In the last video, you saw how you can learn a neural language model in order to get good word embeddings. In this video, you see the Word2Vec algorithm which is simple and comfortably more efficient way to learn this types of embeddings. Lets take a look. Most of the ideas I'll present in this video are due to Tomas Mikolov, Kai Chen, Greg Corrado, and Jeff Dean. Let's say you're given this sentence in your training set. In the skip-gram model, what we're going to do is come up with a few context to target errors to create our supervised learning problem. So rather than having the context be always the last four words or the last end words immediately before the target word, what I'm going to do is, say, randomly pick a word to be the context word. And let's say we chose the word orange. And what we're going to do is randomly pick another word within some window. Say plus minus five words or plus minus ten words of the context word and we choose that to be target word. So maybe just by chance you might pick juice to be a target word, that's just one word later. Or you might choose two words before. So you have another pair where the target could be glass or, Maybe just by chance you choose the word my as the target. And so we'll set up a supervised learning problem where given the context word, you're asked to predict what is a randomly chosen word within say, a plus minus ten word window, or plus minus five or ten word window of that input context word. And obviously, this is not a very easy learning problem, because within plus minus 10 words of the word orange, it could be a lot of different words. But a girl that's setting up this supervised learning problem, isn't to do well on the supervised learning problem per se, it is that we want to use this learning problem to learn good word embeddings. So, here are the details of the model. Let's say that we'll continue to our vocab of 10,000 words. And some have been on vocab sizes that exceeds a million words. But the basic supervised learning problem we're going to solve is that we want to learn the mapping from some Context c, such as the word orange to some target, which we will call t, which might be the word juice or the word glass or the word my, if we use the example from the previous slide. So in our vocabulary, orange is word 6257, and the word juice is the word 4834 in our vocab of 10,000 words. And so that's the input x that you want to learn to map to that open y. So to represent the input such as the word orange, you can start out with some one hot vector which is going to be write as O subscript C, so there's a one hot vector for the context words. And then similar to what you saw on the last video you can take the embedding matrix E, multiply E by the vector O subscript C, and this gives you your embedding vector for the input context word, so here EC is equal to capital E times that one hot vector. Then in this new network that we formed we're going to take this vector EC and feed it to a softmax unit. So I've been drawing softmax unit as a node in a neural network. That's not an o, that's a softmax unit. And then there's a drop in the softmax unit to output y hat. So to write out this model in detail. This is the model, the softmax model, probability of different tanka words given the input context word as e to the e, theta t transpose, ec. Divided by some over all words, so we're going to say, sum from J equals one to all 10,000 words of e to the theta j transposed ec. So here theta T is the parameter associated with, I'll put t, but really there's a chance of a particular word, t, being the label. So I've left off the biased term to solve mass but we could include that too if we wish. And then finally the loss function for softmax will be the usual. So we use y to represent the target word. And we use a one-hot representation for y hat and y here. Then the lost would be The negative log liklihood, so sum from i equals 1 to 10,000 of yi log yi hat. So that's a usual loss for softmax where we're representing the target y as a one hot vector. So this would be a one hot vector with just 1 1 and the rest zeros. And if the target word is juice, then it'd be element 4834 from up here. That is equal to 1 and the rest will be equal to 0. And similarly Y hat will be a 10,000 dimensional vector output by the softmax unit with probabilities for all 10,000 possible targets words. So to summarize, this is the overall little model, little neural network with basically looking up the embedding and then just a soft max unit. And the matrix E will have a lot of parameters, so the matrix E has parameters corresponding to all of these embedding vectors, E subscript C. And then the softmax unit also has parameters that gives the theta T parameters but if you optimize this loss function with respect to the all of these parameters, you actually get a pretty good set of embedding vectors. So this is called the skip-gram model because is taking as input one word like orange and then trying to predict some words skipping a few words from the left or the right side. To predict what comes little bit before little bit after the context words. Now, it turns out there are a couple problems with using this algorithm. And the primary problem is computational speed. In particular, for the softmax model, every time you want to evaluate this probability, you need to carry out a sum over all 10,000 words in your vocabulary. And maybe 10,000 isn't too bad, but if you're using a vocabulary of size 100,000 or a 1,000,000, it gets really slow to sum up over this denominator every single time. And, in fact, 10,000 is actually already that will be quite slow, but it makes even harder to scale to larger vocabularies. So there are a few solutions to this, one which you see in the literature is to use a hierarchical softmax classifier. And what that means is, instead of trying to categorize something into all 10,000 carries on one go. Imagine if you have one classifier, it tells you is the target word in the first 5,000 words in the vocabulary? Or is in the second 5,000 words in the vocabulary? And lets say this binary cost that it tells you this is in the first 5,000 words, think of second class to tell you that this in the first 2,500 words of vocab or in the second 2,500 words vocab and so on. Until eventually you get down to classify exactly what word it is, so that the leaf of this tree, and so having a tree of classifiers like this, means that each of the retriever nodes of the tree can be just a binding classifier. And so you don't need to sum over all 10,000 words or else it will capsize in order to make a single classification. In fact, the computational classifying tree like this scales like log of the vocab size rather than linear in vocab size. So this is called a hierarchical softmax classifier. I should mention in practice, the hierarchical softmax classifier doesn't use a perfectly balanced tree or this perfectly symmetric tree, with equal numbers of words on the left and right sides of each branch. In practice, the hierarchical software classifier can be developed so that the common words tend to be on top, whereas the less common words like durian can be buried much deeper in the tree. Because you see the more common words more often, and so you might need only a few traversals to get to common words like the and of. Whereas you see less frequent words like durian much less often, so it says okay that are buried deep in the tree because you don't need to go that deep. So there are various heuristics for building the tree how you used to build the hierarchical software spire. So this is one idea you see in the literature, the speeding up the softmax classification. But I won't spend too much more time. And you can read more details of this on the paper that I referenced by Thomas and others, on the first slide. But I won't spend too much more time on this. Because in the next video, where she talk about a different method, called nectar sampling, which I think is even simpler. And also works really well for speeding up the softmax classifier and the problem of needing the sum over the entire cap size in the denominator. So you see more of that in the next video. But before moving on, one quick Topic I want you to understand is how to sample the context C. So once you sample the context C, the target T can be sampled within, say, a plus minus ten word window of the context C, but how do you choose the context C? One thing you could do is just sample uniformly, at random, from your training corpus. When we do that, you find that there are some words like the, of, a, and, to and so on that appear extremely frequently. And so, if you do that, you find that in your context to target mapping pairs just get these these types of words extremely frequently, whereas there are other words like orange, apple, and also durian that don't appear that often. And maybe you don't want your training site to be dominated by these extremely frequently or current words, because then you spend almost all the effort updating ec, for those frequently occurring words. But you want to make sure that you spend some time updating the embedding, even for these less common words like e durian. So in practice the distribution of words pc isn't taken just entirely uniformly at random for the training set purpose, but instead there are different heuristics that you could use in order to balance out something from the common words together with the less common words. So that's it for the Word2Vec skip-gram model. If you read the original paper by that I referenced earlier, you find that that paper actually had two versions of this Word2Vec model, the skip gram was one. And the other one is called the CBow, the continuous backwards model, which takes the surrounding contexts from middle word, and uses the surrounding words to try to predict the middle word, and that algorithm also works, it has some advantages and disadvantages. But the key problem with this algorithm with the skip-gram model as presented so far is that the softmax step is very expensive to calculate because needing to sum over your entire vocabulary size into the denominator of the soft packs. In the next video I show you an algorithm that modifies the training objective that makes it run much more efficiently therefore lets you apply this in a much bigger fitting set as well and therefore learn much better word embeddings. Lets go onto the next video.
