MMiDS 3.6: Self-Assessment Quiz

What is the primary goal of logistic regression?

• a) To predict a continuous outcome variable.
• b) To classify data points into multiple categories.
• c) To model the probability of a binary outcome.
• d) To find the optimal linear combination of features.

Which loss function is commonly used in logistic regression?

• a) Mean squared error
• b) Mean absolute error
• c) Hinge loss
• d) Cross-entropy loss

What is the relationship between maximizing the likelihood function and minimizing the cross-entropy loss in logistic regression?

• a) They are unrelated concepts.
• b) Maximizing the likelihood is equivalent to minimizing the cross-entropy loss.
• c) Minimizing the cross-entropy loss is a first step towards maximizing the likelihood.
• d) Maximizing the likelihood is a special case of minimizing the cross-entropy loss.

Which of the following is the correct expression for the sigmoid function, denoted by \( \sigma(z) \) in the text?

• a) \( \sigma(z) = \frac{1}{1 + e^z} \)
• b) \( \sigma(z) = \frac{e^z}{1 + e^z} \)
• c) \( \sigma(z) = \frac{1}{1 - e^{-z}} \)
• d) \( \sigma(z) = \frac{1}{1 + e^{-z}} \)

Which of the following is the correct formula for the logistic regression objective function (cross-entropy loss)?

• a) \( \ell(x; A, b) = \frac{1}{n} \sum_{i=1}^n \log(1 + \exp(-b_i \alpha_i^T x)) \)
• b) \( \ell(x; A, b) = \frac{1}{n} \sum_{i=1}^n (b_i - \sigma(\alpha_i^T x))^2 \)
• c) \( \ell(x; A, b) = \frac{1}{n} \sum_{i=1}^n \{-b_i \log(\sigma(\alpha_i^T x)) - (1 - b_i) \log(1 - \sigma(\alpha_i^T x))\} \)
• d) \( \ell(x; A, b) = \frac{1}{n} \sum_{i=1}^n b_i \alpha_i^T x \)

Which of the following is the correct gradient of the logistic regression objective function?

• a) \( \nabla \ell(x; A, b) = -\frac{1}{n} \sum_{i=1}^n (b_i - \sigma(\alpha_i^T x)) \alpha_i \)
• b) \( \nabla \ell(x; A, b) = -\frac{1}{n} \sum_{i=1}^n b_i \alpha_i \)
• c) \( \nabla \ell(x; A, b) = -\frac{1}{n} \sum_{i=1}^n \sigma(\alpha_i^T x) \alpha_i \)
• d) \( \nabla \ell(x; A, b) = -\frac{1}{n} \sum_{i=1}^n (1 - b_i) \alpha_i \)

What is the purpose of standardizing the input features in the airline customer satisfaction dataset example?

• a) To ensure the objective function is convex.
• b) To speed up the convergence of gradient descent.
• c) To allow comparison of the influence of different features on the prediction.
• d) To handle missing data in the dataset.

Which of the following is true about the convergence of gradient descent for logistic regression?

• a) Gradient descent is guaranteed to converge to the global minimum of the objective function.
• b) Gradient descent may converge to a local minimum of the objective function.
• c) Gradient descent may not converge at all.
• d) The convergence of gradient descent depends on the choice of the initial point.

Which of the following is a valid reason for adding a column of 1's to the feature matrix in logistic regression?

• a) To ensure the objective function is convex.
• b) To allow for an intercept term in the model.
• c) To standardize the input features.
• d) To handle missing data in the dataset.