Solved–HOMEWORK 1 V0 –Solution

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1. Nearest Neighbors and the Curse of Dimensionality – 30 pts. 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.

(a) [10 pts] 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) [10 pts] 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) [10 pts] 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”?

2. Decision Trees – 30 pts. 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. The data were cleaned by removing words from titles not part of the headlines, removing special characters and restricting real news headlines after October 2016 using the word “trump”. The cleaned data are available as clean_real.txt and clean_fake.txt on the course webpage. It is expected that you use these cleaned data sources for this assignment.

You will build a decision tree to classify real vs. fake news headlines. Instead of coding the deci-sion trees yourself, you will do what we normally do in practice | use an existing implementation.

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You should use the DecisionTreeClassifier included in sklearn. Note that guring out how to use this implementation, its corresponding attributes and methods is a part of the assignment.

All code should be included in your pdf submission le.

(a) [6 pts] 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, we suggest you use CountVectorizer as it is the simplest in nature), and splits the entire dataset randomly into 70% training, 15% validation, and 15% test examples.

(b) [6 pts] 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 (information 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.

(c) [6 pts] 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.

(d) [12 pts] 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. Regression – 40 pts. In this question, you will derive certain properties of linear regres-sion, and experiment with cross validation to tune its regularization parameter.

3.1. Linear regression – 10 pts. Suppose that X 2 Rn m with n m and Y 2 Rn, and that
j N 2 ^
Y X; (X ; I). We know that the maximum likelihood estimate of is given by

^ T X) 1 X T Y:
= (X

(a) Write the log-likelihood implied by the model above, and compute its gradient w.r.t. . By
^
setting it equal to 0, derive the above estimator .
^
(b) Find the distribution of , its expectation and covariance matrix. Hint: Read the property of multivariate Gaussian random vectors in preliminaries.pdf on course webpage.

3.2. Ridge regression and MAP – 10 pts. Suppose that we have Y jX; N (X ; 2I) and we place a normal prior on , i.e., N (0; 2I). Show that the MAP estimate of given Y in this context is

(3.1) ^ T X+ I) 1 X T Y
MAP = (X

where = 2= 2.

3.3. Cross validation – 20 pts. In this problem, you will write a function that performs K-fold cross validation (CV) procedure to tune the penalty parameter in Ridge regression. CV 2

procedure is one of the most commonly used methods for tuning hyperparameters. In this question, you shouldn’t use the package scikit-learn to perform CV. You should implement all of the below functions yourself. You may use numpy and scipy for basic math operations such as linear algebra, sampling etc.

In class we learned training, test, and validation procedures which assumes that you have enough data and you can set aside a validation set and a test set to use it for assessing the performance of your machine learning algorithm. However in practice, this may be problematic since we may not have enough data. A remedy to this issue is K-fold cross- validation which uses a part of the available data to t the model, and a di erent part to test it. K-fold CV procedure splits the data into K equal-sized parts; for example, when K = 5, the scenario looks like this:

Fig 1: credit: Elements of Statistical Learning

1. We rst set aside a test dataset and never use it until the training and parameter tuning procedures are complete. We will use this data for nal evaluation. In this question, test data is provided to you as a separate dataset.

2. CV error estimates the test error of a particular hyperparameter choice. For a particular hyperparameter value, we split the training data into K blocks (See the gure), and for

k = 1; 2; :::; K we use the k-th block for validation and the remaining K 1 blocks are for training. Therefore, we train and validate our algorithm K times. Our CV estimate for the test error for that particular hyperparameter choice is given by the average validation error across these K blocks.

3. We repeat the above procedure for several hyperparameter choices and choose the one that provides us with the smalles CV error (which is an estimate for the test error).

Below, we will code the above procedure for tuning the regularization parameter in linear regression which is a hyperparameter. Your cross_validation function will rely on 6 short functions which are de ned below along with their variables.

data is a variable and refers to a (y; X) pair (can be test, training, or validation) where y

is the target (response) vector, and X is the feature matrix.
^
model is a variable and refers to the coe cients of the trained model, i.e. .
data_shf = shuffle_data(data) is a function and takes data as an argument and returns its randomly permuted version along the samples. Here, we are considering a uniformly random permutation of the training data. Note that y and X need to be permuted the same way preserving the target-feature pairs.

data_fold, data_rest = split_data(data, num_folds, fold) is a function that takes data, number of partitions as num_folds and the selected partition fold as its arguments

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and returns the selected partition (block) fold as data_fold, and the remaining data as data_rest. If we consider 5-fold cross validation, num_folds=5, and your function splits the data into 5 blocks and returns the block fold (2 f1; 2; 3; 4; 5g) as the validation fold and the remaining 4 blocks as data_rest. Note that data_rest [ data_fold = data, and data_rest \ data_fold = ;.
model = train_model(data, lambd) is a function that takes data and lambd as its argu-ments, and returns the coe cients of ridge regression with penalty level . For simplicity, you may ignore the intercept and use the expression in equation (3.1).

predictions = predict(data, model) is a function that takes data and model as its arguments, and returns the predictions based on data and model.

error = loss(data, model) is a function which takes data and model as its arguments

and returns the average squared error loss based on model. This means if data is composed
of y 2 Rn and X 2 Rn p, and model is ^, then the return value is ky X ^k2=n.

cv_error = cross_validation(data, num_folds, lambd_seq) is a function that takes the training data, number of folds num_folds, and a sequence of ’s as lambd_seq as its arguments and returns the cross validation error across all ’s. Take lambd_seq as evenly spaced 50 numbers over the interval (0.02, 1.5). This means cv_error will be a vector of 50 errors corresponding to the values of lambd_seq. Your function will look like:

data = shuffle_data(data)

for i = 1,2,…,length(lambd_seq) lambd = lambd_seq(i) cv_loss_lmd = 0.

for fold = 1,2, …,num_folds

val_cv, train_cv = split_data(data, num_folds, fold)

model = train_model(train_cv, lambd)

cv_loss_lmd += loss(val_cv, model)

cv_error(i) = cv_loss_lmd / num_folds

return cv_error

(a) Download the dataset from the course webpage hw1_data.zip and place and extract in your working directory, or note its location file_path. For example, le path could be /Users/yourname/Desktop/

In Python:

import numpy as np

data_train = {’X’: np.genfromtxt(’data_train_X.csv’, delimiter=’,’),

’y’: np.genfromtxt(’data_train_y.csv’, delimiter=’,’)}

data_test = {’X’: np.genfromtxt(’data_test_X.csv’, delimiter=’,’),

’y’: np.genfromtxt(’data_test_y.csv’, delimiter=’,’)}

(b) Write the above 6 functions, and identify the correct order and arguments to do cross validation.

(c) Find the training and test errors corresponding to each in lambd_seq. This part does not use the cross_validation function but you may nd the other functions helpful.

(d) Plot training error, test error, and 5-fold and 10-fold cross validation errors on the same plot for each value in lambd_seq. What is the value of proposed by your cross validation

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procedure? Comment on the shapes of the error curves.

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