Dive into Deep Learning
Table Of Contents
Dive into Deep Learning
Table Of Contents

Implementation of Word2vec

This section is a practice exercise for the two previous sections. We use the skip-gram model from the “Word Embedding (word2vec)” section and negative sampling from the “Approximate Training” section as examples to introduce the implementation of word embedding model training on a corpus. We will also introduce some implementation tricks, such as subsampling and mask variables.

First, import the packages and modules required for the experiment.

In [1]:
import collections
import gluonbook as gb
import math
from mxnet import autograd, gluon, nd
from mxnet.gluon import data as gdata, loss as gloss, nn
import random
import sys
import time
import zipfile

Process the Data Set

Penn Tree Bank (PTB) is a small commonly-used corpus[1]. It takes samples from Wall Street Journal articles and includes training sets, validation sets, and test sets. We will train the word embedding model on the PTB training set. Each line of the data set acts as a sentence. All the words in a sentence are separated by spaces.

In [2]:
with zipfile.ZipFile('../data/ptb.zip', 'r') as zin:
    zin.extractall('../data/')

with open('../data/ptb/ptb.train.txt', 'r') as f:
    lines = f.readlines()
    # St is the abbreviation of "sentence" in the loop.
    raw_dataset = [st.split() for st in lines]

'# sentences: %d' % len(raw_dataset)
Out[2]:
'# sentences: 42068'

For the first three sentences of the data set, print the number of words and the first five words of each sentence. The end character of this data set is “<eos>”, uncommon words are all represented by “<unk>”, and numbers are replaced with “N”.

In [3]:
for st in raw_dataset[:3]:
    print('# tokens:', len(st), st[:5])
# tokens: 24 ['aer', 'banknote', 'berlitz', 'calloway', 'centrust']
# tokens: 15 ['pierre', '<unk>', 'N', 'years', 'old']
# tokens: 11 ['mr.', '<unk>', 'is', 'chairman', 'of']

Create Word Index

For the sake of simplicity, we only keep words that appear at least 5 times in the data set.

In [4]:
# Tk is an abbreviation for "token" in the loop.
counter = collections.Counter([tk for st in raw_dataset for tk in st])
counter = dict(filter(lambda x: x[1] >= 5, counter.items()))

Then, map the words to the integer indexes.

In [5]:
idx_to_token = [tk for tk, _ in counter.items()]
token_to_idx = {tk: idx for idx, tk in enumerate(idx_to_token)}
dataset = [[token_to_idx[tk] for tk in st if tk in token_to_idx]
           for st in raw_dataset]
num_tokens = sum([len(st) for st in dataset])
'# tokens: %d' % num_tokens
Out[5]:
'# tokens: 887100'

Subsampling

In text data, there are generally some words that appear at high frequencies, such “the”, “a”, and “in” in English. Generally speaking, in a context window, it is better to train the word embedding model when a word (such as “chip”) and a lower-frequency word (such as “microprocessor”) appear at the same time, rather than when a word appears with a higher-frequency word (such as “the”). Therefore, when training the word embedding model, we can perform subsampling[2] on the words. Specifically, each indexed word \(w_i\) in the data set will drop out at a certain probability. The dropout probability is given as:

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

Here, \(f(w_i)\) is the ratio of the instances of word \(w_i\) to the total number of words in the data set, and the constant \(t\) is a hyper-parameter (set to \(10^{-4}\) in this experiment). As we can see, it is only possible to drop out the word \(w_i\) in subsampling when \(f(w_i) > t\). The higher the word’s frequency, the higher its dropout probability.

In [6]:
def discard(idx):
    return random.uniform(0, 1) < 1 - math.sqrt(
        1e-4 / counter[idx_to_token[idx]] * num_tokens)

subsampled_dataset = [[tk for tk in st if not discard(tk)] for st in dataset]
'# tokens: %d' % sum([len(st) for st in subsampled_dataset])
Out[6]:
'# tokens: 376131'

As we can see, we have removed about half of the words after the second sampling. The following compares the number of times a word appears in the data set before and after subsampling. The sampling rate of the high-frequency word “the” is less than 1/20.

In [7]:
def compare_counts(token):
    return '# %s: before=%d, after=%d' % (token, sum(
        [st.count(token_to_idx[token]) for st in dataset]), sum(
        [st.count(token_to_idx[token]) for st in subsampled_dataset]))

compare_counts('the')
Out[7]:
'# the: before=50770, after=2081'

But the low-frequency word “join” is completely preserved.

In [8]:
compare_counts('join')
Out[8]:
'# join: before=45, after=45'

Extract Central Target Words and Context Words

We use words with a distance from the central target word not exceeding the context window size as the context words of the given center target word. The following definition function extracts all the central target words and their context words. It uniformly and randomly samples an integer to be used as the context window size between integer 1 and the max_window_size (maximum context window).

In [9]:
def get_centers_and_contexts(dataset, max_window_size):
    centers, contexts = [], []
    for st in dataset:
        if len(st) < 2:  # Each sentence needs at least 2 words to form a "central target word - context word" pair.
            continue
        centers += st
        for center_i in range(len(st)):
            window_size = random.randint(1, max_window_size)
            indices = list(range(max(0, center_i - window_size),
                                 min(len(st), center_i + 1 + window_size)))
            indices.remove(center_i)  # Exclude the central target word from the context words.
            contexts.append([st[idx] for idx in indices])
    return centers, contexts

Next, we create an artificial data set containing two sentences of 7 and 3 words, respectively. Assume the maximum context window is 2 and print all the central target words and their context words.

In [10]:
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 [3, 4, 6]
center 6 has contexts [4, 5]
center 7 has contexts [8, 9]
center 8 has contexts [7, 9]
center 9 has contexts [8]

In the experiment, we set the maximum context window size to 5. The following extracts all the central target words and their context words in the data set.

In [11]:
all_centers, all_contexts = get_centers_and_contexts(subsampled_dataset, 5)

Negative Sampling

We use negative sampling for approximate training. For a central and context word pair, we randomly sample \(K\) noise words (\(K=5\) in the experiment). According to the suggestion in the Word2vec paper, the noise word sampling probability \(\mathbb{P}(w)\) is the ratio of the word frequency of \(w\) to the total word frequency raised to the power of 0.75[2].

In [12]:
def get_negatives(all_contexts, sampling_weights, K):
    all_negatives, neg_candidates, i = [], [], 0
    population = list(range(len(sampling_weights)))
    for contexts in all_contexts:
        negatives = []
        while len(negatives) < len(contexts) * K:
            if i == len(neg_candidates):
                # An index of k words is randomly generated as noise words based on the weight of each word (sampling_weights)
                # . For efficient calculations, k can be set slightly larger.
                i, neg_candidates = 0, random.choices(
                    population, sampling_weights, k=int(1e5))
            neg, i = neg_candidates[i], i + 1
            # Noise words cannot be context words.
            if neg not in set(contexts):
                negatives.append(neg)
        all_negatives.append(negatives)
    return all_negatives

sampling_weights = [counter[w]**0.75 for w in idx_to_token]
all_negatives = get_negatives(all_contexts, sampling_weights, 5)

Reading Data

We extract all central target words all_centers, and the context words all_contexts and noise words all_negatives of each central target word from the data set. We will read them in random mini-batches.

In a mini-batch of data, the \(i\)-th example includes a central word and its corresponding \(n_i\) context words and \(m_i\) noise words. Since the context window size of each example may be different, the sum of context words and noise words, \(n_i+m_i\), will be different. When constructing a mini-batch, we concatenate the context words and noise words of each example, and add 0s for padding until the length of the concatenations are the same, that is, the length of all concatenations is \(\max_i n_i+m_i\)(max_len). In order to avoid the effect of padding on the loss function calculation, we construct the mask variable masks, each element of which corresponds to an element in the concatenation of context and noise words, contexts_negatives. When an element in the variable contexts_negatives is a padding, the element in the mask variable masks at the same position will be 0. Otherwise, it takes the value 1. In order to distinguish between positive and negative examples, we also need to distinguish the context words from the noise words in the contexts_negatives variable. Based on the construction of the mask variable, we only need to create a label variable labels with the same shape as the contexts_negatives variable and set the elements corresponding to context words (positive examples) to 1, and the rest to 0.

Next, we will implement the mini-batch reading function batchify. Its mini-batch input data is a list whose length is the batch size, each element of which contains central target words center, context words context, and noise words negative. The mini-batch data returned by this function conforms to the format we need, for example, it includes the mask variable.

In [13]:
def batchify(data):
    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 (nd.array(centers).reshape((-1, 1)), nd.array(contexts_negatives),
            nd.array(masks), nd.array(labels))

We use the batchify function just defined to specify the mini-batch reading method in the DataLoader instance. Then, we print the shape of each variable in the first batch read.

In [14]:
batch_size = 512
num_workers = 0 if sys.platform.startswith('win32') else 4
dataset = gdata.ArrayDataset(all_centers, all_contexts, all_negatives)
data_iter = gdata.DataLoader(dataset, batch_size, shuffle=True,
                             batchify_fn=batchify, num_workers=num_workers)
for batch in data_iter:
    for name, data in zip(['centers', 'contexts_negatives', 'masks',
                           'labels'], batch):
        print(name, 'shape:', data.shape)
    break
centers shape: (512, 1)
contexts_negatives shape: (512, 60)
masks shape: (512, 60)
labels shape: (512, 60)

The Skip-Gram Model

We will implement the skip-gram model by using embedding layers and mini-batch multiplication. These methods are also often used to implement other natural language processing applications.

Embedding Layer

The layer in which the obtained word is embedded is called the embedding layer, which can be obtained by creating an nn.Embedding instance in Gluon. The weight of the embedding layer is a matrix whose number of rows is the dictionary size (input_dim) and whose number of columns is the dimension of each word vector (output_dim). We set the dictionary size to 20 and the word vector dimension to 4.

In [15]:
embed = nn.Embedding(input_dim=20, output_dim=4)
embed.initialize()
embed.weight
Out[15]:
Parameter embedding0_weight (shape=(20, 4), dtype=float32)

The input of the embedding layer is the index of the word. When we enter the index \(i\) of a word, the embedding layer returns the \(i\)th row of the weight matrix as its word vector. Below we enter an index of shape (2,3) into the embedding layer. Because the dimension of the word vector is 4, we obtain a word vector of shape (2,3,4).

In [16]:
x = nd.array([[1, 2, 3], [4, 5, 6]])
embed(x)
Out[16]:

[[[ 0.01438687  0.05011239  0.00628365  0.04861524]
  [-0.01068833  0.01729892  0.02042518 -0.01618656]
  [-0.00873779 -0.02834515  0.05484822 -0.06206018]]

 [[ 0.06491279 -0.03182812 -0.01631819 -0.00312688]
  [ 0.0408415   0.04370362  0.00404529 -0.0028032 ]
  [ 0.00952624 -0.01501013  0.05958354  0.04705103]]]
<NDArray 2x3x4 @cpu(0)>

Mini-batch Multiplication

We can multiply the matrices in two mini-batches one by one, by the mini-batch multiplication operation batch_dot. Suppose the first batch contains \(n\) matrices \(\boldsymbol{X}_1, \ldots, \boldsymbol{X}_n\) with a shape of \(a\times b\), and the second batch contains \(n\) matrices \(\boldsymbol{Y}_1, \ldots, \boldsymbol{Y}_n\) with a shape of \(b\times c\). The output of matrix multiplication on these two batches are \(n\) matrices \(\boldsymbol{X}_1\boldsymbol{Y}_1, \ldots, \boldsymbol{X}_n\boldsymbol{Y}_n\) with a shape of \(a\times c\). Therefore, given two NDArrays of shape (\(n\), \(a\), \(b\)) and (\(n\), \(b\), \(c\)), the shape of the mini-batch multiplication output is (\(n\), \(a\), \(c\)).

In [17]:
X = nd.ones((2, 1, 4))
Y = nd.ones((2, 4, 6))
nd.batch_dot(X, Y).shape
Out[17]:
(2, 1, 6)

Skip-gram Model Forward Calculation

In forward calculation, the input of the skip-gram model contains the central target word index center and the concatenated context and noise word index contexts_and_negatives. In which, the center variable has the shape (batch size, 1), while the contexts_and_negatives variable has the shape (batch size, max_len). These two variables are first transformed from word indexes to word vectors by the word embedding layer, and then the output of shape (batch size, 1, max_len) is obtained by mini-batch multiplication. Each element in the output is the inner product of the central target word vector and the context word vector or noise word vector.

In [18]:
def skip_gram(center, contexts_and_negatives, embed_v, embed_u):
    v = embed_v(center)
    u = embed_u(contexts_and_negatives)
    pred = nd.batch_dot(v, u.swapaxes(1, 2))
    return pred

To train a model

Before training the word embedding model, we need to define the loss function of the model.

Binary Cross Entropy Loss Function

According to the definition of the loss function in negative sampling, we can directly use Gluon’s binary cross entropy loss function SigmoidBinaryCrossEntropyLoss.

In [19]:
loss = gloss.SigmoidBinaryCrossEntropyLoss()

It is worth mentioning that we can use the mask variable to specify the partial predicted value and label that participate in loss function calculation in the mini-batch: when the mask is 1, the predicted value and label of the corresponding position will participate in the calculation of the loss function; When the mask is 0, the predicted value and label of the corresponding position do not participate in the calculation of the loss function. As we mentioned earlier, mask variables can be used to avoid the effect of padding on loss function calculations.

In [20]:
pred = nd.array([[1.5, 0.3, -1, 2], [1.1, -0.6, 2.2, 0.4]])
# 1 and 0 in the label variables label represent context words and the noise words, respectively.
label = nd.array([[1, 0, 0, 0], [1, 1, 0, 0]])
mask = nd.array([[1, 1, 1, 1], [1, 1, 1, 0]])  # Mask variable.
loss(pred, label, mask) * mask.shape[1] / mask.sum(axis=1)
Out[20]:

[0.8739896 1.2099689]
<NDArray 2 @cpu(0)>

Next, as a comparison, we will implement binary cross-entropy loss function calculation from scratch and calculate the predicted value with a mask of 1 and the loss of the label based on the mask variable mask.

In [21]:
def sigmd(x):
    return -math.log(1 / (1 + math.exp(-x)))

print('%.7f' % ((sigmd(1.5) + sigmd(-0.3) + sigmd(1) + sigmd(-2)) / 4))
print('%.7f' % ((sigmd(1.1) + sigmd(-0.6) + sigmd(-2.2)) / 3))
0.8739896
1.2099689

Initialize Model Parameters

We construct the embedding layers of the central and context words, respectively, and set the hyper-parameter word vector dimension embed_size to 100.

In [22]:
embed_size = 100
net = nn.Sequential()
net.add(nn.Embedding(input_dim=len(idx_to_token), output_dim=embed_size),
        nn.Embedding(input_dim=len(idx_to_token), output_dim=embed_size))

Training

The training function is defined below. Because of the existence of padding, the calculation of the loss function is slightly different compared to the previous training functions.

In [23]:
def train(net, lr, num_epochs):
    ctx = gb.try_gpu()
    net.initialize(ctx=ctx, force_reinit=True)
    trainer = gluon.Trainer(net.collect_params(), 'adam',
                            {'learning_rate': lr})
    for epoch in range(num_epochs):
        start_time, train_l_sum = time.time(), 0
        for batch in data_iter:
            center, context_negative, mask, label = [
                data.as_in_context(ctx) for data in batch]
            with autograd.record():
                pred = skip_gram(center, context_negative, net[0], net[1])
                # Use the mask variable to avoid the effect of padding on loss function calculations.
                l = (loss(pred.reshape(label.shape), label, mask) *
                     mask.shape[1] / mask.sum(axis=1))
            l.backward()
            trainer.step(batch_size)
            train_l_sum += l.mean().asscalar()
        print('epoch %d, train loss %.2f, time %.2fs'
              % (epoch + 1, train_l_sum / len(data_iter),
                 time.time() - start_time))

Now, we can train a skip-gram model using negative sampling.

In [24]:
train(net, 0.005, 8)
epoch 1, train loss 0.46, time 8.64s
epoch 2, train loss nan, time 8.32s
epoch 3, train loss nan, time 8.32s
epoch 4, train loss nan, time 8.30s
epoch 5, train loss nan, time 8.30s
epoch 6, train loss nan, time 8.31s
epoch 7, train loss nan, time 8.36s
epoch 8, train loss nan, time 8.31s

Applying the Word Embedding Model

After training the word embedding model, we can represent similarity in meaning between words based on the cosine similarity of two word vectors. As we can see, when using the trained word embedding model, the words closest in meaning to the word “chip” are mostly related to chips.

In [25]:
def get_similar_tokens(query_token, k, embed):
    W = embed.weight.data()
    x = W[token_to_idx[query_token]]
    cos = nd.dot(W, x) / nd.sum(W * W, axis=1).sqrt() / nd.sum(x * x).sqrt()
    topk = nd.topk(cos, k=k+1, ret_typ='indices').asnumpy().astype('int32')
    for i in topk[1:]:  # Remove the input words.
        print('cosine sim=%.3f: %s' % (cos[i].asscalar(), (idx_to_token[i])))

get_similar_tokens('chip', 3, net[0])
cosine sim=nan: <unk>
cosine sim=nan: N
cosine sim=nan: years

Summary

  • We can use Gluon to train a skip-gram model through negative sampling.
  • Subsampling attempts to minimize the impact of high-frequency words on the training of a word embedding model.
  • We can pad examples of different lengths to create mini-batches with examples of all the same length and use mask variables to distinguish between padding and non-padding elements, so that only non-padding elements participate in the calculation of the loss function.

Problems

  • We use the batchify function to specify the mini-batch reading method in the DataLoader instance and print the shape of each variable in the first batch read. How should these shapes be calculated?
  • Try to find synonyms for other words.
  • Tune the hyper-parameters and observe and analyze the experimental results.
  • When the data set is large, we usually sample the context words and the noise words for the central target word in the current mini-batch only when updating the model parameters. In other words, the same central target word may have different context words or noise words in different epochs. What are the benefits of this sort of training? Try to implement this training method.

Reference

[1] Penn Tree Bank. https://catalog.ldc.upenn.edu/ldc99t42

[2] Mikolov, T., Sutskever, I., Chen, K., Corrado, G. S., & Dean, J. (2013). Distributed representations of words and phrases and their compositionality. In Advances in neural information processing systems (pp. 3111-3119).

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