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CS 534: Homework #1 SOLUTION

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CS 534: Homework #1
Submission Instructions: The homework is due on Sept 19th at 11:59 PM ET on Gradescope.
A part of your homework will be automatically graded by a Python autograder. The autograder
will support Python 3.10. Additional packages and their versions can be found in the
requirements.txt. Please be aware that the use of other packages and/or versions outside of
those in the file may cause your homework to fail some test cases due to incompatible method calls
or the inability to import the module. We have split homework 1 into 2 parts on Gradescope, the
autograded portion and the written answer portion. If either of the two parts is late, then your
homework is late.
1. Upload PDF to HW1-Written Assignment: Create a single high-quality PDF with your
solutions to the non-coding problems. The solutions must be typed (e.g., Word, Google Docs,
or LaTeX) and each problem appropriately tagged on Gradescope. If we must search through
your entire PDF for each problem, you may lose points! Note that you must submit the code
used to generate your results in the Code part of the assignment, otherwise you may not get
any points for the results.
2. Submit code to the HW1-Code Assignment: Your submitted code must contain the
following files: ‘q2.py’, ‘elastic.py’, ‘README.txt’. You must submit ALL files you used to
generate the results but the autograder will only copy these files when running the test cases
so make sure they are self-contained (i.e., capable of running standalone). Make sure you
always upload ALL of these files when you (re)submit.
1. (Written) Ridge Regression (10 pts): Show that the ridge regression estimates can be
obtained by ordinary least squares regression on an augmented data set. We augment the
centered matrix X with k additional rows with the value √
λI and augment y with k zeros.
The idea is that by introducing artificial data having response value zero, the fitting procedure
is forced to shrink the coefficients toward zero.
2. Predicting Appliance Energy Usage using Linear Regression (2 + 3 + 3 + 3 + 2 +
3 + 3 + 2 + 4 + 2 + 10 + 3 = 40 pts)
Consider the Appliances energy prediction dataset (energydata.zip), which contains
measurements of temperature and humidity sensors from a wireless network, weather from a
nearby airport station, and the recorded energy use of lighting fixtures to predict the energy
consumption of appliances (Appliances attribute) in a low energy house. The data has been
split into three subsets: training data from measurements up to 3/20/16 5:30, validation data
from measurements between 3/20/16 5:30 and 5/7/16 4:30, and test data from measurements
after 5/7/16 4:30. There are 26 attributes1
for each 10-minute interval, which is described
1We have removed the last two random variables as they aren’t relevant for this class.
1
in detail on the UCL ML repository, Applicances energy prediction dataset. Your goal is
to predict the Appliances For this problem, you will use scikit-learn for linear regression,
ridge regression, and lasso regression. All the specified functions should be in the file ‘q2.py’.
The functions in ‘q2.py’ will be tested against a different training, validation, and test set,
so it should work for a variety of datasets and assume that the data has been appropriately
pre-processed (i.e., do not do any standardization or scaling or anything to the data prior
to training the model). Any additional work such as loading the file, plotting, and required
analysis with the data (e.g., parts 2e, 2h, 2j, etc.) should be done in a separate file and
submitted with the Code.
(a) (Written) How did you preprocess the data? Explain your reasoning for using this
pre-processing.
(b) (Code) Write a Python function preprocess data(trainx, valx, testx) that does
what you specified in 2a above. If you do any feature extraction, you should do it
outside of this function. Your function should return the preprocessed trainx, valx, and
testx.
(c) (Code) Write a Python function eval linear1(trainx, trainy, valx, valy,
testx, testy) that takes in a training set, validation set, and test set, (in the form
of a numpy arrays), respectively, and trains a standard linear regression model only on
the training data and reports the RMSE and R2 on the training set, validation set, and
test set. Your function must return a dictionary with 6 keys, ‘train-rmse’, ‘train-r2’,
‘val-rmse’, ‘val-r2’, ‘test-rmse’, ‘test-r2’ and the associated values are the numeric values
(e.g., {‘train-rmse’: 10.2, ‘train-r2’: 0.3, ‘val-rmse’: 7.2, ‘val-r2’: 0.2, ‘test-rmse’: 12.1,
‘test-r2’: 0.4}).
(d) (Code) Write a Python function eval linear2(trainx, trainy, valx, valy,
testx, testy) that takes in a training set, validation set, and test set, respectively,
and trains a standard linear regression model using the training and validation data
together and reports the RMSE and R2 on the training set, validation set, and test set.
Your function should follow the same output format specified above.
(e) (Written) Report (using a table) the RMSE and R2 between 2c and 2d on the energydata.
How do the performances compare and what do the numbers suggest?
(f) (Code) Write a Python function eval ridge1(trainx, trainy, valx, valy, testx,
testy, alpha) that takes the regularization parameter, alpha, and trains a ridge
regression model only on the training data. Your function should follow the same output
format specified in (a) and (b).
(g) (Code) Write a Python function eval lasso1(trainx, trainy, valx, valy, testx,
testy, alpha) that takes the regularization parameter, alpha, and trains a lasso
regression model only on the training data. Your function should follow the same output
format specified in (a), (b), and (d).
(h) (Written) Report (using a table) the RMSE and R2
for training, validation, and test for
all the different (λ) values you tried. What would be the optimal parameter you would
select based on the validation data performance?
(i) (Code) Similar to part 2d, write the Python functions, eval ridge2(trainx,
trainy, valx, valy, testx, testy, alpha) and eval lasso2(trainx, trainy,
valx, valy, testx, testy, alpha) that train ridge and lasso using the training and
validation set.
2
(j) (Written) Use the optimal regularization parameter from 2h and report the RMSE and
R2 on the training set, validation set, and test set for the functions you wrote on 2i?
How does this compare to the results from 2h? What do the numbers suggest?
(k) (Written) Generate the coefficient path plots (regularization value vs. coefficient
value) for both ridge and lasso. Also, note (line or point or star) where the optimal
regularization parameters from 2h are on their respective plots. Make sure that your
plots encompass all the expected behavior (coefficients should shrink towards 0).
(l) (Written) What are 3 observations you can draw from looking at the coefficient path
plots, and the metrics? This should be different from your observations from 2e, 2h, and
2j.
3. (4 + 5 + 10 + 2 + 4 + 10 + 5 + 5 + 5 = 50 pts) Predicting Appliance Energy
Usage using SGD
Consider the Appliances energy prediction Data set from the previous problem. A template
file, elastic.py, defines a class ElasticNet that takes in the regularization parameters, el
(λ),2 alpha (α), eta (η) or the learning rate, the batch size (batch ∈ [1, N]), and epoch
or the maximum number of epochs as parameters when creating the object (i.e., elastic =
new ElasticNet(el, alpha, eta, batch, epoch)). You will implement ElasticNet using
stochastic gradient descent to train your model. The functions in ‘elastic.py’ will be tested
against a different training, validation, and test set, so it should work for a variety of
datasets and assume that the data has been appropriately pre-processed (i.e., do not do
any standardization or scaling or anything to the data prior to training the model). For
this problem, you ARE NOT allowed to use any existing toolbox/implementation (e.g.,
scikit-learn). Similar to problem 2, any additional work outside of (Code) should be
done in a separate file and submitted with the Code for full credit.
(a) (Code) Implement the loss objective helper function in elastic.py. As a reminder, the
optimization problem is:
min fo(x) = 1
2
||y − Xβ||2
2 + λ

α||β||2
2 + (1 − α)||β||1

, 0 ≤ α ≤ 1 (1)
In other words, given the coefficients, data and the regularization parameters, your
function will calculate the loss fo(x) as shown in Eq (1).
(b) (Code) Implement the gradient helper function in elastic.py. You may find it helpful
to derive the update for a single training sample and to consider proximal gradient
descent for the ||β||1 portion of the objective function. As a reminder, given step size η
and regularization parameter λ such that f(x) = g(x) + λ||x||1, the proximal update is:
prox(xi) =



xi − λη if xi > λη
0 if − λη ≤ xi ≤ λη
xi + λη if xi < −λη
(c) (Code) Implement the Python function train(self, x, y) for your class that trains
an elastic net regression model using stochastic gradient descent. Your function should
return a dictionary where the key denotes the epoch number and the value of the loss
associated with that epoch.
2
lambda is not used since it is a Python function and can cause confusion.

(d) (Code) Implement the Python function coef(self) for your class that returns the
learned coefficients as a numpy array.
(e) (Code) Implement the Python function predict(self, x) that predicts the label for
each training sample in x. If x is a numpy m × d array, then y is a numpy 1-d array of
size m × 1.
(f) (Written) For the optimal regularization parameters from ridge (λridge) and lasso (λlasso)
from 2h, and α =
1
2
, what are good learning rates for the dataset? Justify the selection
by trying various learning rates and illustrating the objective value (fo(x)) on a graph
for a range of epochs (one epoch = one pass through the training data)3
. For the chosen
learning rate you identified, what are the RMSE and R2
for the elastic net model trained
on the entire training set on the training, validation, and test sets?
(g) (Written) Using the learning rate from the previous part, train elastic net (using only
training data) for different values of α (it should encompass the entire range and include
α = 0, 1). Report the RMSE and R2
for the models on training, validation, and test set.
(h) (Written) Based on the results from (c) and 2(a) and 2(c), what conclusions can you
draw in terms of RMSE and R2
? Which model is the best? Also, discuss the differences
between the SGD-variants of Ridge and LASSO and the standard implementations
(Problem 2).
(i) (Written) What are the final coefficients that yield the best elastic net model on the test
data? Compare these with the final coefficients for the best-performing model on the
validation dataset. Are there noticeable differences? If so, discuss the differences with
respect to the impact on the performance.
3You do not need to use the entire training set. SGD, in theory, is not sensitive to the dataset. Thus, you can
subsample a reasonable percentage of data to tune the learning rate.
4

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