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IIT M FOUNDATION DIPLOMA FN EXAM - 01 Sep 2024

MLT End Term30 questions50 marks14 SA · 13 MCQ · 3 MSQFoundation / Diploma / forenoon

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Q1MCQ3 marks

Q2MCQ3 marks

Q3MCQ3 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)

Q4MCQ3 marks

The predicted labels for x1 and x2 are 1 and −1 respectively.
The predicted labels for x1 and x2 are −1 and 1 respectively.
The predicted label for both data-points is 1.
The predicted label for both data-points is −1.

Q5MCQ3 marks

Q6MCQ3 marks

2x1 + x2 = 0
x1 − 2x2 = 0
x1 + x2 = 0
x1 + 2x2 = 0

Q7SA3 marks

Your numeric answer:

Q8SA3 marks

Your numeric answer:

Q9SA3 marks

Your numeric answer:

Q10MSQ3 marksselect all that apply

Q11MSQ3 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.

Q12MSQ3 marksselect all that apply

Comprehension passage - the questions below it refer to this

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

Q13MCQ1 mark

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

TRUE
FALSE

Q14MCQ1 mark

TRUE
FALSE

Q15MCQ1 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

Q16SA0.5 marks

Your numeric answer:

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:

Comprehension passage - the questions below it refer to this

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

Q22SA0.5 marks

Your numeric answer:

Q23SA0.5 marks

Your numeric answer:

Q24SA0.5 marks

Your numeric answer:

Q25SA0.5 marks

Enter the number corresponding to the squared loss._______________

Your numeric answer:

Q26MCQ1 mark

Comprehension passage - the questions below it refer to this

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

Q27SA1 mark

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

Your numeric answer:

Q28MCQ1 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

Q29MCQ1 mark

ReLU
Sigmoid

Q30MCQ2 marks

Answer key

Q1 (MCQ, 3 marks): A
Q2 (MCQ, 3 marks): A
Q3 (MCQ, 3 marks): A
Q4 (MCQ, 3 marks): A
Q5 (MCQ, 3 marks): A
Q6 (MCQ, 3 marks): A
Q7 (SA, 3 marks): 0.4
Q8 (SA, 3 marks): 30
Q9 (SA, 3 marks): 0.26 to 0.37
Q10 (MSQ, 3 marks): A, B
Q11 (MSQ, 3 marks): A, B
Q12 (MSQ, 3 marks): A, B
Q13 (MCQ, 1 marks): A
Q14 (MCQ, 1 marks): B
Q15 (MCQ, 1 marks): B
Q16 (SA, 0.5 marks): 2
Q17 (SA, 0.5 marks): 1
Q18 (SA, 0.5 marks): 0
Q19 (SA, 0.5 marks): 3
Q20 (SA, 0.5 marks): 0
Q21 (SA, 0.5 marks): -1
Q22 (SA, 0.5 marks): 3
Q23 (SA, 0.5 marks): 2
Q24 (SA, 0.5 marks): 4
Q25 (SA, 0.5 marks): 1
Q26 (MCQ, 1 marks): A
Q27 (SA, 1 marks): 160
Q28 (MCQ, 1 marks): A
Q29 (MCQ, 1 marks): A
Q30 (MCQ, 2 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 Diploma Level: Machine Learning Techniques exam, along with Mermaid knowledge graphs to visualize relationships.

1. Core Topics & Key Concepts

A. Dimensionality Reduction (PCA)

Key Idea: Reduce dimensionality while preserving variance.
Eigenvalues of Covariance Matrix:
- Represent variance along principal components (PCs).
- Total Variance = Sum of all eigenvalues.
- Variance Captured by Top-k PCs = (Sum of top-k eigenvalues) / (Total variance) × 100%.
Equation:

\text{Variance \%} = \frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^d \lambda_i} \times 100\%

(Example: For eigenvalues [15, 5, 5, 0, 0], top-2 PCs capture (15+5)/25 = 80% variance.)

B. Regularization (Ridge Regression)

Key Idea: Penalize large weights to prevent overfitting.
Loss Function:

L(\beta) = \text{SSE} + \lambda \|\beta\|_2^2 \quad (\text{SSE} = \sum (y_i - \hat{y}_i)^2)

Plot Interpretation:
- Training Loss: Decreases with model complexity (lower λ).
- Test Loss: U-shaped curve; optimal λ balances bias-variance tradeoff.
- Best f(λ): Typically log(λ) or 1/λ to visualize tradeoff clearly.

C. Ensemble Methods

Technique	Key Characteristics	Use Case
Bagging	- Uses deep trees (high variance).	Reduces variance (e.g., Random Forest).
	- Parallel execution (independent models).
Boosting	- Uses decision stumps (weak learners).	Reduces bias (e.g., AdaBoost, Gradient Boosting).
	- Sequential execution (corrects errors iteratively).

D. Decision Trees & Information Gain

Entropy (measure of impurity):

H(S) = -\sum p_i \log_2 p_i

Information Gain:

\text{IG} = H(\text{parent}) - \sum \left( \frac{|S_v|}{|S|} H(S_v) \right)

(Example: Parent entropy = 0.811, children entropy = 0.5 → IG = 0.311.)

E. Clustering (k-Means)

Cluster Boundary: Perpendicular bisector between centroids.
- For centroids at $(a, b)$ and $(c, d)$ , boundary is:

2(c - a)x + 2(d - b)y = c^2 + d^2 - a^2 - b^2

(Example: Centroids at (1,0) and (-1,0) → boundary is $x = 0$ .)

F. Kernels (Polynomial Kernel)

Polynomial Kernel:

K(x, z) = (x^T z + c)^p

Non-Negative Output: Only possible for even-degree (p = 2, 4, ...) due to $(x^T z)^p \geq 0$ .

G. Linear Separability & Perceptron

Conditions for Linear Separability:
- Dataset is separable if ∃ a hyperplane $w^T x + b = 0$ with $y_i(w^T x_i + b) > 0$ ∀ $i$ .
- Perceptron Convergence: Guaranteed if data is linearly separable.
Margin: Distance from hyperplane to closest point. Positive margin implies no points lie on the hyperplane.

H. Logistic Regression

Probability Prediction:

P(y=1|x) = \frac{1}{1 + e^{-w^T x}}

Weight Interpretation:
- If $P(y=1|x) = \frac{1}{1 + e^2}$ for $x = [1, 1]$ , then $w^T x = -2$ .
- Possible weight vectors: Any $w$ satisfying $w_1 + w_2 = -2$ (e.g., $[-1, -1]$ , $[-2, 0]$ ).

I. Support Vector Machines (SVM)

Hard-Margin SVM:
- Only support vectors (SVs) have $\alpha_i > 0$ .
- False Statement: "If $\alpha_i^* \geq 0$ , $x_i$ is an SV" (correct: $\alpha_i^* > 0$ ).
Soft-Margin SVM:
- Allows misclassification via slack variables $\xi_i$ .
- Optimal Weight Vector: Depends on $C$ ; not necessarily identical to hard-margin SVM.
Lagrange Multipliers ( $\alpha_i$ ):
- $\alpha_i = 0$ : Non-SV.
- $0 < \alpha_i < C$ : SV on margin boundary.
- $\alpha_i = C$ : Misclassified SV.
Hinge Loss:

L(y) = \max(0, 1 - y \cdot (w^T x))

(Graph: Red curve (3) in the loss function plot.)

J. Loss Functions

Loss	Equation	Graph Color
Logistic	$\log(1 + e^{-y \cdot (w^T x)})$	Yellow (1)
Hinge (SVM)	$\max(0, 1 - y \cdot (w^T x))$	Red (3)
Squared	$(y - (w^T x))^2$	Green (2)
0-1 Loss	$1$ if $y \cdot (w^T x) < 0$ , else $0$	Blue (4)

Intersection of Logistic & Hinge Loss: Occurs when $(w^T x) y = \ln(\varepsilon - 1)$ (for some $\varepsilon > 1$ ).

K. Neural Networks

Parameter Count (ignoring biases):
- For layer $L$ with $n_{in}$ inputs and $n_{out}$ neurons: $n_{in} \times n_{out}$ weights.
(Example: 10-20-40-1 network → 10×20 + 20×40 + 40×1 = 1600 parameters.)
Activation Functions:
- Output Layer:
  - Binary Cross-Entropy: Sigmoid.
  - Multi-Class Cross-Entropy: Softmax.
- Hidden Layers: ReLU (avoids vanishing gradients).

Graph 2: Regularization & Ridge Regression

Graph 3: Ensemble Methods

Graph 4: SVM & Loss Functions

Graph 5: Neural Networks

3. Key Equations Summary

Topic	Equation
PCA Variance	$\text{Variance \%} = \frac{\sum_{i=1}^k \lambda_i}{\sum_{i=1}^d \lambda_i} \times 100\%$
Ridge Loss	$L(\beta) = \sum (y_i - \hat{y}_i)^2 + \lambda \\|\beta\\|_2^2$
Entropy	$H(S) = -\sum p_i \log_2 p_i$
Information Gain	$\text{IG} = H(\text{parent}) - \sum \frac{\|S_v\|}{\|S\|} H(S_v)$
k-Means Boundary	$2(c - a)x + 2(d - b)y = c^2 + d^2 - a^2 - b^2$
Polynomial Kernel	$K(x, z) = (x^T z + c)^p$
Logistic Regression	$P(y=1\|x) = \frac{1}{1 + e^{-w^T x}}$
SVM Hinge Loss	$L(y) = \max(0, 1 - y \cdot (w^T x))$
Perceptron Update	$w \leftarrow w + \eta \cdot y_i x_i$ (if misclassified)
Neural Network Params	$\text{Total} = \sum_{l=1}^L n_{in}^l \times n_{out}^l$

4. Exam Tips

PCA: Always normalize data first. Variance calculation is eigenvalue-based.
Ridge Regression: Optimal λ is where test loss is minimized (not training loss).
Ensemble Methods:
- Bagging = parallel, Boosting = sequential.
- Deep trees → Bagging; Stumps → Boosting.
SVM:
- Hard margin: No misclassification allowed.
- Soft margin: $\xi_i$ permits misclassification.
- $\alpha_i$ determines SV status.
Loss Functions: Memorize graphs (hinge = red, logistic = yellow, squared = green).
Neural Networks:
- Binary output → sigmoid + cross-entropy.
- Hidden layers → ReLU (avoid sigmoid/tanh for deep networks).