3. Text classification

We will start our journey with a simple NLP problem, text classification. You might have already gone through the core techniques if you have taken an ML course, however, we hope to provide some new insights from the NLP perspective.

Let’s consider the binary sentiment classification task. Our input is a document (e.g. a review), and we want to predict whether it is positive or negative.

We will use the IMDB movie review dataset as our running example.

from d2l import mxnet as d2l
from mxnet import gluon, np, npx
import gluonnlp as nlp
npx.set_np()
train_dataset = nlp.data.IMDB(root='data/imdb', segment='train')

3.1. An intuitive approach

Let’s pretend for a second that we don’t know anything about ML or NLP. How should we approach this problem? Let’s take a look at the data to have a better sense of the problem.

print('# training examples:', len(train_dataset))
print(train_dataset[0])
print('labels:', set([d[1] for d in train_dataset]))

# training examples: 25000
['Bromwell High is a cartoon comedy. It ran at the same time as some other programs about school life, such as "Teachers". My 35 years in the teaching profession lead me to believe that Bromwell High's satire is much closer to reality than is "Teachers". The scramble to survive financially, the insightful students who can see right through their pathetic teachers' pomp, the pettiness of the whole situation, all remind me of the schools I knew and their students. When I saw the episode in which a student repeatedly tried to burn down the school, I immediately recalled ......... at .......... High. A classic line: INSPECTOR: I'm here to sack one of your teachers. STUDENT: Welcome to Bromwell High. I expect that many adults of my age think that Bromwell High is far fetched. What a pity that it isn't!', 9]
labels: {1, 2, 3, 4, 7, 8, 9, 10}

While the review itself can be quite complex, we don’t really need to understand everything in it. To separate positive and negative reviews, we might be able to just look at individual words. Intuitively, we expect to see more nice words in positive reviews such as “fantastic” and “wonderful”. Let’s test our intuition against the data.

Before we start to play with the data, we need to tokenize the text, i.e. separating the string into a list of tokens. The definition of a token can be language- and task-dependent. One common choise is to split the string into words. For example, we can use a simple regex to tokenize English text, however, it doesn’t work for Chinese which doesn’t have word boundary markers such as spaces. Similarly, in German there is no spaces in compound nouns, which can get long. A more general solution is to discard the notion of words and tokenize by subwords strings, e.g. characters (n-grams) or byte pair encoding. For more information on tokenization, read [JM 2.4.2]. In most cases we can use existing tokenizers in text processing libraries such as NLTK or SpaCy.

Let’s tokenize the data and map the ratings to binary labels. Also, for efficiency we randomly sample a subset.

tokenizer = nlp.data.SpacyTokenizer('en')
preprocess = lambda d : (tokenizer(d[0].lower()), 1 if int(d[1]) > 5 else 0)

import random
mini_dataset = [preprocess(d) for d in random.choices(train_dataset, k=1000)]
print(mini_dataset[0][1], mini_dataset[0][0][:20])

1 ['i', 'loved', 'that', 'this', 'film', 'recognizes', 'the', 'intelligence', 'of', 'the', 'viewer', ',', 'allowing', 'the', 'layers', 'to', 'peel', 'from', 'the', 'characters']

import itertools
pos_tokens = list(itertools.chain.from_iterable([d[0] for d in mini_dataset if d[1] == 1]))
neg_tokens = list(itertools.chain.from_iterable([d[0] for d in mini_dataset if d[1] == 0]))
print(len(pos_tokens), len(neg_tokens))

145268 130978

Now we can check the counts of a word in positive and negative examples.

token = 'wonderful'
print(pos_tokens.count(token))
print(neg_tokens.count(token))

61
8

So a simple heuristic approach is to count the frequency of occurence of each word in positive and negative examples, and classifiy an example as positive if it contains more words of high-frequency in the positive examples. The key takeaway is that we can do a reasonable job classifying the text based on individual words without understanding its meaning.

3.2. Naive Bayes model

Now, let’s take a more principled approach, following the key steps we talked about last time: design a model, specify a loss function, and minimize the average loss.

Consider a probabilistic model for binary classification:

(3.2.1)\[\begin{split}f_w(x) = \begin{cases} 1 & \text{if $p_w(y\mid x) > 0.5$} \\ 0 & \text{otherwise} \end{cases} .\end{split}\]

How do we parameterize \(p_w(y\mid x)\)? Recall the Bayes rule from probability:

(3.2.2)\[p(y\mid x) = \frac{p(x\mid y) p(y)}{p(x)} = \frac{p(x\mid y) p(y)}{\sum_{y\in\mathcal{Y}}p(x\mid y) p(y)} .\]

Given that \(y\) is binary, it’s reasonable to model it as a Bernoulli random variable. For more than two classes, we can use a categorical distribution.

What about \(p(x\mid y)\)? Note that it’s the joint probability of all words in the sentence \(x\): \(p(x\mid y) = p(x_1, \ldots, x_n \mid y)\) (\(n\) is the document length), where \(x_i\) is the token at the \(i\)-th position in the document. Directly modeling the joint probability will require a huge number of parameters (and can only be learned if we have a huge corpus). Following our intuition, we can ignore the interaction among all words and assume that they are conditionally independt. This is the key assumption in Naive Bayes models:

(3.2.3)\[p(x_1, \ldots, x_n \mid y) = \prod_{i=1}^n p(x_i\mid y) .\]

Let’s think about the data generating process. To generate a review, we first flip a coin to decide its sentiment, then roll a \(|V|\)-sided die for \(n\) steps to decide the word at each position (\(|V|\) denotes the vocabulary size). Therefore, \(p(x_1, \ldots, x_n \mid y)\) is given by a multinomial distribution.

3.3. Maximum likelihood estimation

Now that we have specified the model, the next step is to estimate parameters. For probabilistic models, we can use maximum likelihood estimation (MLE), corresponding to the negative log-likelihood (NLL) loss.

Our goal is to maximize the log-likelihood of the dataset \(D = \{x^{(i)}, y^{(i)}\}_{i=1}^N\):

(3.3.1)\[\log \prod_{i=1}^N p(x^{(i)}, y^{(i)}) = \log \prod_{i=1}^N \prod_{j=1}^{L_i} p(x_j^{(i)}\mid y^{(i)}) p(y^{(i)}),\]

where \(L_i\) is the length of the \(i\)-th example. We can plug in the probability mass function (pmf) of Bernoulli and multinomial distribution and carry out the optimization.

In this case, we have a closed-form solution (and a very intuitive one). The conditionaly probability of each word given a label is simply the percentage of time it occurs in documents with that label:

(3.3.2)\[p(x_j \mid y) = \frac{\text{count}(x_j, y)}{\sum_{x\in V} \text{count}(x, y)},\]

where \(V\) is the vocabulary and \(\text{count}(x_j, y)\) is the count of word \(x_j\) in documents with label \(y\). So we have re-discovered our intuitive method through the Naive Bayes model!

3.3.1. Laplace smoothing

One potential problem of our estimator is that a word that doesn’t occur in the training set will have zero probability. For example, if none of the positive reviews contain the word “awwwwwwwesome”, any review containing that word will have zero probability given the positive label. This is undesirable because due to the sparsity of language it’s not uncommon to have unknown words at test time. Even if we ignore unknown words in the input, we will have problems with rare words (low frequency in the training set), because their estimated conditional probabilities might be off.

To remedy this problem, we use a technique called “smoothing”, which add pseudocounts to the actual count of each word:

(3.3.3)\[p(x_j \mid y) = \frac{\alpha + \text{count}(x_j, y)}{\alpha |V| + \sum_{x\in V}\text{count}(x, y)} .\]

This means that even before we have seen any data, we believe that all words should occur \(\alpha\) times. For Laplace smoothing \(\alpha=1\).

Question: what happens when we increase/decrease \(\alpha\)?

3.4. Logistic regression

You might have wondered in Naive Bayes modeling why did we take the trouble to rewrite \(p(y\mid x)\) using the Bayes’ rule instead of directy modeling the conditional distribution. After all, that’s the distribution we use for prediction. Also, we don’t have to make assumptions on how \(x\) is generated (e.g. conditional independence). In fact, models like Naive Bayes that models \(p(x\mid y)\) are called generative models and they usually assume a generative story of how the data is generated. Models that directly model \(p(y\mid x)\) are called discriminative models. Both approaches have merits. However, if you have lots of data, empirically discriminative models may be better since it makes less assumptions about the data distribution.

How should we model \(p(y\mid x)\)? Similar to the Naive Bayes model, since \(y\) is binary, we can model it by a Bernoulli distribution: \(p(y\mid x) = h(x)^y(1-h(x))^{1-y}\). Here \(h(x)\) is \(p(y=1\mid x)\). Ideally, we would like \(x\) to enter the equation through a score \(w\cdot \phi(x)\) (think linear regression), where \(w\) is the weight vector we want to learn and \(\phi\colon \mathcal{X} \rightarrow \mathbb{R}^d\) is a feature extractor that maps a piece of text to a vector. Note that here we can ignore the bias term in \(w\cdot\phi(x) + b\), assuming that a dimension of constant value 1 is incorporated in \(\phi\).

To map the score to a probability, we use the logistic function:

%matplotlib inline
from IPython import display
from matplotlib import pyplot as plt

display.set_matplotlib_formats('svg')
x = np.arange(-10, 10, 0.01)
plt.plot(x, 1/(1+np.exp(-x)))
plt.ylabel('$1/(1+e^{-w\cdot\phi(x)})$')
plt.xlabel('$w\cdot\phi(x)$')

Text(0.5, 0, '$w\cdot\phi(x)$')
../_images/output_text_classification_e2cfc8_10_1.svg

Note that larger score corresponds to higher \(p(y=1\mid x)\). This gives us the logistic regression model:

(3.4.1)\[p(y=1\mid x) = \frac{1}{1+e^{-w\cdot\phi(x)}} .\]

For multiclass classification, \(p(y\mid x)\) is modeled by a multinomial distribution and we transform the scores \(w_k\cdot \phi(x)\) (k:raw-latex:in `:raw-latex:mathcal{Y}`) using the softmax function:

(3.4.2)\[p(y=k\mid x) = \frac{e^{w_k\cdot\phi(x)}}{\sum_{i\in\mathcal{Y}} e^{w_i\cdot\phi(x)}} .\]

Similar to Naive Bayes, we can use MLE to estimate \(w\). But for logistic regression we don’t have a closed-form solution (try it) and need to use (stochastic) gradient descent. The objective function is concave, so we can always reach a global optimal solution.

Exercise: Show that MLE for logistic regression is equivalent to minimizing the average logistic loss.

3.5. Bag-of-words (BoW) representation

We have ignored one question above in logistic regression: what is the feature extractor \(\phi\)? We need to represent the document as a vector that can work with our linear model.

How are we going to represent a piece of text? If we want the representation the whole sequence it’s going to be challenging (an exponentially large set). Following our intuition, it may be sufficient to just consider individual words for our task. The BoW representation is a vector \(x = (1, x_1, \ldots, x_d)\) where \(x_i \in \mathbb{N}\). Each coordinate corresponds to a unique word in the vocabulary, thus \(d\) is the vocabulary size here. Note that the first dimensition corresponds to the bias term. The value \(x_i\) is the count of the \(i\)-th word in the input text. It’s called a bag of words because we are throwing away the position information.

Example. To map a sequence of tokens to the BoW vector, first we need to build the vocabulary.

vocab = nlp.Vocab(nlp.data.count_tokens(pos_tokens + neg_tokens))
print(len(vocab))

20406

Convert an example to BoW vector representation:

# map words to ints
x = np.array(vocab(mini_dataset[0][0]))
# convert to vector of counts
x = npx.one_hot(x, len(vocab)).sum(axis=0)
print('feature vector size:', x.shape)
# show counts
print('top word counts:')
ids = [int(i) for i in np.argsort(x)[::-1][:10]]
print([(vocab.idx_to_token[i], int(x[i])) for i in ids])

feature vector size: (20406,)
top word counts:
[('the', 8), ('a', 4), ('of', 4), ('to', 4), ('with', 4), ('.', 3), ('and', 3), ('is', 3), ('i', 3), ('that', 3)]

Side note. Note that here the feature vector is different from the one we used for the Naive Bayes model. In Section 3.2, our effective feature vector (we didn’t need to explicit represent it then) is a sequence of words in the input, thus its length varies across examples, whereas here the feature vector has a fixed dimension of the vocabulary size. One can show that the Naive Bayes model we described above corresponds to assuming a multinomial distribution of the count vector \(\phi(x)\), thus it’s also called a Multinomial Naive Bayes model.

3.6. Feature extractor

Looking at the word counts in our BoW feature vector above, clearly, we don’t have very informative features. In addition, only considering single words (unigram) breaks compound nouns (e.g. “ice cream”) and proper nouns (e.g. “New York”). It’s also hard to represent negation (e.g. “not good at all”), which is important especially in sentiment classification. One easy fix is to consider an n-gram (\(n\) consecutive words in a document) as a single word. For text classification, bi-gram is commonly used.

But what if we want to use even richer features, such as whether the suffix of a word contains repeated letters (e.g. “yayyyyy”). One advantage of using logistic regression instead of Naive Bayes is that it allows for rich features using the feature extractor \(\phi\).

For example, we can define the following functions given an input \(x\) of \(n\) tokens with label \(y\):

(3.6.1)\[\begin{split}\phi_1(x) &= \begin{cases} 1 & \text{$\exists x_i\in \{x_1,\ldots x_n\}$ such that $x_i=$"happy"} \\ 0 & \text{otherwise} \end{cases} , \\ \phi_2(x) &= \begin{cases} 1 & \text{$\exists x_i\in \{x_1,\ldots x_n\}$ such that $\text{suffix}(x_i, 4)=$"yyyy"} \\ 0 & \text{otherwise} \end{cases} ,\end{split}\]

so on and so forth.

Side note. In practice, we may have a huge number of features (e.g. all n-grams in a corpus). However, for NLP problems the features are often sparse, meaning that we only have a handful of non-zero features. Thus it’s common to represent the feature value as a string and hash it to integers (feature index), e.g. \(\text{hash}(\text{"w=happy"}) = 1\). See FeatureHasher from sklearn.

3.7. Evaluation

TODO

3.8. Additional readings