Unsupervised Machine Translation Using Monolingual Corpora Only

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.

The general approach of the methodology is to first use a unsupervised word-by-word translation model proposed by [Conneau, 2017], then iteratively improve on the translation by utilizing 2 architectures:

1. A denoising auto-encoder to reconstruct noisy versions of sentences for both source and target languages.
2. A discriminator to align the distributions of the source and target languages in a latent space.

Methodology

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 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 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.

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