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IIT M FOUNDATION DIPLOMA FN EXAM - 22 Dec 2024

MLT End Term22 questions50 marks12 SA · 7 MCQ · 3 MSQFoundation / Diploma / forenoon

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Q1SA2 marks

NOTE: Enter the answer correct to one decimal place question figure

Your numeric answer:

Q2SA3 marks

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Q3SA3 marks

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Q4SA3 marks

Consider a logistic regression model trained to detect spam emails. Emails containing harmful phishing links are labeled as spam (1), while regular emails are labeled as not spam (0). A good spam detector should correctly identify almost all emails with phishing links. Even a single phishing email incorrectly classified as not spam could expose users to significant risks. However, the detector may classify some regular emails as spam. This trade-off is acceptable to ensure safety. To ensure user safety, if the threshold is set to a low value to prioritize the detection of phishing emails, type 1. If the threshold is set to a high value to minimize false positives (misclassifying regular emails as spam), type 0.

Your numeric answer:

Q5SA5 marks

Your numeric answer:

Q6MCQ3 marks

3, 5, 1, 4, 2
5, 3, 1, 2, 4
3, 5, 2, 4, 1
5, 3, 1, 4, 2
None of these

Q7MCQ3 marks

Q8MCQ4 marks

Q9MCQ4 marks

Q10MCQ2 marks

If we remove all the non-support vectors from the dataset, what will be the impact on the model using SVM algorithm?

Model will overfit.
The model will not be changed.
Model will underfit.
Accuracy of the model will increase.

Q11MCQ2 marks

In each round of AdaBoost, the weight for a particular training observation is decreased from round t to round t + 1 if the observation was:

classified incorrectly by the weak learner trained in the round.
classified correctly by the weak learner trained in the round.
classified incorrectly by a majority of the weak learners trained up to theround.
classified correctly by a majority of the weak learners trained up to the round.

Q12MCQ1 mark

Is the following statement true or false? For a fixed size of the training and the test set, increasing the complexity of the model always leads to an increment in the test error.

TRUE
FALSE

Q13MSQ3 marksselect all that apply

Consider a supervised ML problem where a model is trained using a training dataset, and its performance is evaluated on both the training and validation datasets. During training, the model’s loss is plotted over time for both the datasets. Assume that the loss L(t) is decreasing as the model learns. The following graph shows two curves, Curve (A) and Curve (B), representing the loss for the training and validation datasets, respectively: question figure

Curve (A) represents the training loss.
Curve (B) represents the training loss.
Curve (A) represents the validation loss.
Curve (B) represents the validation loss.

Q14MSQ3 marksselect all that apply

Which of the following options are correct? Select all that apply.

Underfitted models generally have less bias and high variance.
A decision tree with the maximum possible length may lead to overfitting.
Weak learners are those whose performance is slightly better than therandom classifier.
All the estimators in bagging can be trained parallelly.

Comprehension passage - the questions below it refer to this

Q15SA2 marks

Your numeric answer:

Q16SA2 marks

Your numeric answer:

Q17SA2 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Q18SA0.5 marks

Your numeric answer:

Q19SA0.5 marks

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Q20SA0.5 marks

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Q21SA0.5 marks

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Q22MSQ1 markselect all that apply

Which of the following is correct ?

Answer key

Q1 (SA, 2 marks): 2.5
Q2 (SA, 3 marks): 0
Q3 (SA, 3 marks): 55
Q4 (SA, 3 marks): 1
Q5 (SA, 5 marks): 2
Q6 (MCQ, 3 marks): D
Q7 (MCQ, 3 marks): B
Q8 (MCQ, 4 marks): D
Q9 (MCQ, 4 marks): A
Q10 (MCQ, 2 marks): B
Q11 (MCQ, 2 marks): B
Q12 (MCQ, 1 marks): B
Q13 (MSQ, 3 marks): A, D
Q14 (MSQ, 3 marks): B, C, D
Q15 (SA, 2 marks): 1
Q16 (SA, 2 marks): 2
Q17 (SA, 2 marks): 1
Q18 (SA, 0.5 marks): 1
Q19 (SA, 0.5 marks): 0
Q20 (SA, 0.5 marks): 0
Q21 (SA, 0.5 marks): 2
Q22 (MSQ, 1 marks): A, B

Concepts this paper tests

Auto-generated summary from the source site. Handy as a checklist; the lessons above explain each idea from scratch.

Here’s a structured summary of core topics, concepts, principles, and equations essential for the exam, followed by Mermaid knowledge graphs to visualize relationships.

Core Topics & Key Concepts

1. Principal Component Analysis (PCA)

Purpose: Dimensionality reduction by projecting data onto orthogonal axes (principal components) that maximize variance.
Steps:
1. Center the data (subtract mean).
2. Compute covariance matrix: $\Sigma = \frac{1}{n} X^T X$ .
3. Eigenvalue decomposition: $\Sigma = U \Lambda U^T$ , where $\Lambda$ contains eigenvalues, $U$ contains eigenvectors.
4. Sort eigenvectors by descending eigenvalues; top $k$ eigenvectors form the projection matrix.
5. Project data: $X_{\text{new}} = X U_k$ .
Variance along PC1: For a dataset aligned along a single direction (e.g., scalar multiples of a vector), the variance along the first PC is the largest eigenvalue of the covariance matrix.
- Equation: If data points are $\{ \alpha_i \mathbf{u} \}$ , the covariance matrix is $\sigma^2 \mathbf{u} \mathbf{u}^T$ , and the top eigenvalue is $\sigma^2 \sum \alpha_i^2$ .
- Example (Q2): For $\{ -2\mathbf{u}, -\mathbf{u}, \mathbf{u}, 2\mathbf{u} \}$ , variance along PC1 = 2.5.

Reconstruction Error: If you retain only the top $k$ PCs, the error is the sum of the discarded eigenvalues.
- Equation: $\text{Error} = \sum_{i=k+1}^d \lambda_i$ .
- Example (Q3): If the data lies in a 2D subspace, retaining 2 PCs gives 0 error.

2. Maximum A Posteriori (MAP) Estimation

Purpose: Estimate parameters by maximizing the posterior distribution.
Equation: $\hat{\theta}_{\text{MAP}} = \arg \max_\theta \, p(\theta | \mathbf{X}) = \arg \max_\theta \, p(\mathbf{X} | \theta) p(\theta)$ .
Uniform Prior: For $X_i \sim \text{Uniform}[0, \theta]$ and prior $\theta \sim \text{Uniform}[0, 60]$ , the posterior is:

p(\theta | \mathbf{X}) \propto \begin{cases} \theta^{-n} & \text{if } \theta \geq \max(X_i), \\ 0 & \text{otherwise}. \end{cases}

MAP Estimate: $\hat{\theta}_{\text{MAP}} = \max(X_i)$ .
Example (Q4): For data $\{10, 55, \ldots\}$ , MAP estimate = 55.

3. Logistic Regression & Class Imbalance

Threshold Adjustment: Lowering the decision threshold (e.g., from 0.5 to 0.3) increases recall (reduces false negatives) but may increase false positives.
- Example (Q5): For spam detection, prioritize recall (detect all phishing emails) → threshold = 1 (low).

4. Leave-One-Out Cross-Validation (LOOCV)

Purpose: Estimate model generalization error by leaving out one data point at a time.
Error Formula:

\text{Error}_{\text{LOOCV}} = \frac{1}{n} \sum_{i=1}^n (y_i - \hat{f}^{-i}(x_i))^2,

where $\hat{f}^{-i}$ is the model trained without the $i$ -th point.

Example (Q6): For $D = \{(1,2), (2,2), (0,1)\}$ and model $\hat{f} = \beta x$ , LOOCV error = 2.

5. K-Means Clustering

Algorithm Steps:
1. Initialize: Randomly assign cluster centers.
2. Assign: Assign each point to the nearest center.
3. Update: Recompute centers as the mean of assigned points.
4. Repeat: Steps 2–3 until convergence.
Correct Sequence (Q7): 3 (initialize) → 1 (assign) → 4 (update) → 2 (reassign).

6. Lagrangian & Constrained Optimization

Lagrangian Function: For a problem $\min f(x)$ s.t. $g(x) \geq 0$ , the Lagrangian is:

L(x, \lambda) = f(x) - \lambda g(x).

Dual Problem: $\max_{\lambda \geq 0} L(x', \lambda)$ .
- If $x'$ violates constraints, the dual tends to $+\infty$ .
- Example (Q8): For $x' = [1, 4]$ and constraint $x_1 - x_2 \geq 3$ , $\max L = +\infty$ . But the correct answer is 89 (likely a miscalculation in the problem).

7. Gradient Descent for Linear Regression

Objective: Minimize $f(w) = \|X^T w - y\|^2$ .
Update Rule:

w_{t+1} = w_t - \eta \nabla f(w_t),

where $\nabla f(w) = 2 X (X^T w - y)$ .

Example (Q9):
- Given $X = \begin{bmatrix} 1 & -2 & 0 \\ 2 & 5 & 1 \end{bmatrix}$ , $y = [1, 6, 1]^T$ , $w_0 = [1, 0.5]^T$ , $\eta = 0.05$ .
- After one iteration: $w_1 = [-0.2, 3.1]^T$ .

8. Perceptron Algorithm

Update Rule: For misclassified point $(\mathbf{x}_i, y_i)$ :

\mathbf{w}^{t+1} = \mathbf{w}^t + \eta y_i \mathbf{x}_i.

Example (Q10):
- $\mathbf{w}^t = [0, 1, 2]^T$ , $(\mathbf{x}_1, y_1) = ([1, -1, 0]^T, 1)$ .
- $\mathbf{w}^{t+1} = [1, 0, 2]^T$ .
- Next point $(\mathbf{x}_2, y_2) = ([-1, 1, 1]^T, -1)$ .
- $\mathbf{w}^{t+2} = [2, -1, 1]^T$ .

9. Support Vector Machines (SVM)

Key Concepts:
- Support Vectors: Points lying on the margin boundaries ( $w^T x = \pm 1$ ).
- Slack Variables ( $\xi_i$ ): Measure violation of margin constraints.
- Soft-Margin SVM: Minimize $\|w\|^2 + C \sum \xi_i$ .
Impact of Removing Non-Support Vectors (Q11): No change to the model (only support vectors define the decision boundary).
Slack Variables (Q19–Q23):
- $\xi_i = 0$ if point is correctly classified and outside the margin.
- $\xi_i =$ distance from the margin if misclassified or inside the margin.
- Examples:
  - $\xi_1 = 0$ (Q20), $\xi_2 = 0$ (Q21), $\xi_6 = 2$ (Q22).

10. AdaBoost

Weight Update: Increase weights for misclassified points, decrease for correctly classified.
- Example (Q12): Weights decrease if classified correctly in round $t$ .

11. Bias-Variance Tradeoff & Model Complexity

Underfitting: High bias, low variance.
Overfitting: Low bias, high variance.
Key Points (Q13, Q15):
- Increasing model complexity does not always increase test error (it may decrease initially).
- Decision Trees: Maximum depth can lead to overfitting (B).
- Weak Learners: Slightly better than random (C).
- Bagging: Estimators can be trained in parallel (D).

12. Naive Bayes

Assumption: Features are conditionally independent given the class.
Parameter Estimation: For binary features $x_i \in \{0,1\}$ , estimate $P(x_i = 1 | y)$ .
- Example (Q16–Q18):
  - Dataset: 4 points with $y=1$ , 4 with $y=0$ .
  - $a + b = 1$ (Q16), $b + d + f = 2$ (Q17).
  - Predicted label for $[0, ?, ?]^T$ : 1 (Q18, likely due to higher $P(y=1)$ ).

13. Loss Curves (Training vs. Validation)

Training Loss: Decreases monotonically.
Validation Loss: May increase if overfitting.
Example (Q14): Curve A (steep decrease) = training loss; Curve B (levels off) = validation loss.

IIT M FOUNDATION DIPLOMA FN EXAM - 22 Dec 2024

Answer key

Concepts this paper tests

Core Topics & Key Concepts

1. Principal Component Analysis (PCA)

2. Maximum A Posteriori (MAP) Estimation

3. Logistic Regression & Class Imbalance

4. Leave-One-Out Cross-Validation (LOOCV)

5. K-Means Clustering

6. Lagrangian & Constrained Optimization

7. Gradient Descent for Linear Regression

8. Perceptron Algorithm

9. Support Vector Machines (SVM)

10. AdaBoost

11. Bias-Variance Tradeoff & Model Complexity

12. Naive Bayes

13. Loss Curves (Training vs. Validation)

2. MAP Estimation & Bayesian Learning

3. Logistic Regression & Class Imbalance

4. SVM & Slack Variables

5. Ensemble Methods (AdaBoost, Bagging)

6. Bias-Variance Tradeoff

7. K-Means Algorithm

8. Gradient Descent