Decoding : Challenge Problem 2
Decoding is process of taking input that looks like this:
honorables sÃ©nateurs , que se est  il passÃ© ici , mardi dernier ?
...And turning into output that looks like this:
honourable senators , what happened here on Tuesday ?
In order to decode we need a probability model over pairs of English and French sentences. You did most of the work of creating such a model in Challenge 1. In this assignment, we will give you a fixed model consisting of a (phrasebased) translation model and a language model. Your challenge is to find the most probable translation.
Getting Started
If you already have a clone of the repository from homework 1, you can update it from anywhere in your project directory by running the command:
git pull origin master
Alternatively, clone a fresh version of the repository by running:
git clone https://github.com/alopez/dreamt.git
Under the new decode directory, We have provided you with a very simple decoder written in Python (v.2.62.7; older versions will not work!) The decoder translates monotonically — that is, without reordering the English phrases — and by default it also uses very strict pruning limits, which you can vary on the command line. There is also a data directory containing a translation model, a language model, and a set of input sentences to translate. Run the decoder using this command:
decode > output
This loads the models and decodes the input sentences, storing the result in output. You can see the translations simply by looking at the file. To calculate their true model score, run the command:
grade < output
This command computes the probability of the output sentences according to the model. It works by summing over all possible ways that the model could have generated the English from the French. In general this is intractable, but because the phrase dictionary is fixed and sparse, the specific instances here can be computed in a few minutes. It is still easier to do this exactly than it is to find the optimal translation. In fact, if you look at the grade script you may get some hints about how to do the assignment!
Improving the search algorithm in the decoder — for instance by enabling it to search over permutations of English phrases — should permit you to find more probable translations of the input French sentences than the ones found by the baseline system. This assignment differs from Homework 1, in that there is no hidden evaluation measure. The grade program will tell you the probability of your output, and whoever finds the most probable output will receive the most points.
The Challenge
Your task for this assignment is to find the English sentence with the highest possible probability. Formally, this means your goal is to solve the problem: \( \mathop{\arg\,\max}\limits_e~ p(fe) \times p(e) \), where \(f\) is a French sentence and \(e\) is an English sentence. In the model we have provided you, \( p(e) = p(e_1START) \times \prod_{j=2}^J p(e_je_{j1}e_{j2}) \times p(ENDe_J,e_{J1}) \) and \( p(fe) = p(segmentation) \times p(reordering) \times p(phrase~translation) \). We will make the simplifying assumption that segmentation and reordering probabilities are uniform across all sentences, and hence constant. This results in a model whose probability density function does not sum to one. But from a practical perspective, it slightly simplifies the implementation without substantially harming empirical accuracy. This means that you only need consider the product of the phrase translation probabilities \( p(fe) = \prod_{\langle i,i',j,j' \rangle \in a} p(f_{\langle i,i' \rangle}e_{\langle j,j' \rangle}) \) where \( \langle i,i' \rangle \) and \( \langle j,j' \rangle \) index phrases in \(f\) and \(e\), respectively. Unfortunately, even with all of these simplifications, finding the most probable English sentence is completely intractable! To compute it exactly, for each English sentence you would need to compute \( p(fe) \) as a sum over all possible alignments with the French sentence: \( p(fe) = \sum_a p(f,ae) \). A nearly universal approximation is to instead search for the English string together with a single alignment, \(\mathop{\arg\,\max}\limits_{e,a}~ p(f,ae) \times p(e) \). This is the approach taken by the monotone baseline decoder.
Since this involves multiplying together many small probabilities, it is helpful to work in logspace to avoid numerical underflow. We instead solve for \(e,a\) that maximizes: \( \log p(f,ae) + \log p(e) = \log p(e_1START) + \sum_{j=2}^J \log p(e_je_{j1}e_{j2}) + \log p(ENDe_J) + \sum_{\langle i,i',j,j' \rangle \in a} \log p(f_{\langle i,i' \rangle}e_{\langle j,j' \rangle}) \). The baseline decoder already works with log probabilities, so it is not necessary for you to perform any additional conversion; you can simply work with the sum of the scores that the model provides for you. Note that since probabilities are always less than or equal to one, their equivalent values in logspace will always be negative or zero, respectively (you may notice that grade sometimes reports positive translation model scores; this is because the sum it computes does not include the large negative constant associated with the log probabilities of segmentation and reordering). Hence your translations will always have negative scores, and you will be looking for the one with the smallest absolute value. In other words, we have transformed the problem of finding the most probable translation into a problem of finding the shortest path through a large graph of possible outputs.
Under the phrasebased model we've given you, the goal is to find a phrase segmentation, translation of each resulting phrase, and permutation of those phrases such that the product of the phrase translation probabilities and the language model score of the resulting sentence is as high as possible. Arbitrary permutations of the English phrases are allowed, provided that the phrase translations are onetoone and exactly cover the input sentence. Even with all of the simplifications we have made, this problem is still NPComplete, so we recommend that you solve it using an approximate method like stack decoding, discussed in Chapter 6 of the textbook. You can trade efficiency for search effectiveness by implementing histogram pruning or threshold pruning, or by playing around with reordering limits as described in the textbook. Or, you might consider implementing other approaches to solving the search problem:
 Implement a greedy decoder.
 Reduce the problem to a traveling salesman problem (TSP) and decode using an offtheshelf TSP solver.
 Reformulate the problem as a shortestpath problem through a finitestate lattice and decode using finitestate devices.
 Decode using Lagrangian relaxation (often finds exact solutions!)
Several techniques used for the IBM Models (which have very similar search problems as phrasebased models) could also be adapted:
 Implement A* Search.
 Implement a bidirectional decoder.
 Decode using an integer linear programming (ILP) solver.
But the sky's the limit! There are many, many ways to try to solve the decoding problem, and you can try anything you want as long as you follow the ground rules:
Ground Rules

You may work in independently or in groups of any size, under these
conditions:
 You must notify us by posting a public note to piazza.
 Everyone in the group will receive the same grade on the assignment.
 You can add people or merge groups at any time before you post your final submission. HOWEVER, you cannot drop people from your group once you've added them. Collaboration is fine with us, but adjudicating Rashomonstyle stories about who did or did not contribute is not.
 You must turn in three things:
 Your translations of the entire dataset, uploaded to BASE_URL/assignment2.txt following the Assignment 0 instructions. You can upload new output as often as you like. There is no hidden metric for this assignment; the grade program will tell you exactly how probable your output it. Whoever has the most probable output at the deadline will receive the most bonus points.
 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 selfcontained, selfdocumenting, and easy to use.
 A description of your algorithm, posted to piazza. Brevity is encouraged, as long as it is clear what you did; a paragraph or even bullet points is fine. You should tell us not only about your final algorithm, but also things that you tried that didn't work, experiments that you did, or other interesting things that you observed while working on it. What did you learn? What did you think about the quality of the translations? Do you think that more probable translations are better? Please post your response within two days of submitting your final solution to the leaderboard; we will withold your grade until we receive it.
 You do not need any other data than what is provided. You should feel free to use additional codebases and libraries except for those expressly intended to decode machine translation models. You must write your own decoder. If you would like to base your solution on finitestate toolkits or generic solvers for traveling salesman problems or integer linear programming, that is fine. But machine translation software including (but not limited to) Moses, cdec, Joshua, or phrasal is offlimits. You may of course inspect these systems if you want to 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 quite modest amount of python code. If you aren't sure whether something is permitted, ask us.
If you have any questions or you're confused about anything, just ask.