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Assignment 2: Ridge Regression and kNN

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Assignment 2: Ridge Regression and kNN
Machine Learning
1 KNN and Model Selection (K) (programming)
• Purpose 1: To implement a very simple classifier, k-nearest neighbors (KNN), from scratch.
• Purpose 2: To implement k-folds CV for classification.
This problem provides an introduction to classification using the KNN algorithm. When creating a
classification algorithm, one has to make an assumption about the nature of the data. For example, if you
are classifying cats vs dogs, you would probably not make the assumption that cats have the same colors
as other cats. In our case, KNN makes the assumption that a data point has the same label as the most
popular label of the k labeled data points that are closest to itself. The measurement of closeness that we
will use is euclidean distance. We will test our implementation on ”Movie Review Data.txt”
• Please use the ”knn.py” template to guide your work. Please use the following instructions and follow
the function names and descriptions in the template. Please use Numpy or other related package to
implement the knn algorithm. Feel free to cross-check your implementation against sci-kit’s. Other
requirements or recommendations are the same as Homework1.
• 2.1 Please download the ”knn.py” file and implement the read csv method. The last column includes
a 0 or 1 label.
• 2.2 Implement the fold method:
training, testing = fold(data, i, kfold)
Where data is the full data matrix, i is the iteration of cross validation, and kfold is the total number
of folds. This method will be as simple as splitting the data matrix into training and testing sets.
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• 2.3 Implement the classify method:
predictions = classify(training, testing, k)
Where training is the training set of data, testing is the testing set, and k is the number of data points
to take into consideration when labeling an unlabeled data point. Note how the training data is part of
the algorithm. In KNN, we label new data points based on the k points that are closest in our dataset.
For each testing point, we find the k points in the training set that have the closest Euclidian distance
to the testing point. The most popular label of the k closest points is the prediction.
• 2.4 Implement the calc accuracy method:
acc = calc accuracy(predictions, labels)
Where predictions is the list of 0 or 1 predictions given from the classify method and labels is the true
label for the testing points (the last column in the data matrix).
(Hint1: If your accuracy is below 50% look at the data, and consider how the order of the samples are
dictated by the class)
• 2.5 Run the code with k = (3, 5, 7, 9, 11, 13). Report the accuracy and the best k. Discuss why some
k values work better than others.
• A bar graph is recommended to show the change of accuracy with k. By using k as x-axis and accuracy
as y-axis
• Att: we will not consider the speed in grading your kNN codes.
• Att: please remember to shuffle the whole data before performing the CV.
• Att: there are many ways to read in the reference datasets, e.g., our template reads in the whole file
and put it into one numpy array. (But in HW1, our template actually read the file into two numpy
array, one for Xval, the other for Yval. Both ways are correct.)
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2 Ridge Regression (programming and QA)
• Purpose 1: To emphasize the importance of selecting the right model through k-folds Cross Validation
(CV) when using supervised regression.
• Purpose 2: To show a real case in which linear regression learns badly and adding regularization is
necessary.
This problem provides a case study in which just using a linear regression model for data fitting is not
enough. Adding regularization like ridge estimator is necessary for certain cases.
• Here we assume Xn×p represents a data sample matrix which has p features and n samples. Yn×1
includes target variable’s value of n samples. We use β to represent the coefficient. (Just a different
notation. We had used θ for representing coefficient before.)
• 1.1 Please provide the math derivation procedure for ridge regression (shown in Figure)
Figure 1: Ridge Regression / Solution Derivation / 1.1
(Hint1: provide a procedure similar to how linear regression gets the normal equation through minimizing its loss function. )
(Hint2: λ|β|2 = λβT β = λβT
Iβ = β
T
(λI)β)
(Hint3: Linear Algebra Handout Page 24, first two equations after the line “To recap,”)
• 1.2 Suppose X =


1 2
3 6
5 10

 and Y = [1, 2, 3]T
, could this problem be solved through linear regression?
Please provide your reasons.
(Hint: just use the normal equation to explain)
• 1.3 If you have the prior knowledge that the coefficient β should be sparse, which regularized linear
regression method should be chosen to use ? (Hint: sparse vector)
• A data file named “RRdata.txt” is provided. For this data, you are expected to write programs to
compare between linear regression and ridge regression.
• Please submit your python code as “ridgeRegression.py” . Please use the following instructions and use
required function names. Please use Numpy or other related package to implement the ridge regression.
Other requirements or recommendations are the same as Homework1.
• Notation: The format of each row in data file is [1, x1, x2, y], where x1, x2 are two features and y is
the target value.
• 1.4 For “ridgeReregression.py”,
– Load the data file and assume the last column is the target value. You should use xV al to
represent the data sample matrix and yV al to represent the target value vector.
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– 1.4.1 The first function is to implement the ridge regression and return the coefficient β with the
hyperparameter λ = 0. (i.e. when λ = 0, it’s just the standard linear regression). Please plot
the data points and the learned plane 1
. Please submit the result into the writing part of this
assignment. You are required to provide the following function (and module) for grading:
betaLR = ridgeRegression.ridgeRegress(xV al, yV al, lambdaV = 0)
– (Extra and not required Hint: In the state-of-the-art ridge regression implementations, the tools
actually don’t regularize the β0. If you want to implement this strategy, you can estimate the
un-regularized version β0 through centering the input(i.e. βˆ
0 =
Pyi
n
) OR using the trick provided
in the last EXTRA slide of our ”ridge-regression” lecture.
– 1.4.2 The second function is to find the best λ by using a k = 10 cross validation procedure (please
feel free to try other k like k = 4-fold). The function should be,
lambdaBest = ridgeRegression.cv(xV al, yV al)
– (Hint1: you should implement a function to split the data into ten folds; then loop over the folds;
use one as test, the rest train )
– (Hint2: for each fold, on the train part, perform ridgeRegress to learn βk; Then use this βk on all
samples in the test fold to get predicted ˆy; Then calculate the error (difference) between true y
and ˆy, sum over all testing points in the current fold k. )
– 1.4.3 Please try all the λ values from a set of values: {0.02, 0.04, 0.06, . . . , 1} (i.e. {0.02i|i ∈
1, 2, . . . , 50}). Pick the λ achieving the best objective criterion from the 10-fold cross validation
procedure. Our objective criterion is just the value of the loss function (i.e. J(θ) MSE in the
slides) on each test fold. Please plot the λ versus J(β) graph (which is also called path of finding
the best λ) and provide it into the writing. (ATT: the MSE is roughly in the range of e − 2. )
– Note : To constrain the randomness, please set seed to be 37. 2
– Then run the ridge regression again by using the best λ calculated from 1.4.2. Please include the
result into writing.
betaRR = ridgeRegression.ridgeRegress(xV al, yV al, lambdaBest)
– Please plot the data points and the learned plane from best ridge regression. Please include the
result into writing. 3
.
– Att: there are many ways to read in the reference dataset, e.g., the ridge regression template
reads in the file and put into two numpy array, one for Xval, the other for Yval. )
• 1.5 If assuming the true coefficient in problem 1.4 is β = (3, 1, 1)T
, could you compare and conclude
whether linear regression or ridge regression performs better ? Explain why this happens based on the
data we give.
– (Hint: 1. Please implement a standard linear regression between x1, x2 and plot the x1 versus x2
graph;)
– (Hint: 2. Guess the relationship between the two features and consider the problem 1.2.)
– Please feel free to reuse your standRegress code from HW1
1http://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html#surface-plots
2More about random in python, please see, https://docs.python.org/2/library/random.html
3http://matplotlib.org/mpl_toolkits/mplot3d/tutorial.html#surface-plots
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3 Sample Exam Questions:
Each assignment covers a few sample exam questions to help you prepare for the midterm and the final.
(Please do not bother by the information of points in some the exam questions.)
Question 1. Short Answer
True or False? If true, explain why in at most two sentences. If false, explain why or give a brief
counterexample in at most two sentences.
• (True/False). Ridge regression model increases the bias but reduces the variance comparing to the
linear regression model.
• (True or False?) The error of a hypothesis measured over its training set provides a pessimistically
biased estimate of the true error of the hypothesis.
• (True or False?) If you are given m data points, and use half for training and half for testing, the
difference between training error and test error decreases as m increases.
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• (True or False?) Overfitting is more likely when the set of training data is small.
• (True or False?) Overfitting is more likely when the hypothesis space is small.
• (True/False) When the tuning parameter λ increases its value, the parameter β in the ridge regression
will not converge to zero vector, since Lasso enforces sparsity on β (assuming no bias term here).
• (True/False). Ridge regression can fit the data well even if its feature variables have certain linearly
dependent relationships among each other.
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Question 2. Bayes Rule (fake points)
(a) (4 points) I give you the following fact:
P(A|B) = 2/3
Do you have enough information to compute P(B|A)? If not, write ”not enough info”. If so, computer
the value of P(B|A).
(b) (5 points) Instead, I give you the following facts:
P(A|B) = 2/3
P(A|∼B) = 1/3
Do you now have enough information to computer P(B|A)? If not, write ”not enough info”. If so,
compute the value of P(B|A).
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(c) (5 points) Instead, I give you the following facts:
P(A|B) = 2/3
P(A|∼B) = 1/3
P(B) = 1/3
Do you now have enough information to computer P(B|A)? If not, write ”not enough info”. If so,
compute the value of P(B|A).
(d) (5 points) Instead, I give you the following facts:
P(A|B) = 2/3
P(A|∼B) = 1/3
P(B) = 1/3
P(A) = 4/9
Do you now have enough information to computer P(B|A)? If not, write ”not enough info”. If so,
compute the value of P(B|A).
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