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CSC2515  Homework 1

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Homework 1
1. [3pts] 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 Chapter 6 of Mathematics for Machine
Learning2
, or the Metacademy resources3
.
(a) [2pts] First, consider two independent univariate random variables X and Y sampled
uniformly from the unit interval [0, 1]. Determine the expectation and variance of the
random variable Z, defined as the squared distance Z = (X − Y )
2
. You are allowed to
evaluate integrals numerically (e.g. using scipy.integrate.quad or scipy.integrate.dblquad),
but you should explain what integral(s) you are evaluating, and why.
(b) [1pt] Now suppose we sample two points independently from a unit cube in d dimensions.
Observe that each coordinate is sampled independently from [0, 1], i.e. we can view this as
sampling random variables X1, . . . , Xd, Y1, . . . , Yd independently from [0, 1]. The squared
Euclidean distance can be written as R = Z1 + · · · + Zd, where Zi = (Xi − Yi)
2
. Using
the properties of expectation and variance, determine E[R] and Var[R]. You may give
your answer in terms of the dimension d, and E[Z] and Var[Z] (the answers from part
(a)).
(c) [for your own benefit, not to be handed in] Based on your answer to part (b),
compare the mean and standard deviation of R to the maximum possible squared Euclidean distance (i.e. the distance between opposite corners of the cube). Why does this
support the claim that in high dimensions, “most points are far away, and approximately
the same distance”?
1
https://markus.teach.cs.toronto.edu/csc2515-2019-09
2
https://mml-book.github.io/
3
https://metacademy.org/graphs/concepts/expectation_and_variance
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CSC2515 Fall 2019 Homework 1
2. [3pts] Decision Trees. 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 a sense for the sort of functionality it provides. You may use
any scikit-learn functionality, even if you feel like it trivializes the question. (This is not
meant to be a hard question.)
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.
All code should be included in the file hw1_code.py which you submit through MarkUs.
(a) [1pt] 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) [1pt] 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. 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
CSC2515 Fall 2019 Homework 1
This question is taken from a project by Lisa Zhang and Michael Guerzhoy.
3. [4pts] Information Theory. The goal of this question is to help you become more familiar
with the basic equalities and inequalities of information theory. They appear in many contexts
in machine learning and elsewhere, so having some experience with them is quite helpful. We
review some concepts from information theory, and ask you a few questions.
Recall the definition of the entropy of a discrete random variable X with probability mass
function p: H(X) = P
x
p(x) log2

1
p(x)

. Here the summation is over all possible values of
x ∈ X , which (for simplicity) we assume is finite. For example, X might be {1, 2, . . . , N}.
(a) [1pt] Prove that the entropy H(X) is non-negative.
An important concept in information theory is the relative entropy or the KL-divergence of
two distributions p and q. It is defined as
KL(p||q) = X
x
p(x) log2
p(x)
q(x)
.
The KL-divergence is one of the most commonly used measure of difference (or divergence)
between two distributions, and it regularly appears in information theory, machine learning,
and statistics. For this question, you may assume p(x) > 0 and q(x) > 0 for all x.
If two distributions are close to each other, their KL divergence is small. If they are exactly
the same, their KL divergence is zero. KL divergence is not a true distance metric (since it
isn’t symmetric and doesn’t satisfy the triangle inequality), but we often use it as a measure
of dissimilarity between two probability distributions.
(b) [2pt] Prove that KL(p||q) is non-negative. Hint: you may want to use Jensen’s Inequality, which is described in the Appendix.
3
CSC2515 Fall 2019 Homework 1
(c) [1pt] The Information Gain or Mutual Information between X and Y is I(Y ; X) =
H(Y ) − H(Y |X). Show that
I(Y ; X) = KL(p(x, y)||p(x)p(y)),
where p(x) = P
y
p(x, y) is the marginal distribution of X.
Appendix: Convexity and Jensen’s Inequality. Here, we give some background on convexity which you may find useful for some of the questions in this assignment. You may assume
anything given here.
Convexity is an important concept in mathematics with many uses in machine learning. We
briefly define convex set and function and some of their properties here. Using these properties are
useful in solving some of the questions in the rest of this homework. If you are interested to know
more about convexity, refer to Boyd and Vandenberghe, Convex Optimization, 2004.
A set C is convex if the line segment between any two points in C lies within C, i.e., if for any
x1, x2 ∈ C and for any 0 ≤ λ ≤ 1, we have
λx1 + (1 − λ)x2 ∈ C.
For example, a cube or sphere in R
d are convex sets, but a cross (a shape like X) is not.
A function f : R
d → R is convex if its domain is a convex set and if for all x1, x2 in its domain,
and for any 0 ≤ λ ≤ 1, we have
f(λx1 + (1 − λ)x2) ≤ λf(x1) + (1 − λ)f(x2).
This inequality means that the line segment between (x1, f(x1)) and (x2, f(x2)) lies above the
graph of f. A convex function looks like `. We say that f is concave if −f is convex. A concave
function looks like a.
Some examples of convex and concave functions are (you do not need to use most of them in
your homework, but knowing them is useful):
• Powers: x
p
is convex on the set of positive real numbers when p ≥ 1 or p ≤ 0. It is concave
for 0 ≤ p ≤ 1.
• Exponential: e
ax is convex on R, for any a ∈ R.
• Logarithm: log(x) is concave on the set of positive real numbers.
• Norms: Every norm on R
d
is convex.
• Max function: f(x) = max{x1, x2, . . . , xd} is convex on R
d
.
• Log-sum-exp: The function f(x) = log(e
x1 + . . . + e
xd ) is convex on R
d
.
4
CSC2515 Fall 2019 Homework 1
An important property of convex and concave functions, which you may need to use in your
homework, is Jensen’s inequality. Jensen’s inequality states that if φ(x) is a convex function of x,
we have
φ(E [X]) ≤ E [φ(X)] .
In words, if we apply a convex function to the expectation of a random variable, it is less than or
equal to the expected value of that convex function when its argument is the random variable. If
the function is concave, the direction of the inequality is reversed.
Jensen’s inequality has a physical interpretation: Consider a set X = {x1, . . . , xN } of points
on R. Corresponding to each point, we have a probability p(xi). If we interpret the probability as
mass, and we put an object with mass p(xi) at location (xi
, φ(xi)), then the centre of gravity of
these objects, which is in R
2
, is located at the point (E [X] , E [φ(X)]). If φ is convex `, the centre
of gravity lies above the curve x 7→ φ(x), and vice versa for a concave function a.
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