Unsupervised Neural Machine Translation

Introduction

The paper presents an unsupervised method to machine translation using only monoligual corpora without any alignment between sentences or documents. Monoligual corpora are text corpora that is made up of one language only. This contrasts with the usual translation approach that uses parallel corpora, where two corpora are the direct translation of each other and the translations are aligned by words or sentences. This problem is important as there are a large number of languages that lack parallel pairing, e.g. for German-Russian.

The general approach of the methodology is to:

1. Using monolingual corpora in the source and target languages to learn source and target word embeddings.
2. Align the 2 sets of word embeddings in the same latent space.

Then iteratively perform:

1. Train an auto-encoder to reconstruct noisy versions of sentence embeddings for both source and target language, where the encoder is shared and the decoder is different in each language.
2. Tune the decoder in each language by back-translating between the source and target language.

The paper employs methodology developed in a few other papers to do the first 2 steps to preprocess the corpora. Then for iteratively training the translation model in the last 2 steps, it uses a standard encoder-decoder with attention mechanism as shown in Figure 1.

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Background

Word Embedding Alignment

The paper uses word2vec [Mikolov, 2013] to convert each monoligual corpora to vector enbeddings. These embeddings have been shown to contain the contextual and syntactic features independent of language, and so in theory there could exist a linear map that maps the embeddings from language L1 to language L2.

Figure 2 shows the word embeddings in English and French (a & b), and (c) shows the aligned word embeddings after some linear transformation.

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The paper uses the methodology proposed by [Artetxe, 2018] to do cross-lingual embedding aligning in an unsupervised manner and without parallel data. Without going into the details, the general approach of this paper is starting from a seed dictionary of numeral pairings (e.g. 1-1, 2-2, etc.), to iteratively learn the mapping between 2 language embeddings, while concurrently improving the dictionary at each iteration.

Methodology

The model uses a sequence to sequence model with attention, without input-feeding. Both the encoder and decoder are 3 layer LSTMs, and the encoder is bidirectional. The encoder and decoder are invariant to the language being used, as there is only 1 set of parameters for the encoder, and another set for the decoder.

The objective function that proposed by the paper is a combination of 3 component objective functions:

1. Reconstruction loss of the denoising auto-encoder
2. Cross domain translation loss of the auto-encoder
3. Adversarial cross entropy loss of the discriminator

Notations

[math]\displaystyle{ \mathcal{W}_S, \mathcal{W}_T }[/math] are the sets of words in the source language domain.

[math]\displaystyle{ \mathcal{Z}^S , \mathcal{Z}^T }[/math] are the sets of word embeddings in the source and target language domain.

[math]\displaystyle{ \ell \in \{src, tgt\} }[/math] denote the source or target language

[math]\displaystyle{ x \in \mathbb{R}^m }[/math] is a vector of m words in a particular language [math]\displaystyle{ \ell }[/math]

[math]\displaystyle{ e_{\theta_{enc},\mathcal{Z}}(x, \ell) }[/math] is the encoder parameterized by [math]\displaystyle{ \theta_{enc} }[/math], it takes as input [math]\displaystyle{ x }[/math] and [math]\displaystyle{ \ell }[/math] and computes [math]\displaystyle{ z \in \mathbb{R}^m }[/math], which is a sequence of m hidden states using embedding [math]\displaystyle{ \mathcal{Z}^{\ell} }[/math]

[math]\displaystyle{ d_{\theta_{dec},\mathcal{Z}}(z, \ell) }[/math] is the decoder parameterized by [math]\displaystyle{ \theta_{dec} }[/math], it takes as input [math]\displaystyle{ z }[/math] and [math]\displaystyle{ \ell }[/math] and computes [math]\displaystyle{ y \in \mathbb{R}^k }[/math], which a sequence of k words from vocabulary [math]\displaystyle{ \mathcal{W}^{\ell} }[/math]

Noise Model

The Noise model used throughout the paper [math]\displaystyle{ C(x) }[/math] is a randomly sampled noisy version of sentence [math]\displaystyle{ x }[/math]. Noise is added in 2 ways:

1. Randomly dropping each word in the sentence with probability [math]\displaystyle{ p_{wd} }[/math].
2. Slightly shuffling the words in the sentence where each word can be at most [math]\displaystyle{ k }[/math] positions away from its original position.

The authors found in practice [math]\displaystyle{ p_{wd}= 0.1 }[/math] and [math]\displaystyle{ k=3 }[/math] to be good parameters.

Loss Component 1: Reconstruction Loss

This component captures the expected cross entropy loss between [math]\displaystyle{ x }[/math] and the reconstructed [math]\displaystyle{ \hat{x} }[/math], where [math]\displaystyle{ \hat{x} }[/math] is constructed as follows:

1. Construct [math]\displaystyle{ C(x) }[/math], noisy version of [math]\displaystyle{ x }[/math] from a language [math]\displaystyle{ \ell }[/math]
2. Input [math]\displaystyle{ C(x) }[/math] and language [math]\displaystyle{ \ell }[/math] into the encoder parameterized with [math]\displaystyle{ \theta_{enc} }[/math], to get [math]\displaystyle{ e(C(x),\ell) }[/math].
3. Input the [math]\displaystyle{ e(C(x),\ell) }[/math] and [math]\displaystyle{ \ell }[/math] into the decoder parameterized with [math]\displaystyle{ \theta_{dec} }[/math], to get [math]\displaystyle{ \hat{x} \sim d(e(C(x),\ell),\ell) }[/math].

\begin{align} \mathcal{L}_{auto}(\theta_{enc}, \theta_{dec}, \mathcal{Z}, \ell) = E_{x\sim D_\ell, \hat{x}\sim d(e(C(x),\ell),\ell)}[\Delta(\hat{x},x)] \end{align}

Loss Component 2: Cross Domain Translation Loss

This component captures the expected cross entropy loss between [math]\displaystyle{ x }[/math] and the reconstructed [math]\displaystyle{ \hat{x} }[/math] from the translation of [math]\displaystyle{ x }[/math], where [math]\displaystyle{ \hat{x} }[/math] is constructed as follows:

1. Using the current iteration of the translation model [math]\displaystyle{ M }[/math], construct translation [math]\displaystyle{ M(x) }[/math] in [math]\displaystyle{ \ell_2 }[/math], where [math]\displaystyle{ x }[/math] is from a language [math]\displaystyle{ \ell_1 }[/math]. (Initialization of M is using a different translation model discussed later)
2. Construct [math]\displaystyle{ C(M(x)) }[/math], noisy version of translation [math]\displaystyle{ M(x) }[/math].
3. Input [math]\displaystyle{ C(M(x)) }[/math] and language [math]\displaystyle{ \ell_2 }[/math] into the encoder parameterized with [math]\displaystyle{ \theta_{enc} }[/math], to get [math]\displaystyle{ e(C(M(x)),\ell_2) }[/math].
4. Input [math]\displaystyle{ e(C(M(x)),\ell_2) }[/math] and [math]\displaystyle{ \ell_1 }[/math] into the decoder parameterized with [math]\displaystyle{ \theta_{dec} }[/math], to get [math]\displaystyle{ \hat{x} \sim d(e(C(M(x)),\ell_2),\ell_1) }[/math].

\begin{align} \mathcal{L}_{cd}(\theta_{enc}, \theta_{dec}, \mathcal{Z}, \ell_1,\ell_2) = E_{x\sim D_{\ell_1}, \hat{x}\sim d(e(C(M(x)),\ell_2),\ell_1)}[\Delta(\hat{x},x)] \end{align}

Loss Component 3: Adversarial Loss

A discriminator parameterized with [math]\displaystyle{ \theta_D }[/math] is trained to to distinguish the language [math]\displaystyle{ \ell }[/math] given a vector [math]\displaystyle{ z }[/math] in the latent space. It is trained by minimizing the cross entropy loss [math]\displaystyle{ \mathcal{L}_D }[/math] of the predicted language and the ground truth language, given the language produced the vector [math]\displaystyle{ z }[/math].

The enconder is trained to fool the discriminator, and the adversarial loss is minimized when given an encoding of [math]\displaystyle{ x }[/math] in language [math]\displaystyle{ \ell_i }[/math], the discriminator predicts that it comes from [math]\displaystyle{ \ell_j }[/math].

The end result at convergence is that the representation in the latent space for language [math]\displaystyle{ \ell_1 }[/math] is indistinguishable from language [math]\displaystyle{ \ell_2 }[/math].

\begin{align} \mathcal{L}_{adv}(\theta_{enc}, \mathcal{Z}|\theta_D) = -E_{x_i,\ell_i}[log p_D (\ell_j|e(x_i,\ell_i))] \end{align} with [math]\displaystyle{ \ell_j=\ell_1 }[/math] if [math]\displaystyle{ \ell_i=\ell_2 }[/math], and vice versa.

Final Objective Loss Function

Combining all 3 components the following objective function is obtained:

\begin{align*} \mathcal{L}_{auto}(\theta_{enc}, \theta_{dec}, \mathcal{Z}) &= \lambda_{auto}[\mathcal{L}_{auto}(\theta_{enc}, \theta_{dec}, \mathcal{Z},src)+\mathcal{L}_{auto}(\theta_{enc}, \theta_{dec}, \mathcal{Z},tgt)]\\ &+ \lambda_{cd}[\mathcal{L}_{cd}(\theta_{enc}, \theta_{dec}, \mathcal{Z},src,tgt) +\mathcal{L}_{cd}(\theta_{enc}, \theta_{dec}, \mathcal{Z},tgt,src)]\\ &+\lambda_{adv}\mathcal{L}_{adv}(\theta_{enc}, \mathcal{Z}|\theta_D) \end{align*}

[math]\displaystyle{ \lambda_{auto} }[/math], [math]\displaystyle{ \lambda_{cd} }[/math], and [math]\displaystyle{ \lambda_{adv} }[/math] are hyperparameters that represent the weights of each component. The discriminator loss [math]\displaystyle{ \mathcal{L}_D }[/math] is minimized in parallel, because its parameter [math]\displaystyle{ \theta_{D} }[/math] is used in the last component.

• • insert Figure2 here**

Training

The training is iterative, where the translation model [math]\displaystyle{ M^{(t)} }[/math]improves at each time step [math]\displaystyle{ t }[/math]. To seed the training, [math]\displaystyle{ M^{(1)} }[/math] is a translation from a different unsupervised word-by-word translation as proposed by [Conneau, 2017]. Each iteration of the training is as follows:

1. Use [math]\displaystyle{ M^{(t)} }[/math] to obtain a translation [math]\displaystyle{ M^{(t)}(x) }[/math].
2. Use [math]\displaystyle{ M^{(t)}(x) }[/math] and [math]\displaystyle{ x }[/math] to train auto-encoder, training discriminator at the same time (i.e. minimizing final objective function)
3. Update [math]\displaystyle{ M^{(t+1)} }[/math], repeat.

• • inser Algorithm1 here**

Model Selection Criterion

In Machine Translation, the Bilingual Evaluation Understudy(BLEU) Score is typically used to evaluate the quality of the translation, using a reference (groud-truth) translation. However since the training is unsupervised without parallel copora, BLEU cannot be used during training to select hyper-parameters.

The paper proposes a scoring method that correlates with the BLEU. The main idea is to assess BLEU score between [math]\displaystyle{ x }[/math] and the back-translated version using the model(i.e. translate [math]\displaystyle{ x }[/math] to the target language then translate it back to source language). With this it is possible to score the quality of the translation model without supervision.

References

1. [Lample, 2018] Lample, G., Conneau, A., Ranzato, M., Denoyer, L., "Unsupervised Machine Translation Using Monolingual Corpora Only". arXiv:1711.00043

1. [Conneau, 2017] Conneau, A., Lample, G., Ranzato, M., Denoyer, L., Jégou, H., "Word Translation without Parallel Data". arXiv:1710.04087