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Text Classification

+

Machine Learning Review

CS 287

Contents

Text Classification

Preliminaries: Machine Learning for NLP

Features and Preprocessing

Output

Classification

Linear Models

Linear Model 1: Naive Bayes

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DAYS

Sentiment

Good Sentences

I A thoughtful, provocative, insistently humanizing film.

I Occasionally melodramatic, it’s also extremely effective.

I Guaranteed to move anyone who ever shook, rattled, or rolled.

Bad Sentences

I A sentimental mess that never rings true.

I This 100-minute movie only has about 25 minutes of decent

material.

I Here, common sense flies out the window, along with the hail of

bullets, none of which ever seem to hit Sascha.

Multiclass Sentiment

I ? ? ??

I visited The Abbey on several occasions on a visit to Cambridge

and found it to be a solid, reliable and friendly place for a meal.

I ??

However, the food leaves something to be desired. A very obvious

menu and average execution

I ? ? ? ? ?

Fun, friendly neighborhood bar. Good drinks, good food, not too

pricey. Great atmosphere!

Text Categorization

I Straightforward setup.

I Lots of practical applications:

I Spam Filtering

I Sentiment

I Text Categorization

I e-discovery

I Twitter Mining

I Author Identification

I . . .

I Introduces machine learning notation.

However, a relatively solved problem these days.

Contents

Text Classification



Output

Classification

Linear Models


Preliminary Notation

I b,m; bold letters for vectors.

I B,M; bold capital letters for matrices.

I B,M; script-case for sets.

I B,M; capital letters for random variables.

I bi , xi ; lower case for scalars or indexing into vectors.

I δ(i); one-hot vector at position i

δ(2) = [0; 1; 0; . . . ]

I 1(x = y); indicator 1 if x = y , o.w. 0

Text Classification

1. Extract pertinent information from the sentence.

2. Use this to construct an input representation.

3. Classify this vector into an output class.

Input Representation:

I Conversion from text into a mathematical representation?

I Main focus of this class, representation of language

I Point in coming lectures: sparse vs. dense representations

Sparse Features (Notation from YG)

I F ; a discrete set of features values.

I f1 ∈ F , . . . , fk ∈ F ; active features for input.

For a given sentence, let f1, . . . fk be the relevant features.

Typically k << |F |.

I Sparse representation of the input defined as,

x =k

∑i=1

δ(fi )

I x ∈ R1×din ; input representation

Features 1: Sparse Bag-of-Words Features

Representation is counts of input words,

I F ; the vocabulary of the language.

I x = ∑i δ(fi )

Example: Movie review input,

A sentimental mess

x = δ(word:A) + δ(word:sentimental) + δ(word:mess)

x> =

1...

0

0

+

0...

0

1

+

0...

1

0

=

1...

1

1

word:A...

word:mess

word:sentimental

Features 2: Sparse Word Properties

Representation can use specific aspects of text.

I F ; Spelling, all-capitals, trigger words, etc.

I x = ∑i δ(fi )

Example: Spam Email

Your diploma puts a UUNIVERSITY JOB PLACEMENT COUNSELOR

at your disposal.

x = δ(misspelling) + δ(allcapital) + δ(trigger:diploma) + . . .

x> =

0...

1

0

+

0...

0

1

+

1...

0

0

=

1...

1

1

misspelling...

capital

word:diploma

Text Classification: Output Representation

1. Extract pertinent information from the sentence.

2. Use this to construct an input representation.

3. Classify this vector into an output class.

Output Representation:

I How do encode the output classes?

I We will use a one-hot output encoding.

I In future lectures, efficiency of output encoding.

Output Class Notation

I C = {1, . . . , dout}; possible output classes

I c ∈ C; always one true output class

I y = δ(c) ∈ R1×dout ; true one-hot output representation

Output Form: Binary Classification

Examples: spam/not-spam, good review/bad review, relevant/irrelevant

document, many others.

I dout = 2; two possible classes

I In our notation,

bad c = 1 y =[

1 0]vs.

good c = 2 y =[

0 1]

I Can also use a single output sign representation with dout = 1

Output Form: Multiclass Classification

Examples: Yelp stars, etc.

I dout = 5; for examples

I In our notation, one star, two star...

? c = 1 y =[

1 0 0 0 0]vs.

? ? c = 2 y =[

0 1 0 0 0]

. . .

Examples: Word Prediction (Unit 3)

I dout > 100, 000;

I In our notation, C is vocabulary and each c is a word.

the c = 1 y =[

1 0 0 0 . . . 0]vs.

dog c = 2 y =[

0 1 0 0 . . . 0]

. . .

Evaluation

I Consider evaluating accuracy on outputs y1, . . . , yn.

I Given a predictions c1 . . . cn we measure accuracy as,

n

∑i=1

1(δ(ci ) = yi )

n

I Simplest of several different metrics we will explore in the class.

Contents

Text Classification



Output

Classification

Linear Models


Supervised Machine Learning

Let,

I (x1, y1), . . . , (xn, yn); training data

I xi ∈ R1×din ; input representations

I yi ∈ R1×dout ; true output representations (one-hot vectors)

Goal: Learn a classifier from input to output classes.

Note:

I xi is an input vector xi ,j is element of the vector, or just xj when

there is a clear single input .

I Practically, store design matrix X ∈ Rn×din and output classes.

Experimental Setup

I Data is split into three parts training, validation, and test.

I Experiments are all run on training and validation, test is final

output.

I For assignments, full training and validation data, and only inputs

for test.

For very small text classification data sets,

I Use K-fold cross-validation.

1. Split into K folds (equal splits).

2. For each fold, train on other K-1 folds, test on current fold.

Linear Models for Classification

Linear model,

y = f (xW+ b)

I W ∈ Rdin×dout ,b ∈ R1×dout ; model parameters

I f : Rdout 7→ Rdout ; activation function

I Sometimes z = xW+ b informally “score” vector.

I Note z and y are not one-hot.

Class prediction,

c = arg maxi∈C

yi = arg maxi∈C

(xW+ b)i

Interpreting Linear Models

Parameters give scores to possible outputs,

I Wf ,i is the score for sparse feature f under class i

I bi is a prior score for class i

I yi is the total score for class i

I c is highest scoring class under the linear model.

Example:

I For single feature score,

[β1, β2] = δ(word:dreadful)W,

Expect β1 > β2 (assuming 2 is class good).

Probabilistic Linear Models

Can estimate a linear model probabilistically,

I Let output be a random variable Y , with sample space C.

I Representation be a random vector X .

I (Simplified frequentist representation)

I Interested in estimating parameters θ,

P(Y |X ; θ)

Informally we use p(y = δ(c)|x) for P(Y = c |X = x).

Generative Model. Joint Log-Likelihood as Loss

I (x1, y1), . . . , (xn, yn); supervised data

I Select parameters to maximize likelihood of training data.

L(θ) = −n

∑i=1

log p(xi , yi ; θ)

For linear models θ = (W,b)

I Do this by minimizing negative log-likelihood (NLL).

arg minθL(θ)

Multinomial Naive Bayes

Reminder, joint probability chain rule,

p(x, y) = p(x|y)p(y)

For a sparse features, with observed classes we can write as,

p(x, y) = p(xf1 = 1, . . . , xfk = 1|y = δ(c))p(y = δ(c)) =

=k

∏i=1

p(xfi = 1|xf1 = 1, . . . , xfi−1 = 1, y = δ(c))p(y = δ(c)) ≈

=k

∏i=1

p(xfi = 1|y)p(y)

First is by chain-rule, second is by independence assumption.

Estimating Multinomial Distributions

Let S be a random variable with sample space S and we have

observations s1, . . . , sn,

I P(S = s; θ) = Cat(s; θ); parameterized as a multinomial

distribution.

I Minimizing NLL for multinomial (MLE) for data has a closed-form.

P(S = s; θ) = Cat(s; θ) =n

∑i=1

1(si = s)

n

I Exercise: Derive this by minimizing L.

I Also called categorical or multinoulli (in Murphy).

Multinomial Naive Bayes

I Both p(y) and p(x|y) are parameterized as multinomials.

I Fit first as,

p(y = δ(c)) =n

∑i=1

1(yi = c)

n

I Fit second using count matrix F ,

I Let

Ff ,c =n

∑i=1

1(yi = c)1(xi ,f = 1) forall c ∈ C, f ∈ F

I Then,

p(xf = 1|y = δ(c)) =Ff ,c

∑f ′∈F

Ff ′,c

Alternative: Multivariate Bernoulli Naive Bayes

I Both p(y) is multinomial as above and p(xf |y) is Bernoulli over

each features .

I Fit class as Categorical,

p(y = δ(c)) =n

∑i=1

1(yi = c)

n

I Fit features using count matrix F ,

I Let

Ff ,c =n

∑i=1

1(yi = c)1(xi ,f = 1) forall c ∈ C, f ∈ F

I Then,

p(xf |y = δ(c)) =Ff ,c

n

∑i=1

1(yi = c)

Prediction with Naive Bayes

I For prediction, last term is constant, so

arg maxc

log p(y = δ(c)|x) = log p(x|y = δ(c)) + log p(y = δ(c))

I Can write as linear model,

Wf ,c = log p(xf = 1|y = c) for all c ∈ C, f ∈ F

bc = log p(y = δ(c)) for all c ∈ C

Getting a Conditional Distribution

I What if we want conditional probabilities?

p(y|x) = p(x|y)p(y)p(x)

=p(x|y)p(y)

∑c ′∈C p(x, y = δ(c ′))

I Denominator is acquired by renormalizing,

p(y|x) ∝ p(x|y)p(y)

Practical Aspects: Calculating Log-Sum-Exp

I Because of numerical issues, calculate in log-space,

f (z) = log p(y = δ(c)|x) = log zc − log ∑c ′∈C

exp(zc ′)

where for naive Bayes

z = xW+ b

I However hard to calculate,

log ∑c ′∈C

exp(zc ′)

I Instead

log ∑c ′∈C

exp(yc ′ −M) +M

where M = maxc ′∈C zc ′

Digression: Zipf’s Law

Laplace Smoothing

Method for handling the long tail of words by distributing mass,

I Add a value of α to each element in the sample space before

normalization.

θs =α + ∑n

i=1 1(si = s)

α|S|+ n

I (Similar to Dirichlet prior in a Bayesian interpretation.)

For naive Bayes:

F = α + F

Naive Bayes In Practice

I Very fast to train

I Relatively interpretable.

I Performs quite well on small datasets

(RT-S [movie review], CR [customer reports], MPQA [opinion polarity],

SUBJ [subjectivity])

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