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COMP9417 - Machine Learning

创建日期：2024-06-19 15:04:41

所属分类：计算机computer science

标签： COMP9417

Machine Learning

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COMP9417 - Machine Learning

Homework 2: Bias, Variance and an application of Gradient
Descent
Introduction In this homework we revisit the notion of bias and variance as metrics for characterizing the
behaviour of an estimator. We then take a look at a new gradient descent based algorithm for combining
different machine learning models into a single, more complex, model.
Points Allocation There are a total of 28 marks.
• Question 1 a): 1 mark
• Question 1 b): 3 marks
• Question 1 c): 3 marks
• Question 1 d): 1 mark
• Question 1 e): 1 mark
• Question 2 a): 3 marks
• Question 2 b): 2 marks
• Question 2 c): 6 marks
• Question 2 d): 2 marks
• Question 2 e): 3 marks
• Question 2 f): 2 marks
• Question 2 g): 1 mark
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.
1
• 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.
• 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 about 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 Moodle announcements for updates to this spec. It is your responsibility to check for
announcements 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(s) 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.
• You may not use SymPy or any other symbolic programming toolkits to answer the derivation ques-
tions. This will result in an automatic grade of zero for the relevant question. You must do the
derivations manually.
When and Where to Submit
• Due date: Week 5, Friday June 28th, 2024 by 5pm. Please note that the forum will not be actively
monitored on weekends.
• Late submissions will incur a penalty of 5% per day from the maximum achievable grade. For ex-
ample, if you achieve a grade of 80/100 but you submitted 3 days late, then your final grade will be
80− 3× 5 = 65. Submissions that are more than 5 days late will receive a mark of zero.
• Submission must be made on Moodle, no exceptions.
Page 2
Question 1. Bias of Estimators
Let γ > 0 and suppose that X1, . . . , Xn
i.i.d.∼ N(γ, γ2). We define:
X =
1
n
n∑
i=1
Xi,
S2 =
1
n− 1
n∑
i=1
(Xi −X)2.
You may use the following two facts without proof:
(F1) X and S2 are independent.
(F2) X and c∗S are both unbiased estimators of γ.1
What to submit: for all parts (a)-(e), include your working out, either typed or handwritten. For all parts, you
must show all working for full credit. Answers without working will receive a grade of zero.
(a) Consider the estimator:
T1 = aX + (1− a)c∗S.
Show that for any choice of constant a, T1 is unbiased for γ.
(b) What choice of a gives you the best (in the sense of MSE) possible estimator? Derive an explicit
expression for this optimal choice of a. We refer to this estimator as T ∗1 .
(c) Consider now a different estimator:
T2 = a1X¯ + a2(c∗S),
and we do not make any assumptions about the relationship between a1 and a2. Find the con-
stants a1, a2 explicitly that make T2 best (from the MSE perspective), i.e. choose a1, a2 to minimize
MSE(T2) = E(T2 − γ)2. We refer to this estimator as T ∗2 .
(d) Show that T ∗2 has MSE that is less than or equal to the MSE of T ∗1 .
(e) Consider the estimator V+ = max{0, T ∗2 }. Show that the MSE of V+ is smaller than or equal to the
MSE of T ∗2 .
Question 2. Gradient Descent for Learning Combinations of Models
In this question, we discuss and implement a gradient descent based algorithm for learning combina-
tions of models, which are generally termed ’ensemble models’. The gradient descent idea is a very
powerful one that has been used in a large number of creative ways in machine learning beyond direct
minimization of loss functions.
The Gradient-Combination (GC) algorithm can be described as follows: Let F be a set of base learning
algorithms2. The idea is to combine the base learners in F in an optimal way to end up with a good
learning algorithm. Let `(y, yˆ) be a loss function, where y is the target, and yˆ is the predicted value.3
Suppose we have data (xi, yi) for i = 1, . . . , n, which we collect into a single data set D0. We then set
the number of desired base learners to T and proceed as follows:
1You do not need to worry about knowing or calculating c∗ for this question, it is just some constant.
2For example, you could take F to be the set of all regression models with a single feature, or alternatively the set of all regression
models with 4 features, or the set of neural networks with 2 layers etc.
3Note that this set-up is general enough to include both regression and classification algorithms.
Page 3
(I) Initialize f0(x) = 0 (i.e. f0 is the zero function.)
(II) For t = 1, 2, . . . , T :
(GC1) Compute:
rt,i = − ∂
∂f(xi)
n∑
j=1
`(yj , f(xj))
∣∣∣∣
f(xj)=ft−1(xj), j=1,...,n
for i = 1, . . . , n. We refer to rt,i as the i-th pseudo-residual at iteration t.
(GC2) Construct a new pseudo data set, Dt, consisting of pairs: (xi, rt,i) for i = 1, . . . , n.
(GC3) Fit a model to Dt using our base class F . That is, we solve
ht = arg min
f∈F
n∑
i=1
`(rt,i, f(xi))
(GC4) Choose a step-size. This can be done by either of the following methods:
(SS1) Pick a fixed step-size αt = α
(SS2) Pick a step-size adaptively according to
αt = arg min
α
n∑
i=1
`(yi, ft−1(xi) + αht(xi)).
(GC5) Take the step
ft(x) = ft−1(x) + αtht(x).
(III) return fT .
We can view this algorithm as performing (functional) gradient descent on the base class F . Note that
in (GC1), the notation means that after taking the derivative with respect to f(xi), set all occurences
of f(xj) in the resulting expression with the prediction of the current model ft−1(xj), for all j. For
example:
∂
∂x
log(x+ 1)
∣∣∣∣
x=23
=
1
x+ 1
∣∣∣∣
x=23
=
1
24
.
(a) Consider the regression setting where we allow the y-values in our data set to be real numbers.
Suppose that we use squared error loss `(y, yˆ) = 12 (y− yˆ)2. For round t of the algorithm, show that
rt,i = yi − ft−1(xi). Then, write down an expression for the optimization problem in step (GC3)
that is specific to this setting (you don’t need to actually solve it).
What to submit: your working out, either typed or handwritten.
(b) Using the same setting as in the previous part, derive the step-size expression according to the
adaptive approach (SS2).
What to submit: your working out, either typed or handwritten.
(c) We will now implement the gradient-combination algorithm on a toy dataset from scratch, and we
will use the class of decision stumps (depth 1 decision trees) as our base class (F), and squared error
loss as in the previous parts.4. The following code generates the data and demonstrates plotting
the predictions of a fitted decision tree (more details in q1.py):
4In your implementation, you may make use of sklearn.tree.DecisionTreeRegressor, but all other code must be your
own. You may use NumPy and matplotlib, but do not use an existing implementation of the algorithm if you happen to find one.
Page 4
1 np.random.seed(123)
2 X, y = f_sampler(f, 160, sigma=0.2)
3 X = X.reshape(-1,1)
4
5 fig = plt.figure(figsize=(7,7))
6 dt = DecisionTreeRegressor(max_depth=2).fit(X,y) # example model
7 xx = np.linspace(0,1,1000)
8 plt.plot(xx, f(xx), alpha=0.5, color=’red’, label=’truth’)
9 plt.scatter(X,y, marker=’x’, color=’blue’, label=’observed’)
10 plt.plot(xx, dt.predict(xx.reshape(-1,1)), color=’green’, label=’dt’) # plotting
example model
11 plt.legend()
12 plt.show()
13
The figure generated is
Your task is to generate a 5 x 2 figure of subplots showing the predictions of your fitted gradient-
combination model. There are 10 subplots in total, the first should show the model with 5 base
learners, the second subplot should show it with 10 base learners, etc. The last subplot should be
the gradient-combination model with 50 base learners. Each subplot should include the scatter of
data, as well as a plot of the true model (basically, the same as the plot provided above but with
your fitted model in place of dt). Comment on your results, what happens as the number of base
learners is increased? You should do this two times (two 5x2 plots), once with the adaptive step
size, and the other with the step-size taken to be α = 0.1 fixed throughout. There is no need to
split into train and test data here. Comment on the differences between your fixed and adaptive
step-size implementations. How does your model perform on the different x-ranges of the data?
What to submit: two 5 x 2 plots, one for adaptive and one for fixed step size, some commentary, and a screen
shot of your code and a copy of your code in your .py file.
Page 5
(d) Repeat the analysis in the previous question but with depth 2 decision trees as base learners in-
stead. Provide the same plots. What do you notice for the adaptive case? What about the non-
adaptive case? What to submit: two 5 x 2 plots, one for adaptive and one for fixed step size, some commen-
tary, and a copy of your code in your .py file.
(e) Now, consider the classification setting where y is taken to be an element of {−1, 1}. We consider
the following classification loss: `(y, yˆ) = log(1 + e−yyˆ). For round t of the algorithm, what is the
expression for rt,i? Write down an expression for the optimization problem in step (GC3) that is
specific to this setting (you don’t need to actually solve it).
What to submit: your working out, either typed or handwritten.
(f) Using the same setting as in the previous part, write down an expression for αt using the adaptive
approach in (SS2). Can you solve for αt in closed form? Explain.
What to submit: your working out, either typed or handwritten, and some commentary.
(g) In practice, if you cannot solve for αt exactly, explain how you might implement the algorithm.
Assume that using a constant step-size is not a valid alternative. Be as specific as possible in your
answer. What, if any, are the additional computational costs of your approach relative to using a
constant step size ?
What to submit: some commentary.

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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 MAS316/414 ACCT2522 MSIN0181 ATMS 201 COMP3425 MATH7861 SOLA2060 MA 325 7CCEMCTH ECON 432 INFT2051 SOLA 2060 ECOS2002 Stats141XP BIOA02H3 S2022 MA4601 K5 COMP5329 EG 284 IB3A70 WECO1020 COMP3027 Energy ACS61011 MAST20026 STAT 533 MATA31 ENVS257 BEM2031 EC 486 COMP0172 CS3214 EECE5644 IB2B20 COSC2536 COSC1147 COMP6214 5CCM242A COMP0130 AMME2000 MTMG50: EG238 ACF6006 QBUS2810 COSC2391 ECON7560 MATH377 STA305 CS 486 COMP1117B EL648 ELEC3909 ECON 241 COMP 4901W BUSI4428 ENEN20002 ENGN4524 ELEC3115 CSC 413 PSYC 100B ECON1002 MATH256 ANSC20005 MTHS2008 MATH40082 COSC 2406 FIT3174 STA 238 CMT202 Differential Equations MTHM017 GARCH 101 MATH2069 CSC336S ACSC12 ETF2121 CEME 2001 DB B474F ARHT1001 EECS 3311 3220 STATS 786 CS 484 COMP20008 MATH4091 ECE 455 CSCI 204 BUS B200F BA 875 BU5562 COSC 1147 X11 GEOG 101 IMM250F CSE 584A MATH38172 LAB 4 ELEC5882M FINS5523 EE 430 COMP 465 6CCS3PRE 7CCSMNSE FINANCE ACMB02H3 ENGG2112 ELEC5616 SCM B371 CHEM3120 CS22 ECMT2130 MAST30011 ADSPBF706 CENG0013 ACCOUNTING BUSS1040 CHEM 181 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