting the assumption of sentence independence. As a consequence, enumerating the entire neighbourhood becomes too expensive, which is why we resort to a “first-choice” strategy that non-deterministically generates states and accepts the first one encountered that meets the acceptance criterion. More recently, Gibbs sampling was proposed as a way to generate samples from the posterior distribution of a phrase-based SMT decoder (Arun et al., 2009; Arun et al., 2010), a process that resembles local search in its use of a set of state-modifying operators to generate a sequence of decoder states. Where local search seeks for the best state attainable from a given initial state, Gibbs sampling produces a representative sample from the posterior. Like all work on SMT decoding that we know of, the Gibbs sampler presented by Arun et al. (2010) assumes independence of sentences and considers the complete neighbourhood of each state before taking a sample. 6 Conclusion In the last twenty years of SMT research, there has been a strong assumption that sentences in a text newstest2009 newstest2010 newstest201 1 BLEU NIST BLEU NIST BLEU NIST 22.56 6.513 27.27 7.034 24.94 7.170 + hill climbing 22.60 6.518 27.33 7.046 24.97 7.169 with semantic LM 22.71 6.549 27.53 7.087 24.90 7.199 DP search only DP Table 1: Experimental results with a cross-sentence semantic language model are independent of one another, and discourse context has been largely neglected. Several factors have contributed to this. Developing good discourse-level models is difficult, and considering the modest translation quality that has long been achieved by SMT, there have been more pressing problems to solve and lower hanging fruit to pick. However, we argue that the popular DP beam search algorithm, which delivers excellent decoding performance, but imposes a particular kind of local dependency structure on the feature models, has also had its share in driving researchers away from discourse-level problems. In this paper, we have presented a decoding procedure for phrase-based SMT that makes it possible to define feature models with cross-sentence dependencies. Our algorithm can be combined with DP beam search to leverage the quality of the traditional approach with increased flexibility for models at the discourse level. We have presented preliminary results on a cross-sentence semantic language model addressing the problem of lexical cohesion to demonstrate that this kind of models is worth exploring further. Besides lexical cohesion, cross-sentence models are relevant for other linguistic phenomena such as pronominal anaphora or verb tense selection. 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