### Read-Again
Two options:
* GRU: run a pass of regular GRU on the input text $x_1,\ldots,x_n$. Use its hidden states $h_1,\ldots,h_n$ to compute weights vector for every step $i$ :
$\alpha_i = \tanh \left( W_e h_i  + U_e h_n + V_e x_i\right)$ and then runs a second GRU pass on the same input text. In the second pass the weights $\alpha_i$, from the first pass, are multiplied with the internal $z_i$ GRU gatting (controlling if hidden state is directly copied) of the second pass.
* LSTM: concatenate the hidden states from the first pass with the input text
$\left[ x_i, h_i, h_n \right]$ and run a second pass on this new input.

In case of multiple sentences the above passes are done per sentence. In addition the $h^s_n$ of each sentence $s$ is concatenated with the $h^{s'}_n$ of the other sentences or with $\tanh \left( \sum_s V_s h_s  + v\right)$

### Decoder with copy mechanism
LSTM with hidden state $s_t$. Input is previously generated word $y_{t-1}$ and context computed with attention mechanism: $c_t = \sum_i^n \beta_{it} h_i$. Here $h_i$ are the hidden states of the 2nd pass of the encoder. The weights are $\beta_{it} = \text{softmax}  \left(  v_a^T \tanh \left( W_a s_{t-1}  + U_a h_i\right) \right)$

The decoder vocabulary $Y$ used is small. If $y_{t-1}$ does not appear in $Y$ but does appear in the input at $x_i$ then its embedding is replaced with $p_t = \tanh \left( W_c h_i + b_c\right)$ and <UNK> otherwise.

$p_t$ is also used to copy the input to the output (details not given)

### Experiments
abstractive summarization [DUC2003 and DUC2004 competitions](http://www-nlpir.nist.gov/projects/duc/data.html).

For all animals, the taste sense is crucial to detect and avoid ingesting toxic molecules. Many toxins are synthesized by plants as a defense mechanism against insect predation. One example of such a natural toxic molecule is L-canavanine, a nonprotein amino acid found in the seeds of many legumes. Whether and how insects are informed that some plants contain L-canavanine remains to be elucidated. In insects, the taste sense relies on gustatory receptors forming the gustatory receptor (Gr) family. Gr proteins display highly divergent sequences, suggesting that they could cover the entire range of tastants. However, one cannot exclude the possibility of evolutionarily independent taste receptors. Here, we show that L-canavanine is not only toxic, but is also a repellent for Drosophila. Using a pharmacogenetic approach, we find that flies sense food containing this poison by the DmX receptor. DmXR is an insect orphan G-protein-coupled receptor that has partially diverged in its ligand binding pocket from the metabotropic glutamate receptor family. Blockade of DmXR function with an antagonist lowers the repulsive effect of L-canavanine. In addition, disruption of the DmXR encoding gene, called mangetout (mtt), suppresses the L-canavanine repellent effect. To avoid the ingestion of L-canavanine, DmXR expression is required in bitter-sensitive gustatory receptor neurons, where it triggers the premature retraction of the proboscis, thus leading to the end of food searching. These findings show that the DmX receptor, which does not belong to the Gr family, fulfills a gustatory function necessary to avoid eating a natural toxin.

## Task
Add '**rejection**' output to an existing classification model with softmax layer.

## Method
1. Choose some threshold $\delta$ and temperature $T$
2. Add a perturbation to the input x (eq 2),
let $\tilde x = x - \epsilon \text{sign}(-\nabla_x \log S_{\hat y}(x;T))$
3. If $p(\tilde x;T)\le \delta$, rejects
4. If not, return the output of the original classifier

$p(\tilde x;T)$ is the max prob with temperature scailing for input $\tilde x$

$\delta$ and $T$ are manually chosen.

As per the “holistic” in the paper title, the goal of this work is to take a suite of existing work within semi-supervised learning, and combine many of its ideas into one training pipeline that can (with really impressive empirical success) leverage the advantages of those different ideas. 

The core premise of supervised learning is that, given true-label training signal from a small number of labels, you can leverage large amounts of unsupervised data to improve your model. A central intuition of many of these methods is that, even if you don’t know the class of a given sample, you know it *has* a class, and you can develop a loss by pushing your model to predict the class for an example and a modified or perturbed version of that example, since, if you have a prior belief that that modification should not change your true class label, then your unlabeled data point should have the same class prediction both times. Entropy minimization is built off similar notions: although we don’t know a point’s class, we know it must have one, and so we’d like our model to make a prediction that puts more of its weight on a single class, rather than be spread out, since we know the “correct model” will be a very confident prediction of one class, though we don’t know which it is. These methods will give context and a frame of mind for understanding the techniques merged together into the MixMatch approach. 

At its very highest level, MixMatch’s goal is to take in a dataset of both labeled and unlabeled data, and produce a training set of inputs, predictions, and (occasionally constructed or modified labels) to calculate a model update loss from. 
https://i.imgur.com/6lHQqMD.png
- First, for each unlabeled example in the dataset, we produce K different augmented versions of that image (by cropping it, rotating it, flipping it, etc). This is in the spirit of the consistency loss literature, where you want your model to make the same prediction across augmentations
- Do the same augmentation for each labeled example, but only once per input, rather than k times 
- Run all of your augmented examples through your model, and take the average of their predictions. This is based on the idea that the average of the predictions will be a lower variance, more stable pseudo-target to pull each of the individual predictions towards. Also in the spirit of making something more shaped like a real label, they undertake a sharpening step, turning down the temperature of the averaged distribution. This seems like it would have the effect of more confidently pulling the original predictions towards a single “best guess” label 
- At this point, we have a set of augmented labeled data, with a true label, and also a set of augmented unlabeled data, with a label based off of an averaged and sharpened best guess from the model over different modifications. At this point, the pipeline uses something called “MixUp” (on which there is a previous paper, so I won’t dive into it too much here), which takes pairs of data points, calculates a convex combination of the inputs, runs it through the model, and uses as the loss-function target a convex combination of the outputs. So, in the simple binary case, if you have a positive and negatively labeled image and sample a combination parameter of 0.75, you have an image that is 0.75 positive, 0.25 negative, and the new label that you’re calculating cross entropy loss against is 0.75. 
- MixMatch generates pairs for its MixUp calculation by mixing (heh) labeled and unlabeled data together, and pairing each labeled and unlabeled pair with one observation from the merged set. 
At this point, we have combined inputs, and we have combined labels, and we can calculate loss between them 

With all of these methods combined, this method takes the previous benchmark of 38% error, for a CIFAR dataset with only 250 labels, and drops that to 11%, which is a pretty astonishing improvement in error rate. After performing an ablation study, they find that MixUp itself, temperature sharpening, and calculating K>1 augmentations of unlabeled data rather than K=1 are the strongest value-adds; it doesn’t appear like there’s that much difference that comes from mixing between unlabeled and labeled for the MixUp pairs.  

The core goal of this paper is to perform in an unsupervised (read: without parallel texts) way what other machine translation researchers had previously only effectively performed in a supervised way: the creation of a word-to-word translational mapping between natural languages. To frame the problem concretely: the researchers start with word embeddings learned in each language independently, and their desired output is a set of nearest neighbors for a source word that contains the true target (i.e. translated) word as often a possible. 

An interesting bit of background for this paper is that Mikilov, who was the initial progenitor of the word embedding approach, went on to posit, based on experiments he’d conducted, that the embeddings produced by different languages share characteristics in vector space, such that one could expect a linear translation (i.e. taking a set of points and rotating, shifting, and/or scaling them) to be able to map from one language to another. This assumption is relied on heavily in this paper. A notional note: when I refer to “a mapped source embedding” or “mapped source”, that just means that a matrix transformation, captured in a weight matrix W, is being used to do some form of rotation, scaling, or shifting, to “map” between the source embedding space and the shared space. 

The three strategies this paper employs are: 
1. Using adversarial training to try to force the distributions of the embeddings in source and target languages to be similar to one another 
2. Taking examples where method (1) has high confidence, and borrowing a method from supervised word-to-word translation, called the Procrustes method, to further optimize the mapping into the shared vector space 
3. Calculating the nearest neighbors of a source word using an approach they develop called “Cross-Domain Similarity Local Scaling”. At a high level, this conducts nearest neighbors, but “normalizes” for density, so that, on an intuitive level, it’s basically scaling distances up in dense regions of the space, and scaling them down in sparse regions 

Focusing on (1) first, the notion here goes back to that assumption I mentioned earlier: that internal relationships within embedding space are similar across languages, such that if you able to align the overall distributions of target embedding with a mapped source embedding, then you might - if you take Mikilov’s assumption seriously -  reasonably expect this to push words in the mapped-source space close to their corresponding words in target space. And this does work, to some degree, but the researchers found that this approach on it’s own didn’t get them to where they wanted to be in terms of accuracy. 

To further refine the mapping created by the adversarial training, the authors use something called the “Procrustes Method”. They go into it in more detail in the paper, but at a high level, it turns out that if you’re trying to solve the problem of minimizing the sum of squared distances between a mapped-source embedding and a target embedding, assuming that that mapping is linear, and that you want the weight matrix to be orthogonal, that problem reduces to doing the singular value decomposition of the matrix of source embeddings multiplied by the (transposed) matrix of target embeddings, for a set of ground truth shared words. Now, you may reasonably note: this is an unsupervised method, we don’t have access to ground truth embeddings across languages. And you would be correct. So, here, what the authors do is take words that are *mutual* nearest neighbors (according to the CSLS metric of nearest neighbors I’ll describe in (3)  ) after conducting their adversarially-learned rotation, and take that mutual-nearest-neighbor-dom as a marker of high confidence in that word pair. They took these mutually-nearest-neighbor pairs, and used those as “ground truth” to conduct this singular value decomposition, which was applied on top of the adversarially-learned rotation to get to their final mapping. 

(3) is described well in equation form in the paper itself, and is just a way of constructing a similarity metric between a mapped-source embedding and a target embedding that does some clever normalization. Specifically, it takes two times the (cosine) distance between Ws (mapped source) and t (target), and subtracts out the average (cosine) distance of Ws to its k nearest target words, as well as the (average) cosine distance of t to its k nearest source words. In this way, it normalizes the distance between Ws and t based on how dense each of their neighborhoods is. 

Using all of these approaches together, the authors really do get quite impressive performance. For EN-ES, ES-EN, EN-FR, FR-EN, EN-DE, DE-EN, and EO (Esperanto)-EN, the performance of the adversarial method is within 0.5 accuracy score of the supervised method, with the adversarial method being higher in 5 of those 7 cases (note: I read this as "functionally equivalent"). Interestingly, though, for EN-RU, RU-EN, EN-CHN, and CHN-EN, the adversarial method was dramatically less effective, with accuracy deltas ranging from 5 to 10 points between the adversarial and the supervised method, with the supervised method prevailing in all cases. This suggests that the assumption of a simple linear mapping between the vector spaces of different languages may be a more valid one when the languages are more closely related, and thus closer in their structure. I'd be really interested in any experiments that try to actually confirm this by testing on a wider array of languages, or testing on subgroups of languages that are closer or farther (i.e. you would expect ES-FR to do even better than EN-FR, and you would expect ES-DE to do worse than EN-DE).