{ "paper_id": "P12-1003", "header": { "generated_with": "S2ORC 1.0.0", "date_generated": "2023-01-19T09:27:05.259694Z" }, "title": "Prediction of Learning Curves in Machine Translation", "authors": [ { "first": "Prasanth", "middle": [], "last": "Kolachina", "suffix": "", "affiliation": { "laboratory": "", "institution": "IIIT-Hyderabad", "location": { "settlement": "Hyderabad", "country": "India" } }, "email": "" }, { "first": "Nicola", "middle": [], "last": "Cancedda", "suffix": "", "affiliation": { "laboratory": "Xerox Research Centre Europe", "institution": "", "location": { "addrLine": "6 chemin de Maupertuis", "postCode": "38240", "settlement": "Meylan", "country": "France" } }, "email": "" }, { "first": "Marc", "middle": [], "last": "Dymetman", "suffix": "", "affiliation": { "laboratory": "Xerox Research Centre Europe", "institution": "", "location": { "addrLine": "6 chemin de Maupertuis", "postCode": "38240", "settlement": "Meylan", "country": "France" } }, "email": "" }, { "first": "Sriram", "middle": [], "last": "Venkatapathy", "suffix": "", "affiliation": { "laboratory": "Xerox Research Centre Europe", "institution": "", "location": { "addrLine": "6 chemin de Maupertuis", "postCode": "38240", "settlement": "Meylan", "country": "France" } }, "email": "" } ], "year": "", "venue": null, "identifiers": {}, "abstract": "Parallel data in the domain of interest is the key resource when training a statistical machine translation (SMT) system for a specific purpose. Since ad-hoc manual translation can represent a significant investment in time and money, a prior assesment of the amount of training data required to achieve a satisfactory accuracy level can be very useful. In this work, we show how to predict what the learning curve would look like if we were to manually translate increasing amounts of data. We consider two scenarios, 1) Monolingual samples in the source and target languages are available and 2) An additional small amount of parallel corpus is also available. We propose methods for predicting learning curves in both these scenarios.", "pdf_parse": { "paper_id": "P12-1003", "_pdf_hash": "", "abstract": [ { "text": "Parallel data in the domain of interest is the key resource when training a statistical machine translation (SMT) system for a specific purpose. Since ad-hoc manual translation can represent a significant investment in time and money, a prior assesment of the amount of training data required to achieve a satisfactory accuracy level can be very useful. In this work, we show how to predict what the learning curve would look like if we were to manually translate increasing amounts of data. We consider two scenarios, 1) Monolingual samples in the source and target languages are available and 2) An additional small amount of parallel corpus is also available. We propose methods for predicting learning curves in both these scenarios.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Abstract", "sec_num": null } ], "body_text": [ { "text": "Parallel data in the domain of interest is the key resource when training a statistical machine translation (SMT) system for a specific business purpose. In many cases it is possible to allocate some budget for manually translating a limited sample of relevant documents, be it via professional translation services or through increasingly fashionable crowdsourcing. However, it is often difficult to predict how much training data will be required to achieve satisfactory translation accuracy, preventing sound provisional budgetting. This prediction, or more generally the prediction of the learning curve of an SMT system as a function of available in-domain parallel data, is the objective of this paper.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "We consider two scenarios, representative of realistic situations.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "1. In the first scenario (S1), the SMT developer is given only monolingual source and target samples from the relevant domain, and a small test parallel corpus.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "2. In the second scenario (S2), an additional small seed parallel corpus is given that can be used to train small in-domain models and measure (with some variance) the evaluation score at a few points on the initial portion of the learning curve.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In both cases, the task consists in predicting an evaluation score (BLEU, throughout this work) on the test corpus as a function of the size of a subset of the source sample, assuming that we could have it manually translated and use the resulting bilingual corpus for training.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this paper we provide the following contributions:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "1. An extensive study across six parametric function families, empirically establishing that a certain three-parameter power-law family is well suited for modeling learning curves for the Moses SMT system when the evaluation score is BLEU. Our methodology can be easily generalized to other systems and evaluation scores (Section 3); 2. A method for inferring learning curves based on features computed from the resources available in scenario S1, suitable for both the scenarios described above (S1) and (S2) (Section 4); 3. A method for extrapolating the learning curve from a few measurements, suitable for scenario S2 (Section 5); 4. A method for combining the two approaches above, achieving on S2 better prediction accuracy than either of the two in isolation (Section 6).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "In this study we limit tuning to the mixing parameters of the Moses log-linear model through MERT, keeping all meta-parameters (e.g. maximum phrase length, maximum allowed distortion, etc.) at their default values. One can expect further tweaking to lead to performance improvements, but this was a necessary simplification in order to execute the tests on a sufficiently large scale.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Our experiments involve 30 distinct language pair and domain combinations and 96 different learning curves. They show that without any parallel data we can predict the expected translation accuracy at 75K segments within an error of 6 BLEU points (Table 4), while using a seed training corpus of 10K segments narrows this error to within 1.5 points (Table 6).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Introduction", "sec_num": "1" }, { "text": "Learning curves are routinely used to illustrate how the performance of experimental methods depend on the amount of training data used. In the SMT area, Koehn et al. (2003) used learning curves to compare performance for various meta-parameter settings such as maximum phrase length, while Turchi et al. (2008) extensively studied the behaviour of learning curves under a number of test conditions on Spanish-English. In Birch et al. (2008) , the authors examined corpus features that contribute most to the machine translation performance. Their results showed that the most predictive features were the morphological complexity of the languages, their linguistic relatedness and their word-order divergence; in our work, we make use of these features, among others, for predicting translation accuracy (Section 4).", "cite_spans": [ { "start": 154, "end": 173, "text": "Koehn et al. (2003)", "ref_id": "BIBREF7" }, { "start": 291, "end": 311, "text": "Turchi et al. (2008)", "ref_id": "BIBREF18" }, { "start": 422, "end": 441, "text": "Birch et al. (2008)", "ref_id": "BIBREF0" } ], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "In a Machine Learning context, Perlich et al. (2003) used learning curves for predicting maximum performance bounds of learning algorithms and to compare them. In Gu et al. (2001) , the learning curves of two classification algorithms were modelled for eight different large data sets. This work uses similar a priori knowledge for restricting the form of learning curves as ours (see Section 3), and also similar empirical evaluation criteria for comparing curve families with one another. While both application and performance metric in our work are different, we arrive at a similar conclusion that a power law family of the form y = c \u2212 a x \u2212\u03b1 is a good model of the learning curves.", "cite_spans": [ { "start": 31, "end": 52, "text": "Perlich et al. (2003)", "ref_id": "BIBREF14" }, { "start": 163, "end": 179, "text": "Gu et al. (2001)", "ref_id": "BIBREF6" } ], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "Learning curves are also frequently used for determining empirically the number of iterations for an incremental learning procedure.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "The crucial difference in our work is that in the previous cases, learning curves are plotted a posteriori i.e. once the labelled data has become available and the training has been performed, whereas in our work the learning curve itself is the object of the prediction. Our goal is to learn to predict what the learning curve will be a priori without having to label the data at all (S1), or through labelling only a very small amount of it (S2).", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "In this respect, the academic field of Computational Learning Theory has a similar goal, since it strives to identify bounds to performance measures 1 , typically including a dependency on the training sample size. We take a purely empirical approach in this work, and obtain useful estimations for a case like SMT, where the complexity of the mapping between the input and the output prevents tight theoretical analysis.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Related Work", "sec_num": "2" }, { "text": "The first step in our approach consists in selecting a suitable family of shapes for the learning curves that we want to produce in the two scenarios being considered.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "We formulate the problem as follows. For a certain bilingual test dataset d, we consider a set of observations", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "O d = {(x 1 , y 1 ), (x 2 , y 2 )...(x n , y n )},", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "where y i is the performance on d (measured using BLEU (Papineni et al., 2002) ) of a translation model trained on a parallel corpus of size x i . The corpus size x i is measured in terms of the number of segments (sentences) present in the parallel corpus.", "cite_spans": [ { "start": 55, "end": 78, "text": "(Papineni et al., 2002)", "ref_id": "BIBREF12" } ], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "We consider such observations to be generated by a regression model of the form:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "y i = F (x i ; \u03b8) + i 1 \u2264 i \u2264 n (1)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "where F is a function depending on a vector parameter \u03b8 which depends on d, and i is Gaussian noise of constant variance. Based on our prior knowledge of the problem, we limit the search for a suitable F to families that satisfies the following conditions-monotonically increasing, concave and bounded. The first condition just says that more training data is better. The second condition expresses a notion of \"diminishing returns\", namely that a given amount of additional training data is more advantageous when added to a small rather than to a big amount of initial data. The last condition is related to our use of BLEUwhich is bounded by 1 -as a performance measure; It should be noted that some growth patterns which are sometimes proposed, such as a logarithmic regime of the form y a + b log x, are not compatible with this constraint. We consider six possible families of functions satisfying these conditions, which are listed in Table 1 . Preliminary experiments indicated that curves from the \"Power\" and \"Exp\" family with only two parameters underfitted, while those with five or more parameters led to overfitting and solution instability. We decided to only select families with three or four parameters.", "cite_spans": [], "ref_spans": [ { "start": 942, "end": 949, "text": "Table 1", "ref_id": "TABREF0" } ], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "Model Formula Exp 3 y = c \u2212 e \u2212ax+b Exp 4 y = c \u2212 e \u2212ax \u03b1 +b ExpP 3 y = c \u2212 e (x\u2212b) \u03b1 Pow 3 y = c \u2212 ax \u2212\u03b1 Pow 4 y = c \u2212 (\u2212ax + b) \u2212\u03b1 ILog 2 y = c \u2212 (a/ log x)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "Curve fitting technique Given a set of observations Table 1 , we compute a best fit\u03b8 where:\u03b8", "cite_spans": [], "ref_spans": [ { "start": 52, "end": 59, "text": "Table 1", "ref_id": "TABREF0" } ], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "{(x 1 , y 1 ), (x 2 , y 2 )...(x n , y n )} and a curve fam- ily F (x; \u03b8) from", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "= arg min \u03b8 n i=1 [y i \u2212 F (x i ; \u03b8)] 2 ,", "eq_num": "(2)" } ], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "through use of the Levenberg-Marquardt method (Mor\u00e9, 1978) for non-linear regression. For selecting a learning curve family, and for all other experiments in this paper, we trained a large number of systems on multiple configurations of training sets and sample sizes, and tested each on multiple test sets; these are listed in Table 2 . All experiments use Moses (Koehn et al., 2007 (Neubig, 2011) Jp, En En, Jp 2 EMEA (Tiedemann, 2009) Da, De En 4 News (Callison-Burch et al., 2011) Cz,En,Fr,De,Es Cz,En,Fr,De,Es 3 The goodness of fit for each of the families is eval- 2 The settings used in training the systems are those described in http://www.statmt.org/wmt11/ baseline.html uated based on their ability to i) fit over the entire set of observations, ii) extrapolate to points beyond the observed portion of the curve and iii) generalize well over different datasets .", "cite_spans": [ { "start": 46, "end": 58, "text": "(Mor\u00e9, 1978)", "ref_id": "BIBREF10" }, { "start": 364, "end": 383, "text": "(Koehn et al., 2007", "ref_id": "BIBREF8" }, { "start": 384, "end": 398, "text": "(Neubig, 2011)", "ref_id": "BIBREF11" }, { "start": 420, "end": 437, "text": "(Tiedemann, 2009)", "ref_id": "BIBREF17" }, { "start": 455, "end": 484, "text": "(Callison-Burch et al., 2011)", "ref_id": "BIBREF1" }, { "start": 571, "end": 572, "text": "2", "ref_id": null } ], "ref_spans": [ { "start": 328, "end": 335, "text": "Table 2", "ref_id": "TABREF2" } ], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "We use a recursive fitting procedure where the curve obtained from fitting the first i points is used to predict the observations at two points: x i+1 , i.e. the point to the immediate right of the currently observed x i and x n , i.e. the largest point that has been observed.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "The following error measures quantify the goodness of fit of the curve families:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "1. Average root mean-squared error (RMSE):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "1 N c\u2208S t\u2208Tc 1 n n i=1 [y i \u2212 F (x i ;\u03b8)] 2 1/2 ct", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "where S is the set of training datasets, T c is the set of test datasets for training configuration c, \u03b8 is as defined in Eq. 2, N is the total number of combinations of training configurations and test datasets, and i ranges on a grid of training subset sizes.The expressions n, x i , y i ,\u03b8 are all local to the combination ct.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "2. Average root mean squared residual at next point X = x i+1 (NPR):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "1 N c\u2208S t\u2208Tc 1 n \u2212 k \u2212 1 n\u22121 i=k [y i+1 \u2212 F (x i+1 ;\u03b8 i )] 2 1/2 ct", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "where\u03b8 i is obtained using only observations up to x i in Eq. 2 and where k is the number of parameters of the family. 3 3. Average root mean squared residual at the last point X = x n (LPR):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "1 N c\u2208S t\u2208Tc 1 n \u2212 k \u2212 1 n\u22121 i=k [y n \u2212 F (x n ;\u03b8 i )] 2 1/2 ct", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "Curve fitting evaluation The evaluation of the goodness of fit for the curve families is presented in Table 3 . The average values of the root meansquared error and the average residuals across all the learning curves used in our experiments are shown in this table. The values are on the same scale as the BLEU scores. Figure 1 shows the curve fits obtained Loooking at the values in Table 3 , we decided to use the Pow 3 family as the best overall compromise. While it is not systematically better than Exp 4 and Pow 4 , it is good overall and has the advantage of requiring only 3 parameters.", "cite_spans": [], "ref_spans": [ { "start": 102, "end": 109, "text": "Table 3", "ref_id": "TABREF4" }, { "start": 320, "end": 328, "text": "Figure 1", "ref_id": null }, { "start": 385, "end": 392, "text": "Table 3", "ref_id": "TABREF4" } ], "eq_spans": [], "section": "Selecting a parametric family of curves", "sec_num": "3" }, { "text": "In this section we address scenario S1: we have access to a source-language monolingual collection (from which portions to be manually translated could be sampled) and a target-language in-domain monolingual corpus, to supplement the target side of a parallel corpus while training a language model. The only available parallel resource is a very small test corpus. Our objective is to predict the evolution of the BLEU score on the given test set as a function of the size of a random subset of the training data that we manually translate 4 . The intuition behind this is that the source-side and target-side monolingual data already convey significant information about the difficulty of the translation task. We proceed in the following way. We first train models to predict the BLEU score at m anchor sizes s 1 , . . . , s m , based on a set of features globally characterizing the configuration of interest. We restrict our attention to linear models:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "\u00b5 j = w j \u03c6, j \u2208 {1 . . . m}", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "where w j is a vector of feature weights specific to predicting at anchor size j, and \u03c6 is a vector of sizeindependent configuration features, detailed below. We then perform inference using these models to predict the BLEU score at each anchor, for the test case of interest. We finally estimate the parameters of the learning curve by weighted least squares regression using the anchor predictions.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "Anchor sizes can be chosen rather arbitrarily, but must satisfy the following two constraints:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "1. They must be three or more in number in order to allow fitting the tri-parameter curve.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "2. They should be spread as much as possible along the range of sample size.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "For our experiments, we take m = 3, with anchors at 10K, 75K and 500K segments. The feature vector \u03c6 consists of the following features:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "1. General properties: number and average length of sentences in the (source) test set.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "2. Average length of tokens in the (source) test set and in the monolingual source language corpus.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Inferring a learning curve from mostly monolingual data", "sec_num": "4" }, { "text": "(a) type-token ratios for n-grams of order 1 to 5 in the monolingual corpus of both source and target languages (b) perplexity of language models of order 2 to 5 derived from the monolingual source corpus computed on the source side of the test corpus.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "4. Features capturing divergence between languages in the pair:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "(a) average ratio of source/target sentence lengths in the test set. (b) ratio of type-token ratios of orders 1 to 5 in the monolingual corpus of both source and target languages.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "5. Word-order divergence: The divergence in the word-order between the source and the target languages can be captured using the part-ofspeech (pos) tag sequences across languages. We use cross-entropy measure to capture similarity between the n-gram distributions of the pos tags in the monolingual corpora of the two languages. The order of the n-grams ranges between n = 2, 4 . . . 12 in order to account for long distance reordering between languages. The pos tags for the languages are mapped to a reduced set of twelve pos tags (Petrov et al., 2012) in order to account for differences in tagsets used across languages.", "cite_spans": [ { "start": 534, "end": 555, "text": "(Petrov et al., 2012)", "ref_id": "BIBREF15" } ], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "These features capture our intuition that translation is going to be harder if the language in the domain is highly variable and if the source and target languages diverge more in terms of morphology and word-order. The weights w j are estimated from data. The training data for fitting these linear models is obtained in the following way. For each configuration (combination of language pair and domain) c and test set t in Table 2 , a gold curve is fitted using the selected tri-parameter power-law family using a fine grid of corpus sizes. This is available as a byproduct of the experiments for comparing different parametric families described in Section 3. We then compute the value of the gold curves at the m anchor sizes: we thus have m \"gold\" vectors \u00b5 1 , . . . , \u00b5 m with accurate estimates of BLEU at the anchor sizes 5 . We construct the design matrix \u03a6 with one column for each feature vector \u03c6 ct corresponding to each combination of training configuration c and test set t.", "cite_spans": [], "ref_spans": [ { "start": 426, "end": 433, "text": "Table 2", "ref_id": "TABREF2" } ], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "We then estimate weights w j using Ridge regression (L 2 regularization):", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "w j = arg min w ||\u03a6 w \u2212 \u00b5 j || 2 + C||w|| 2", "eq_num": "(3)" } ], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "where the regularization parameter C is chosen by cross-validation. We also run experiments using Lasso (L 1 ) regularization (Tibshirani, 1994) instead of Ridge. As baseline, we take a constant mean model predicting, for each anchor size s j , the average of all the \u00b5 jct . We do not assume the difficulty of predicting BLEU at all anchor points to be the same. To allow for this, we use (non-regularized) weighted leastsquares to fit a curve from our parametric family through the m anchor points 6 . Following (Croarkin and Tobias, 2006, Section 4.4.5 .2), the anchor confidence is set to be the inverse of the cross-validated mean square residuals:", "cite_spans": [ { "start": 126, "end": 144, "text": "(Tibshirani, 1994)", "ref_id": "BIBREF16" }, { "start": 514, "end": 555, "text": "(Croarkin and Tobias, 2006, Section 4.4.5", "ref_id": null } ], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "\u03c9 j = 1 N c\u2208S t\u2208Tc (\u03c6 ct w \\c j \u2212 \u00b5 jct ) 2 \u22121", "eq_num": "(4)" } ], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "where w \\c j are the feature weights obtained by the regression above on all training configurations except c, \u00b5 jct is the gold value at anchor j for training/test combination c, t, and N is the total number of such combinations 7 . In other words, we assign to each anchor point a confidence inverse to the crossvalidated mean squared error of the model used to predict it.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "For a new unseen configuration with feature vector \u03c6 u , we determine the parameters \u03b8 u of the corresponding learning curve as:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "\u03b8 u = arg min \u03b8 j \u03c9 j F (s j ; \u03b8) \u2212 \u03c6 u w j 2 (5)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "5 Extrapolating a learning curve fitted on a small parallel corpus Given a small \"seed\" parallel corpus, the translation system can be used to train small in-domain models and the evaluation score can be measured at a few initial sample sizes", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "{(x 1 , y 1 ), (x 2 , y 2 )...(x p , y p )}.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "The performance of the system for these initial points provides evidence for predicting its performance for larger sample sizes. In order to do so, a learning curve from the family Pow 3 is first fit through these initial points. We assume that p \u2265 3 for this operation to be welldefined. The best fit\u03b7 is computed using the same curve fitting as in Eq. 2.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "At each individual anchor size s j , the accuracy of prediction is measured using the root mean-squared error between the prediction of extrapolated curves and the gold values:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "1 N c\u2208S t\u2208Tc [F (s j ;\u03b7 ct ) \u2212 \u00b5 ctj ] 2 1/2 (6)", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "where\u03b7 ct are the parameters of the curve fit using the initial points for the combination ct.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "In general, we observed that the extrapolated curve tends to over-estimate BLEU for large samples.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Lexical diversity features:", "sec_num": "3." }, { "text": "In scenario S2, the models trained from the seed parallel corpus and the features used for inference (Section 4) provide complementary information. In this section we combine the two to see if this yields more accurate learning curves.", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Combining inference and extrapolation", "sec_num": "6" }, { "text": "For the inference method of Section 4, predictions of models at anchor points are weighted by the inverse of the model empirical squared error (\u03c9 j ). We extend this approach to the extrapolated curves. Let u be a new configuration with seed parallel corpus of size x u , and let x l be the largest point in our grid for which x l \u2264 x u . We first train translation models and evaluate scores on samples of size x 1 , . . . , x l , fit parameters\u03b7 u through the scores, and then extrapolate BLEU at the anchors s j : F (s j ;\u03b7 u ), j \u2208 {1, . . . , m}. Using the models trained for the experiments in Section 3, we estimate the squared extrapolation error at the anchors s j when using models trained on size up to x l , and set the confidence in the extrapolations 8 for u to its inverse:", "cite_spans": [], "ref_spans": [], "eq_spans": [], "section": "Combining inference and extrapolation", "sec_num": "6" }, { "text": "EQUATION", "cite_spans": [], "ref_spans": [], "eq_spans": [ { "start": 0, "end": 8, "text": "EQUATION", "ref_id": "EQREF", "raw_str": "\u03be " }, "TABREF1": { "text": "). 2", "num": null, "type_str": "table", "html": null, "content": "
DomainSource LanguageTarget Language# Test sets
Europarl (Koehn, 2005)Fr, De, Es EnEn Fr, De, Es4
KFTT
" }, "TABREF2": { "text": "The translation systems used for the curve fitting experiments, comprising 30 language-pair and domain combinations for a total of 96 learning curves.", "num": null, "type_str": "table", "html": null, "content": "
Language codes: Cz=Czech, Da=Danish, En=English,
De=German, Fr=French, Jp=Japanese, Es=Spanish
" }, "TABREF4": { "text": "Evaluation of the goodness of fit for the six families.", "num": null, "type_str": "table", "html": null, "content": "" }, "TABREF6": { "text": "Root mean squared error of the linear regression models for each anchor size", "num": null, "type_str": "table", "html": null, "content": "
" }, "TABREF8": { "text": "Root mean squared error of the combined curves at the three anchor sizes", "num": null, "type_str": "table", "html": null, "content": "
" } } } }