# Decoding Challenge Problem 2

Decoding is process of taking input in French:

honorables sénateurs , que se est - il passé ici , mardi dernier ?

…And finding its best English translation under your model:

honourable senators , what happened here last Tuesday ?

To decode, we need a model of English sentences conditioned on the French sentence. You did most of the work of creating such a model in Homework 1. In this assignment, we will give you some French sentences and a probabilistic model consisting of a phrase-based translation model $p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e})$ and an n-gram language model $p_{\textrm{LM}}(\textbf{e})$. Your challenge is to find the most probable English translation under the model. We assume a noisy channel decomposition.

\begin{align*} \textbf{e}^* & = \arg \max_{\textbf{e}} p(\textbf{e} \mid \textbf{f}) \\ & = \arg \max_{\textbf{e}} \frac{p_{\textrm{TM}}(\textbf{f} \mid \textbf{e}) \times p_{\textrm{LM}}(\textbf{e})}{p(\textbf{f})} \\ &= \arg \max_{\textbf{e}} p_{\textrm{TM}}(\textbf{f} \mid \textbf{e}) \times p_{\textrm{LM}}(\textbf{e}) \\ &= \arg \max_{\textbf{e}} \sum_{\textbf{a}} p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e}) \times p_{\textrm{LM}}(\textbf{e}) \end{align*}

## Getting Started

If you have a clone of the repository from homework 1, you can update it from your working directory:

git pull origin master

Alternatively, get a fresh copy:

git clone https://github.com/alopez/en600.468.git

Under the decode directory, you now have simple decoder. Test it out!

python decode > output

This creates the file output with translations of data/input. You can compute $p(\textbf{e} \mid \textbf{f})$ using compute-model-score.

python compute-model-score < output

This command sums over all possible ways that the model could have generated the English from the French, including translations that permute the phrases. This sum is intractable, but the phrase dictionary is fixed and sparse, so we can compute it in a few minutes. It is still easier to do this than it is to find the optimal translation. But if you look at this command you may get some hints about how to do the assignment!

The decoder generates the most probable translations that it can find, using three common approximations.

First, it seeks the Viterbi approximation to the most probable translation. Instead of computing the intractable sum over all alignments for each sentence, we simply find the best single alignment and use its translation.

\begin{align*} \textbf{e}^* &= \arg \max_{\textbf{e}} \sum_{\textbf{a}} p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e}) \times p_{\textrm{LM}}(\textbf{e}) \\ &\approx \arg \max_{\textbf{e}} \max_{\textbf{a}} p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e}) \times p_{\textrm{LM}}(\textbf{e}) \end{align*}

Second, it translates French phrases into English without changing their order. So, it only reorders words if the reordering has been memorized as a phrase pair. For example, in the first sentence, we see that mardi dernier is correctly translated as last Tuesday. If we consult data/tm, we will find that the model has memorized the phrase pair mardi dernier ||| last Tuesday. But in the second sentence, we see that Comité de sélection is translated as committee selection, rather than the more correct selection committee. To show that this is a search problem rather than a modeling problem, we can generate the latter output by hand and check that the model really prefers it.

head -2 data/input | tail -1 > example
python decode -i example | python compute-model-score -i example
echo a selection committee was achievement . | python compute-model-score -i example

The scores are reported as log-probabilities, and higher scores (with lower absolute value) are better. We see that the model prefers selection committee, but the decoder does not consider this word order.

Finally, our decoder uses strict pruning. As it consumes the input sentence from left to right, it keeps only the highest-scoring output up to that point. You can vary the number of number of outputs kept at each point in the translation using the -s parameter. See how this affects the resulting model score.

python decode | python compute-model-score
python decode -s 10000 | python compute-model-score

## The Challenge

Your task is to find the most probable English translation. Our model assumes that any segmentation of the French sentence into phrases followed by a one-for-one substitution and permutation of those phrases is a valid translation. We make the simplifying assumption that segmentation and ordering probabilities are uniform across all sentences, hence constant. This means that $p(\textbf{e},\textbf{a} \mid \textbf{f})$ is proportional to the product of the n-gram probabilities in $p_{\textrm{LM}}(\textbf{e})$ and the phrase translation probabilities in $p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e})$. To avoid numerical underflow we work in logspace, seeking $\arg \max_{\textbf{e}} \max_{\textbf{a}} \log p_{\textrm{TM}}(\textbf{f},\textbf{a} \mid \textbf{e}) + \log p_{\textrm{LM}}(\textbf{e})$. The baseline decoder works with log probabilities, so you can simply follow what it does.

To pass, you must implement a beam-search decoder like the one we have given you that is also capable of swapping adjacent phrases. To get full credit, you must additionally experiment with another decoding algorithm. Any permutation of phrases is a valid translation, so we strongly suggest searching over all or some part of this larger space. This search is NP-Hard, so it will not be easy. You can trade efficiency for search effectiveness by implementing histogram pruning or threshold pruning, or by using reordering limits as described in the textbook (Chapter 6). Or, you might consider implementing other approaches to solving this combinatorial optimization problem:

These methods all attempt to approximate or solve the Viterbi approximation to decoding. You can also try to approximate $p(\textbf{e} \mid \textbf{f})$ directly.

But the sky’s the limit! There are many ways to decode. You can try anything you want as long as you follow the ground rules:

## Ground Rules

• You can work in independently or in groups of up to three, under these conditions:
1. You must announce the group publicly on piazza.
2. You agree that everyone in the group will receive the same grade on the assignment.
3. You can add people or merge groups at any time before the assignment is due. You cannot drop people from your group once you’ve added them. We encourage collaboration, but we will not adjudicate Rashomon-style stories about who did or did not contribute.
• You must turn in three things:
1. Your translations of the entire dataset, uploaded to the leaderboard submission site according to the Assignment 0 instructions. You can upload new output as often as you like, up until the assignment deadline.
2. Your code. Send us a URL from which we can get the code and git revision history (a link to a tarball will suffice, but you’re free to send us a github link if you don’t mind making your code public). This is due at the deadline: when you upload your final answer, send us the code. You are free to extend the code we provide or roll your own in whatever langugage you like, but the code should be self-contained, self-documenting, and easy to use.
3. A clear, mathematical description of your algorithm and its motivation written in scientific style. This needn’t be long, but it should be clear enough that one of your fellow students could re-implement it exactly. We will review examples in class before the due date.
• You do not need any other data than what we provide. You can free to use any code or software you like, except for those expressly intended to decode machine translation models. You must write your own decoder. If you want to use finite-state libraries, solvers for traveling salesman problems, or integer linear programming toolkits, that is fine. But machine translation software including (but not limited to) Moses, cdec, Joshua, or phrasal is off-limits. You may of course inspect these systems if it helps you understand how they work. But be warned: they are generally quite complicated because they provide a great deal of other functionality that is not the focus of this assignment. It is possible to complete the assignment with a modest amount of python code. If you aren’t sure whether something is permitted, ask us. If you want to do system combination, join forces with your classmates.

Credits: Chris Dyer made many improvements to this assignment.