# Assignment 1.4: Negative sampling (15 points)¶

You may have noticed that word2vec is really slow to train. Especially with big (> 50 000) vocabularies. Negative sampling is the solution.

The task is to implement word2vec with negative sampling.

This is what was discussed in Stanford lecture. The main idea is in the formula:

$$L = \log\sigma(u^T_o \cdot u_c) + \sum^k_{i=1} \mathbb{E}_{j \sim P(w)}[\log\sigma(-u^T_j \cdot u_c)]$$

Where $\sigma$ - sigmoid function, $u_c$ - central word vector, $u_o$ - context (outside of the window) word vector, $u_j$ - vector or word with index $j$.

The first term calculates the similarity between positive examples (word from one window)

The second term is responsible for negative samples. $k$ is a hyperparameter - the number of negatives to sample. $\mathbb{E}_{j \sim P(w)}$ means that $j$ is distributed accordingly to unigram distribution.

Thus, it is only required to calculate the similarity between positive samples and some other negatives. Not across all the vocabulary.