15.3. The Dataset for Pretraining Word Embeddings
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Now that we know the technical details of the word2vec models and approximate training methods, let’s walk through their implementations. Specifically, we will take the skip-gram model in Section 15.1 and negative sampling in Section 15.2 as an example. In this section, we begin with the dataset for pretraining the word embedding model: the original format of the data will be transformed into minibatches that can be iterated over during training.

pytorchmxnet
import collections
import math
import os
import random
import torch
from d2l import torch as d2l

import collections
import math
import os
import random
from mxnet import gluon, np
from d2l import mxnet as d2l

15.3.1. Reading the Dataset

The dataset that we use here is Penn Tree Bank (PTB). This corpus is sampled from Wall Street Journal articles, split into training, validation, and test sets. In the original format, each line of the text file represents a sentence of words that are separated by spaces. Here we treat each word as a token.

pytorchmxnet
#@save
d2l.DATA_HUB['ptb'] = (d2l.DATA_URL + 'ptb.zip',
                       '319d85e578af0cdc590547f26231e4e31cdf1e42')

#@save
def read_ptb():
    """Load the PTB dataset into a list of text lines."""
    data_dir = d2l.download_extract('ptb')
    # Read the training set
    with open(os.path.join(data_dir, 'ptb.train.txt')) as f:
        raw_text = f.read()
    return [line.split() for line in raw_text.split('\n')]

sentences = read_ptb()
f'# sentences: {len(sentences)}'

Downloading ../data/ptb.zip from http://d2l-data.s3-accelerate.amazonaws.com/ptb.zip...

'# sentences: 42069'

#@save
d2l.DATA_HUB['ptb'] = (d2l.DATA_URL + 'ptb.zip',
                       '319d85e578af0cdc590547f26231e4e31cdf1e42')

#@save
def read_ptb():
    """Load the PTB dataset into a list of text lines."""
    data_dir = d2l.download_extract('ptb')
    # Read the training set
    with open(os.path.join(data_dir, 'ptb.train.txt')) as f:
        raw_text = f.read()
    return [line.split() for line in raw_text.split('\n')]

sentences = read_ptb()
f'# sentences: {len(sentences)}'

Downloading ../data/ptb.zip from http://d2l-data.s3-accelerate.amazonaws.com/ptb.zip...

'# sentences: 42069'

After reading the training set, we build a vocabulary for the corpus, where any word that appears less than 10 times is replaced by the “<unk>” token. Note that the original dataset also contains “<unk>” tokens that represent rare (unknown) words.

pytorchmxnet
vocab = d2l.Vocab(sentences, min_freq=10)
f'vocab size: {len(vocab)}'

'vocab size: 6719'

vocab = d2l.Vocab(sentences, min_freq=10)
f'vocab size: {len(vocab)}'

'vocab size: 6719'

15.3.2. Subsampling

Text data typically have high-frequency words such as “the”, “a”, and “in”: they may even occur billions of times in very large corpora. However, these words often co-occur with many different words in context windows, providing little useful signals. For instance, consider the word “chip” in a context window: intuitively its co-occurrence with a low-frequency word “intel” is more useful in training than the co-occurrence with a high-frequency word “a”. Moreover, training with vast amounts of (high-frequency) words is slow. Thus, when training word embedding models, high-frequency words can be subsampled (Mikolov et al., 2013). Specifically, each indexed word \(w_i\) in the dataset will be discarded with probability

(15.3.1)\[P(w_i) = \max\left(1 - \sqrt{\frac{t}{f(w_i)}}, 0\right),\]

where \(f(w_i)\) is the ratio of the number of words \(w_i\) to the total number of words in the dataset, and the constant \(t\) is a hyperparameter (\(10^{-4}\) in the experiment). We can see that only when the relative frequency \(f(w_i) > t\) can the (high-frequency) word \(w_i\) be discarded, and the higher the relative frequency of the word, the greater the probability of being discarded.

pytorchmxnet
#@save
def subsample(sentences, vocab):
    """Subsample high-frequency words."""
    # Exclude unknown tokens ('<unk>')
    sentences = [[token for token in line if vocab[token] != vocab.unk]
                 for line in sentences]
    counter = collections.Counter([
        token for line in sentences for token in line])
    num_tokens = sum(counter.values())

    # Return True if `token` is kept during subsampling
    def keep(token):
        return(random.uniform(0, 1) <
               math.sqrt(1e-4 / counter[token] * num_tokens))

    return ([[token for token in line if keep(token)] for line in sentences],
            counter)

subsampled, counter = subsample(sentences, vocab)

#@save
def subsample(sentences, vocab):
    """Subsample high-frequency words."""
    # Exclude unknown tokens ('<unk>')
    sentences = [[token for token in line if vocab[token] != vocab.unk]
                 for line in sentences]
    counter = collections.Counter([
        token for line in sentences for token in line])
    num_tokens = sum(counter.values())

    # Return True if `token` is kept during subsampling
    def keep(token):
        return(random.uniform(0, 1) <
               math.sqrt(1e-4 / counter[token] * num_tokens))

    return ([[token for token in line if keep(token)] for line in sentences],
            counter)

subsampled, counter = subsample(sentences, vocab)

The following code snippet plots the histogram of the number of tokens per sentence before and after subsampling. As expected, subsampling significantly shortens sentences by dropping high-frequency words, which will lead to training speedup.

pytorchmxnet
d2l.show_list_len_pair_hist(['origin', 'subsampled'], '# tokens per sentence',
                            'count', sentences, subsampled);
../_images/output_word-embedding-dataset_f77071_39_0.svg
d2l.show_list_len_pair_hist(['origin', 'subsampled'], '# tokens per sentence',
                            'count', sentences, subsampled);
../_images/output_word-embedding-dataset_f77071_42_0.svg

For individual tokens, the sampling rate of the high-frequency word “the” is less than 1/20.

pytorchmxnet
def compare_counts(token):
    return (f'# of "{token}": '
            f'before={sum([l.count(token) for l in sentences])}, '
            f'after={sum([l.count(token) for l in subsampled])}')

compare_counts('the')

'# of "the": before=50770, after=2010'

def compare_counts(token):
    return (f'# of "{token}": '
            f'before={sum([l.count(token) for l in sentences])}, '
            f'after={sum([l.count(token) for l in subsampled])}')

compare_counts('the')

'# of "the": before=50770, after=2007'

In contrast, low-frequency words “join” are completely kept.

pytorchmxnet
compare_counts('join')

'# of "join": before=45, after=45'

compare_counts('join')

'# of "join": before=45, after=45'

After subsampling, we map tokens to their indices for the corpus.

pytorchmxnet
corpus = [vocab[line] for line in subsampled]
corpus[:3]

[[], [4127, 3228, 1773], [3922, 1922, 4743, 2696]]

corpus = [vocab[line] for line in subsampled]
corpus[:3]

[[], [3228, 4060], [3922, 1922, 4743]]

15.3.3. Extracting Center Words and Context Words

The following get_centers_and_contexts function extracts all the center words and their context words from corpus. It uniformly samples an integer between 1 and max_window_size at random as the context window size. For any center word, those words whose distance from it does not exceed the sampled context window size are its context words.

pytorchmxnet
#@save
def get_centers_and_contexts(corpus, max_window_size):
    """Return center words and context words in skip-gram."""
    centers, contexts = [], []
    for line in corpus:
        # To form a "center word--context word" pair, each sentence needs to
        # have at least 2 words
        if len(line) < 2:
            continue
        centers += line
        for i in range(len(line)):  # Context window centered at `i`
            window_size = random.randint(1, max_window_size)
            indices = list(range(max(0, i - window_size),
                                 min(len(line), i + 1 + window_size)))
            # Exclude the center word from the context words
            indices.remove(i)
            contexts.append([line[idx] for idx in indices])
    return centers, contexts

#@save
def get_centers_and_contexts(corpus, max_window_size):
    """Return center words and context words in skip-gram."""
    centers, contexts = [], []
    for line in corpus:
        # To form a "center word--context word" pair, each sentence needs to
        # have at least 2 words
        if len(line) < 2:
            continue
        centers += line
        for i in range(len(line)):  # Context window centered at `i`
            window_size = random.randint(1, max_window_size)
            indices = list(range(max(0, i - window_size),
                                 min(len(line), i + 1 + window_size)))
            # Exclude the center word from the context words
            indices.remove(i)
            contexts.append([line[idx] for idx in indices])
    return centers, contexts

Next, we create an artificial dataset containing two sentences of 7 and 3 words, respectively. Let the maximum context window size be 2 and print all the center words and their context words.

pytorchmxnet
tiny_dataset = [list(range(7)), list(range(7, 10))]
print('dataset', tiny_dataset)
for center, context in zip(*get_centers_and_contexts(tiny_dataset, 2)):
    print('center', center, 'has contexts', context)

dataset [[0, 1, 2, 3, 4, 5, 6], [7, 8, 9]]
center 0 has contexts [1]
center 1 has contexts [0, 2]
center 2 has contexts [0, 1, 3, 4]
center 3 has contexts [1, 2, 4, 5]
center 4 has contexts [2, 3, 5, 6]
center 5 has contexts [3, 4, 6]
center 6 has contexts [5]
center 7 has contexts [8, 9]
center 8 has contexts [7, 9]
center 9 has contexts [7, 8]

tiny_dataset = [list(range(7)), list(range(7, 10))]
print('dataset', tiny_dataset)
for center, context in zip(*get_centers_and_contexts(tiny_dataset, 2)):
    print('center', center, 'has contexts', context)

dataset [[0, 1, 2, 3, 4, 5, 6], [7, 8, 9]]
center 0 has contexts [1, 2]
center 1 has contexts [0, 2, 3]
center 2 has contexts [0, 1, 3, 4]
center 3 has contexts [2, 4]
center 4 has contexts [2, 3, 5, 6]
center 5 has contexts [4, 6]
center 6 has contexts [4, 5]
center 7 has contexts [8, 9]
center 8 has contexts [7, 9]
center 9 has contexts [7, 8]

When training on the PTB dataset, we set the maximum context window size to 5. The following extracts all the center words and their context words in the dataset.

pytorchmxnet
all_centers, all_contexts = get_centers_and_contexts(corpus, 5)
f'# center-context pairs: {sum([len(contexts) for contexts in all_contexts])}'

'# center-context pairs: 1503420'

all_centers, all_contexts = get_centers_and_contexts(corpus, 5)
f'# center-context pairs: {sum([len(contexts) for contexts in all_contexts])}'

'# center-context pairs: 1497612'

15.3.4. Negative Sampling

We use negative sampling for approximate training. To sample noise words according to a predefined distribution, we define the following RandomGenerator class, where the (possibly unnormalized) sampling distribution is passed via the argument sampling_weights.

pytorchmxnet
#@save
class RandomGenerator:
    """Randomly draw among {1, ..., n} according to n sampling weights."""
    def __init__(self, sampling_weights):
        # Exclude
        self.population = list(range(1, len(sampling_weights) + 1))
        self.sampling_weights = sampling_weights
        self.candidates = []
        self.i = 0

    def draw(self):
        if self.i == len(self.candidates):
            # Cache `k` random sampling results
            self.candidates = random.choices(
                self.population, self.sampling_weights, k=10000)
            self.i = 0
        self.i += 1
        return self.candidates[self.i - 1]

#@save
class RandomGenerator:
    """Randomly draw among {1, ..., n} according to n sampling weights."""
    def __init__(self, sampling_weights):
        # Exclude
        self.population = list(range(1, len(sampling_weights) + 1))
        self.sampling_weights = sampling_weights
        self.candidates = []
        self.i = 0

    def draw(self):
        if self.i == len(self.candidates):
            # Cache `k` random sampling results
            self.candidates = random.choices(
                self.population, self.sampling_weights, k=10000)
            self.i = 0
        self.i += 1
        return self.candidates[self.i - 1]

For example, we can draw 10 random variables \(X\) among indices 1, 2, and 3 with sampling probabilities \(P(X=1)=2/9, P(X=2)=3/9\), and \(P(X=3)=4/9\) as follows.

mxnet
generator = RandomGenerator([2, 3, 4])
[generator.draw() for _ in range(10)]

[3, 3, 1, 3, 1, 2, 3, 3, 2, 1]

For a pair of center word and context word, we randomly sample K (5 in the experiment) noise words. According to the suggestions in the word2vec paper, the sampling probability \(P(w)\) of a noise word \(w\) is set to its relative frequency in the dictionary raised to the power of 0.75 (Mikolov et al., 2013).

pytorchmxnet
#@save
def get_negatives(all_contexts, vocab, counter, K):
    """Return noise words in negative sampling."""
    # Sampling weights for words with indices 1, 2, ... (index 0 is the
    # excluded unknown token) in the vocabulary
    sampling_weights = [counter[vocab.to_tokens(i)]**0.75
                        for i in range(1, len(vocab))]
    all_negatives, generator = [], RandomGenerator(sampling_weights)
    for contexts in all_contexts:
        negatives = []
        while len(negatives) < len(contexts) * K:
            neg = generator.draw()
            # Noise words cannot be context words
            if neg not in contexts:
                negatives.append(neg)
        all_negatives.append(negatives)
    return all_negatives

all_negatives = get_negatives(all_contexts, vocab, counter, 5)

#@save
def get_negatives(all_contexts, vocab, counter, K):
    """Return noise words in negative sampling."""
    # Sampling weights for words with indices 1, 2, ... (index 0 is the
    # excluded unknown token) in the vocabulary
    sampling_weights = [counter[vocab.to_tokens(i)]**0.75
                        for i in range(1, len(vocab))]
    all_negatives, generator = [], RandomGenerator(sampling_weights)
    for contexts in all_contexts:
        negatives = []
        while len(negatives) < len(contexts) * K:
            neg = generator.draw()
            # Noise words cannot be context words
            if neg not in contexts:
                negatives.append(neg)
        all_negatives.append(negatives)
    return all_negatives

all_negatives = get_negatives(all_contexts, vocab, counter, 5)

15.3.5. Loading Training Examples in Minibatches

After all the center words together with their context words and sampled noise words are extracted, they will be transformed into minibatches of examples that can be iteratively loaded during training.

In a minibatch, the \(i^\textrm{th}\) example includes a center word and its \(n_i\) context words and \(m_i\) noise words. Due to varying context window sizes, \(n_i+m_i\) varies for different \(i\). Thus, for each example we concatenate its context words and noise words in the contexts_negatives variable, and pad zeros until the concatenation length reaches \(\max_i n_i+m_i\) (max_len). To exclude paddings in the calculation of the loss, we define a mask variable masks. There is a one-to-one correspondence between elements in masks and elements in contexts_negatives, where zeros (otherwise ones) in masks correspond to paddings in contexts_negatives.

To distinguish between positive and negative examples, we separate context words from noise words in contexts_negatives via a labels variable. Similar to masks, there is also a one-to-one correspondence between elements in labels and elements in contexts_negatives, where ones (otherwise zeros) in labels correspond to context words (positive examples) in contexts_negatives.

The above idea is implemented in the following batchify function. Its input data is a list with length equal to the batch size, where each element is an example consisting of the center word center, its context words context, and its noise words negative. This function returns a minibatch that can be loaded for calculations during training, such as including the mask variable.

pytorchmxnet
#@save
def batchify(data):
    """Return a minibatch of examples for skip-gram with negative sampling."""
    max_len = max(len(c) + len(n) for _, c, n in data)
    centers, contexts_negatives, masks, labels = [], [], [], []
    for center, context, negative in data:
        cur_len = len(context) + len(negative)
        centers += [center]
        contexts_negatives += [context + negative + [0] * (max_len - cur_len)]
        masks += [[1] * cur_len + [0] * (max_len - cur_len)]
        labels += [[1] * len(context) + [0] * (max_len - len(context))]
    return (torch.tensor(centers).reshape((-1, 1)), torch.tensor(
        contexts_negatives), torch.tensor(masks), torch.tensor(labels))

#@save
def batchify(data):
    """Return a minibatch of examples for skip-gram with negative sampling."""
    max_len = max(len(c) + len(n) for _, c, n in data)
    centers, contexts_negatives, masks, labels = [], [], [], []
    for center, context, negative in data:
        cur_len = len(context) + len(negative)
        centers += [center]
        contexts_negatives += [context + negative + [0] * (max_len - cur_len)]
        masks += [[1] * cur_len + [0] * (max_len - cur_len)]
        labels += [[1] * len(context) + [0] * (max_len - len(context))]
    return (np.array(centers).reshape((-1, 1)), np.array(
        contexts_negatives), np.array(masks), np.array(labels))

Let’s test this function using a minibatch of two examples.

pytorchmxnet
x_1 = (1, [2, 2], [3, 3, 3, 3])
x_2 = (1, [2, 2, 2], [3, 3])
batch = batchify((x_1, x_2))

names = ['centers', 'contexts_negatives', 'masks', 'labels']
for name, data in zip(names, batch):
    print(name, '=', data)

centers = tensor([[1],
        [1]])
contexts_negatives = tensor([[2, 2, 3, 3, 3, 3],
        [2, 2, 2, 3, 3, 0]])
masks = tensor([[1, 1, 1, 1, 1, 1],
        [1, 1, 1, 1, 1, 0]])
labels = tensor([[1, 1, 0, 0, 0, 0],
        [1, 1, 1, 0, 0, 0]])

x_1 = (1, [2, 2], [3, 3, 3, 3])
x_2 = (1, [2, 2, 2], [3, 3])
batch = batchify((x_1, x_2))

names = ['centers', 'contexts_negatives', 'masks', 'labels']
for name, data in zip(names, batch):
    print(name, '=', data)

centers = [[1.]
 [1.]]
contexts_negatives = [[2. 2. 3. 3. 3. 3.]
 [2. 2. 2. 3. 3. 0.]]
masks = [[1. 1. 1. 1. 1. 1.]
 [1. 1. 1. 1. 1. 0.]]
labels = [[1. 1. 0. 0. 0. 0.]
 [1. 1. 1. 0. 0. 0.]]
[22:01:00] ../src/storage/storage.cc:196: Using Pooled (Naive) StorageManager for CPU

15.3.6. Putting It All Together

Last, we define the load_data_ptb function that reads the PTB dataset and returns the data iterator and the vocabulary.

pytorchmxnet
#@save
def load_data_ptb(batch_size, max_window_size, num_noise_words):
    """Download the PTB dataset and then load it into memory."""
    num_workers = d2l.get_dataloader_workers()
    sentences = read_ptb()
    vocab = d2l.Vocab(sentences, min_freq=10)
    subsampled, counter = subsample(sentences, vocab)
    corpus = [vocab[line] for line in subsampled]
    all_centers, all_contexts = get_centers_and_contexts(
        corpus, max_window_size)
    all_negatives = get_negatives(
        all_contexts, vocab, counter, num_noise_words)

    class PTBDataset(torch.utils.data.Dataset):
        def __init__(self, centers, contexts, negatives):
            assert len(centers) == len(contexts) == len(negatives)
            self.centers = centers
            self.contexts = contexts
            self.negatives = negatives

        def __getitem__(self, index):
            return (self.centers[index], self.contexts[index],
                    self.negatives[index])

        def __len__(self):
            return len(self.centers)

    dataset = PTBDataset(all_centers, all_contexts, all_negatives)

    data_iter = torch.utils.data.DataLoader(dataset, batch_size, shuffle=True,
                                      collate_fn=batchify,
                                      num_workers=num_workers)
    return data_iter, vocab

#@save
def load_data_ptb(batch_size, max_window_size, num_noise_words):
    """Download the PTB dataset and then load it into memory."""
    sentences = read_ptb()
    vocab = d2l.Vocab(sentences, min_freq=10)
    subsampled, counter = subsample(sentences, vocab)
    corpus = [vocab[line] for line in subsampled]
    all_centers, all_contexts = get_centers_and_contexts(
        corpus, max_window_size)
    all_negatives = get_negatives(
        all_contexts, vocab, counter, num_noise_words)
    dataset = gluon.data.ArrayDataset(
        all_centers, all_contexts, all_negatives)
    data_iter = gluon.data.DataLoader(
        dataset, batch_size, shuffle=True,batchify_fn=batchify,
        num_workers=d2l.get_dataloader_workers())
    return data_iter, vocab

Let’s print the first minibatch of the data iterator.

pytorchmxnet
data_iter, vocab = load_data_ptb(512, 5, 5)
for batch in data_iter:
    for name, data in zip(names, batch):
        print(name, 'shape:', data.shape)
    break

centers shape: torch.Size([512, 1])
contexts_negatives shape: torch.Size([512, 60])
masks shape: torch.Size([512, 60])
labels shape: torch.Size([512, 60])

data_iter, vocab = load_data_ptb(512, 5, 5)
for batch in data_iter:
    for name, data in zip(names, batch):
        print(name, 'shape:', data.shape)
    break

centers shape: (512, 1)
contexts_negatives shape: (512, 60)
masks shape: (512, 60)
labels shape: (512, 60)

15.3.7. Summary

  • High-frequency words may not be so useful in training. We can subsample them for speedup in training.

  • For computational efficiency, we load examples in minibatches. We can define other variables to distinguish paddings from non-paddings, and positive examples from negative ones.

15.3.8. Exercises

  1. How does the running time of code in this section changes if not using subsampling?

  2. The RandomGenerator class caches k random sampling results. Set k to other values and see how it affects the data loading speed.

  3. What other hyperparameters in the code of this section may affect the data loading speed?

pytorchmxnet

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