Cross-Lingual Word Embedding Refinement by $\boldsymbol{\ell}_{1}$ Norm Optimisation

Cross-Lingual Word Embedding (CLWE) techniques have recently received significant attention as an effective means to support Natural Language Processing applications for low-resource languages, e.g., machine translation (Artetxe et al., 2018b) and transfer learning (Peng et al., 2021).

The most successful CLWE models are the so-called projection-based methods, which learn mappings between monolingual word vectors with very little, or even zero, cross-lingual supervision (Lample et al., 2018; Artetxe et al., 2018a; Glavaš et al., 2019). Mainstream projection-based CLWE models typically identify orthogonal mappings by minimising the topological dissimilarity between source and target embeddings based on $\ell_{2}$ loss (aka. Frobenius loss or squared error) Glavaš et al. (2019); Ruder et al. (2019). This learning strategy has two advantages. First, adding the orthogonality constraint to the mapping function has been demonstrated to significantly enhance the quality of CLWEs (Xing et al., 2015). Second, the existence of a closed-form solution to the $\ell_{2}$ optima (Schönemann, 1966) greatly simplifies the computation required (Artetxe et al., 2016; Smith et al., 2017).

Despite its popularity, work in various application domains has noted that $\ell_{2}$ loss is not robust to noise and outliers. It is widely known in computer vision that $\ell_{2}$-loss-based solutions can severely exaggerate noise, leading to inaccurate estimates (Aanaes et al., 2002; De La Torre and Black, 2003). In data mining, Principal Component Analysis (PCA) using $\ell_{2}$ loss has been shown to be sensitive to the presence of outliers in the input data, degrading the quality of the feature space produced (Kwak, 2008). Previous studies have demonstrated that the processes used to construct monolingual and cross-lingual embeddings may introduce noise (e.g. via reconstruction error (Allen and Hospedales, 2019) and structural variance (Ruder et al., 2019)), making the presence of outliers more likely. Empirical analysis of CLWEs also demonstrates that more distant word pairs (which are more likely to be outliers) have more influence on the behaviour of $\ell_{2}$ loss than closer pairs. This raises the question of the appropriateness of $\ell_{2}$ loss functions for CLWEs.

Compared to the conventional $\ell_{2}$ loss, $\ell_{1}$ loss (aka. Manhattan distance) has been mathematically demonstrated to be less affected by outliers (Rousseeuw and Leroy, 1987) and empirically proven useful in computer vision and data mining (Aanaes et al., 2002; De La Torre and Black, 2003; Kwak, 2008). Motivated by this insight, our paper proposes a simple yet effective post-processing technique to improve the quality of CLWEs: adjust the alignment of any cross-lingual vector space to minimise the $\ell_{1}$ loss without violating the orthogonality constraint. Specifically, given existing CLWEs, we bidirectionally retrieve bilingual vectors and optimise their Manhattan distance using a numerical solver. The approach can be applied to any CLWEs, making the post-hoc refinement technique generic and applicable to a wide range of scenarios. We believe this to be the first application of $\ell_{1}$ loss to the CLWE problem.

To demonstrate the effectiveness of our method, we select four state-of-the-art baselines and conduct comprehensive evaluations in both supervised and unsupervised settings. Our experiments involve ten languages from diverse branches/families and embeddings trained on corpora of different domains. In addition to the standard Bilingual Lexicon Induction (BLI) benchmark, we also investigate a downstream task, namely cross-lingual transfer for Natural Language Inference (NLI). In all setups tested, our algorithm significantly improves the performance of strong baselines. Finally, we provide an intuitive visualisation illustrating why $\ell_{1}$ loss is more robust than it $\ell_{2}$ counterpart when refining CLWEs (see Fig. 1). Our code is available at https://github.com/Pzoom522/L1-Refinement.

Our contribution is three-fold: (1) we propose a robust refinement technique based on the $\ell_{1}$ norm training objective, which can effectively enhance CLWEs; (2) our approach is generic and can be directly coupled with both supervised and unsupervised CLWE models; (3) our $\ell_{1}$ refinement algorithm achieves state-of-the-art performance for both BLI and cross-lingual transfer for NLI tasks.

A common characteristic of CLWE methods that apply the orthogonality constraint is that they optimise using $\ell_{2}$ loss (see § 2). However, outliers have disproportionate influence in $\ell_{2}$ since the penalty increases quadratically and this can be particularly problematic with noisy data since the solution can ‘‘shift’’ towards them (Rousseeuw and Leroy, 1987). The noise and outliers present in real-world word embeddings may affect the performance of $\ell_{2}$-loss-based CLWEs.

The $\ell_{1}$ norm cost function is more robust than $\ell_{2}$ loss as it is less affected by outliers (Rousseeuw and Leroy, 1987). Therefore, we propose a refinement algorithm for improving the quality of CLWEs based on $\ell_{1}$ loss. This novel method, which we refer to as $\ell_{1}$ refinement, is generic and can be applied post-hoc to improve the output of existing CLWE models. To our knowledge, the use of alternatives to $\ell_{2}$-loss-based optimisation has never been explored by the CLWE community.

To begin with, analogous to $\ell_{2}$ OPA (cf. Eq. (1)), $\ell_{1}$ OPA can be formally defined and rewritten as

where $\operatorname{tr}(\cdot)$ returns the matrix trace, $\operatorname{sgn}(\cdot)$ is the signum function, and $\in\mathcal{O}$ denotes that $M$ is subject to the orthogonal constraint. Compared to $\ell_{2}$ OPA which has a closed-form solution, solving Eq. (3) is much more challenging due to the discontinuity of $\operatorname{sgn}(\cdot)$. This issue can be addressed by replacing $\operatorname{sgn}(\cdot)$ with $\tanh(\alpha(\cdot))$, a smoothing function parameterised by $\alpha$, such that

Larger values for $\alpha$ lead to closer approximations to $\operatorname{sgn}(\cdot)$ but reduce the smoothing effect. This approach has been used in many applications, such as the activation function of long short-term memory networks (Hochreiter and Schmidhuber, 1997).

However, in practice, we find that Eq. (4) remains unsolvable in our case with standard gradient-based frameworks for two reasons. First, $\alpha$ has to be sufficiently large in order to achieve a good approximation of $\operatorname{sgn}(\cdot)$. Otherwise, relatively small residuals will be down-weighted during fitting and the objective will become biased towards outliers, just similar to $\ell_{2}$ loss. However, satisfying this requirement (i.e., large $\alpha$) will lead to the activation function $\tanh(\alpha(\cdot))$ becoming easily saturated, resulting in an optimisation process that becomes trapped during the early stages. In other words, the optimisation can only reach an unsatisfactory local optimum. Second, the orthogonality constraint (i.e., $\mathbf{M}\in\mathcal{O}$) also makes the optimisation more problematic for these methods.

We address these challenges by adopting the approaches proposed by Trendafilov (2003). This method explicitly encourages the solver to only explore the desired manifold $\mathcal{O}$ thereby reducing the $\ell_{1}$ solver’s search space and difficulty of the optimisation problem. We begin by calculating the gradient $\nabla$ w.r.t. the objective in Eq. (4) through matrix differentiation:

where $\mathbf{Z}$$=$$\alpha(\mathbf{A}\mathbf{M}$$-$$\mathbf{B})$ and $\odot$ is the Hadamard product. Next, to find the steepest descent direction while ensuring that any $\mathbf{M}$ produced is orthogonal, we project $\nabla$ onto $\mathcal{O}$, yielding¹¹1See Chu and Trendafilov (2001) for derivation details.

Here $\mathbf{I}$ is an identity matrix with the shape of $\mathbf{M}$. With Eq. (6) defining the optimisation flow, our $\ell_{1}$ loss minimisation problem reduces to an integration problem, as

where $\mathbf{M_{0}}$ is a proper initial solution of Eq. (3) (e.g., $\ell_{2}$-optimal mapping obtained via Eq. (2)).

Empirically, unlike the aforementioned standard gradient-based methods, by following the established policy of Eq. (6), the optimisation process of Eq. (7) will not violate the orthogonality restriction or get trapped during early stages. However, this $\ell_{1}$ OPA solver requires extremely small step size to generate reliable solutions (Trendafilov, 2003), making it computationally expensive²²2It takes averagely 3 hours and up to 12 hours to perform Eq. (7) on an Intel Core i9-9900K CPU. In comparison, the time required to solve Eq. (2) in each training loop is less than 1 second and the iterative $\ell_{2}$-norm-based training takes 1 to 5 hours in total.. Therefore, it is impractical to perform $\ell_{1}$ refinement in an iterative fashion like $\ell_{2}$ refinement without significant computational resources.

Previous work has demonstrated that applying the $\ell_{1}$-loss-based algorithms from a good initial state can speed up the optimisation. For instance, Kwak (2008) found that feature spaces created by $\ell_{2}$ PCA were severely affected by noise. Replacing the cost function with $\ell_{1}$ loss significantly reduced this problem, but required expensive linear programming. To reduce the convergence time, Brooks and Jot (2013) exploited the first principal component from the $\ell_{2}$ solution as an initial guess. Similarly, when reconstructing corrupted pixel matrices, $\ell_{2}$-loss-based results are far from satisfactory; using $\ell_{1}$ norm estimators can improve the quality, but are too slow to handle large-scale datasets Aanaes et al. (2002). However, taking the $\ell_{2}$ optima as the starting point allowed less biased reconstructions to be learned in an acceptable time De La Torre and Black (2003).

Inspired by these works, we make use of $\ell_{1}$ refinement to carry out post-hoc enhancement of existing CLWEs. Our full pipeline is described in Algorithm 1 (see § 4.3 for implemented configurations). In common with $\ell_{2}$ refinement (cf. § 2), steps 1-4 bootstrap a synthetic dictionary $D$ and compose bilingual word vector matrices $\mathbf{A}$ and $\mathbf{B}$ which have reliable row-wise correspondence. Taking them as the starting state, in step 5 an identity matrix naturally serves as our initial solution $\mathbf{M_{0}}$.

During the execution of Eq. (7), we record $\ell_{1}$ loss per iteration and see if either of the following two stopping criteria have been satisfied: (1) the updated $\ell_{1}$ loss exceeds that of the previous iteration; (2) on-the-fly $\mathbf{M}$ has non-negligibly departed from the orthogonal manifold, which can be indicated by the maximum value of the disparity matrix as

where $\epsilon$ is a sufficiently small threshold. The resulting $\mathbf{M^{\star}}$ can be used to adjust the word vectors of $\mathtt{L_{A}}$ and output refined CLWEs.

A significant advantage of our algorithm is its generality: it is fully independent of the method used for creating the original CLWEs and can therefore be used to enhance a wide range of models, both in supervised and unsupervised settings.

We evaluate the effectiveness of the proposed $\ell_{1}$ refinement technique on two benchmarks: Bilingual Lexicon Induction (BLI), the de facto standard for measuring the quality of CLWEs, and a downstream natural language inference task based on cross-lingual transfer. In addition to comparison against state-of-the-art CLWE models, we also report the performance of the single-iteration $\ell_{2}$ refinement method which follows steps 1-4 of Algorithm 1 then minimises $\ell_{2}$ loss in the final step.

To reduce randomness, we executed each model in each setup three times and the average accuracy (ACC, aka. precision at rank 1) is reported. Following Glavaš et al. (2019), by comparing scores achieved before and after $\ell_{1}$ refinement, statistical significance is indicated via the $p$-value of two-tailed t-tests with Bonferroni correction (Dror et al., 2018) (note that $p$-values are not recorded for Tab. 2(b) given the small number of runs).

This paper proposes a generic post-processing technique to enhance CLWE performance based on optimising $\ell_{1}$ loss. This algorithm is motivated by successful applications in other research fields (e.g. computer vision and data mining) which exploit the $\ell_{1}$ norm cost function since it has been shown to be more robust to noisy data than the commonly-adopted $\ell_{2}$ loss. The approach was evaluated using ten diverse languages and word embeddings from different domains on the popular BLI benchmark, as well as a downstream task of cross-lingual transfer for NLI. Results demonstrated that our algorithm can significantly improve the quality of CLWEs in both supervised and unsupervised setups. It is therefore recommended that this straightforward technique be applied to improve performance of CLWEs.

The convergence speed of the optimiser prevented us from performing $\ell_{1}$ loss optimisation over multiple iterations. Future work will focus on improving the efficiency of our $\ell_{1}$ OPA solver, as well as exploring the application of other robust loss functions within CLWE training strategies.

This work provides an effective post-hoc method to improve CLWEs, advancing the state-of-the-art in both supervised and unsupervised settings. Our comprehensive empirical studies demonstrate that the proposed algorithm can facilitate researches in machine translation, cross-lingual transfer learning, etc, which have deep societal impact of bridging cultural gaps across the world.

Besides, this paper introduces and solves an optimisation problem based on an under-explored robust cost function, namely $\ell_{1}$ loss. We believe it could be of interest for the wider community as outlier is a long-standing issue in many artificial intelligence applications.

One caveat with our method, as is the case for all word-embedding-based systems, is that various biases may exist in vector spaces. We suggest this problem should always be looked at critically. In addition, our implemented solver can be computationally expensive, leading to increased electricity consumption and the associated negative environmental repercussions.

This work is supported by the award made by the UK Engineering and Physical Sciences Research Council (Grant number: EP/P011829/1) and Baidu, Inc. We would also like to express our sincerest gratitude to Guanyi Chen, Ruizhe Li, Xiao Li, Shun Wang, and the anonymous reviewers for their insightful and helpful comments.