This very new paper, is currently receiving quite a bit of attention by the [community](https://www.reddit.com/r/MachineLearning/comments/5qxoaz/r_170107875_wasserstein_gan/).

The paper describes a new training approach, which solves the two major practical problems with current GAN training:

1) The training process comes with a meaningful loss. This can be used as a (soft) performance metric and will help debugging, tune parameters and so on.

2) The training process does not suffer from all the instability problems. In particular the paper reduces mode collapse significantly.

On top of that, the paper comes with quite a bit mathematical theory, explaining why there approach works and other approachs have failed. This paper is a must read for anyone interested in GANs.

Here is a video overview:
https://www.youtube.com/watch?v=t-fow6GJepQ

Here is an image of the poster:
https://i.imgur.com/Ti9btj9.png

This paper describes an architecture designed for generating class predictions based on a set of features in situations where you may only have a few examples per class, or, even where you see entirely new classes at test time. Some prior work has approached this problem in ridiculously complex fashion, up to and including training a network to predict the gradient outputs of a meta-network that it thinks would best optimize loss, given a new class. The method of Prototypical Networks prides itself on being much simpler, and more intuitive, so I hope I’ll be able to convey that in this explanation.  In order to think about this problem properly, it makes sense to take a few steps back, and think about some fundamental assumptions that underly machine learning. 
https://i.imgur.com/Q45w0QT.png

One very basic one is that you need some notion of similarity between observations in your training set, and potential new observations in your test set, in order to properly generalize.  To put it very simplistically, if a test example is very similar to examples of class A that we saw in training, we might predict it to be of class A at testing. But what does it *mean* for two observations to be similar to one another? If you’re using a method like K Nearest Neighbors, you calculate a point’s class identity based on the closest training-set observations to it in Euclidean space, and you assume that nearness in that space corresponds to likelihood of two data points having come the same class. This is useful for the use case of having new classes show up after training, since, well, there isn’t really a training period: the strategy for KNN is just carrying your whole training set around, and, whenever a new test point comes along, calculating it’s closest neighbors among those training-set points. If you see a new class in the wild, all you need to do is add the examples of that class to your group of training set points, and then after a few examples, if your assumptions hold, you’ll be able to predict that class by (hopefully) finding those two or three points as neighbors. But what if some dimensions of your feature space matter much more than others for differentiating between classes? In a simplistic example, you could have twenty features, but, unbeknownst to you, only one is actually useful for separating out your classes, and the other 19 are random. If you use the naive KNN assumption, you wouldn’t expect to perform well here, because you will have distances in these 19 meaningless directions spreading out your points, due to randomness, more than the meaningful dimension spread them out due to belonging to different classes. And what if you want to be able to learn non-linear relationships between your features, which the composability of multi-layer neural networks lends itself well to? In cases like those, the features you were handed may be a woefully suboptimal metric space in which to calculate a kind of similarity that corresponds to differences in class identity, so you’ll just have to strike out for the territories and create a metric space for yourself. 

That is, at a very high level, what this paper seeks to do: learn a transformation between input features and some vector space, such that distances in that vector space correspond as well as possible to probabilities of belonging to a given output class. You may notice me using “vector space” and “embedding” similarity; they are the same idea: the result of that learned transformation, which represents your input observations as dense vectors in some p-dimensional space, where p is a chosen hyperparameter. 

What are the concrete learning steps this architecture goes through?
 
1. During each training episode, sample a subset of classes, and then divide those classes into training examples, and query examples 
2. Using a set of weights that are being learned by the network, map the input features of each training example into a vector space. 
3. Once all training examples are mapped into the space, calculate a “mean vector” for class A by averaging all of the embeddings of training examples that belong to class A. This is the “prototype” for class A, and once we have it, we can forget the values of the embedded examples that were averaged to create it. This is a nice update on the KNN approach, since the number of parameters we need to carry around to evaluate is only (num-dimensions) * (num-classes), rather than (num-dimensions) * (num-training-examples). 
4. Then, for each query example, map it into the embedding space, and use a distance metric in that space to create a softmax over possible classes. (You can just think of a softmax as a network’s predicted probability, it’s a set of floats that add up to 1). 
5. Then, you can calculate the (cross-entropy) error between the true output and that softmax prediction vector in the same way as you would for any classification network 
6. Add up the prediction loss for all the query examples, and then backpropogate through the network to update your weights 

The overall effect of this process is to incentivize your network to learn, not necessarily a good prediction function, but a good metric space. The idea is that, if the metric space is good enough, and the classes are conceptually similar to each other (i.e. car vs chair, as opposed to car vs the-meaning-of-life), a space that does well at causing similar observed classes to be close to one another will do the same for classes not seen during training. 

I admit to not being sufficiently familiar with the datasets used for testing to have a sense for how well this method compares to more fully supervised classification schemes; if anyone does, definitely let me know! But the paper claims to get state of the art results compared to other approaches in this domain of few-shot learning (matching networks, and the aforementioned meta-learning). 

One interesting note is that the authors found that squared Euclidean distance, when applied within the embedded space, worked meaningfully better than cosine distance (which is a more standard way of measuring distances between vectors, since it measures only angle, rather than magnitude). They suspect that this is because Euclidean distance, but not cosine distance belongs to a category of divergence/distance metrics (called Bregman Divergences) that have a special set of properties such that the point closest on aggregate to all points in a cluster is the average of all those points. If you want to dive way deep into the minutia on this point, I found this blog post quite good: http://mark.reid.name/blog/meet-the-bregman-divergences.html

This paper’s approach goes a step further away from the traditional word embedding approach - of training embeddings as the lookup-table first layer of an unsupervised monolingual network - and proposes a more holistic form of transfer learning that involves not just transferring over learned knowledge contained in a set of vectors, but a fully trained model. 

Transfer learning is the general idea of using part or all of a network trained on one task to perform a different task. The most common kind of transfer learning is in the image domain, where models are first trained on the enormous ImageNet dataset, and then several of the lower layers of the network (where more local, small-pixel-range patterns are detected) are transferred, with their weights fixed in place to a new network. The modeler then attaches a few more layers to the top, connects it to a new target, and then is able to much more quickly learn their new target, because the pre-training has gotten them into a useful region of parameter-space. 

https://i.imgur.com/wjloHdi.png
Within NLP, the most common form of transfer learning is initializing the lookup table of vectors that’s used to convert discrete words in to vectors (also known as an embedding) with embeddings pre-trained on huge unsupervised datasets, like GloVe, trained on all of English Wikipedia. Again, this makes your overall task easier to train, because you’ve already converted words from their un-useful binary representation (where the word cat is just as far from Peru as it is from kitten) to a meaningful real-valued representation.

The approach suggested in this paper goes beyond simply learning the vector input representation of words. Instead, the authors suggest using as word vectors the sequence of encodings produced by an encoder-decoder bi-directional recurrent model. An encoder-decoder model means that you have one part of the network that maps from input sentence to an “encoded” representation of the sentence, and then another part that maps that encoded representation into the proper tokens in the target language. Historically, this encoding had been a single vector for the whole sentence, which tried to conceptually capture all of the words into one vector. More recently, a different approach has grown popular, where the RNN produces a number of encodings equal to the number of input words. Then, when the decoder is producing words in the target sentence, it uses something called “attention” to select a weighted combination of these encodings at each point in time. Under this scheme, the decoder might pull out information about verbs when its own hidden state suggests it needs a verb, and might pull out information about pronoun referents when its own hidden state asks for that. The upshot of all of this is that you end up with a sequence of encoded vectors equal in length to your number of inputs. Because the RNN is bidirectional, which means the encoding is a concatenation of the forward RNN and backward RNN, that means that each of these encodings captures both information about its corresponding word, and contextual information about the rest of the sentence. 

The proposal of the authors is to train the encoder-decoder outlined above, and, once it is trained, lop off the decoder, and use the encoded sequence of words as your representation of the input sequence of words. An important note in all this is that recurrent encoder-decoder model was itself trained using a lookup table initialized with learned GloVe vectors, so in a sense they’re not substituting for the unsupervised embeddings so much as learning marginal information on top of them.

The authors went on to test this approach on a few problems - question answering, logical entailment, and sentiment classification. They compared their use of the RNN encoded word vectors (which they call Context Vectors, or CoVE) with models initialized just using the fixed GloVE word vectors. One important note here is that, because each word vector is learned fully in context, the same word will have a different vector in each sentence it appears in. That’s why you can’t transfer one single vector per word, but instead have to transfer the recurrent model that can produce the vectors. All in all, the authors found that concatenating CoVe vectors to GloVe vectors, and using the concatenated version as input, produced sizable gains on the problems where it was tried. 

That said, it’s a pretty heavy lift to integrate someone else’s learned weights into your own model, just in terms of getting all the code to play together nicely. I’m not sure if this is a compelling enough result, a la ImageNet pretraining, for practitioners to want to go to the trouble of tacking a non-training RNN onto the bottom of all their models. If I ever get a chance, I’d be interested to play with the vectors you get out of this model, and look at how much variance you see in the vectors learned for different words across different sentences. Do you see clusters that correspond to sense disambiguation, (a la state of mind, vs a rogue state)? And, how does this contextual approach to the paper I reviewed yesterday, that also learns embeddings on a machine translation task, but does so in terms of training a lookup table, rather than using trained encodings? All in all, I enjoyed this paper: it was a simple idea, and I’m not sure whether it was a compelling one, but it did leave me with some interesting questions. 

Normal RL agents in multi-agent scenarios treat their opponents as a static part of the environment, not taking into account the fact that other agents are learning as well. This paper proposes LOLA, a learning rule that should take the agency and learning of opponents into account by optimizing "return under one step look-ahead of opponent learning"

So instead of optimizing under the current parameters of agent 1 and 2 
$$V^1(\theta_i^1, \theta_i^2)$$

LOLA proposes to optimize taking into account one step of opponent (agent 2) learning
$$V^1(\theta_i^1, \theta_i^2 + \Delta \theta^2_i)$$

where we assume the opponent's naive learning update $\Delta \theta^2_i = \nabla_{\theta^2} V^2(\theta^1, \theta^2) \cdot \eta$ and we add a second-order correction term

on top of this, the authors propose
- a learning rule with policy gradients in the case that the agent does not have access to exact gradients
- a way to estimate the parameters of the opponent, $\theta^2$, from its trajectories using maximum likelihood in the case you can't access them directly
$$\hat \theta^2 = \text{argmax}_{\theta^2} \sum_t \log \pi_{\theta^2}(u_t^2|s_t)$$

LOLA is tested on iterated prisoner's dilemma and converges to a tit-for-tat strategy more frequently than the naive RL learning algorithm, and outperforms it. LOLA is tested on iterated matching pennies (similar to prisoner's dilemma) and stably converges to the Nash equilibrium whereas the naive learners do not. In testing on coin game (a higher dimensional version of prisoner's dilemma) they find that naive learners generally choose the defect option whereas LOLA agents have a mostly-cooperative strategy.

As well, the authors show that LOLA is a dominant learning rule in IPD, where both agents always do better if either is using LOLA (and even better if both are using LOLA).

Finally, the authors also propose second order LOLA, which instead of assuming the opponent is a naive learner, assumes the opponent uses a LOLA learning rule. They show that second order LOLA does not lead to improved performance so there is no need to have a $n$th order LOLA arms race.