CS 189/289A Machine Learning

CS 189/289A Introduction to Machine Learning
Due: Wednesday, February 26 at 11:59 pm
• Homework 3 consists of coding assignments and math problems.
• We prefer that you typeset your answers using LATEX or other word processing software. If
you haven’t yet learned LATEX, one of the crown jewels of computer science, now is a good
time! Neatly handwritten and scanned solutions will also be accepted.
• In all of the questions, show your work, not just the final answer.
• The assignment covers concepts on Gaussian distributions and classifiers. Some of the ma terial may not have been covered in lecture; you are responsible for finding resources to
understand it.
• Start early; you can submit models to Kaggle only twice a day!
Deliverables:

1. Submit your predictions for the test sets to Kaggle as early as possible. Include your Kaggle
   scores in your write-up. The Kaggle competition for this assignment can be found at
   • MNIST: https://www.kaggle.com/t/ca07d5e39d9b49cd946deb02583ad31f
   • SPAM: https://www.kaggle.com/t/3fb20b97254049f8acbf189a75830627
2. Write-up: Submit your solution in PDF format to “Homework 3 Write-Up” in Gradescope.
   • On the same page as the honor code, please list students and their SIDs with whom you
   collaborated.
   • Start each question on a new page. If there are graphs, include those graphs on the
   same pages as the question write-up. DO NOT put them in an appendix. We need each
   solution to be self-contained on pages of its own.
   • Only PDF uploads to Gradescope will be accepted. You are encouraged use LATEX
   or Word to typeset your solution. You may also scan a neatly handwritten solution to
   produce the PDF.
   • Replicate all your code in an appendix. Begin code for each coding question in a fresh
   page. Do not put code from multiple questions in the same page. When you upload this
   PDF on Gradescope, make sure that you assign the relevant pages of your code from
   appendix to correct questions.
   • While collaboration is encouraged, everything in your solution must be your (and only
   your) creation. Copying the 代写CS 189/289A Introduction to Machine Learninganswers or code of another student is strictly forbidden.
   Furthermore, all external material (i.e., anything outside lectures and assigned readings,
   HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 1
   including figures and pictures) should be cited properly. We wish to remind you that
   consequences of academic misconduct are particularly severe!
3. Code: Submit your code as a .zip file to “Homework 3 Code”. The code must be in a form that
   enables the readers to compile (if necessary) and run it to produce your Kaggle submissions.
   • Set a seed for all pseudo-random numbers generated in your code. This ensures
   your results are replicated when readers run your code. For example, you can seed
   numpy with np.random.seed(42).
   • Include a README with your name, student ID, the values of random seed you used,
   and instructions for compiling (if necessary) and running your code. If the data files
   need to be anywhere other than the main directory for your code to run, let us know
   where.
   • Do not submit any data files. Supply instructions on how to add data to your code.
   • Code requiring exorbitant memory or execution time might not be considered.
   • Code submitted here must match that in the PDF Write-up. The Kaggle score will not
   be accepted if the code provided a) does not compile/run or b) runs but does not produce
   the file submitted to Kaggle.
   HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 2
4. Honor Code
5. Please list the names and SIDs of all students you have collaborated with below.
6. Declare and sign the following statement (Mac Preview, PDF Expert, and FoxIt PDF Reader,
   among others, have tools to let you sign a PDF file):
   “I certify that all solutions are entirely my own and that I have not looked at anyone else’s
   solution. I have given credit to all external sources I consulted.”
   Signature:
   HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 3
7. Gaussian Classification
   Let fX|Y=Ci
   (x) ∼ N(µi
   , σ2
   ) for a two-class, one-dimensional (d = 1) classification problem with
   classes C1 and C2, P(Y = C1) = P(Y = C2) = 1/2, and µ2 > µ1.
8. Find the Bayes optimal decision boundary and the corresponding Bayes decision rule by
   finding the point(s) at which the posterior probabilities are equal. Use the 0-1 loss function.
9. Suppose the decision boundary for your classifier is x = b. The Bayes error is the probability
   of misclassification, namely,
   Pe = P((C1 misclassified as C2) ∪ (C2 misclassified as C1)).
   Show that the Bayes error associated with this decision rule, in terms of b, is
   Pe(b) =
   1
   2
   √
   2πσ
   
   
   Z
   b
   −∞
   exp
   
   −
   (x − µ2)
   2
   2σ2
   
   
   dx +
   Z
   b
   ∞
   exp
   
   −
   (x − µ1)
   2
   2σ2
   
   
   dx
   
   
   .
10. Using the expression above for the Bayes error, calculate the optimal decision boundary b
    ∗
    that minimizes Pe(b). How does this value compare to that found in part 1? Hint: Pe(b) is
    convex for µ1 < b < µ2.
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11. Classification and Risk
    Suppose we have a classification problem with classes labeled 1, . . . , c and an additional “doubt”
    category labeled c + 1. Let r : R
    d → {1, . . . , c + 1} be a decision rule. Define the loss function
    L(r(x) = i, y = j) =
    
    
    
12. if i = j i, j ∈ {1, . . . , c},
    λc
    if i , j i ∈ {1, . . . , c},
    λd if i = c + 1
    (1)
    where λc ≥ 0 is the loss incurred for making a misclassification and λd ≥ 0 is the loss incurred for
    choosing doubt. In words this means the following:
    • When you are correct, you should incur no loss.
    • When you are incorrect, you should incur some penalty λc for making the wrong choice.
    • When you are unsure about what to choose, you might want to select a category correspond ing to “doubt” and you should incur a penalty λd.
    The risk of classifying a new data point x as a class i ∈ {1, 2, . . . , c + 1} is
    R(r(x) = i | x) =
    cX
    j=1
    L(r(x) = i, y = j) P(Y = j | x).
    To be clear, the actual label Y can never be c + 1.
13. First, we will simplify the risk function using our specific loss function separately for when
    r(x) is or is not the doubt category.
    (a) Prove that R(r(x) = i | x) = λc
      1 − P(Y = i | x)
    
    when i is not the doubt category (i.e.
    i , c + 1).
    (b) Prove that R(r(x) = c + 1 | x) = λd.
14. Show that the following policy ropt(x) obtains the minimum risk:
    • (R1) Find the non-doubt class i such that P(Y = i | x) ≥ P(Y = j | x) for all j, meaning
    you pick the class with the highest probability given x.
    • (R2) Choose class i if P(Y = i | x) ≥ 1 −
    λd
    λc
    • (R3) Choose doubt otherwise.
15. How would you modify your optimum decision rule if λd = 0? What happens if λd > λc?
    Explain why this is or is not consistent with what one would expect intuitively.
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 5
16. Maximum Likelihood Estimation and Bias
    Let X1, . . . , Xn ∈ R be n sample points drawn independently from univariate normal distributions
    such that Xi ∼ N(µ, σ2
    i
    ), where σi = σ/ √
    i for some parameter σ. (Every sample point comes
    from a distribution with a different variance.) Note the word “univariate”; we are working in
    dimension d = 1, and each “point” is just a real number.
17. Derive the maximum likelihood estimates, denoted ˆµ and ˆσ, for the mean µ and the pa rameter σ. (The formulae from class don’t apply here, because every point has a different
    variance.) You may write an expression for ˆσ
    2
    rather than ˆσ if you wish—it’s probably
    simpler that way. Show all your work.
18. Given the true value of a statistic θ and an estimator θˆ of that statistic, we define the bias of
    the estimator to be the the expected difference from the true value. That is,
    bias(θˆ) = E[θˆ] − θ.
    We say that an estimator is unbiased if its bias is 0.
    Either prove or disprove the following statement: The MLE sample estimator µˆ is unbiased.
    Hint: Neither the true µ nor true σ
19. are known when estimating sample statistics, thus we
    need to plug in appropriate estimators.
20. Either prove or disprove the following statement: The MLE sample estimator σˆ
    2
    is unbiased.
    Hint: Neither the true µ nor true σ
21. are known when estimating sample statistics, thus we
    need to plug in appropriate estimators.
22. Suppose the Variance Fairy drops by to give us the true value of σ
    2
    , so that we only have
    to estimate µ. Given the loss function L( ˆµ, µ) = ( ˆµ − µ)
    2
    , what is the risk of our MLE
    estimator ˆµ?
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 6

23. Covariance Matrices and Decompositions
    As described in lecture, the covariance matrix Var(R) ∈ R
    d×d
    for a random variable R ∈ R
    d with
    mean µ ∈ R
    d
    is
    Var(R) = Cov(R, R) = E[(R − µ) (R − µ)

    ] =
    Var(R1) Cov(R1, R2) . . . Cov(R1, Rd)
    Cov(R2, R1) Var(R2) Cov(R2, Rd)
    Cov(Rd, R1) Cov(Rd, R2) . . . Var(Rd)
    where Cov(Ri
    , Rj) = E[(Ri − µi) (Rj − µj)] and Var(Ri) = Cov(Ri
    , Ri).
    If the random variable R is sampled from the multivariate normal distribution N(µ, Σ) with the
    then as you proved in Homework 2, Var(R) = Σ.
    Given n points X1, X2, . . . , Xn sampled from N(µ, Σ), we can estimate Σ with the maximum likeli hood estimator
    Σ =ˆ
    1
    n
    nX
    i=1
    (Xi − µˆ) (Xi − µˆ)
    ,
    which is also known as the sample covariance matrix.

24. The estimate Σˆ makes sense as an approximation of Σ only if Σˆ is invertible. Under what cir cumstances is Σˆ not invertible? Express your answer in terms of the geometric arrangement
    of the sample points Xi
    . We want a geometric characterization, not an algebraic one. Make
    sure your answer is complete; i.e., it includes all cases in which the covariance matrix of the
    sample is singular. (No proof is required.)
25. Suggest a way to fix a singular covariance matrix estimator Σˆ by replacing it with a similar but
    invertible matrix. Your suggestion may be a kludge, but it should not change the covariance
    matrix too much. Note that infinitesimal numbers do not exist; if your solution uses a very
    small number, explain how to calculate a number that is sufficiently small for your purposes.
26. Consider the normal distribution N(0, Σ) with mean µ = 0. Consider all vectors of length 1;
    i.e., any vector x for which k xk = 1. Which vector(s) x of length 1 maximizes the PDF f(x)?
    Which vector(s) x of length 1 minimizes f(x)? Your answers should depend on the properties
    of Σ. Explain your answer.

27. Suppose we have X ∼ N(0, Σ), X ∈ R
    n
    and a unit vector y ∈ R
    n
    . We can compute the
    projection of the random vector X onto a unit direction vector y as p = y

    X. First, compute
    the variance of p. Second, with this information, what does the largest eigenvalue λmax of the
    covariance matrix tell us about the variances of expressions of the form y
    X?
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 7

28. Isocontours of Normal Distributions
    Let f(µ, Σ) be the probability density function of a normally distributed random variable in R
    2
    .
    Write code to plot the isocontours of the following functions, each on its own separate figure. Make
    sure it is clear which figure belongs to which part. You’re free to use any plotting libraries or stats
    utilities you like; for instance, in Python you can use Matplotlib and SciPy. Choose the boundaries
    of the domain you plot large enough to show the interesting characteristics of the isocontours (use
    your judgment). Make sure we can tell what isovalue each contour is associated with—you can do
    this with labels or a colorbar/legend.
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 8
29. Eigenvectors of the Gaussian Covariance Matrix
    Consider two one-dimensional random variables X1 ∼ N(3, 9) and X2 ∼
    1
    2
    X1 + N(4, 4), where
    N(µ, σ2
    ) is a Gaussian distribution with mean µ and variance σ
    2
    . (This means that you have to
    draw X1 first and use it to compute a random X2. Be mindful that most packages for sampling from
    a Gaussian distribution use standard deviation, not variance, as input.)
    Write a program that draws n = 100 random two-dimensional sample points from (X1, X2). For
    each sample point, the value of X2 is a function of the value of X1 for that same sample point,
    but the sample points are independent of each other. In your code, make sure to choose and set a
    fixed random number seed for whatever random number generator you use, so your simulation is
    reproducible, and document your choice of random number seed and random number generator in
    your write-up. For each of the following parts, include the corresponding output of your program.
30. Compute the mean (in R
    2
    ) of the sample.
31. Compute the 2 × 2 covariance matrix of the sample (based on the sample mean, not the true
    mean—which you would not know given real-world data). Ensure that the sample covariance
    uses the maximum likelihood estimator as described in Question 5.
32. Compute the eigenvectors and eigenvalues of this covariance matrix.
33. On a two-dimensional grid with a horizonal axis for X1 with range [−15, 15] and a vertical
    axis for X2 with range [−15, 15], plot
    (i) all n = 100 data points, and
    (ii) arrows representing both covariance eigenvectors. The eigenvector arrows should orig inate at the mean and have magnitudes equal to their corresponding eigenvalues.
    Hint: make sure your plotting software is set so the figure is square (i.e., the horizontal and
    vertical scales are the same). Not doing that may lead to hours of frustration!

34. Let U = [v1 v2] be a 2×2 matrix whose columns are the unit eigenvectors of the covariance
    matrix, where v1 is the eigenvector with the larger eigenvalue. We use U

    as a rotation
    matrix to rotate each sample point from the (X1, X2) coordinate system to a coordinate system
    aligned with the eigenvectors. (As U
    = U
    −1
    , the matrix U reverses this rotation, moving
    back from the eigenvector coordinate system to the original coordinate system). Center your
    sample points by subtracting the mean µ from each point; then rotate each point by U
    ,
    giving xrotated = U
    (x − µ). Plot these rotated points on a new two dimensional-grid, again
    with both axes having range [−15, 15]. (You are not required to plot the eigenvectors, which
    would be horizontal and vertical.)
    In your plots, clearly label the axes and include a title. Moreover, make sure the horizontal
    and vertical axis have the same scale! The aspect ratio should be one.
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35. Gaussian Classifiers for Digits and Spam
    In this problem, you will build classifiers based on Gaussian discriminant analysis. Unlike Home work 1, you are NOT allowed to use any libraries for out-of-the-box classification (e.g. sklearn).
    You may use anything in numpy and scipy.
    The training and test data can be found with this homework. Do NOT use the training/test data
    from Homework 1, as they have changed for this homework. The starter code is similar to
    HW1’s; we provide check.py and save csv.py files for you to produce your Kaggle submission
    files. Submit your predicted class labels for the test data on the Kaggle competition website and be
    sure to include your Kaggle display name and scores in your writeup. Also be sure to include an
    appendix of your code at the end of your writeup.
    Reminder: please also select relevant code from the appendix on Gradescope for your answer to
    each question.
36. (Code) Taking pixel values as features (no new features yet, please), fit a Gaussian distri bution to each digit class using maximum likelihood estimation. This involves computing a
    mean and a covariance matrix for each digit class, as discussed in Lecture 9 and Section 4.4
    of An Introduction to Statistical Learning. Attach the relevant code as your answer to this
    part.
    Hint: You may, and probably should, contrast-normalize the images before using their pixel
    values. One way to normalize is to divide the pixel values of an image by the ` 2-norm of its
    pixel values.
37. (Written Answer + Graph) Visualize the covariance matrix for a particular class (digit). Tell
    us which digit and include your visualization in your write-up. How do the diagonal terms
    compare with the off-diagonal terms? What do you conclude from this?

38. Classify the digits in the test set on the basis of posterior probabilities with two different
    approaches.
    (a) (Graph) Linear discriminant analysis (LDA). Model the class conditional probabilities
    as Gaussians N(µC, Σ) with different means µC (for class C) and the same pooled within class covariance matrix Σ, which you compute from a weighted average of the 10 co variance matrices from the 10 classes, as described in Lecture 9.
    In your implementation, you might run into issues of determinants overflowing or under-
    flowing, or normal PDF probabilities underflowing. These problems might be solved by
    learning about numpy.linalg.slogdet and/or scipy.stats.multivariate normal.
    logpdf.
    To implement LDA, you will sometimes need to compute a matrix-vector product of
    the form Σ
    −1
    x for some vector x. You should not compute the inverse of Σ (nor the
    determinant of Σ) as it is not guaranteed to be invertible. Instead, you should find a way
    to solve the implied linear system without computing the inverse.
    Hold out 10,000 randomly chosen training points for a validation set. (You may re use your Homework 1 solution or an out-of-the-box library for dataset splitting only.)
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 10
    Classify each image in the validation set into one of the 10 classes. Compute the error
    rate (1−

    points correctly classified

    total points ) on the validation set and plot it over the following numbers

    of randomly chosen training points: 100, 200, 500, 1,000, 2,000, 5,000, 10,000, 30,000,
    50,000. (Expect unpredictability in your error rate when few training points are used.)
    (b) (Graph) Quadratic discriminant analysis (QDA). Model the class conditional probabili ties as Gaussians N(µC, ΣC), where ΣC is the estimated covariance matrix for class C. (If
    any of these covariance matrices turn out singular, implement the trick you described in
    Q5(b). You are welcome to use validation to choose the right constant(s) for that trick.)
    Repeat the same tests and error rate calculations you did for LDA.
    (c) (Written Answer) Which of LDA and QDA performed better? (Note: We don’t expect
    everybody to get the same answer.) Why?
    (d) (Written Answer + Graphs) Include two plots, one using LDA and one using QDA, of
    validation error versus the number of training points for each digit. Each plot should
    include all the 10 curves on the same graph as shown in Figure 1. Which digit is easiest
    to classify for LDA/QDA? Write down your answer and suggest why you think it’s the
    easiest digit.
    Figure 1: Sample graph with 10 plots

39. (Written Answer) With mnist-data-hw3.npz, train your best classifier for the training data
    and classify the images in the test data. Submit your labels to the online Kaggle com petition. Record your optimum prediction rate in your write-up and include your Kaggle
    username. Don’t forget to use the “submissions” tab or link on Kaggle to select your best
    submission!
    You are welcome to compute extra features for the Kaggle competition, as long as they do
    not use an exterior learned model for their computation (no transfer learning!). If you do so,
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 11
    please describe your implementation in your assignment. Please use extra features only for
    the Kaggle portion of the assignment.
40. (Written Answer) Next, apply LDA or QDA (your choice) to spam (spam-data-hw3.npz).
    Submit your test results to the online Kaggle competition. Record your optimum prediction
    rate in your submission. If you use additional features (or omit features), please describe
    them. We include a featurize.py file (similar to HW1’s) that you may modify to create
    new features.
    Optional: If you use the defaults, expect relatively low classification rates. We suggest using
    a Bag-Of-Words model. You are encouraged to explore alternative hand-crafted features, and
    are welcome to use any third-party library to implement them, as long as they do not use a
    separate model for their computation (no large language models, BERT, or word2vec!).
    HW3, ©UCB CS 189/289A, Spring 2025. All Rights Reserved. This may not be publicly shared without explicit permission. 12
    Submission Checklist
    Please ensure you have completed the following before your final submission.
    At the beginning of your writeup...
41. Have you copied and hand-signed the honor code specified in Question 1?
42. Have you listed all students (Names and SIDs) that you collaborated with?
    In your writeup for Question 8...
43. Have you included your Kaggle Score and Kaggle Username for both questions 8.4 and
    8.5?
    At the end of the writeup...
44. Have you provided a code appendix including all code you wrote in solving the homework?
45. Have you included featurize.py in your code appendix if you modified it?
    Executable Code Submission
46. Have you created an archive containing all “.py” files that you wrote or modified to generate
    your homework solutions (including featurize.py if you modified it)?
47. Have you removed all data and extraneous files from the archive?
48. Have you included a README file in your archive briefly describing how to run your code
    on the test data and reproduce your Kaggle results?
    Submissions
49. Have you submitted your test set predictions for both MNIST and SPAM to the appropriate
    Kaggle challenges?
50. Have you submitted your written solutions to the Gradescope assignment titled HW3 Write Up and selected pages appropriately?
51. Have you submitted your executable code archive to the Gradescope assignment titled HW3
    Code?
    Congratulations! You have completed Homework 3.
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