## Description

Problem 1: Linear regression learns a linear function of feature variables X to fit the

responses y. In this problem, you will derive the closed-form solution for linear regression

formulations.

1. The standard linear regression can be formulated as solving a least square problem

minimize

w

φ(w) = kXw − yk

2

2 = hXw − y, Xw − yi

where X ∈ R

n×m (n ≥ m) represents the feature matrix, y ∈ R

n×1

represents the

response vector and w ∈ R

m×1

is the vector variable of the linear coefficients. Here the

i − j-th element of X, denoted xij , is the j-th attribute value for the i-th data sample

(observation) and yi

is the true response for the i-th data sample. This is a convex

objective function of w. Derive the optimal w by setting the gradient of the function

wrt w to zero to minimize the objective function. To find the gradient, you can use

the following formula

φ(w + δ) = [X(w + δ)]T X(w + δ) − 2 [X(w + δ)]T

y + y

T y

= φ(w) + 2 [Xδ]

T

[Xw − y] + (Xδ)

T Xδ,

and note that w must be determined so that φ(w + δ) ≥ φ(w) for any possible vector

δ (why?). Here XT denotes the transpose of X.

2. In practice, a L2-norm regularizer is often introduced with the least squares, called

Ridge Regression, to overcome ill-posed problems where the hessian matrix is not

positive definite. The objective function of ridge regression is defined as

minimize

w

φ˜(w) = kXw − yk

2 + λkwk

2 =

X

√

λI

w −

y

0

2

where λ > 0 and I is an m × m identity matrix. This objective function is strictly

convex. Derive the solution of the ridge regression problem to find the optimal w.

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Problem 2: Consider a coin with probability of heads equal to Pr(H) = p and probability

of tails Pr(T) = 1 − p. You toss it 5 times and get outcomes H,H,T,T,H.

1. What is the probability of observing the sequence H,H,T,T,H in five tosses. Also give

the formula for the natural logarithm of this probability. Your formulas should be a

function of p.

2. You have a box containing exactly 2 coins, one fair with p = 1/2 and one biased with

p = 2/3. You choose one of these two coins at random with equal probability, toss it

5 times and get the outcome H,H,T,T,H.

(a) Give the joint probability that the coin chosen was the fair coin (p = 1/2) and

the outcome was H,H,T,T,H.

(b) Give the joint probability that the coin chosen was the biased coin (p = 2/3) and

the outcome was H,H,T,T,H.

3. What should the bias p = Pr(H) be to maximize the probability of observing H,H,T,T,H,

and what is the corresponding probability of observing H,H,T,T,H (i.e., what is the

maximum likelihood estimate for p), assuming p were unknown? Show the derivation.

Hint: maximize the log of the function.

Problem 3: Below is the pseudo-code of perceptron algorithm for binary classification,

where (x

t

, rt

) is the t-th data sample: x

t

is the vector of attribute values (real numbers) and

r

t = ±1 is the class label for the t-th sample:

1. w = w0.

2. Do Iterate until convergence

3. For each sample (x

t

, rt

), t = 1, 2, · · ·

4. If r

t

hw, x

t

i ≤ 0

5. w = w + r

tx

t

Here “convergence” means w does not change at all over one pass through the entire training

dataset in the loop starting in step 3.

1. Implement the perceptron algorithm and test it on the provided data. To begin, do

“load data1.mat” to load the file the data file into MATLAB. X ∈ R

40×2

is the

feature matrix of 40 samples in 2 dimensions and r ∈ R

40×1

is the label vector (±1).

Use initial value w0 = [1; −1]T

. Now, run your perceptron algorithm on the given data.

How many iterations does it take to converge?

2. Visualize all the samples (use 2 different colors for the 2 different classes) and plot the

decision boundary defined by the initial w0. Plot the decision boundary defined by the

w returned by the perceptron program.

Hint: To visualize the samples you could use the MATLAB function call

scatter(X(:,1), X(:,2), 50, y, ’*’);

2

Type help scatter for more information. Plotting the boundary is equivalent to plotting the line wT x = w1x1 + w2x2 = 0. Since all the sample points are located within

the square {(x1, x2), −1 ≤ x1, x2 ≤ +1}, choose two points (a, b) and (c, d) by setting

a = −1, c = +1 and solving for b, d, or else set b = −1, d = +1 and solving for a, c, and

then draw the line between the two points (a, b) and (c, d) with the command

hold on; plot([a,c],[b,d]); hold off;

Use the hold function to add the line to the existing scatter plot, and axis to adjust

the axes, if needed. Draw both the initial boundary and the final boundary on the

same plot.

Submission

• Things to submit:

1. hw0 sol.pdf: a document contains all the derivations of Problem 1 & 2 and the

plot asked by Problem 3.

2. MyPerceptron.m: a MATLAB function defined with header function [w, step]=MyPerceptroy, w0), where X is the feature matrix, y is a label vector (±1) and w0 is the initial value for the parameter vector w. In the output, w is the parameter found

by perceptron and step represents the number of steps the algorithm takes to

converge. The function should also display the plot of samples and boundary.

3. Zip both files into a single zipped file and name it as your lastname.zip.

• Submit: All material must be submitted electronically via Canvas. This homework

will not be graded but required as a proof of satisfying the prerequisites for taking the

class.

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