Paraphrases do not explain word analogies

02/23/2021 ∙ by Louis Fournier, et al. ∙ Ecole Polytechnique 0

Many types of distributional word embeddings (weakly) encode linguistic regularities as directions (the difference between "jump" and "jumped" will be in a similar direction to that of "walk" and "walked," and so on). Several attempts have been made to explain this fact. We respond to Allen and Hospedales' recent (ICML, 2019) theoretical explanation, which claims that word2vec and GloVe will encode linguistic regularities whenever a specific relation of paraphrase holds between the four words involved in the regularity. We demonstrate that the explanation does not go through: the paraphrase relations needed under this explanation do not hold empirically.

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1 Introduction

The study of linguistic regularities in distributional word embeddings—that the difference vector calculated between the vectors

jump and jumped shows a similar direction to that of walk and walked, and so on—has been both stimulating and controversial. While a number of such regularities appear to hold, across a number of different kinds of embeddings, the standard 3CosAdd analogy test used to measure the presence of these regularities has come under fire for confounding analogical regularities with unrelated properties of semantic embeddings. It is thus important to note that several papers have proposed theoretical explanations for why linguistic regularities should hold in distributional word embeddings. Particularly in light of the controversies over linguistic regularities, it is important to examine the soundness of these arguments.

Allen and Hospedales (2019) develop such an explanation by linking the semantic definition of an analogy to paraphrases. In the sense of Gittens et al. (2017), paraphrases are sets of words which are semantically and distributionally closely equivalent to another word or set of words—for example, king may be paraphrased by {man, royal}. Allen and Hospedales argue that the standard analogy criterion, that king - man + woman = queen, is equivalent to a criterion whereby {king, woman} paraphrases {man, queen}. With this in mind, it becomes possible to rewrite the arithmetic analogy criterion in terms of vectors encoding the pointwise mutual information (PMI) between words and their contexts, and to decompose the error in the analogy equality into several components, including a paraphrase error

term measuring the degree to which the critical paraphrase holds. Making use of an assumption that the word2vec embedding is a linear transformation of the PMI matrix, they argue that results in terms of PMI apply to word vectors. Thus, under their explanation, a major part of success on an analogy

is due to and being close distributional paraphrases.

We first review the literature on the analogy test itself, underlining known pitfalls which any explanation of linguistic regularities must navigate. We then show empirically that the relation between the PMI matrix and word2vec embeddings is to some degree linear, which may be enough to satisfy the assumption of Allen and Hospedales (2019). We further examine the proposed decomposition into error terms. We demonstrate that, empirically, these error terms tend to be undefined due to data sparseness, undermining their explanatory force. Most importantly, examining a number of analogies which pass the standard test, we show that the critical paraphrase error term is, contrary to the proposed explanation, very large.111Code is available at www.github.com/bootphon/paraphrases_do_not_explain_analogies.

2 Related work

Early works proposing explanations of the analogical properties of word embeddings include Mikolov et al. (2013b) and Pennington et al. (2014). A geometrical explanation is proposed by Arora et al. (2016), but this explanation relies on very strong preconditions, notably, that the word vectors be distributed uniformly in space. Ethayarajh et al. (2019) also propose an explanation, providing a link between the PMI and the norm of word embeddings. However, as pointed out by Allen and Hospedales (2019), this explanation, too, rests on strong assumptions. Notably, the words involved in the analogy are required to be coplanar, a property that seems unlikely in light of the lack of parallelism we discuss in the next section.

3 Issues with the test

Issues have arisen with the standard way of measuring linguistic analogies. Levy and Goldberg (2014), Vylomova et al. (2016), Rogers et al. (2017), and Fournier et al. (2020) all demonstrate that the standard 3CosAdd measure conflates several very different properties of embeddings, simultaneously measuring not only the directional regularities suggested by typical illustrations of vectors in a parallelogram, but also the similarity of individual matched pairs such as king, man, as well as the global arrangement of vectors in semantic fields, such as king, queen, prince, … versus man, woman, child, … in distinct regions of the space. These issues undermine the construct validity of the standard analogy test. This conflation of properties explains certain pathological behaviours of the test Linzen (2016); Rogers et al. (2017). In spite of these issues, Fournier et al. (2020) demonstrate, using alternative measures, that linguistic regularities are nevertheless coded by directional similarities. This parallelism is weak, with directions tending to be closer, in the absolute, to being orthogonal than to being parallel, but is present above chance level (unmatched word pairs).

Thus, before turning to Allen and Hospedales (2019), one of a number of theoretical attempts to explain performance on the 3CosAdd objective, we underscore that such demonstrations run the risk of explaining properties of the test which may be of secondary interest, or, conversely, of placing undue emphasis on the role of directional regularities, which have been shown to play only a small role in success on 3CosAdd.

4 Explaining analogies through paraphrases

For a word and a word which can appear in the context of , the pairwise mutual information is defined as . As shown by Levy and Goldberg (2014), skip-gram word2vec with negative sampling factorizes the PMI: PMI , with and the word and context embedding matrices of a word2vec model.

For two pairs of words and from the same semantic relation, the standard arithmetic analogy test criterion is that . Writing , and the PMI vector of , Allen and Hospedales (2019) show that is possible to rewrite the arithmetic analogy formula with PMI vectors, and to decompose the error in the equality into five terms as follows:

(1)

The error terms are vectors of length (vocabulary size), with each element defined as:

(2)

The authors claim that these terms can be embedded linearly into a word2vec embedding space by multiplying them by the Moore-Penrose pseudo-inverse of the context matrix . Then with the word2vec embedding of , . Thus we get the final decomposition:

(3)

The paraphrase error term is claimed to be small for successful analogies. Elaborating on the notation, is taken to paraphrase if, wherever all appear together, we observe the same distribution of surrounding words as for . The paraphrase error assesses the similarity of the distributions of words in the context of (all words in appearing together) versus .

5 Linearity of the link between PMI and word2vec

Though it is true that there is a relation between the word2vec matrices and the PMI matrix, in practice the link is more complicated than simple linear matrix factorization, due in part to the training tricks described in Mikolov et al. (2013a). The result of Allen and Hospedales (2019) requires that the embedding from PMI vectors to word2vec embeddings be “linear enough” for PMI to approximate .

To assess this, we use the text8 corpus 222 A text dataset composed of 100 million characters from Wikipedia: Mahoney (2006). both to train word2vec embeddings 333Skip-gram architecture with negative sampling (1 word), negative sampling exponent equal to 1, no undersampling of common words, and a high dimension size of 500. These parameters allow us to be as close as possible to a direct factorization of the PMI matrix.

and to estimate a PMI matrix. We replace infinite values in the PMI matrix by 0. In Figure

0(a), we show the distribution of the Pearson correlation coefficient (assessing the presence of a linear relation) between the word2vec embedding and the corresponding row of PMI for the top ten thousand words in the corpus. As can be seen from the figure, the correlation tends to be between 0.5 and 0.8. For instance in Figure 0(b), the word2vec embedding for king is plotted against the row of PMI corresponding to king.

While the relation is not perfectly linear—many words have a correlation of around 0.55, far lower than that of king—the empirical relations shown here leave open the possibility that it may indeed be “sufficiently linear” to be taken for granted. However, while linearity is necessary for the result of Allen and Hospedales (2019) to go through, it is not sufficient. In the next section, we assess the critical question of whether the paraphrase error is small enough to serve as an explanation for the success of linguistic analogies.

(a) Figure 1a: Histogram of the Pearson correlations between true and approximated word2vec embeddings for the top ten thousand words in the text8 corpus. The mean value is

and the variance is

.
(b) Figure 1b: Plot of the values of the word2vec embedding for king, versus coefficients for the row of PMI corresponding to king, for word2vec matrices trained on the same corpus (text8). The Pearson correlation is one of the best possible at 0.825.

6 Empirical analysis of the error terms

We now seek to examine the proposed explanation by calculating the proposed error terms empirically. However, in practice, many of the terms are undefined, since they rely on cooccurrences unattested in practical corpora. The most extreme situation occurs when the two words of a paraphrase are never present in the same context window in the corpus. We found that only 16% of the paraphrase sets associated with the BATS analogy set Gladkova et al. (2016)—for example, king, woman—were present together in the text8 corpus in a context window of length five. We refer to such paraphrase sets as “well-defined” with respect to the corpus. The problem of zero co-occurrence counts was anticipated by Allen and Hospedales (2019), who propose to restrict their analysis to the case where the context window is sufficiently large that all relevant terms are well defined. We stress that our trained word2vec vectors are also trained with a context window of five, and yield expected levels of performance on the BATS analogy test, despite having access to little training data on which to model co-occurrences such as king, woman, queen, man, and so on.

Category I01 I02 I05 I06 I07 I08 I09 I10 D02 D03 D05 D08 D10 E01 E02
Paraphrase error norm 177 153 111 127 126 124 138 97 102 122 130 110 107 124 176
Dependence errors sum norm 1006 938 867 903 957 883 952 908 856 893 514 585 699 749 848
All errors sum norm 1032 957 878 917 970 897 966 916 864 905 539 602 710 765 875
Category E03 E04 E05 E08 E09 E10 L02 L03 L04 L05 L06 L07 L08 L09 L10
Paraphrase error norm 162 176 155 229 179 190 197 189 209 206 133 169 185 175 432
Dependence errors sum norm 866 797 519 739 910 833 642 982 907 1103 921 995 1044 1017 1302
All errors sum norm 889 822 553 778 933 865 683 1007 939 1131 937 1016 1066 1040 1416
Table 1: L2 norms of the error terms in 1, following our implementation.
Category I01 I02 I05 I06 I07 I08 I09 I10 D02 D03 D05 D08 D10 E01 E02
Average rank 7762K 7589K 7759K 8744K 8160K 6454K 7028K 11889K 31952K 19558K 7857K 1506K 2556K 4394K 9507K
Median rank 1630K 2195K 3055K 3239K 2530K 4090K 3004K 4535K 6754K 3564K 3260K 1506K 2556K 2117K 1622K
Category E03 E04 E05 E08 E09 E10 L02 L03 L04 L05 L06 L07 L08 L09 L10
Average rank 1305K 5611K 9192K 727K 8421K 11946K 52183K 1857K 12687K 6747K 2475K 7727K 4502K 4679K 16871K
Median rank 695K 1703K 1426K 854K 1908K 169K 52182K 1261K 2460K 1343K 2255K 2136K 1549K 1 739K 785K
Table 2: For an analogy equivalent to two paraphrases and , the rank of in the list of the closest paraphrases to with respect to the L2 norm of the paraphrase error vector. 7762K means a rank of 7762000, rounded to the nearest thousand.

At a minimum, if the proposed explanation holds, the cases for which the error terms are empirically well-defined should show signs of the paraphrase error being relatively small. We now detail how we implemented the error terms in cases for which they were well-defined. We count co-occurrences in text8 for all triplets of words , with at the center of the context window, and any paraphrase, both occurring anywhere within a context window of width five. We restrict analysis to the ten thousand most frequent word types and , yielding possible paraphrases.444 is allowed to vary over all of the types included in the training for word2vec, of which there are 71290. Thus, for each paraphrase, the error vectors have 71290 elements, one for each vocabulary word. We use the relative frequencies as estimators of and , and marginalize to obtain , and . The error terms follow. Since this can still lead to ill-defined elements, we replace and by , with (within reason, the value of is immaterial). We also replace with 0.

Table 1 shows the mean and median values of the L2 norms of the paraphrase error vectors across several categories of the BATS dataset. We compare them with the sum of the four dependence error terms (the dependence error reflects statistical dependencies within and irrelevant to the analogy), as well as the sum of all five error terms (equal to the difference between the PMI of and ).The paraphrase error is indeed smaller than the other error terms. However, as we now show, the paraphrase error is not small enough to contribute substantially to the success of analogies.555We note also that the error values seem relatively consistent between categories, while success on the analogy test varies differ greatly between categories.

Take the norm of the paraphrase error vector as a measure of the divergence in the PMI between two paraphrases. For an analogy with associated paraphrases and , we assess how many paraphrases are closer to than to by calculating the rank of the norm of among all , where spans over all pairs of words constructible from the top ten thousand most frequent words in the corpus. To do so, we define a Paraphrase Conditional Information matrix (PCI). For and , we define , the value at column and row to be , where with is a unique index associate with tuple . We compute only the positive PCI, to obtain a sparse matrix. The difference between two PCI columns is a paraphrase error vector, and their Euclidean distance is the norm of the paraphrase error.

We now compute, for each analogy, the distance between the PCI column of and every other column (paraphrase) of the PCI matrix. We calculate the rank of the true analogy pair . Given that the analogy test generally succeeds in picking out as being the most similar to out of the entire vocabulary (modulo Linzen 2016), we would expect that, for successful analogies, the paraphrase error for the true analogy would be among the highest, if small paraphrase error were the explanation for success. Table 2 displays the mean of this rank within each BATS category. The rank is extremely low (in the millions), making the paraphrase error in true analogies far too high to be the explanation for their success.666Limiting the search to the paraphrases composed by at least one of the words of still results in a very low rank for .

7 Conclusion

Recent work has shown that, in spite of the standard analogy test’s confound with simple vector similarity, distributional word vectors genuinely do encode linguistic regularities as directional regularities above and beyond vector similarity (Fournier et al., 2020). Further research is warranted into the mechanisms by which distributional word embeddings come to show these regularities. However, the analysis of analogies as paraphrases does not hold up as an explanation of performance on the analogy test—nor would an explanation of performance on the 3CosAdd analogy test be a satisfying result, since the test is not a useful measure to begin with.

Acknowledgments

This work was funded in part by the European Research Council (ERC-2011-AdG-295810 BOOTPHON), the Agence Nationale pour la Recherche (ANR-17-EURE-0017 Frontcog, ANR-17-CE28-0009 GEOMPHON, ANR-10-IDEX-0001-02 PSL*, ANR-19-P3IA-0001 PRAIRIE 3IA Institute, ANR-18-IDEX-0001 U de Paris, ANR-10-LABX-0083 EFL) and grants from CIFAR (Learning in Machines and Brains), Facebook AI Research (Research Grant), Google (Faculty Research Award), Microsoft Research (Azure Credits and Grant), and Amazon Web Service (AWS Research Credits).

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