Advice for this homework:

Words are simply strings separated by whitespace. Note that words which
only differ in capitalization are considered separate (e.g.
great and Great are considered different words).

You might find some useful functions in
util.py
. Have a look
around in there before you start coding.
Problem 1: Building intuition
Here are two reviews of Perfect Blue, from
Rotten Tomatoes:
Rotten Tomatoes has classified these reviews as "positive" and "negative,",
respectively, as indicated by the intact tomato on the left and the
splattered tomato on the right. In this assignment, you will create a simple
text classification system that can perform this task automatically.
We'll warm up with the following set of four minireviews, each labeled
positive $(+1)$ or negative $(1)$:
 $(1)$ pretty bad
 $(+1)$ good plot
 $(1)$ not good
 $(+1)$ pretty scenery
Each review $x$ is mapped onto a feature vector $\phi(x)$, which maps each
word to the number of occurrences of that word in the review. For example,
the first review maps to the (sparse) feature vector $\phi(x) =
\{\text{pretty}:1, \text{bad}:1\}$. Recall the definition of the hinge loss:
$$\text{Loss}_{\text{hinge}}(x, y, \mathbf{w}) = \max \{0, 1  \mathbf{w}
\cdot \phi(x) y\},$$ where $x$ is the review text, $y$ is the correct label,
$\mathbf{w}$ is the weight vector.

Suppose we run stochastic gradient descent once for each of the 4 samples
in the order given above, updating the weights according to $$\mathbf{w}
\leftarrow \mathbf{w}  \eta \nabla_\mathbf{w}
\text{Loss}_{\text{hinge}}(x, y, \mathbf{w}).$$ After the updates, what
are the weights of the six words ("pretty", "good", "bad", "plot", "not",
"scenery") that appear in the above reviews?
 Use $\eta = 0.1$ as the step size.
 Initialize $\mathbf{w} = [0, 0,0,0,0, 0]$.

The gradient $\nabla_\mathbf{w} \text{Loss}_{\text{hinge}}(x, y,
\mathbf{w}) = 0$ when margin is exactly 1.
A weight vector that contains a numerical value for each of the tokens
in the reviews ("pretty", "good", "bad","plot", "not", "scenery"),
in this order. For example: $[0.1, 0.2,0.3,0.4,0.5, 0.6]$.

Given the following dataset of reviews:
 ($1$) bad
 ($+1$) good
 ($+1$) not bad
 ($1$) not good
Prove that no linear classifier using word features can get zero error on
this dataset. Remember that this is a question about classifiers, not
optimization algorithms; your proof should be true for any linear
classifier, regardless of how the weights are learned.
Propose a single additional feature for your dataset that we could augment
the feature vector with that would fix this problem.
 a short written proof (~35 sentences).

a viable feature that would allow a linear classifier to have zero
error on the dataset (classify all examples correctly).
Problem 2: Predicting Movie Ratings
Suppose that we are now interested in predicting a numeric rating for
movie reviews. We will use a nonlinear predictor that takes a movie
review $x$ and returns $\sigma(\mathbf w \cdot \phi(x))$, where $\sigma(z)
= (1 + e^{z})^{1}$ is the logistic function that squashes a real number
to the range $(0, 1)$. For this problem, assume that the movie rating $y$
is a realvalued variable in the range $[0, 1]$.
Do not use math software such as Wolfram Alpha to solve this
problem.

Suppose that we wish to use
squared loss. Write out the expression for $\text{Loss}(x, y,
\mathbf w)$ for a single datapoint $(x,y)$.
A mathematical expression for the loss. Feel free to use $\sigma$ in the
expression.

Given $\text{Loss}(x, y, \mathbf w)$ from the previous part, compute the
gradient of the loss with respect to w, $\nabla_w \text{Loss}(x, y,
\mathbf w)$. Write the answer in terms of the predicted value $p =
\sigma(\mathbf w \cdot \phi(x))$.
A mathematical expression for the gradient of the loss.

Suppose there is one datapoint $(x, y)$ with some arbitrary $\phi(x)$ and
$y = 1$. Specify conditions for $\mathbf w$ to make the magnitude of the
gradient of the loss with respect to $\mathbf w$ arbitrarily small (i.e.
minimize the magnitude of the gradient). Can the magnitude of the gradient
with respect to $\mathbf w$ ever be exactly zero? You are allowed to make
the magnitude of $\mathbf w$ arbitrarily large but not infinity.
Hint: try to understand intuitively what is going on and what each part
of the expression contributes. If you find yourself doing too much
algebra, you're probably doing something suboptimal.
Motivation: the reason why we're interested in the magnitude of the
gradients is because it governs how far gradient descent will step. For
example, if the gradient is close to zero when $\mathbf w$ is very far
from the optimum, then it could take a long time for gradient descent to
reach the optimum (if at all). This is known as the
vanishing gradient problem when training neural networks.
12 sentences describing the conditions for $\mathbf w$ to minimize the
magnitude of the gradient, 12 sentences explaining whether the gradient
can be exactly zero.
Problem 3: Sentiment Classification
In this problem, we will build a binary linear classifier that reads movie
reviews and guesses whether they are "positive" or "negative."
Do not import any outside libraries (e.g. numpy) for any of the coding
parts.
Only standard python libraries and/or the libraries imported in the starter
code are allowed. In this problem, you must implement the functions without
using libraries like Scikitlearn.

Implement the function
extractWordFeatures
, which takes a
review (string) as input and returns a feature vector $\phi(x)$, which is
represented as a dict
in Python.

Implement the function
learnPredictor
using stochastic
gradient descent and minimize hinge loss. Print the training error and
validation error after each epoch to make sure your code is working. You
must get less than 4% error rate on the training set and less than 30%
error rate on the validation set to get full credit.

Write the
generateExample
function (nested in the
generateDataset
function) to generate artificial data
samples.
Use this to double check that your
learnPredictor
works! You can do this by using
generateDataset()
to generate training and validation
examples. You can then pass in these examples as
trainExamples
and
validationExamples
respectively to
learnPredictor
with the identity function
lambda x: x
as a featureExtractor.

When you run the grader.py on test case
3b2
, it should
output a weights
file and a erroranalysis
file.
Find 3 examples of incorrected predictions. For each example, give
a onesentence explanation on why the classifier got it wrong. State what
additional information the classifier would need to get these examples
correct.
Note: The main point is to convey intuition about the problem. There isn't
always a single correct answer. You do not need to pick 3 different types
of errors and explain each. It suffices to show 3 instances of the same
type of error, and for each explain why the classification was incorrect.

3 sample incorrect predictions, each with one sentence explaining
why the classifications for these sentences was incorrect.

a single separate paragraph (35 sentences) outlining what
information the classifier would need to get these predictions
correct.

Some languages are written without spaces between words, so is splitting
the words really necessary or can we just naively consider strings of
characters that stretch across words? Implement the function
extractCharacterFeatures
(by filling in the
extract
function), which maps each string of $n$ characters
to the number of times it occurs, ignoring whitespace (spaces and tabs).

Run your linear predictor with feature extractor
extractCharacterFeatures
. Experiment with different values of
$n$ to see which one produces the smallest validation error. You should
observe that this error is nearly as small as that produced by word
features. Why is this the case?
Construct a review (one sentence max) in which character $n$grams
probably outperform word features, and briefly explain why this is so.
Note: There is code in submission.py
that will help
you test different values of $n$. Remember to write your final written
solution in sentiment.pdf.

a short paragraph (~46) sentences. In the paragraph state which
value of $n$ produces the smallest validation error, why this is
likely the value that produces the smallest error.

a onesentence review and explanation for when character $n$grams
probably outperform word features.
Problem 4: Kmeans clustering
Suppose we have a feature extractor $\phi$ that produces 2dimensional feature
vectors, and a toy dataset $\mathcal D_\text{train} = \{x_1, x_2, x_3, x_4\}$
with
 $\phi(x_1) = [10, 0]$
 $\phi(x_2) = [30, 0]$
 $\phi(x_3) = [10, 20]$
 $\phi(x_4) = [20, 20]$

Run 2means on this dataset until convergence. Please show your work. What
are the final cluster assignments $z$ and cluster centers $\mu$? Run this
algorithm twice with the following initial centers:
 $\mu_1 = [20, 30]$ and $\mu_2 = [20, 10]$
 $\mu_1 = [0, 10]$ and $\mu_2 = [30, 20]$
Show the cluster centers and assignments for each step.

Implement the
kmeans
function. You should initialize your $k$
cluster centers to random elements of examples
.
After a few iterations of kmeans, your centers will be very dense
vectors. In order for your code to run efficiently and to obtain full
credit, you will need to precompute certain quantities. As a reference,
our code runs in under a second on cardinal, on all test cases. You might
find generateClusteringExamples
in util.py useful for testing your code.
Do not use libraries such as Scikitlearn.

Sometimes, we have prior knowledge about which points should belong in the
same cluster. Suppose we are given a set $G$ of disjoint set of points
that must be assigned to the same cluster.
For example, suppose we have 6 examples; then $G = \{ (1,5), (2,3,4), (6)
\}$ says that examples 2, 3, and 4 must be in the same cluster and that
examples 1 and 5 must be in the same cluster. 6 is in its own group and is
unbounded, so it can be freely assigned to its own cluster, or to a
cluster with any other group, depending on initialization and the value of
$K$ in kmeans.
All examples must appear in $G$ exactly once.
Provide the modified kmeans algorithm that performs alternating
minimization on the reconstruction loss: $$\sum \limits_{i=1}^n \
\mu_{z_i}  \phi(x_i) \^2,$$ where $\mu_{z_i}$ is the assigned centroid
for the feature vector $\phi(x_i)$.
Hint 1: recall that alternating minimization is when we are optimizing
two variables jointly by alternating which variable we keep constant. We
recommend starting by first keeping $z$ fixed and optimizing over $\mu$
and then keeping $\mu$ fixed and optimizing over $z$.
A mathematical expression representing the modified cluster assignment
update rule for the kmeans steps, and a brief explanation for each
step. Do not modify the problem setup or make additional assumptions on
the inputs.

What is the advantage of running Kmeans multiple times on the same
dataset with the same K, but different random initializations?
A ~13 sentences explanation.

If we scale all dimensions in our initial centroids and data points by
some factor, are we guaranteed to retrieve the same clusters after running
Kmeans (i.e. will the same data points belong to the same cluster before
and after scaling)? What if we scale only certain dimensions? If your
answer is yes, provide a short explanation; if not, give a counterexample.
counterexample.
This response should have two parts. The first should be a yes/no
response and explanation or counterexample for the first subquestion
(scaling all dimensions). The second should be a yes/no response and
explanation or counterexample for the second subquestion (scaling only
certain dimensions).