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IIT M FOUNDATION DIPLOMA AN EXAM - 01 Sep 2024

MLT End Term30 questions50 marks14 MCQ · 14 SA · 2 MSQFoundation / Diploma / afternoon

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Q1MCQ2 marks

Q2MCQ3 marks

Q3MCQ3 marks

Q4MCQ3 marks

Match the characteristics given below with the appropriate ensemble technique. 1. Decision stumps 2. Deep decision trees 3. Parallel execution 4. Sequential execution

Bagging → (2), (3); Boosting → (1), (4)
Bagging → (1), (4); Boosting → (2), (3)
Bagging → (1), (3); Boosting → (2), (4)
Bagging → (2), (4); Boosting → (1), (3)

Q5MCQ3 marks

Q6MCQ3 marks

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Q8MCQ3 marks

0.75
0.25
0.5
0.15

Q9MSQ3 marksselect all that apply

Q10MSQ3 marksselect all that apply

The dataset is linearly separable with a positive margin.
The perceptron algorithm will terminate after a finite number of iterationswhen trained on this dataset.
The dataset is linearly separable, but the margin may be zero.
The dataset is not linearly separable.

Q11SA3 marks

Your numeric answer:

Q12SA3 marks

Your numeric answer:

Q13SA3 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Q14MCQ1 mark

The weight vector output by the perceptron algorithm can be expressed as a linear combination of the data-points where the coefficients of the linear combination are integers.

TRUE
FALSE

Q15MCQ1 mark

TRUE
FALSE

Q16MCQ1 mark

If a soft-margin SVM is trained on this dataset, the optimal weight vector it returns will be the same as the one returned by a hard-margin SVM, irrespective of the value of the hyperparameter C.

TRUE
FALSE

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q17SA0.5 marks

Your numeric answer:

Q18SA0.5 marks

Your numeric answer:

Q19SA0.5 marks

Your numeric answer:

Q20SA0.5 marks

Your numeric answer:

Q21SA0.5 marks

Your numeric answer:

Q22SA0.5 marks

Your numeric answer:

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q23SA0.5 marks

Enter the number corresponding to the logistic loss.

Your numeric answer:

Q24SA0.5 marks

Enter the number corresponding to the (SVM) hinge loss.

Your numeric answer:

Q25SA0.5 marks

Enter the number corresponding to the perceptron loss.

Your numeric answer:

Q26SA0.5 marks

Enter the number corresponding to the squared loss.

Your numeric answer:

Q27MCQ1 mark

Comprehension passage - the questions below it refer to this

Based on the above data, answer the given subquestions. question figure

Q28SA1 mark

How many learnable parameters does this network have? Ignore the biases in the computation.

Your numeric answer:

Q29MCQ1 mark

What is the most appropriate choice of activation function for the output layer if the binary cross-entropy loss is used?

Sigmoid
Linear
ReLU

Q30MCQ1 mark

ReLU
Sigmoid

Answer key

Q1 (MCQ, 2 marks): A
Q2 (MCQ, 3 marks): C
Q3 (MCQ, 3 marks): A
Q4 (MCQ, 3 marks): A
Q5 (MCQ, 3 marks): B
Q6 (MCQ, 3 marks): A
Q7 (MCQ, 3 marks): A
Q8 (MCQ, 3 marks): A
Q9 (MSQ, 3 marks): A, B
Q10 (MSQ, 3 marks): A, B
Q11 (SA, 3 marks): 10
Q12 (SA, 3 marks): 0.75
Q13 (SA, 3 marks): 0.50 to 0.60
Q14 (MCQ, 1 marks): A
Q15 (MCQ, 1 marks): B
Q16 (MCQ, 1 marks): B
Q17 (SA, 0.5 marks): 0
Q18 (SA, 0.5 marks): 1
Q19 (SA, 0.5 marks): 2
Q20 (SA, 0.5 marks): 0
Q21 (SA, 0.5 marks): 5
Q22 (SA, 0.5 marks): -1
Q23 (SA, 0.5 marks): 3
Q24 (SA, 0.5 marks): 2
Q25 (SA, 0.5 marks): 4
Q26 (SA, 0.5 marks): 1
Q27 (MCQ, 1 marks): A
Q28 (SA, 1 marks): 205
Q29 (MCQ, 1 marks): A
Q30 (MCQ, 1 marks): A

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.

1. Core Topics & Concepts

A. Linear Regression & Gradient Descent

Sum of Squared Errors (SSE):

\text{SSE} = \sum_{i=1}^n (y_i - \hat{y}_i)^2 = \sum_{i=1}^n (y_i - w^T x_i)^2

Gradient of SSE w.r.t. $w$ (for a single data point):

\nabla_w \text{SSE} = -2x(y - w^T x)

(Simplified for single point: $\nabla_w = x(y - \hat{y})$ )

Key Insight: The gradient is linear in $x$ and the error term.

B. Principal Component Analysis (PCA)

Covariance Matrix $C$ :

C = \frac{1}{n} X^T X \quad (\text{for centered data})

Variance along Principal Component $w$ :

\text{Variance} = w^T C w

Eigenvalue Decomposition:

C = V \Lambda V^T

(Eigenvectors $V$ are principal components; eigenvalues $\Lambda$ are variances.)

C. Regularization (Ridge/Lasso)

Ridge Regression (L2):

\text{Loss} = \text{SSE} + \lambda \|w\|_2^2

Effect of $\lambda$ :
- Small $\lambda$ (e.g., $10^{-4}$ ) → Overfitting.
- Large $\lambda$ (e.g., $10^2$ ) → Underfitting.
- Lasso (L1): Encourages sparsity (feature selection).

D. Support Vector Machines (SVM)

Hard-Margin SVM:

\min_w \frac{1}{2} \|w\|^2 \quad \text{s.t.} \quad y_i(w^T x_i) \geq 1

Margin: Distance between supporting hyperplanes $\frac{2}{\|w\|}$ .
Decision Boundary: $w^T x = 0$ .
Supporting Hyperplanes: $w^T x = \pm 1$ .
- Soft-Margin SVM:

\min_w \frac{1}{2} \|w\|^2 + C \sum \xi_i \quad \text{s.t.} \quad y_i(w^T x_i) \geq 1 - \xi_i, \xi_i \geq 0

Slack Variables $\xi_i$ : Measure misclassification.
Lagrange Multipliers $\alpha_i$ :
- $\alpha_i > 0$ → Support vector.
- $\alpha_i = C$ → Margin violator.
- Kernel Trick:
Valid kernels: Linear ( $x^T y$ ), Polynomial ( $(x^T y)^d$ ), Gaussian ( $\exp(-\|x-y\|^2)$ ), Sigmoid.

E. Naive Bayes

Class Conditional Independence:

p(x_1, x_2, \dots, x_d | y) = \prod_{i=1}^d p(x_i | y)

Posterior Probability:

p(y | x) \propto p(y) \prod_{i=1}^d p(x_i | y)

F. Decision Trees & Ensemble Methods

Decision Stumps: Trees with 1 split (e.g., $x < 0$ ).
Information Gain:

\text{Gain} = H(S) - \sum \frac{|S_v|}{|S|} H(S_v)

( $H(S)$ : Entropy of parent node; $H(S_v)$ : Entropy of child nodes.)

Bagging vs. Boosting: | Bagging (e.g., Random Forest) | Boosting (e.g., AdaBoost) | |-----------------------------------|--------------------------------| | Parallel execution | Sequential execution | | Deep trees | Decision stumps | | Reduces variance | Reduces bias |

G. Clustering (k-Means)

Objective:

\min_{\mu, C} \sum_{i=1}^n \|x_i - \mu_{c_i}\|^2

( $\mu$ : Cluster centroids; $c_i$ : Cluster assignment.)

Cluster Boundary: Perpendicular bisector between centroids (e.g., $x_1 + x_2 = 0$ ).

H. Logistic Regression

Sigmoid Function:

\sigma(z) = \frac{1}{1 + e^{-z}}, \quad z = w^T x

Probability Interpretation:

p(y=1 | x) = \sigma(w^T x), \quad p(y=0 | x) = 1 - \sigma(w^T x)

Symmetry: $p(y=1 | -x) = 1 - p(y=1 | x)$ .

I. Neural Networks

Learnable Parameters (ignoring biases):

\text{Params} = \sum_{l=1}^{L-1} n_l n_{l+1}

( $n_l$ : Neurons in layer $l$ .)

Activation Functions:
- Output Layer (Binary Classification): Sigmoid (for cross-entropy loss).
- Hidden Layers: ReLU ( $\max(0, x)$ ), Sigmoid, Tanh.
Loss Functions:
- Logistic Loss: $\log(1 + e^{-z})$ .
- Hinge Loss (SVM): $\max(0, 1 - z)$ .
- Perceptron Loss: $\max(0, -z)$ .
- Squared Loss: $(y - \hat{y})^2$ .

J. Bayesian Learning

Beta-Binomial Conjugacy:
- Prior: $p \sim \text{Beta}(\alpha, \beta)$ .
- Posterior: $p | \text{data} \sim \text{Beta}(\alpha + \text{successes}, \beta + \text{failures})$ .
- Point Estimate: $\hat{p} = \frac{\alpha + \text{successes}}{\alpha + \beta + n}$ .

Graph 2: Unsupervised Learning & Bayesian Methods

Graph 3: Neural Networks

Graph 4: SVM & Kernels

Graph 5: Ensemble Methods

3. Key Equations Summary

Topic	Equation
Linear Regression	$\nabla_w \text{SSE} = x(y - w^T x)$
PCA Variance	$\text{Variance} = w^T C w$
Ridge Regression	$\text{Loss} = \text{SSE} + \lambda \\|w\\|_2^2$
SVM Margin	$\text{Margin} = \frac{2}{\\|w\\|}$
Soft-Margin SVM	$\min_w \frac{1}{2} \\|w\\|^2 + C \sum \xi_i$
Naive Bayes	$p(x\|y) = \prod_i p(x_i\|y)$
Information Gain	$\text{Gain} = H(S) - \sum \frac{\|S_v\|}{\|S\|} H(S_v)$
Logistic Regression	$p(y=1\|x) = \sigma(w^T x)$
Neural Networks	$\text{Params} = \sum_{l=1}^{L-1} n_l n_{l+1}$
Beta Posterior	$p \| \text{data} \sim \text{Beta}(\alpha + \text{successes}, \beta + \text{failures})$

4. Exam Tips

SVM:
- Hard-margin: No slack variables ( $\xi_i = 0$ ).
- Soft-margin: $\xi_i > 0$ for misclassified points.
- $\alpha_i > 0$ → Support vector; $\alpha_i = C$ → Margin violator.
PCA: Eigenvectors of $C$ are principal components; eigenvalues are variances.
Regularization:
- $\lambda \uparrow$ → Underfitting (high bias).
- $\lambda \downarrow$ → Overfitting (high variance).
Decision Trees: Information gain measures purity (entropy reduction).
Neural Networks: ReLU is common in hidden layers; sigmoid for binary output.
Bayesian Learning: Beta prior + Binomial likelihood → Beta posterior.