代写COMP9417 - Machine Learning Homework 2

COMP9417 - Machine Learning
Homework 2
Introduction In this homework we first take a closer look at feature maps induced by kernels. We then ex-
plore a creative use of the gradient descent method introduced in homework 1. We will show that gradient
descent techniques can be used to construct combinations of models from a base set of models such that the
combination can outperform any single base model.
Points Allocation There are a total of 28 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.
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 7, Monday July 10th, 2023 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. 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
algorithms1. The idea is to combine the base learners in F in an optimal way to end up with a good
learning algorithm. Let `(y, y?) be a loss function, where y is the target, and y? is the predicted value.2
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:
(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 = argmin
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 = argmin
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
.
1For 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.
2Note that this set-up is general enough to include both regression and classification algorithms.
Page 3
(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, y?) = 12 (y? y?)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.3. The following code generates the data and demonstrates plotting
the predictions of a fitted decision tree (more details in q1.py):
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
3In 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
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.
(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, y?) = log(1 + e?yy?). 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.
Page 5
(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.
Question 2. Test Set Linear Regression
Recall that the rMSE (root-MSE) of a hypothesis4 h on a test set {(xi, yi)}ni=1, where xi is a p-dimensional
vector and yi is a real number, is defined as
rMSE(h) =
√√√√ 1
n
n∑
i=1
(yi ? h(xi))2.
For all parts, assume that the xi’s are known to you, but the yi’s are not. Suppose however that you are
permitted to query rMSE(h). What this means is that you can query, for any hypothesis h, the rMSE
of h on the test set. Suppose further that you know that rMSE(z) = c0, where z is the hypothesis that
returns zero for any input, i.e. z(xi) = 0 for each i = 1, . . . , n, and c0 is some arbitrary positive number.
(a) Assume you have a set of T hypothesesH = {h1, h2, . . . , hT }. You are told that for each i, there is a
hypothesis h inH, such that h(xi) = yi. In other words, for any point in the test set, there is at least
one hypothesis in H that predicts that point correctly5. Suppose that you are permitted to blend
predictions of different hypotheses in H6. Design a naive, brute-force algorithm that constructs a
hypothesis g from the elements of H such that rMSE(g) = 0. How many queries of rMSE do you
need to make? How does your algorithm scale (in the worst case) with the test size n? Describe
your algorithm in detail.
What to submit: a description of your algorithm and the number of queries required, either typed or hand-
written.
(b) We now consider a better approach than brute-force. For a given hypothesis h, define
h(X) = (h(x1), h(x2), . . . , h(xn))
>
y = (y1, y2, . . . , yn)
>.
We can always compute h(X), but we do not know y. What is the smallest number of queries
required to compute y>h(X)? Describe your approach in detail.
What to submit: a description of your approach and the number of queries required, either typed or hand-
written.
(c) Given a set of K hypothesesH = {h1, . . . , hK}, use your result in the previous part to solve
min
α1,...,αK
rMSE
(
K∑
k=1
αkhk
)
,
and obtain the optimal weights α1, . . . , αK . Describe your approach in detail, and be sure to detail
how many queries are needed and the exact values of the α’s, in terms of X, y and the elements of
H.
4The term hypothesis just means a function or model that takes as input x and returns as output a prediction h(x) = y?.
5Note that this does not imply that there is some hypothesis inH that predicts all points correctly.
6For example, you could construct a blended hypothesis g that returns the predictions of h2 on test points 1 to 5, and the predictions
of h5 on points 6 to n.
Page 6
What to submit: a description of your algorithm and the number of queries required, either typed or hand-
written.
Question 3. Newton’s Method
Note: throughout this question do not use any existing implementations of any of the algorithms
discussed unless explicitly asked to in the question. Using existing implementations can result in
a grade of zero for the entire question. In homework 1 we studied gradient descent (GD), which is
usually referred to as a first order method. Here, we study an alternative algorithm known as Newton’s
algorithm, which is generally referred to as a second order method. Roughly speaking, a second order
method makes use of both first and second derivatives. Generally, second order methods are much
more accurate than first order ones. Given a twice differentiable function g : R→ R, Newton’s method
generates a sequence {x(k)} iteratively according to the following update rule:
x(k+1) = x(k) ? g
′(x(k))
g′′(x(k))
, k = 0, 1, 2, . . . , (1)
For example, consider the function g(x) = 12x
2 ? sin(x) with initial guess x(0) = 0. Then
g′(x) = x? cos(x), and g′′(x) = 1 + sin(x),
and so we have the following iterations:
x(1) = x(0) ? x
(0) ? cos(x0)
1 + sin(x(0))
= 0? 0? cos(0)
1 + sin(0)
= 1
x(2) = x(1) ? x
(1) ? cos(x1)
1 + sin(x(1))
= 1? 1? cos(1)
1 + sin(1)
= 0.750363867840244
x(3) = 0.739112890911362
...
and this continues until we terminate the algorithm (as a quick exercise for your own benefit, code this
up, plot the function and each of the iterates). We note here that in practice, we often use a different
update called the dampened Newton method, defined by:
x(k+1) = x(k) ? α g
′(xk)
g′′(xk)
, k = 0, 1, 2, . . . . (2)
Here, as in the case of GD, the step size α has the effect of ‘dampening’ the update.
(a) Consider the twice differentiable function f : Rn → R. The Newton steps in this case are now:
x(k+1) = x(k) ? (H(x(k)))?1?f(x(k)), k = 0, 1, 2, . . . , (3)
where H(x) = ?2f(x) is the Hessian of f . Explain heuristically (in a couple of sentences) how the
above formula is a generalization of equation (1) to functions with vector inputs. what to submit:
Some commentary
(b) Consider the function f : R2 → R defined by
f(x, y) = 100(y ? x2)2 + (1? x)2.
Create a 3D plot of the function using mplot3d (see lab0 for example). Further, compute the
gradient and Hessian of f . what to submit: A single plot, the code used to generate the plot, the gradient
and Hessian calculated along with all working. Add a copy of the code to solutions.py
Page 7
(c) Using NumPy only, implement the (undampened) Newton algorithm to find the minimizer of the
function in the previous part, using an initial guess of x(0) = (?1.2, 1)T . Terminate the algorithm
when
∥∥?f(x(k))∥∥
2
≤ 10?6. Report the values of x(k) for k = 0, 1, . . . ,K where K is your final
iteration. what to submit: your iterations, and a screen shot of your code. Add a copy of the code to
solutions.py
Page 8