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COMP9417 - Machine Learning
创建日期：2024-04-13 18:15:17
标签： COMP9417
Machine Learning

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COMP9417 - Machine Learning

Homework 2: Logistic Regression & Optimization
Introduction In this homework, we will first explore some aspects of Logistic Regression and performing
inference for model parameters. We then turn our attention to gradient based optimisation, the workhorse
of modern machine learning methods.
Points Allocation There are a total of 25 marks. The available marks are:
• Question 1 a): 2 marks
• Question 1 b): 3 marks
• Question 1 c): 2 marks
• Question 1 d): 3 marks
• Question 1 e): 1 mark
• Question 2 a): 2 marks
• Question 2 b): 3 marks
• Question 2 c): 1 mark
• Question 2 d): 0 marks
• Question 2 e): 3 marks
• Question 2 f): 3 marks
• Question 2 g): 2 marks
What to Submit
• A single PDF file which contains solutions to each question. For each question, provide your solution
in the form of text and requested plots. For some questions you will be requested to provide screen
shots of code used to generate your answer — only include these when they are explicitly asked for.
• .py file(s) containing all code you used for the project, which should be provided in a separate .zip
file. This code must match the code provided in the report.
• You may be deducted points for not following these instructions.
• You may be deducted points for poorly presented/formatted work. Please be neat and make your
solutions clear. Start each question on a new page if necessary.
1
• You cannot submit a Jupyter notebook; this will receive a mark of zero. This does not stop you from
developing your code in a notebook and then copying it into a .py file though, or using a tool such as
nbconvert or similar.
• We will set up a Moodle forum for questions on this homework. Please read the existing questions
before posting new questions. Please do some basic research online before posting questions. Please
only post clarification questions. Any questions deemed to be fishing for answers will be ignored
and/or deleted.
• Please check the Moodle forum for updates to this spec. It is your responsibility to check for an-
nouncements about the spec.
• Please complete your homework on your own, do not discuss your solution with other people in the
course. General discussion of the problems is fine, but you must write out your own solution and
acknowledge if you discussed any of the problems in your submission (including their name and
zID).
• As usual, we monitor all online forums such as Chegg, StackExchange, etc. Posting homework ques-
tions on these site is equivalent to plagiarism and will result in a case of academic misconduct.
When and Where to Submit
• Due date: Week 7, Sunday July 18th, 2021 by 11:55pm. Please note that the forum will not be actively
monitored on weekends.
• Late submissions will incur a penalty of 20% per day (from the ceiling, i.e., total marks available for
the homework) for the first 5 days. For example, if you submit 2 days late, the maximum possible
mark is 60% of the available 25 marks.
• Submission must be done through Moodle, no exceptions.
Page 2
Question 1. Regularized Logistic Regression & the Bootstrap
In this problem we will consider the dataset provided in Q1.csv, with binary response variable Y ,
and 45 continuous features X1, . . . , X45. Recall that Regularized Logistic Regression is a regression
model used when the response variable is binary valued. Instead of using mean squared error loss as
in standard regression problems, we instead minimize the log-loss, also referred to as the cross entropy
loss. For a parameter vector β = (β1, . . . , βp) ∈ Rp, yi ∈ {0, 1}, xi ∈ Rp for i = 1, . . . , n, the log-loss is
L(β, β0) =
n∑
i=1
yi ln
(
1
s(β0 + βTxi)
)
+ (1− yi) ln
(
1
1− s(β0 + βTxi)
)
,
where s(z) = (1 + e−z)−1 is the logistic sigmoid (see Homework 0 for a refresher.) In practice, we will
usually add a penalty term, and consider the optimisation:
(βˆ0, βˆ1) = argmin
β0,β
{CL(β, β0) + penalty(β)} (1)
where the penalty is usually not applied to the bias term β0, and C is a hyper-parameter. For example,
in the `1 regularisation case, we take penalty(β) = ‖β‖1 (a LASSO for logistic regression).
(a) Consider the sklearn logistic regression implementation (section 1.1.11), which claims to mini-
mize the following objective:
wˆ, cˆ = argmin
w,c
{
‖w‖1 + C
n∑
i=1
log(1 + exp(−y˜i(wTxi + c)))
}
. (2)
It turns out that this objective is identical to our objective above, but only after re-coding the binary
variables to be in {−1, 1} instead of binary values {0, 1}. That is, y˜i ∈ {−1, 1}, whereas yi ∈ {0, 1}.
Argue rigorously that the two objectives (1) and (2) are identical, in that they give us the same
solutions (βˆ0 = cˆ and βˆ = wˆ). Further, describe the role of C in the objectives, how does it compare
to the standard LASSO parameter λ? What to submit: some commentary/your working.
(b) Take the first 500 observations to be your training set, and the rest as the test set. In this part, we
will perform cross validation over the choice of C from scratch (Do not use existing cross valida-
tion implementations here, doing so will result in a mark of zero.)
Create a grid of 100 C values ranging from C = 0.0001 to C = 0.6 in equally sized increments,
inclusive. For each value of C in your grid, perform 10-fold cross validation (i.e. split the data
into 10 folds, fit logistic regression (using the LogisticRegression class in sklearn) with the
choice of C on 9 of those folds, and record the log-loss on the 10th, repeating the process 10 times.)
For this question, we will take the first fold to be the first 50 rows of the training data, the second
fold to be the next 50 rows, etc. Be sure to use `1 regularisation, and the liblinear solver when
fitting your models.
To display the results, we will produce a plot: the x-axis should reflect the choice of C values, and
for each C, plot a box-plot over the 10 CV scores. Report the value of C that gives you the best CV
performance. Re-fit the model with this chosen C, and report both train and test accuracy using this
model. Note that we do not need to use the y˜ coding here (the sklearn implementation is able to
handle different coding schemes automatically) so no transformations are needed before applying
logistic regression to the provided data. What to submit: a single plot, train and test accuracy of your
final model, a screen shot of your code for this section, a copy of your python code in solutions.py
Page 3
(c) In this part we will compare our results in the previous section to the sklearn implementation of
gridsearch, namely, the GridSearchCV class. My initial code for this section looked like:
1 grid_lr = GridSearchCV(estimator=
2 LogisticRegression(penalty=’l1’,
3 solver=’liblinear’),
4 cv=10,
5 param_grid=param_grid)
6 grid_lr.fit(Xtrain, Ytrain)
7
However, this gave me a very different answer to the result in (b). Provide two reasons for why
this is the case, and then, if it is possible, re-run the code with some changes to give consistent
results to those in (b), and if not, explain why. It may help to read through the documentation.
What to submit: some commentary, a screen shot of your code for this section, a copy of your python code in
solutions.py
We next explore the idea of inference. To motivate the difference between prediction and inference,
see some of the answers to this stats.stachexchange post. Needless to say, inference is a much
more difficult problem than prediction in general. In the next parts, we will study some ways
of quantifying the uncertainty in our estimates of the logistic regression parameters. Assume for
the remainder of this question that C = 1, and work only with the training data set (n = 500
observations) constructed earlier.
(d) In this part, we will consider the nonparametric bootstrap for building confidence intervals for
each of the parameters β1, . . . , βp. (Do not use existing Bootstrap implementations here, doing
so will result in a mark of zero.) To describe this method, let’s first focus on the case of βˆ1. The
idea behind the nonparametric bootstrap is as follows:
1. Generate B bootstrap samples from the original dataset. Each bootstrap sample consists of n
points sampled with replacement from the original dataset, where n is the size of the original
dataset.
2. On each of theB bootstrap samples, compute an estimate of β1, giving us a total ofB estimates
which we denote β˜(1)1 , . . . , β˜
(B)
1 .
3. Define the bootstrap mean and standard error respectively:
β˜1 =
1
B
B∑
b=1
β˜
(b)
1 s˜.e.(β˜1) =
√√√√ 1
B − 1
B∑
b=1
(β˜
(b)
1 − β˜1)2.
4. A 95% bootstrap confidence interval for β1 is then given by the interval:
((β˜1)L, (β˜1)U ) = (5th quantile of the bootstrap estimates, 95th quantile of the bootstrap estimates)
The idea behind a 95% confidence interval is that it gives us a range of values for which we be-
lieve with 95% probability the true parameter lives in that interval. If the computed 95 % interval
contains the value of zero, then this provides us evidence that β1 = 0, which means that the first
feature should not be included in our model.
Take B = 10000 and set a random seed of 12 (i.e. np.random.seed(12)). Generate a plot where
the x-axis represents the different parameters β1, . . . , βp, and plot a vertical bar that runs from
(β˜p)L to (β˜p)U . For those intervals that contain 0, draw the bar in red, otherwise draw it in blue.
Also indicate on each bar the bootstrap mean. Remember to use C = 1.0.
Page 4
What to submit: a single plot, some commentary, a screen shot of your code for this section, a copy of your
python code in solutions.py
(e) Comment on your results in the previous section, what do the confidence intervals tell you about
the underlying data generating distribution? How does this relate to the choice of C when running
regularized logistic regression on this data? Is regularization necessary?
Question 2. Gradient Based Optimization
In this question we will explore some algorithms for gradient based optimization. These algorithms
have been crucial to the development of machine learning in the last few decades. The most famous
example is the backpropagation algorithm used in deep learning, which is in fact just an application
of a simple algorithm known as (stochastic) gradient descent. The general framework for a gradient
method for finding a minimizer of a function f : Rn → R is defined by
x(k+1) = x(k) − αk∇f(xk), k = 0, 1, 2, . . . , (3)
where αk > 0 is known as the step size, or learning rate. Consider the following simple example of
minimizing g(x) = 2
√
x3 + 1. We first note that g′(x) = 3x2(x3 + 1)−1/2. We then need to choose a
starting value of x, say x(0) = 1. Let’s also take the step size to be constant, αk = α = 0.1. Then we have
the following iterations:
x(1) = x(0) − 0.1× 3(x(0))2((x(0))3 + 1)−1/2 = 0.5757359
x(2) = x(1) − 0.1× 3(x(1))2((x(1))3 + 1)−1/2 = 0.4672197
...
and this continues until we terminate the algorithm (as a quick exercise for your own benefit, code
this up and compare it to the true minimum of the function which is x∗ = −1). This idea works for
functions that have vector valued inputs, which is often the case in machine learning. For example,
when we minimize a loss function we do so with respect to a weight vector, β. When we take the step-
size to be constant at each iteration, this algorithm is called gradient descent. For the entirety of this
question, do not use any existing implementations of gradient methods, doing so will result in an
automatic mark of zero for the entire question.
(a) Consider the following optimisation problem:
min
x∈Rn
f(x),
where
f(x) =
1
2
‖Ax− b‖22,
and where A ∈ Rm×n, b ∈ Rm are defined as
A =
 1 0 1 −1−1 1 0 2
0 −1 −2 1
 , b =
12
3
 .
Run gradient descent on f using a step size of α = 0.1 and starting point of x(0) = (1, 1, 1, 1). You
will need to terminate the algorithm when the following condition is met: ‖∇f(x(k))‖2 < 0.001. In
Page 5
your answer, clearly write down the version of the gradient steps (3) for this problem. Also, print
out the first 5 and last 5 values of x(k), clearly indicating the value of k, in the form:
k = 0, x(k) = [1, 1, 1, 1]
k = 1, x(k) = · · ·
k = 2, x(k) = · · ·
...
What to submit: an equation outlining the explicit gradient update, a print out of the first 5 and last 5
rows of your iterations, a screen shot of any code used for this section and a copy of your python code in
solutions.py.
(b) Note that using a constant step-size is sub-optimal. Ideally we would ideally like to take large
steps at the beginning (when we are far away from the optimum), then take smaller steps as we
move closer towards the minimum. There are many proposals in the literature for how best to
choose the step size, here we will explore just one of them called the method of steepest descent.
This is almost identical to gradient descent, except at each iteration k, we choose
αk = argmin
α≥0
f(x(k) − α∇f(x(k))).
In words, the step size is chosen to minimize an objective at each iteration of the gradient method,
the objective is different at each step since it depends on the current x-value. In this part, we
will run steepest descent to find the minimizer in (a). First, derive an explicit solution for αk
(mathematically, please show your working). Then run steepest descent with the same x(0) as in
(a), and α0 = 0.1. Use the same termination condition. Provide the first and last 5 values of x(k),
as well as a plot of αk over all iterations. What to submit: a derivation of αk, a print out of the first 5
and last 5 rows of your iterations, a single plot, a screen shot of any code used for this section, a copy of your
python code in solutions.py.
(c) Comment on the differences you observed, why would we prefer steepest descent over gradient
descent? Why would you prefer gradient descent over steepest descent? Finally, explain why this
is a reasonable condition to terminate use to terminate the algorithm.
In the next few parts, we will use the gradient methods explored above to solve a real machine
learning problem. Consider the data provided in Q2.csv. It contains 414 real estate records, each
of which contains the following features:
• transactiondate: date of transaction
• age: age of property
• nearestMRT: distance of property to nearest supermarket
• nConvenience: number of convenience stores in nearby locations
• latitude
• longitude
The target variable is the property price. The goal is to learn to predict property prices as a function
of a subset of the above features.
(d) We need to preprocess the data. First remove any rows with missing values. Then, delete all
features except for age, nearestMRT and nConvenience. Then use the sklearn minmaxscaler
to normalize the features. Finally, create a training set from the first half of the resulting dataset,
and a test set from the remaining half. Your end result should look like:
Page 6
• first row X train: [0.73059361, 0.00951267, 1.]
• last row X train: [0.87899543, 0.09926012, 0.3]
• first row X test: [0.26255708, 0.20677973, 0.1]
• last row X test: [0.14840183, 0.0103754, 0.9]
• first row Y train: 37.9
• last row Y train: 34.2
• first row Y test: 26.2
• last row Y test: 63.9
What to submit: a copy of your python code in solutions.py
(e) Consider the loss function
L(x, y) =
√
1
4
(x− y)2 + 1− 1,
and consider the linear model
yˆi = w0 + w1xi1 + w2xi2 + w3xi3, i = 1, . . . , n.
We can write this more succinctly by letting w = (w0, w1, w2, w3)T and xi = (1, xi1, xi2, xi3)T , so
that yˆi = wTxi. The mean loss achieved by our model (w) on a given dataset of n observations is
then
L(w) = 1
n
n∑
i=1
[√
1
4
(yi − wTxi)2 + 1− 1
]
.
We will run gradient descent to compute the optimal weight vector w. The iterations will look like
w(k+1) = w(k) − αk∇L(w).
Instead of computing the gradient directly though, we will rely on an automatic differentiation
library called JAX. Read the first section of the documentation to get an idea of the syntax. Im-
plement gradient descent from scratch and use the JAX library to compute the gradient of the loss
function at each step. You will only need the following import statements:
1 # pip install --upgrade pip
2 # pip install jax
3 # you may/may not also need to run: pip install jaxlib
4 # pip install --upgrade jax[cpu]
5
6 import jax.numpy as jnp
7 from jax import grad
8
Use w(0) = [1, 1, 1, 1]T , and a step size of 1. Terminate your algorithm when the absolute value of
the loss from one iteration to the other is less than 0.0001. Report the number of iterations taken,
and the final weight vector. Further, report the train and test losses achieved by your final model,
and produce a plot of the training loss at each step of the algorithm. What to submit: a single plot, the
final weight vector, the train and test accuracy of your final model, a screen shot of your code for this section,
a copy of your python code in solutions.py
Page 7
(f) Finally, re-do the previous section but with steepest descent instead. In order to compute αk at
each step, you can either use JAX or it might be easier to use the minimize function in scipy (See
lab3). Run the algorithm with the same w(0) as above, and take α0 = 1 as your initial guess when
numerically solving for αk (for each k). Terminate the algorithm when the loss value falls bellow
2.5. Report the number of iterations it took, as well as the final weight vector, and the train and
test losses achieved. Generate a plot of the losses as before and include it. What to submit: a single
plot, the final weight vector, the train and test accuracy of your final model, a screen shot of your code for
this section, a copy of your python code in solutions.py
(g) In this question we have explored the gradient descent and steepest descent variants of the gradi-
ent method. Many other gradient based algorithms exist in the literature. Choose one and describe
it. What to submit: some commentary, any plots or code you used in this section (you don’t need to write
any code, or supply plots, but you may. Also be sure to cite any sources used here.)
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MGMTMFE 405 25579 1ECE 273 M122 COMP4434 1ELEC ELEC5620M STAT0010 BFF2401 MXB361 MATH3811 CS330 CMT307 FIT1033 CptS 583 UP 431 MATH6005 AF5343 CS2435 ECOM30002 ES50061 CS 2506 ECMT6002 CODE 00099F MT4539 ESSAY ICM317 MATH1231 MFIT5010 BMA547 MATH334 CIVL2201 MATH39032 T-04 CIVE97119 EMESTER CAB302 CAB301 59PM STAT802 ISYS3412 RM3 COSC 2123 AcF 311 MATH2022 1FINAL 4CCS1PHC ICM 142 ENVS3023 Math 4023 I1 ACSE8 FIN B385F CS-275 STA 4109-01 A31021 EE140 MT1 EE124 ECO-6004B Inf2-IADS CS126 IS2184 UL18 MATH5092 ECOM009 COMP 6211 ELEC202 MKT3019 IAB230 FIT5046 584-S21 COMP3210 CO250 A29401 MATH 136 MGT6174 ECMT6007 DATA2001 MAT 443 MATH10222 PHDE0071 ELE00123M COSC1295 MAT00045H MATH32032 1004GRC MAS223 COMP10001 DATA3404 CIVL2110 STAT2008 FNCE10002 EEE6431 COM6515 COMP9120 NBS8185 ECTE942 ECON 405 MATH314301 STAT 2011 COMPSCI 350 COMP20007 CMPSC-132 ECON5002 STA 35 MTH6108 FIT5147 MTH6150 100C FINA 5260 DESC9115 ECON4998 NBS8257 GMM ECON 6070 BSA2 ECE599 STATS 102B STA120 INFS601 BFC3241 MATH20602 COMP532 MAS275 COMP5318 COMP4002 FIT5086 MAST20004 FIT3152 FIT5129 CIS2380 CS521 COMP 1005 EECMS CITS5501 ECE 462 ECON 6002 ENG 4051 CSC 648 CMT654 MECHENG 713 PHYS1002 AMME 2000 FP0060 DDES3200 MECHENG 743 BINF90002 FIT2081 ISYS90081 ELECTENG 722 BUS 360 ECOS3010 BUS2810 ETF2100 ENG2048 STAT5002 COMP2017 CEME 2003 MATS3004 COMP9319 PHYS1001 COMP5349 AMME2261 COMP-381 FIT3173 COSC2757 ENG4025 QBUS2820 COMP9414 MATH2021 EOSC118 MFIN6690 PMATH 321 CSE-141L MATH575A COMPSCI5002 COMP1511 management comp2123 FINS3616 COMP 3331 COMS30034 DTSC 71 AcF 702 AcF702 DTSC71 MAT021 ELE8083 ACFI213 EG-127J 127J FIT5163 FIT2001 ELE00113M CMT310 CS 245 ENGL 1101 MTH6115 EE497 CS 325 Math 124A MTH5130 MBA502 MGT 5380 ARCH 1261 F033800 20LLP109 ELE4009 FINANCIAL COMP3331 Computer ECE4179 20T3 COMP9311 COMP9311 UNSW MFIN6210 ACCT 5930 EE590 ENG2029 ECO100 SIT315 ACCT3610 COMP3600 STK2100 FIT2004 COMP30020 ACTL1101 Stat 101B BANA 200 COSC2473 ENGG1400 AI COMP90043 MMF 1914H PHYS 127 CS584 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COMP2013 IB9X60 INVP001 EMATM0051 COMP220 BLAW1002 E20 TCP8001 ECON61001 EART40005 ECONM1022 MATH 365 MATH 367 ELEC6218 COMP 5812M EDUC 5061M ELEC ENG 1101 equation MATH6174 ST334 C4 EC9570 A33648 EBUS603 ECON47815 stata WFMS0001 PEM102 INF6060 ionically ELECTRICAL MATH 375 MMW 12 MKTG860 ECON60401 ALY6010 SOST20131 Introduction COM3524 MME2 GGR305 JNB 538 EECS101 COMM5030 IEOR E4601 2B CSC336 EN2315 LAW 432 CSC263 MATH2001 OLET1143 CSCB63 SIT772 COMM1170 STAT 441 MS930 INFT3050 GLBH0030 Stat 120B MAST10007 MGEC61 Machine GGR 252H1S ECON7030 Financial Marketing MGMT30019 ECON5120 SDSC 8009 CSC165H1S DSCI 551 3081W ACM41000 ECA 5304 MATH 1064 ECSE444 DATA 604 COMP7105 MIE1615 ECON4004 ENV200H1S EC481 MKT 306 EC1B1 MATH08051 ALY6140 CS260C CSE30 MSIN0226 CS4103 CSC3064 AM16 COMP11212 CSE4101 PSTAT 174 Math 380W CNT 5106C STAT 310 FIN2028 CS5014 STAT 425 COMP 4102A ELEC2140 MANF9544 QRM 9 QRM 8 ECON 457 COMPSCI 4091 MATH 523 DNSC-4211 PEAS 6000 MSIN0180 COMP2119 A2A HT 2022 IB93F0 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FN1024 MATH0094 MANG 6554 ECOS3003 ELEC6252 C2511C PHY328 EC1060 ECON32202 COMP4002-E1 MATH3063 EFIMM0124 QBUS2310 ECON4411 FUNDAMENTALS ACCT20001 18YX RPM COMP 2711 QBUS6600 BUSS5221 ST3189 FIT1047 MAT344 EGB375 MATH60013 MATH96011 COMP3557 MAST20029 ECM159 EAE1011 ENSCEN 105 SECTION A STA 141C SOSE 2022 PAM006 COMP0048 M40007 ISOM2500 CMPE 142 A06472 MCC150 ACST 8083 GEOG6087 ECON920 FINM7008 ENVENG 4110 COMPSCI 361 ENGRCEE 21 PHYS 1112 ACF2100 ECMM108 COMPS267F MAT00050M MAST30025 CSIT970 MSCI 224 INT202 PUBL0054 MSDM5004 BMAN 21020 BMAN21020 MGMT1002 MATH11150 LAWS7023 ACTL30002 CIVL3340 MATH1002 ECON 8069 COMPSCI 351 IBUS7302 STAT4528 CHEM1101 STAT3925 BUSN 7008 Comp 3120 ECOS2040 SWEN90010 ENGN6528 ECM158 ECON20022 COMP6452 Calculation CITS1401 S1889112 BUS 351 ACTL30008 FIT2091 ECON6002 IBUS6020 PHI 001 ECON30290 MATH7501 ECON10004 MFIN7017 ISYS90043 LAWS7012 ECON30025 COMP30023 COMM1190 ECOM20001 GSBS6009 STAT 337 EMET8002 ECOM90001 ECOM30001 FINC 2012 BUSINESS ENGCV512 ECO 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