Below is a selected list of problems & exercises that I had worked on when I took Math 156 - Machine Learning with Professor Farolfi in Fall 2025.

Problems & Exercises

Classical Machine Learning

Problem 1. Show the following matrix derivatives 1. First Order: 2. Second Order:

Problem 2. Suppose we have matrices \(\mathbf{Y} \in \mathbb{R}^{d\times n}\) and \(\mathbf{X} \in \mathbb{R}^{d\times r}\). We seek to find a matrix \(\hat{\mathbf{B}} \in \mathbb{R}^{r\times n}\) where: \[||\mathbf{Y} - \mathbf{XB}||_F^2 + \lambda||\mathbf{F}||_F^2 \tag{1}\] Here \(\lambda \geq 0\) is called the \(L_2\)-regularization parameter. (This is an instance of unconstrained quadratic optimization problem).

1. Show that

\[||\mathbf{Y} - \mathbf{XB}||_F^2 + \lambda ||\mathbf{B}||_F^2 = \text{trace}\left[(\mathbf{Y} - \mathbf{XB})^\top(\mathbf{Y} - \mathbf{XB})\right] + \lambda\,\text{trace}\left(\mathbf{B}^\top\mathbf{B}\right)\]

\[= \text{trace}(\mathbf{Y}^\top\mathbf{Y}) - 2\,\text{trace}(\mathbf{Y}^\top \mathbf{X}\mathbf{B}) + \text{trace}(\mathbf{B}^\top\mathbf{X}^\top \mathbf{X}\mathbf{B}) + \lambda\,\text{trace}(\mathbf{B}^\top\mathbf{B})\]

1. Show that
2. Argue that the quadratic function \(\mathbf{B} \longmapsto |\mathbf{Y} - \mathbf{XB}|_F^2\) is convex, and hence its every critical point is a local minimum.
3. Suppose \(\lambda=0\) and \(\mathbf{X}^\top X\) is invertible. Then, from part 2 and 3, conclude that the quadratic function \(\mathbf{B}\longmapsto|\mathbf{Y} - \mathbf{XB}|_F^2 + \lambda|\mathbf{B}|_F^2\) has a unique global minimum \(\hat{\mathbf{B}} = \left(\mathbf{X}^\top \mathbf{X}\right)^{-1} \mathbf{X}^\top \mathbf{Y}\).
4. Suppose \(\lambda > 0\). Then argue that \(\mathbf{X}^\top \mathbf{X} + \lambda \mathbf{I}\) is always invertible, and the quadratic function \(\mathbf{B}\longmapsto |\mathbf{Y} - \mathbf{XB}|_F^2 + \lambda |\mathbf{B}|_F^2\) has a unique global minimum \(\hat{\mathbf{B}} = \left(\mathbf{X}^\top\mathbf{X} + \lambda \mathbf{I}\right)^{-1}\mathbf{X}^\top \mathbf{Y}\).

Hints. \ For 4. The matrix \(\mathbf{X}^\dagger := \left(\mathbf{X}^\top\mathbf{X}\right)^{-1}\mathbf{X}^\top\) is the Moore-Penrose Pseudoinverse of \(\mathbf{X}\). If \(\mathbf{X}\) is square and invertible, then \(\mathbf{X}^\dagger = \mathbf{X}^{-1}\). So the pseudoinverse can be regarded as a generalization of matrix inverse for non-square matrices. \ For 5. Show that \(\mathbf{X}^\top\mathbf{X} + \lambda\mathbf{I}\) is positive definite if \(\lambda > 0\). Use the fact that the eigenvalues of a positive definite matrix \(\mathbf{A}\) have to be positive so \(\mathbf{Ay} \neq \mathbf{0}\) for any \(\mathbf{y}\) so \(\mathbf{A}\) is invertible.

Multiple-Choice Questions for Classical Machine Learning

Question 1. What is the goal of polynomial regression? - (A): To predict categorical labels - (B): To generate random noise - (C): To model exponential growth - (D): To fit a polynomial function to data by minimizing loss

Question 2. In all probabilistic models that were covered in class, what assumption does allow to write the likelihood function as a product of probability functions of the outputs \(y_i\)? - (A): The outputs \(y_i\) are assumed to be independent - (B): The inputs \(x_i\) are assumed to be correlated - (C): The outputs \(y_i\) are assumed to be correlated - (D): The inputs \(x_i\) are assumed to be independent

Question 3. Overfitting occurs when: - (A): The model performs well on both train and test data - (B): The model underfits the data - (C): The model fits noise in the training data - (D): The regularization parameter is too large

Question 4. Bayesian polynomial regression treats the coefficients \(\mathbf{w}\) as: - (A): Fixed parameters - (B): Random variables - (C): Deterministic constants - (D): Complex-valued functions

Question 5. Let \(\mathbf{w}\) represent the vector of parameters of a regression or classification problem. Let \(\mathbf{X}, \mathbf{Y}\) represent the vector/matrix of inputs and the vector of outputs, respectively. The loss function for \(L_2\)-regularization includes which term? - (A): \(\lambda||\mathbf{w}||_1\) - (B): \(\lambda||\mathbf{w}||_2^2\) - (C): \(\lambda||\mathbf{X}||_F\) - (D): \(\lambda||\mathbf{Y}||_2^2\)

Question 6. Which one of the following helps reducing overfitting? - (A): \(L_1\) regularization - (B): \(L_2\) regularization - (C): Increasing the size of the training data - (D): All of the above

Question 7. The mean squared error (MSE) measures: - (A): Model bias - (B): Cross-entropy loss - (C): Variance of residuals only - (D): Average squared difference between predictions and outputs

Question 8. In regression problems, the optimal decision rule is: - (A): \(\hat{y} = \arg \max_y P\left(Y = y \mid X = x\right) \) - (B): \(\hat{y} = \mathbb{E}\left[Y \mid X = x\right]\) - (C): \(\hat{y} = \arg \min L\left(Y, \hat{y}\right)\) - (D): \(\hat{y} = \text{Var}\left[Y \mid X = x\right]\)

Question 9. A decision algorithm maps - (A): Input features to loss functions - (B): Output labels to probabilities - (C): Inputs to predicted outputs - (D): Random variables to parameters

Question 10. The logistic regression model assumes that: - (A): \( Y \mid X \sim \text{Normal}\) - (B): \( Y \mid X \sim \text{Uniform}\) - (C): \( Y \mid X \sim \text{Bernoulli}\) - (D): \( Y \mid X \sim \text{Exponential}\)

Question 11. In classification, the optimal decision rule is: - (A): \(\hat{y} = \mathbb{E}\left[ Y \mid X = x\right]\) - (B): \(\hat{y} = \arg \max_y P\left(Y = y \mid X = x\right)\) - (C): \(\hat{y} = \min_y L\left(Y, \hat{y}\right)\) - (D): \(\hat{y} = P\left(X \mid Y\right)\)

Question 12. Gradient descent updates parameters using which of the following update rules: - (A): \(w_{t+1} \leftarrow w_t - \eta_t\nabla L\left(w_t\right) \) - (B): \(w_{t+1} \leftarrow w_t + \eta_t\nabla L\left(w_t\right) \) - (C): \(w_{t+1} \leftarrow w_t - X^\top Y\) - (D): \(w_{t+1} \leftarrow w_t - H^{-1} \nabla L\left(w_t\right)\)

Question 13. Which one of the following is true about total expected loss minimization and conditional expected loss minimization: - (A): Conditional expected loss minimization equals total expected loss minimization - (B): There exists no relationship between total expected loss minimization and conditional expected loss minimization - (C): The difference between the two is that total expected loss minimization is stochastic - (D): Total expected loss minimization is more efficient than conditional expected loss minimization

Question 14. Regularization adds a term to the loss function with the purpose of: - (A): Adding random noise to the loss function - (B): Penalizing complexity of the model parameters - (C): Correcting the gradient to make it computationally stable - (D): Adding a bias term to the loss function

Question 15. Logistic regression coefficients can be: - (A): Only non-negative real numbers - (B): Only negative real numbers - (C): Both positive and negative real numbers - (D): Only real numbers in the interval \(\left[0, 1\right]\)

Question 16. Logistic regression has no closed form solution because: - (A): The loss function is not differentiable - (B): The sigmoid function is discontinuous - (C): The log-likelihood function is not quadratic - (D): The optimization problem is non-convex

Question 17. The confusion matrix is used to: - (A): Evaluate classification accuracy - (B): Visualize loss convergence - (C): Compute MSE - (D): Normalize features

Question 18. The “naïve” assumption in Naïve Bayes is that: - (A): Classes are independent - (B): Features are conditionally independent on a given class - (C): Classes are not uniformly distributed - (D): Features are dependent

Question 19. The Shannon entropy of a deterministic probability mass function with only one out come having probability 1 is: - (A): 1 - (B): 0 - (C): Infinite - (D): Undefined

Question 20. Which of the following would prevent the use of the Newton-Raphson algorithm: - (A): The Hessian \(H\) is not invertible - (B): The loss function is convex in the predicted output \(\hat{y}\) - (C): The model outputs \(y\) are assumed to be real numbers - (D): All of the above

Neural Networks & Deep Learning

Multiple-Choice Questions for Neural Networks & Deep Learning