C2_W2_Quiz.md

1. About the Gradient Descent method, choose all that are true:

• It always converges to a local minimum.
• The result may vary depending on the initial point.
• If it converges, then it converges to a global minimum.
• It only works for differentiable functions.

2. Given the Initial Point on the following graph, to which point will the Gradient Descent method converge?

• P1.
• P2.
• P3.
• It won’t converge.

3. Given that $f(x, y) = x^3y^2 + 3y^3$, find its derivative with respect to $y$, i.e., find $\frac{\partial f}{\partial y}$.

Note: Please use * to indicate the product in the answer. So, if we wrote the entire function $f$ as an answer, it would be x^3 * y^2 + 3 * y^3.

Answer: $2x^3y + 9y^2$ (2 * x^3 * y + 9 * y^2)

4. Let $f(x, y) = 2x^2 + 3y^2 - 2xy - 10x$, the minimum value of $f(x, y)$ is

• -15
• 3
• 1

5. What are the parameters that the Gradient Descent algorithm has? (check all that apply)

• Initial point
• Final point
• Learning rate
• Number of iterations

6. Let $f(x, y)$ and $\nabla f(x, y)$ and let the initial point $x_0 = (0, 1)$.

$$f(x, y) = x^2 + y^2 - 6x, \nabla f(x, y) = \begin{bmatrix} 2x - 6 \cr 2y \end{bmatrix}$$

Performing the gradient descent algorithm with learning rate = 0.1, the first iteration will lead us the point $x_1$ which is:

• $x_1 = (0.6, 0.8)$
• $x_1 = (-6, 2)$
• $x_1 = (6, -1)$
• $x_1 = (0, 1)$