CBOW (Continuous Bag of Words)

Use the context to predict the probability of current word.

  1. Context words’ vectors are \(\upsilon_{c-n} … \upsilon_{c+m}\) (\(m\) is the window size)
  2. Context vector \(\hat{\upsilon}=\frac{\upsilon_{c-m}+\upsilon_{c-m+1}+…+\upsilon_{c+m}}{2m}\)
  3. Score vector \(z_i = u_i\hat{\upsilon}\), where \(u_i\) is the output vector representation of word \(\omega_i\)
  4. Turn scores into probabilities \(\hat{y}=softmax(z)\)
  5. We desire probabilities \(\hat{y}\) match the true probabilities \(y\).

We use cross entropy \(H(\hat{y},y)\) to measure the distance between these two distributions. \[H(\hat{y},y)=-\sum_{j=1}^{\lvert V \rvert}{y_j\log(\hat{y}_j)}\]

\(y\) and \(\hat{y}\) is accurate, so the loss simplifies to: \[H(\hat{y},y)=-y_j\log(\hat{y})\]

For perfect prediction, \(H(\hat{y},y)=-1\log(1)=0\)

According to this, we can create this loss function:

\[\begin{aligned} minimize\ J &=-\log P(\omega_c\lvert \omega_{c-m},…,\omega_{c-1},…,\omega_{c+m}) \\
&= -\log P(u_c \lvert \hat{\upsilon}) \\
&= -\log \frac{\exp(u_c^T\hat{\upsilon})}{\sum_{j=1}^{\lvert V \rvert}\exp (u_j^T\hat{\upsilon})} \\
&= -u_c^T\hat{\upsilon}+\log \sum_{j=1}^{\lvert V \rvert}\exp (u_j^T\hat{\upsilon}) \end{aligned}\]


Use current word to predict its context.

  1. We get the input word’s vector \(\upsilon_c\)
  2. Generate \(2m\) score vectors, \(u_{c-m},…,u_{c-1},…,u_{c+m}\).
  3. Turn scores into probabilities \(\hat{y}=softmax(u)\)
  4. We desire probabilities \(\hat{y}\) match the true probabilities \(y\).

\[\begin{aligned} minimize J &=-\log P(\omega_{c-m},…,\omega_{c-1},\omega_{c+1},…\omega_{c+m}\lvert \omega_c)\\
&=-\log \prod_{j=0,j\ne m}^{2m}P(\omega_{c-m+j}\lvert \omega_c)\\
&=-\log \prod_{j=0,j\ne m}^{2m}P(u_{c-m+j}\lvert \upsilon_c)\\
&=-\log \prod_{j=0,j\ne m}^{2m}\frac{\exp (u^T_{c-m+j}\upsilon_c)}{\sum_{k=1}^{\lvert V \rvert}{\exp (u^T_k \upsilon_c)}}\\
&=-\sum_{j=0,j\ne m}^{2m}{u^T_{c-m+j}\upsilon_c+2m\log \sum_{k=1}^{\lvert V \rvert} \exp(u^T_k \upsilon_c)} \end{aligned}\]


Minimize \(J\) is expensive, as the summation is over \(\lvert V \rvert\). There are two ways to reduce the computation. Hierarchical Softmax and Negative Sampling.

Hierarchical Softmax

Encode words into a huffman tree, then each word has a Huffman code. The probability of it’s probability \(P(w\lvert Context(\omega))\) can change to choose the right path from root the the leaf node, each node is a binary classification. Suppose code \(0\) is a positive label, \(1\) is negative label. If the probability of a positive classification is \[\sigma(X^T_\omega \theta)=\frac{1}{1+e^{-X^T_\omega}}\]

Then the probability of negative classification is \[1-\sigma(X^T_\omega \theta)\]

doc2vec_hierarchical_softmax.png 足球’s Huffman code is \(1001\), then it’s probability in each node are

\[\begin{aligned} p(d_2^\omega\lvert X_\omega,\theta^\omega_1&=1-\sigma(X^T_\omega \theta^\omega_1))\\
p(d^\omega_3\lvert X_\omega,\theta^\omega_2&=\sigma(X^T_\omega \theta^\omega_2))\\
p(d^\omega_4\lvert X_\omega,\theta^\omega_3&=\sigma(X^T_\omega \theta^\omega_3))\\
p(d^\omega_5\lvert X_\omega,\theta^\omega_4&=1-\sigma(X^T_\omega \theta^\omega_4))\\

where \(\theta\) is parameter in the node.

The probability of the 足球 is the production of these equation.


\[p(\omega\lvert Context(\omega))=\prod_{j=2}^{l\omega}p(d^\omega_j\lvert X_\omega,\theta^\omega_{j-1})\]

Negative Sampling

Choose some negative sample, add the probability of the negative word into loss function. Maximize the positive words’ probability and minimize the negative words’ probability.

Let \(P(D=0 \lvert \omega,c)\) be the probability that \((\omega,c)\) did not come from the corpus data. Then the objective function will be

\[\theta = \text{argmax} \prod_{(\omega,c)\in D} P(D=1\lvert \omega,c,\theta) \prod_{(\omega,c)\in \tilde{D}} P(D=0\lvert \omega,c,\theta)\]

where \(\theta\) is the parameters of the model(\(\upsilon\) and \(u\)).