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CSCI-UA 9473 - Introduction to Machine Learning

创建日期：2024-05-08 14:08:15

所属分类：计算机computer science

标签： CSCI-UA 9473

Introduction to Machine Learning

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CSCI-UA 9473 - Introduction to Machine Learning

Midterm
Augustin Cosse
April 2022
Total: 45 points
Total time: 1h15
General instructions: The exam consists of 3 questions (each question consisting itself of 3
subquestions). In case you have any question, you can
ask those through the chat. Answer as many questions as you can starting with those you feel more
confident with.
Question 1 (Regression and regularization 15pts)
1. Indicate whether the following statements are true or false (5pts)
True / False Gradient descent finds the global minimum of the least squares loss
for linear regression
True / False Gradient descent finds the global minimum of the ridge loss
for linear regression
True / False Least squares regression can be understood as a Maximum a Posteriori (MAP)
approach with a uniform prior on the regression weights
True / False Least squares regression can be understood as a Maximum Likelihood (MLE)
approach with Gaussian prior on the deviations t(i) −
(
β0 +
D∑
p=1
βpxp
)
True / False LASSO regression can be understood as a Maximum a Posteriori (MAP)
approach with a Poisson prior on the regression weights
True / False Ridge regression can be understood as a Maximum a Posteriori (MAP)
approach with a Gaussian prior on the regression weights
True / False The Ridge minimizer corresponds to
increasing the eigenvalues of XTX by λ
2. We consider the simple dataset show in Fig 1 below. We want to learn a linear regression model on the
points shown in red. Explain how you would proceed (all steps + pseudo-code) [7pts]
3. The balls (‖β‖pp =
(∑D
k=1 |βk|p
)
≤ 1) corresponding to the LASSO (p = 1) and ridge (p = 2) formulations
are shown in Fig. 2 below. Sketch the `p balls for p < 1 and p > 2 [3pts]
Question 2 (Neural network 15pts)
1
−4 −2 0 2 4 6 8−20
−10
0
10
Figure 1: Training set for Question 1. The blue curve corresponds to the equation 0.3x3−x2 and the red points
are respectively located at (0, 2), (2,−2.6), (−2,−5.8), (−3.2,−18.8704) (4.1, 4.2663), (4.6, 8.6408)
A A
x
xˆ
x
xˆ
p = 1 p = 2
Figure 2: lp balls for p = 1 (LASSO) and p = 2 (ridge)
1. Indicate whether the following statements are true or false [5pts]
True / False Neural Networks cannot model the XOR dataset
True / False Neural networks cannot be used with the binary cross entropy loss
True / False Neural networks cannot be used in regression
True / False Increasing the number of hidden layers in a network
will increase the bias
True / False Increasing the number of hidden layers in a network
will increase the variance
2. Describe the backpropagation steps (be as exhaustive as possible, no need to provide a python implemen-
tation) [5pts]
3. We consider the dataset shown in Fig. 3. Explain how you would build a neural network for this dataset
(including number of hidden layers and activation functions) and draw the separating planes on top of the
dataset. [5pts]
Question 3 (Kernels, 15pts)
1. Explain how we can learn a classifier based on the Gaussian kernel with the least squares loss (all the steps
+ pseudo code) [5pts]
2. When is a kernel considered valid (give a formal criterion)? We consider the kernel κ(x(i),x(j)) defined
as κ(x(i),x(j)) =
(
(x(i))T (x(j)) + c
)2
where c is a positive constant. Is this a valid kernel? Motivate your
answer with a proof. [5pts]
2
−10 −5 0 5 10
−10
0
10
Figure 3: Training set for Question 2.
3. We consider the dataset shown in Fig 4. we assume that the red points have target +1 and the blue ones
have target −1. On that dataset, we want to learn a kernel classifier of the form
y(x) =
N∑
i=1
λiκ(x,x
(i))
with a Gaussian kernel, κ(x,x(i)) = exp
(
−‖x− x
(i)‖2
σ
)
. What could be appropriate values for λi, σ
(Motivate your answer, associate each lambda λi to each point x
(i) from the dataset by superimposing the
λi on their respective data points in Fig 4). Plot the resulting classifier on top of the data. [5pts]

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