Question 14: Starting at (x_0, y_0), we now turn to momentum based gr…

Based on the above data, answer the given subquestions.

Starting at $(x_0, y_0)$, we now turn to momentum based gradient descent. The momentum updates are given below. Starting with $u_{-1} = (0,0)$ and $\eta = 0.2$, we have:

$\begin{aligned} u_t &= \beta u_{t-1} + \nabla f(x, y)\Big|_{(x_t, y_t)} \\ (x_{t+1}, y_{t+1}) &= (x_t, y_t) - \eta u_t \end{aligned}$

What should be the momentum parameter $\beta$ so that running two steps of momentum based gradient descent takes us to the global minimum? Note that the sequence of iterates will be: $(x_0, y_0) \to (x_1, y_1) \to (x_2, y_2)$, where $(x_2, y_2)$ is the global minimum of $f$. Enter your answer correct to two places after the decimal.