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Homework 1 Nearest Neighbours and the Curse of Dimensionality

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CSC311
Homework 1

Submission: You need to submit three files through MarkUs1
:
• Your answers to Questions 1, 2, and 3, and outputs requested for Question 2, as a PDF file
titled hw1_writeup.pdf. You can produce the file however you like (e.g. LATEX, Microsoft
Word, scanner), as long as it is readable.
• Your code for Question 2, as the Python file hw1_code.py. This should contain the functions
load_data, select_model, and compute_information_gain.
Neatness Point: One point will be given for neatness. You will receive this point as long as we
don’t have a hard time reading your solutions or understanding the structure of your code.
Late Submission: 10% of the marks will be deducted for each day late, up to a maximum of 3
days. After that, no submissions will be accepted.
Computing: To install Python and required libraries, see the instructions on the course web page.
Homeworks are individual work. See the Course Information handout2
for detailed policies.
1. [4pts] Nearest Neighbours and the Curse of Dimensionality. In this question, you
will verify the claim from lecture that “most” points in a high-dimensional space are far away
from each other, and also approximately the same distance. There is a very neat proof of this
fact which uses the properties of expectation and variance. If it’s been a long time since you’ve
studied these, you may wish to review the Tutorial 1 slides3
, or the Metacademy resources4
.
(a) [2pts] For each choice of dimension d ∈ [20
, 2
1
, 2
2
, …, 2
10], sample 100 points from the
unit cube, and record the average squared Euclidean distance between all pairs of points,
as well as the standard deviation of the squared Euclidean distances. (Recall that squared
Euclidean distance is defined as kx − yk
2 =
P
j
(xj − yj )
2
.) Plot both the average and
the standard deviation as a function of d. (You may wish to use np.mean and np.std to
compute the statistics, and matplotlib for plotting. You may find numpy.random.rand
helpful in sampling from the unit cube.) Include the output figure in your solution PDF
(hw1_writeup.pdf).
(b) [2pts] In this question, we aim to verify our simulations in (a) by deriving the analytical
form of averaged distance and variance of distance. Suppose we sample two points X
and Y independently from a unit cube in d dimensions and define the squared Euclidean
distance R = Z1+· · ·+Zd with Zi = (Xi−Yi)
2
. Given that E[Zi
] = 1
6
and Var[Zi
] = 7
180 ,
determine E[R] and Var[R] using the properties of expectation and variance. You may
give your answer in terms of the dimension d.
Basic rule of expectation and variance: E[Zi+Zj ] = E[Zi
]+E[Zj ] and if we further
know Zi and Zj are independent, then Var[Zi + Zj ] = Var[Zi
] + Var[Zj ].
1
https://markus.teach.cs.toronto.edu/csc311-2021-09
2
http://www.cs.toronto.edu/~rgrosse/courses/csc311_f21/syllabus.pdf
3
https://www.cs.toronto.edu/~rgrosse/courses/csc311_f21/tutorials/tut01.pdf
4
https://metacademy.org/graphs/concepts/expectation_and_variance
1
CSC311 Fall 2021 Homework 1
(c) [Optional, 0pts] In probability theory, one can derive that P(|Z − E[Z]| ≥ a) ≤
Var[Z]
a
2
for any random variable Z. (This fact is known as Markov’s Inequality.) Based on your
answer to part (b), explain why does this support the claim that in high dimensions,
“most points are far away, and approximately the same distance”?
2. [8pts] Decision Trees. This question is taken from a project by Lisa Zhang and Michael
Guerzhoy.
In this question, you will use the scikit-learn decision tree classifier to classify real vs. fake
news headlines. The aim of this question is for you to read the scikit-learn API and get
comfortable with training/validation splits.
We will use a dataset of 1298 “fake news” headlines (which mostly include headlines of articles
classified as biased, etc.) and 1968 “real” news headlines, where the “fake news” headlines
are from https://www.kaggle.com/mrisdal/fake-news/data and “real news” headlines are
from https://www.kaggle.com/therohk/million-headlines. The data were cleaned by
removing words from fake news titles that are not a part of the headline, removing special
characters from the headlines, and restricting real news headlines to those after October
2016 containing the word “trump”. For your interest, the cleaning script is available as
clean_script.py on the course web page, but you do not need to run it. The cleaned-up
data are available as clean_real.txt and clean_fake.txt on the course web page.
Each headline appears as a single line in the data file. Words in the headline are separated
by spaces, so just use str.split() in Python to split the headlines into words.
You will build a decision tree to classify real vs. fake news headlines. Instead of coding
the decision trees yourself, you will do what we normally do in practice — use an existing
implementation. You should use the DecisionTreeClassifier included in sklearn. Note
that figuring out how to use this implementation is a part of the assignment.
Here’s a link to the documentation of sklearn.tree.DecisionTreeClassifier: http://
scikit-learn.org/stable/modules/generated/sklearn.tree.DecisionTreeClassifier.
html
All code should be included in the file hw1_code.py which you submit through MarkUs.
(a) [2pt] Write a function load_data which loads the data, preprocesses it using a vectorizer
(http://scikit-learn.org/stable/modules/classes.html#module-sklearn.feature_
extraction.text), and splits the entire dataset randomly into 70% training, 15% validation, and 15% test examples.
(b) [2pt] Write a function select_model which trains the decision tree classifier using
at least 5 different values of max_depth, as well as two different split criteria (information gain and Gini coefficient), evaluates the performance of each one on the
validation set, and prints the resulting accuracies of each model. You should use
DecisionTreeClassifier, but you should write the validation code yourself. Include
the output of this function in your solution PDF (hw1_writeup.pdf).
(c) [1pt] Now let’s stick with the hyperparameters which achieved the highest validation
accuracy. Extract and visualize the first two layers of the tree. Your visualization may
look something like what is shown below, but it does not have to be an image: it is
perfectly fine to display text. It may be hand-drawn. Include your visualization in your
solution PDF (hw1_writeup.pdf).
2
CSC311 Fall 2021 Homework 1
(d) [3pts] Write a function compute_information_gain which computes the information
gain of a split on the training data. That is, compute I(Y, xi), where Y is the random
variable signifying whether the headline is real or fake, and xi
is the keyword chosen for
the split.
Report the outputs of this function for the topmost split from the previous part, and for
several other keywords.
3. [8pts] Regularized Linear Regression. For this problem, we will use the linear regression
model from the lecture:
y =
X
D
j=1
wjxj + b.
In lecture, we saw that regression models with too much capacity can overfit the training data
and fail to generalize. We also saw that one way to improve generalization is regularization:
adding a term to the cost function which favors some explanations over others. For instance,
we might prefer that weights not grow too large in magnitude. We can encourage them to
stay small by adding a penalty:
R(w) = λ
2
w>w =
λ
2
X
D
j=1
w
2
j
to the cost function, for some λ > 0. It is also possible to apply different regularization
penalties in each dimension. The formulation would be:
J
β
reg(w) = 1
2N
X
N
i=1

y
(i) − t
(i)
2
| {z }
=J
+
1
2
X
D
j=1
βjw
2
j
| {z }
=R
,
3
CSC311 Fall 2021 Homework 1
where i indexes the data points, βj ≥ 0 for all j, and J is the same squared error cost
function from lecture. Note that in this formulation, there is no regularization penalty on
the bias parameter. Also note that when βj = 0, you don’t apply any regularization on j-th
dimension. For this question, show your work in detail as most points are allocated in showing
how you obtained your answer.
(a) [3pts] Determine the gradient descent update rules for the regularized cost function
J
β
reg. Your answer should have the form:
wj ← · · ·
b ← · · ·
This form of regularization is sometimes called “weight decay”. Based on this update
rule, why do you suppose that is?
(b) [3pts] It’s also possible to solve the regularized regression problem directly by setting
the partial derivatives equal to zero. In this part, for simplicity, we will drop the bias
term from the model, so our model is:
y =
X
D
j=1
wjxj .
It is possible to derive a system of linear equations of the following form for J
β
reg:
∂J
β
reg
∂wj
=
X
D
j
0=1
Ajj0wj
0 − cj = 0.
Determine formulas for Ajj0 and cj .
(c) [2pts] Based on your answer to part (b), determine formulas for A and c, and derive a
closed-form solution for the parameter w. Note that, as usual, the inputs are organized
into a design matrix X with one row per training example.
4

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