:label:`sec_glove`

First, we should review the skip-gram model in word2vec. The conditional probability $P(w_j\mid w_i)$ expressed in the skip-gram model using the softmax operation will be recorded as $q_{ij}$, that is:

$$q_{ij}=\frac{\exp(\mathbf{u}_j^\top \mathbf{v}_i)}{ \sum_{k \in \mathcal{V}} \text{exp}(\mathbf{u}_k^\top \mathbf{v}_i)},$$where $\mathbf{v}_i$ and $\mathbf{u}_i$ are the vector representations of word $w_i$ of index $i$ as the center word and context word respectively, and $\mathcal{V} = \{0, 1, \ldots, |\mathcal{V}|-1\}$ is the vocabulary index set.

For word $w_i$, it may appear in the dataset for multiple times. We collect all the context words every time when $w_i$ is a center word and keep duplicates, denoted as multiset $\mathcal{C}_i$. The number of an element in a multiset is called the multiplicity of the element. For instance, suppose that word $w_i$ appears twice in the dataset: the context windows when these two $w_i$ become center words in the text sequence contain context word indices $2, 1, 5, 2$ and $2, 3, 2, 1$. Then, multiset $\mathcal{C}_i = \{1, 1, 2, 2, 2, 2, 3, 5\}$, where multiplicity of element 1 is 2, multiplicity of element 2 is 4, and multiplicities of elements 3 and 5 are both 1. Denote multiplicity of element $j$ in multiset $\mathcal{C}_i$ as $x_{ij}$: it is the number of word $w_j$ in all the context windows for center word $w_i$ in the entire dataset. As a result, the loss function of the skip-gram model can be expressed in a different way:

$$-\sum_{i\in\mathcal{V}}\sum_{j\in\mathcal{V}} x_{ij} \log\,q_{ij}.$$We add up the number of all the context words for the central target word $w_i$ to get $x_i$, and record the conditional probability $x_{ij}/x_i$ for generating context word $w_j$ based on central target word $w_i$ as $p_{ij}$. We can rewrite the loss function of the skip-gram model as

$$-\sum_{i\in\mathcal{V}} x_i \sum_{j\in\mathcal{V}} p_{ij} \log\,q_{ij}.$$In the formula above, $\sum_{j\in\mathcal{V}} p_{ij} \log\,q_{ij}$ computes the conditional probability distribution $p_{ij}$ for context word generation based on the central target word $w_i$ and the cross-entropy of conditional probability distribution $q_{ij}$ predicted by the model. The loss function is weighted using the sum of the number of context words with the central target word $w_i$. If we minimize the loss function from the formula above, we will be able to allow the predicted conditional probability distribution to approach as close as possible to the true conditional probability distribution.

However, although the most common type of loss function, the cross-entropy loss
function is sometimes not a good choice. On the one hand, as we mentioned in
:numref:`sec_approx_train`

the cost of letting the
model prediction $q_{ij}$ become the legal probability distribution has the sum
of all items in the entire dictionary in its denominator. This can easily lead
to excessive computational overhead. On the other hand, there are often a lot of
uncommon words in the dictionary, and they appear rarely in the dataset. In the
cross-entropy loss function, the final prediction of the conditional probability
distribution on a large number of uncommon words is likely to be inaccurate.

To address this, GloVe :cite:`Pennington.Socher.Manning.2014`

, a word embedding model that came after word2vec, adopts
squared loss and makes three changes to the skip-gram model based on this loss.

- Here, we use the non-probability distribution variables $p'_{ij}=x_{ij}$ and $q'_{ij}=\exp(\mathbf{u}_j^\top \mathbf{v}_i)$ and take their logs. Therefore, we get the squared loss $\left(\log\,p'_{ij} - \log\,q'_{ij}\right)^2 = \left(\mathbf{u}_j^\top \mathbf{v}_i - \log\,x_{ij}\right)^2$.
- We add two scalar model parameters for each word $w_i$: the bias terms $b_i$ (for central target words) and $c_i$( for context words).
- Replace the weight of each loss with the function $h(x_{ij})$. The weight function $h(x)$ is a monotone increasing function with the range $[0, 1]$.

Therefore, the goal of GloVe is to minimize the loss function.

$$\sum_{i\in\mathcal{V}} \sum_{j\in\mathcal{V}} h(x_{ij}) \left(\mathbf{u}_j^\top \mathbf{v}_i + b_i + c_j - \log\,x_{ij}\right)^2.$$Here, we have a suggestion for the choice of weight function $h(x)$: when $x < c$ (e.g $c = 100$), make $h(x) = (x/c) ^\alpha$ (e.g $\alpha = 0.75$), otherwise make $h(x) = 1$. Because $h(0)=0$, the squared loss term for $x_{ij}=0$ can be simply ignored. When we use minibatch SGD for training, we conduct random sampling to get a non-zero minibatch $x_{ij}$ from each time step and compute the gradient to update the model parameters. These non-zero $x_{ij}$ are computed in advance based on the entire dataset and they contain global statistics for the dataset. Therefore, the name GloVe is taken from "Global Vectors".

Notice that if word $w_i$ appears in the context window of word $w_j$, then word $w_j$ will also appear in the context window of word $w_i$. Therefore, $x_{ij}=x_{ji}$. Unlike word2vec, GloVe fits the symmetric $\log\, x_{ij}$ in lieu of the asymmetric conditional probability $p_{ij}$. Therefore, the central target word vector and context word vector of any word are equivalent in GloVe. However, the two sets of word vectors that are learned by the same word may be different in the end due to different initialization values. After learning all the word vectors, GloVe will use the sum of the central target word vector and the context word vector as the final word vector for the word.

We can also try to understand GloVe word embedding from another perspective. We will continue the use of symbols from earlier in this section, $P(w_j \mid w_i)$ represents the conditional probability of generating context word $w_j$ with central target word $w_i$ in the dataset, and it will be recorded as $p_{ij}$. From a real example from a large corpus, here we have the following two sets of conditional probabilities with "ice" and "steam" as the central target words and the ratio between them:

$w_k$= | solid | gas | water | fashion |
---|---|---|---|---|

$p_1=P(w_k\mid \text{ice})$ | 0.00019 | 0.000066 | 0.003 | 0.000017 |

$p_2=P(w_k\mid\text{steam})$ | 0.000022 | 0.00078 | 0.0022 | 0.000018 |

$p_1/p_2$ | 8.9 | 0.085 | 1.36 | 0.96 |

We will be able to observe phenomena such as:

- For a word $w_k$ that is related to "ice" but not to "steam", such as $w_k=\text{solid}$, we would expect a larger conditional probability ratio, like the value 8.9 in the last row of the table above.
- For a word $w_k$ that is related to "steam" but not to "ice", such as $w_k=\text{gas}$, we would expect a smaller conditional probability ratio, like the value 0.085 in the last row of the table above.
- For a word $w_k$ that is related to both "ice" and "steam", such as $w_k=\text{water}$, we would expect a conditional probability ratio close to 1, like the value 1.36 in the last row of the table above.
- For a word $w_k$ that is related to neither "ice" or "steam", such as $w_k=\text{fashion}$, we would expect a conditional probability ratio close to 1, like the value 0.96 in the last row of the table above.

We can see that the conditional probability ratio can represent the relationship between different words more intuitively. We can construct a word vector function to fit the conditional probability ratio more effectively. As we know, to obtain any ratio of this type requires three words $w_i$, $w_j$, and $w_k$. The conditional probability ratio with $w_i$ as the central target word is ${p_{ij}}/{p_{ik}}$. We can find a function that uses word vectors to fit this conditional probability ratio.

$$f(\mathbf{u}_j, \mathbf{u}_k, {\mathbf{v}}_i) \approx \frac{p_{ij}}{p_{ik}}.$$The possible design of function $f$ here will not be unique. We only need to consider a more reasonable possibility. Notice that the conditional probability ratio is a scalar, we can limit $f$ to be a scalar function: $f(\mathbf{u}_j, \mathbf{u}_k, {\mathbf{v}}_i) = f\left((\mathbf{u}_j - \mathbf{u}_k)^\top {\mathbf{v}}_i\right)$. After exchanging index $j$ with $k$, we will be able to see that function $f$ satisfies the condition $f(x)f(-x)=1$, so one possibility could be $f(x)=\exp(x)$. Thus:

$$f(\mathbf{u}_j, \mathbf{u}_k, {\mathbf{v}}_i) = \frac{\exp\left(\mathbf{u}_j^\top {\mathbf{v}}_i\right)}{\exp\left(\mathbf{u}_k^\top {\mathbf{v}}_i\right)} \approx \frac{p_{ij}}{p_{ik}}.$$One possibility that satisfies the right side of the approximation sign is $\exp\left(\mathbf{u}_j^\top {\mathbf{v}}_i\right) \approx \alpha p_{ij}$, where $\alpha$ is a constant. Considering that $p_{ij}=x_{ij}/x_i$, after taking the logarithm we get $\mathbf{u}_j^\top {\mathbf{v}}_i \approx \log\,\alpha + \log\,x_{ij} - \log\,x_i$. We use additional bias terms to fit $- \log\, \alpha + \log\, x_i$, such as the central target word bias term $b_i$ and context word bias term $c_j$:

$$\mathbf{u}_j^\top \mathbf{v}_i + b_i + c_j \approx \log(x_{ij}).$$By taking the square error and weighting the left and right sides of the formula above, we can get the loss function of GloVe.

- In some cases, the cross-entropy loss function may have a disadvantage. GloVe uses squared loss and the word vector to fit global statistics computed in advance based on the entire dataset.
- The central target word vector and context word vector of any word are equivalent in GloVe.

- If a word appears in the context window of another word, how can we use the
distance between them in the text sequence to redesign the method for
computing the conditional probability $p_{ij}$? Hint: See section 4.2 from the
paper GloVe :cite:
`Pennington.Socher.Manning.2014`

. - For any word, will its central target word bias term and context word bias term be equivalent to each other in GloVe? Why?