Problem Set 2: Linear Classi cation and Logistic Regression Solution

Description

• Linear Model [15 pts]

• 1. Design a two-input linear model w₁X₁ + w₂X₂ + b that computes the following Boolean functions. Assume T = 1 and F = 0. If a valid linear model exists, show that it is not unique by designing another valid linear model. If no such linear model exists, explain why.

• 1. 1. OR [5 pts]

• 1. 2. XOR [5 pts]

• 2. How many distinct Boolean functions are possible with two Boolean variables? Out of them, how many can be represented by a linear model? Justify your answers. [5 pts]

• Logistic Regression and its Variant [45 pts]

Consider the logistic regression model for binary classi cation that takes input features x_n 2 R^m and predicts y_n 2 f1; 0g. As we learned in class, the logistic regression model ts the probability P (y_n = 1) using the sigmoid function:

1
P (yn = 1) = h(xn) = (wT xn + b) =			:	(1)
	1 + exp( wT xn
		b)

Given N training data points, we learn the logistic regression model by minimizing the negative log-likelihood:

	N
	X
J(w; b) =	[yn log h (xn) + (1 yn) log (1 h (xn))]:	(2)

n=1

1. In the following, we derive the stochastic gradient descent algorithm for logistic regression. [20 pts]

i. Partial derivatives ^@J and ^@J , where wj is the j-th element of the weight vector w. [5

@w_j @b

+ 5 pts]

1. 2. Write down the stochastic gradient descent algorithm for minimizing Eq. (2) using the partial derivatives computed above. [5 pts]

1. 2. Compare the stochastic gradient descent algorithm for logistic regression with the Per-ceptron algorithm. What are the similarities and what are the di erences? [5 pts]

2. Instead of using the sigmoid function, we would like to use the following transformation function:

exp(z) exp( z)

^{A(z) = :}

exp(z) + exp( z)

Answer the following questions [25 pts]:

i. Plot _A(z) as a function of z in python using matplotlib and numpy libraries. Consider z 2 [ 10; 10] for the plot. What are the similarities and di erences between and _A? What happens as z ! 1 and z ! . [5 pts]

ii. Prove the following: [5 pts]

3. Can we assume probability P (y_n = 1) = _A(w^T x_n + b)? Why or why not? [2 pts]

3. If we assume

^{1 +} A^{(wT x}n ^{+ b)}

_{P (yn = 1) = hA(xn) = :}

Given N examples, fx_n; y_ng^N_n=1, please write down the corresponding negative log-likelihood function J_A(w; b). [3 pts]

v. Compute the partial derivatives ^@JA and ^@JA . [5 pts]

@w_j @b

vi. Write down the stochastic gradient descent algorithm for minimizing J_A(w; b). [5 pts]

• Implementation: Polynomial Regression [40 pts]

In this exercise, you will work through linear and polynomial regression. Our data consists of inputs x_n 2 R and outputs y_n 2 R; n 2 f1; : : : ; Ng, which are related through a target function

• = f(x). Your goal is to learn a linear predictor h_w(x) that best approximates f(x). But this time, rather than using scikit-learn, we will further open the \black-box”, and you will implement the regression model!

code and data

• code : Fall2020-CS146-HW2.ipynb

• data : train.csv, test.csv

Please use your @g.ucla.edu email id to access the code and data. Similar to HW-1, copy the colab notebook to your drive and make the changes. Mount the drive appropriately and copy the shared data folder to your drive to access via colab. For colab usage demo, check out the Discussion recordings for Week 2 in CCLE. The notebook has marked blocks where you need to code.

### ========= T ODO : ST ART ========= ###

### ========= T ODO : EN D ========== ###

Note: For the questions requiring you to complete a piece of code, you are expected to copy-paste your code as a part of the solution in the submission pdf. Tip: If you are using L^AT_EX, check out the Minted package (example) for code highlighting.

This is likely the rst time that many of you are working with numpy and matrix operations within a programming environment. For the uninitiated, you may nd it useful to work through a numpy tutorial rst.¹ Here are some things to keep in mind as you complete this problem:

• If you are seeing many errors at runtime, inspect your matrix operations to make sure that you are adding and multiplying matrices of compatible dimensions. Printing the dimensions of variables with the X.shape command will help you debug.

• When working with numpy arrays, remember that numpy interprets the * operator as element-wise multiplication. This is a common source of size incompatibility errors. If you want matrix multiplication, you need to use the dot function in Python. For example, A*B does element-wise multiplication while dot(A,B) does a matrix multiply.

• Be careful when handling numpy vectors (rank-1 arrays): the vector shapes 1 N, N 1, and N are all di erent things. For these dimensions, we follow the the conventions of scikit-learn’s LinearRegression class². Most importantly, unless otherwise indicated (in the code documentation), both column and row vectors are rank-1 arrays of shape N, not rank-2 arrays of shape N 1 or shape 1 N.

Visualization [2 pts]

• Try out SciPy’s tutorial (http://wiki.scipy.org/Tentative_NumPy_Tutorial), or use your favorite search en-gine to nd an alternative. Those familiar with Matlab may nd the \Numpy for Matlab Users” documentation (http://wiki.scipy.org/NumPy_for_Matlab_Users) more helpful.

• http://scikit-learn.org/stable/modules/generated/sklearn.linear_model.LinearRegression.html

It is often useful to understand the data through visualizations. For this data set, you can use a scatter plot to visualize the data since it has only two properties to plot (x and y).

1. Visualize the training and test data using the plot_data(…) function. What do you ob-serve? For example, can you make an educated guess on the e ectiveness of linear regression in predicting the data? [2 pts]

Linear Regression [23 pts]

Recall that linear regression attempts to minimize the objective function

N

X

(hw(xn) yn)2:

J(w) =

n=1

In this problem, we will use the matrix-vector form where

0

1

0

y

1

0

xT

1

w1

y1

x1T

B

w0

C

y =

2

;

X =

2

;

w =

w

B

C

B

C

B

..

C

B

y

N

C

B

xT

C

B

w

C

B

C

B

N

C

B

D

C

@

A

@

A

B

C

and each instance xn = 1; xn;1; : : : ; xn;D

T .

@

A

In this instance, the number of input features

D=1.

Rather than working with this fully generalized, multivariate case, let us start by considering a simple linear regression model:

h_w(x) = w^T x = w₀ + w₁x₁

regression.py contains the skeleton code for the class PolynomialRegression. Objects of this class can be instantiated as model = PolynomialRegression (m) where m is the degree of the

2		m		T	. Setting
polynomial feature vector where the feature vector for instance n, 1; xn;1; xn;1; : : : ; xn;1
m = 1 instantiates an object where the feature vector for instance n, 1; xn;1		T .

2. Note that to take into account the intercept term (w₀), we can add an additional \feature” to each instance and set it to one, e.g. x_i;0 = 1. This is equivalent to adding an additional rst column to X and setting it to all ones [2 pts].

Modify PolynomialRegression.generate_polynomial_features(…) to create the matrix X for a simple linear model.

2. Before tackling the harder problem of training the regression model, complete PolynomialRegression.predict(…) to predict y from X and w. [3 pts]

2. One way to solve linear regression is through gradient descent (GD).

Recall that the parameters of our model are the w_j values. These are the values we will adjust

to minimize J(w).

N

X

J(w) = (h_w(x_n) y_n)²

n=1

In gradient descent, each iteration performs the update

N

X

w_j w_j 2 (h_w(x_n) y_n) x_n;j (simultaneously update w_j for all j):

n=1

With each step of gradient descent, we expect our updated parameters w_j to come closer to the parameters that will achieve the lowest value of J(w). [10 pts]

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• (2 pts) As we perform gradient descent, it is helpful to monitor the convergence by com-

puting the cost, i.e., the value of the objective function J. Complete PolynomialRegression.cost(…) to calculate J(w).

If you have implemented everything correctly, then the following code snippet should print the model cost as 230:867214.

model = PolynomialRegression(1)

model.coef_ = np.zeros(2)

c = model.cost (train_data.X, train_data.y)

print(f model_cost:{c} )

4. • (3 pts) Next, implement the gradient descent step in PolynomialRegression.fit_GD(…). The loop structure has been written for you, and you only need to supply the updates to w and the new predictions y^ = h_w(x) within each iteration.

We will use the following speci cations for the gradient descent algorithm: { We run the algorithm for 10; 000 iterations.

{ We terminate the algorithm earlier if the value of the objective function is unchanged across consecutive iterations.

{ We will use a xed learning rate.

4. • (5 pts) Experiment with di erent values of learning rate = 10 ⁶, 10 ⁵, 10 ³, 0.0168 and make a table of the coe cients, number of iterations until convergence (this number will be 10; 000 if the algorithm did not converge in a smaller number of iterations) and the nal value of the objective function. How do the coe cients compare? How quickly does each algorithm converge? Do you observe something strange when you run with

4. • • 0.0168? Explain the observation and causes.

4. In class, we learned that the closed-form solution to linear regression is

w = (X^T X) ¹X^T y:

Using this formula, you will get an exact solution in one calculation: there is no \loop until convergence” like in gradient descent. [4 pts]

5. • (2 pts) Implement the closed-form solution PolynomialRegression.fit(…).

5. • (2 pts) What is the closed-form solution coe cients? How do the coe cients and the cost compare to those obtained by GD? How quickly does the algorithm run compared to GD?

5. Finally, set a learning rate for GD that is a function of k (the number of iterations) (use _k = ₁₊¹_k ) and converges to the same solution yielded by the closed-form optimization (minus possible rounding errors). Update PolynomialRegression.fit_GD(…) with your proposed learning rate. How many iterations does it take the algorithm to converge with your proposed learning rate? [4 pts]

Polynomial Regression[15 pts]

Now let us consider the more complicated case of polynomial regression, where our hypothesis is h_w(x) = w^T (x) = w₀ + w₁x + w₂x² + : : : + w_mx^m:

7. Recall that polynomial regression can be considered as an extension of linear regression in which we replace our input matrix X with

	0	(x1)T			1
		(x2)T
=	B	...			C	;
	B	(x	N	)T	C
	B				C
	@				A

where (x) is a function such that _j(x) = x^j for j = 0; : : : ; m.

Update PolynomialRegression.generate_polynomial_features(…) to create an m + 1 dimensional feature vector for each instance. [4 pts]

(h) Given N training instances, it is always possible to obtain a \perfect t” (a t in which all the data points are exactly predicted) by setting the degree of the regression to N 1. Of course, we would expect such a t to generalize poorly. In the remainder of this problem, you will investigate the problem of over tting as a function of the degree of the polynomial, m. To measure over tting, we will use the Root-Mean-Square (RMS) error, de ned as

p

E_RMS = J(w)=N ;

where N is the number of instances.³

Why do you think we might prefer RMSE as a metric over J(w)?

Implement PolynomialRegression.rms_error(…). [4 pts]

1. For m = 0; : : : ; 10 (where m is the order of the polynomial transformation applied to fea-tures), use the closed-form solver to determine the best- t polynomial regression model on the training data, and with this model, calculate the RMSE on both the training data and the test data. Generate a plot depicting how RMSE varies with model complexity (polynomial degree) { you should generate a single plot with both training and test error, and include this plot in your writeup. Which degree polynomial would you say best ts the data? Was there evidence of under/over tting the data? Use your plot to justify your answer. [7 pts]

Submission instructions for programming problems

• Please export the notebook to a .py le by clicking the \File” ! \Download.py” and upload to CCLE.

Your code should be commented appropriately. The most important things: { Your name and the assignment number should be at the top of each le. { Each class and method should have an appropriate doctsring.

{ If anything is complicated, it should include some comments.

There are many possible ways to approach the programming portion of this assignment, which makes code style and comments very important so that sta can understand what you did. For this reason, you will lose points for poorly commented or poorly organized code.

• Please submit all the plots and the rest of the solutions (other than codes) to Gradescope

• Note that the RMSE as de ned is a biased estimator. To obtain an unbiased estimator, we would have to divide by n k, where k is the number of parameters tted (including the constant), so here, k = m + 1.

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