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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 \(\displaystyle\sum_{i=1}^{n} 1_{\left\{a_{i} \neq y_{i}\right\}}\), square loss means \(\displaystyle\sum_{i=1}^{n} \left(a_{i} - y_{i}\right)^{2}\), cross entropy loss means \(\displaystyle\sum_{i=1}^{n} -y_{i} \log\left(a_{i}\right) + \left(1 - y_{i}\right) \log\left(1 - a_{i}\right)\).
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? \(C = \displaystyle\sum_{i=1}^{n} \left(f\left(x_{i}\right) - y_{i}\right)^{2}, f\left(x_{i}\right) \in \left\{0, 1\right\}, y_{i} \in \left\{0, 1\right\}\). Note: \(I_{S}\) is the indicator notation on \(S\).
📗 Note: the question is asking for the functions that are identical in values.
📗 Choices:
\(\displaystyle\sum_{i=1}^{n}\)
\(\displaystyle\sum_{i=1}^{n}\)
\(\displaystyle\sum_{i=1}^{n}\)
\(\displaystyle\sum_{i=1}^{n}\)
\(\displaystyle\sum_{i=1}^{n}\)
None of the above
📗 [3 points] In one step of gradient descent for a \(L_{2}\) regularized logistic regression, suppose \(w\) = , \(b\) = , and \(\dfrac{\partial C}{\partial w}\) = , \(\dfrac{\partial C}{\partial b}\) = . If the learning rate is \(\alpha\) = and the regularization parameter is \(\lambda\) = , what is \(w\) after one iteration? Use the loss \(C\left(w, b\right)\) and the regularization \(\dfrac{\lambda}{2} \left\|\begin{bmatrix} w \\ b \end{bmatrix}\right\|_{2}^{2}\) = \(\dfrac{\lambda}{2} \left(w^{2} + b^{2}\right)\).
📗 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 \(g\left(z\right) = \dfrac{1}{1 + \exp\left(-z\right)}\). 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 \(g\left(z\right) = \dfrac{1}{1 + \exp\left(-z\right)}\). 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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📗 You can find videos going through the questions on Link.