Consider the neural network shown in the figure with \[ \text{inputs: } u = 2, \, v = 3 \] \[ \text{weights: } a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 \] \[ \text{output: } y \] R denotes the ReLU function, \( R(x) = \max(0, x) \).
Given \( u = 2, v = 3, a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 \), which one of the following is correct?
Consider the neural network shown in the figure with \[ \text{inputs: } u = 2, \, v = 3 \] \[ \text{weights: } a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 \] \[ \text{output: } y \] R denotes the ReLU function, \( R(x) = \max(0, x) \).
Given \( u = 2, v = 3, a = 1, b = 1, c = 1, d = -1, e = 4, f = -1 \), which one of the following is correct?
Show Hint
\( 1 \) if the input is positive,
\( 0 \) if the input is non-positive.
- \( \frac{\partial y}{\partial a} = 8, \frac{\partial y}{\partial f} = 0 \)
- \( \frac{\partial y}{\partial a} = 1, \frac{\partial y}{\partial f} = 0 \)
- \( \frac{\partial y}{\partial a} = 1, \frac{\partial y}{\partial f} = -1 \)
- \( \frac{\partial y}{\partial a} = 2, \frac{\partial y}{\partial f} = -1 \)
The Correct Option is A
Solution and Explanation
1. The first ReLU unit gives an output of \( R(5) = 5 \),
2. The second ReLU unit gives an output of \( R(-1) = 0 \),
3. The final output \( y \) is \( R(19) = 20 \).
Now, we compute the derivatives:
\( \frac{\partial y}{\partial a} = 8 \),
\( \frac{\partial y}{\partial f} = 0 \).
Thus, the correct answer is Option (A).
Top Questions on Machine Learning
- Which of the following statements is/are correct about the rectified linear unit (ReLU) activation function defined as ReLU(x) = max(x, 0), where \( x \in \mathbb{R} \)?
- GATE DA - 2025
- Data Science and Artificial Intelligence
- Machine Learning
- The naive Bayes classifier is used to solve a two-class classification problem with class-labels \( y_1, y_2 \). Suppose the prior probabilities are \( P(y_1) = \frac{1}{3} \) and \( P(y_2) = \frac{2}{3} \). Assuming a discrete feature space with
\[ P(x | y_1) = \frac{3}{4} \quad \text{and} \quad P(x | y_2) = \frac{1}{4} \]
for a specific feature vector \( x \). The probability of misclassifying \( x \) is: (Round off to two decimal places)- GATE DA - 2025
- Data Science and Artificial Intelligence
- Machine Learning
- Let \( C_1 \) and \( C_2 \) be two sets of objects. Let \( D(x, y) \) be a measure of dissimilarity between two objects \( x \) and \( y \). Consider the following definitions of dissimilarity between \( C_1 \) and \( C_2 \): \[ \text{DIS-1}(C_1, C_2) = \max_{x \in C_1, y \in C_2} D(x, y) \] \[ \text{DIS-2}(C_1, C_2) = \min_{x \in C_1, y \in C_2} D(x, y) \] Which of the following statements is/are correct?
- GATE DA - 2025
- Data Science and Artificial Intelligence
- Machine Learning
Consider designing a linear classifier
\[ y = \text{sign}(f(x; w, b)), \quad f(x; w, b) = w^T x + b \]on a dataset \( D = \{(x_1, y_1), (x_2, y_2), \dots, (x_N, y_N)\} \), where \( x_i \in \mathbb{R}^d \), \( y_i \in \{+1, -1\} \), for \( i = 1, 2, \dots, N \).
Recall that the sign function outputs \( +1 \) if the argument is positive, and \( -1 \) if the argument is non-positive. The parameters \( w \) and \( b \) are updated as per the following training algorithm:
\[ w_{\text{new}} = w_{\text{old}} + y_n x_n, \quad b_{\text{new}} = b_{\text{old}} + y_n \]whenever \( \text{sign}(f(x_n; w_{\text{old}}, b_{\text{old}})) \neq y_n \).
In other words, whenever the classifier wrongly predicts a sample \( (x_n, y_n) \) from the dataset, \( w_{\text{old}} \) gets updated to \( w_{\text{new}} \), and likewise \( b_{\text{old}} \) gets updated to \( b_{\text{new}} \).
Consider the case \( (x_n, +1) \), where \( f(x_n; w_{\text{old}}, b_{\text{old}}) < 0 \). Then:
- GATE DA - 2025
- Data Science and Artificial Intelligence
- Machine Learning
- Consider a two-class problem in \( \mathbb{R}^d \) with class labels red and green. Let \( \mu_{\text{red}} \) and \( \mu_{\text{green}} \) be the means of the two classes. Given test sample \( x \in \mathbb{R}^d \), a classifier calculates the squared Euclidean distance (denoted by \( \| \cdot \|^2 \)) between \( x \) and the means of the two classes and assigns the class label that the sample \( x \) is closest to. That is, the classifier computes \[ f(x) = \| \mu_{\text{red}} - x \|^2 - \| \mu_{\text{green}} - x \|^2 \] and assigns the label red to \( x \) if \( f(x)<0 \), and green otherwise. Which of the following statements is/are correct?
- GATE DA - 2025
- Data Science and Artificial Intelligence
- Machine Learning
Questions Asked in GATE DA exam
- Given the equation: $$ 3^{x^2} = 27 \times 9^x $$ Find the value of: $$ \frac{2^{x^2}}{(2^x)^2} $$
- GATE DA - 2025
- Algebra
Consider a directed graph \( G = (V,E) \), where \( V = \{0,1,2,\dots,100\} \) and
\[ E = \{(i,j) : 0 < j - i \leq 2, \text{ for all } i,j \in V \}. \] Suppose the adjacency list of each vertex is in decreasing order of vertex number, and depth-first search (DFS) is performed at vertex 0. The number of vertices that will be discovered after vertex 50 is:- GATE DA - 2025
- Data Structures
- Consider the following pseudocode.
Create empty stack S Set x = 0, flag = 0, sum = 0 Push x onto S while (S is not empty){ if (flag equals 0){ Set x = x + 1 Push x onto S } if (x equals 8): Set flag = 1 if (flag equals 1){ x = Pop(S) if (x is odd): Pop(S) Set sum = sum + x } } Output sumThe value of \( sum \) output by a program executing the above pseudocode is:- GATE DA - 2025
- Data Structures
- Consider the following Python code snippet.
def f(a, b): if (a == 0): return b if (a % 2 == 1): return 2 * f((a - 1) / 2, b) return b + f(a - 1, b) print(f(15, 10))The value printed by the code snippet is 160 (Answer in integer).- GATE DA - 2025
- Recursion
Consider the following tables, Loan and Borrower, of a bank.
Query: \[ \pi_{\text{branch\_name}, \text{customer\_name}} (\text{Loan} \bowtie \text{Borrower}) \div \pi_{\text{branch\_name}}(\text{Loan}) \] where \( \bowtie \) denotes natural join. The number of tuples returned by the above relational algebra query is 1 (Answer in integer).- GATE DA - 2025
- Relational Algebra