Homework 1 Solution

Computing: To install Python and required libraries, see the instructions on the course web page.




Weekly homeworks are individual work. See the Course Information handout http://www.cs.

toronto.edu/~mren/teach/csc411_19s/syllabus.pdf for detailed policies.




[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.



[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, de ned as the squared distance Z = (X Y )2.




(b) [1pts] Now suppose we sample two points independently from a unit cube in d dimen-sions. 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)).




[1pt] 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"?



























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CSC411 Homework 1










[5pts] 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 classi er 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 classi ed 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.




Each headline appears as a single line in the data le. 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 guring out how to use this implementation is a part of the assignment.




All code should be included in the le hw1_code.py which you submit through MarkUs.




[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% vali-dation, and 15% test examples.




[1pt] Write a function select_model which trains the decision tree classi er using at least 5 di erent values of max_depth, as well as two di erent split criteria (in-formation gain and Gini coe cient), 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).



[1pt] Now let’s stick with the hyperparameters which achieved the highest validation accuracy. Extract and visualize the rst two layers of the tree. Your visualization does not have to be an image: it is perfectly ne to display text. It may also be hand-drawn. Include your visualization in your solution PDF (hw1_writeup.pdf).



[2pts] 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.








































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