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# M16 Past Exam Problems

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# Warning: please enter your ID before you start!

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# Question 15

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# Question 16

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# Question 21

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# Question 22

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# Question 24

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# Question 25

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📗 [3 points] Suppose there are three classifiers

f_{1}, f_{2}, f_{3}

to choose from (i.e. the hypothesis space has three elements), and the activation values from these classifiers based on a training set of three items are listed below. Which classifier is the best one if loss is used for comparison? (Enter a number 1 or 2 or 3).

📗 Reminder: zero-one loss means

\sum_{i = 1}^{n} 1_{{a_{i} \neq y_{i}}}

, square loss means

\sum_{i = 1}^{n} {(a_{i} - y_{i})}^{2}

, cross entropy loss means

\sum_{i = 1}^{n} - y_{i} \log (a_{i}) + (1 - y_{i}) \log (1 - a_{i})

Items	1	2	3
$y$
$f_{1}$
$f_{2}$
$f_{3}$

📗 Answer: .

📗 [3 points] Which ones of the following functions are equal to the squared error for deterministic binary classification?

C = \sum_{i = 1}^{n} {(f (x_{i}) - y_{i})}^{2}, f (x_{i}) \in {0, 1}, y_{i} \in {0, 1}

. Note:

I_{S}

is the indicator notation on

S

📗 Note: the question is asking for the functions that are identical in values.

📗 Choices:

\sum_{i = 1}^{n}

\sum_{i = 1}^{n}

\sum_{i = 1}^{n}

\sum_{i = 1}^{n}

\sum_{i = 1}^{n}

None of the above

📗 [3 points] In one step of gradient descent for a

L_{2}

regularized logistic regression, suppose

w

= ,

b

= , and

\frac{\partial C}{\partial w}

= ,

\frac{\partial C}{\partial b}

= . If the learning rate is

α

= and the regularization parameter is

λ

= , what is

w

after one iteration? Use the loss

C (w, b)

and the regularization

\frac{λ}{2} {‖ [\begin{matrix} w \\ b \end{matrix}] ‖}_{2}^{2}

\frac{λ}{2} (w^{2} + b^{2})

📗 Answer: .

📗 [2 points] Consider a single sigmoid perceptron with bias weight

w_{0}

= , a single input

x_{1}

with weight

w_{1}

= , and the sigmoid activation function

g (z) = \frac{1}{1 + \exp (- z)}

. For what input

x_{1}

does the perceptron output value

a

= .

📗 Note: Math.js does not accept "ln(...)", please use "log(...)" instead.

📗 Answer: .

📗 [2 points] Consider a single sigmoid perceptron with bias weight

w_{0}

= , a single input

x_{1}

with weight

w_{1}

= , and the sigmoid activation function

g (z) = \frac{1}{1 + \exp (- z)}

. For what input

x_{1}

does the perceptron output value

a

= .

📗 The red curve is a plot of the activation function, given the y-value of the green point, the question is asking for its x-value.

📗 Note: Math.js does not accept "ln(...)", please use "log(...)" instead.

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Last Updated: April 07, 2025 at 1:55 AM