Can you answer part $2$ and part $3$ multiple choice parts of this question in the image? Thank you

Given a temporal CNN with one hidden layer. The state at time $t$ is calculated as:

$S_t=U_0{x}_{t-1}+U_1{x}_t$

And the output at time $t$ is calculated as:

$y_t=V_0{S}_{t-1}+V_1{S}_t$

Q$1$

Train this temporal CNN with the squared error loss function:

$e_t=\frac{1}{2}(hat(y{)}_t-y_t{)}^2$

on the dataset:

$x_0=1,y_0=3,x_1=2,y_1=6,x_2=3,y_2=9$

Assume that $U_0=U_1=V_0=V_1=1.$ At time $t=2$ what is the gradient of the loss function $e_2,$ evaluated

at the assumed values, with respect to $U_0?$ Please round your answer to one decimal place.

Q$2$

Let's now train multiple layers of temporal convolutions $(10$ in total$)$ with dilation on a large dataset where each

data point has $T=1000$ timesteps. All temporal convolutions are of size $2$ and dilated with $2^k$ for layers

$k=0$dots$9.$ How many matrix multiplications does the gradient go through from $l_T$ to $x_0?$ Consider the most

efficient implementation.

$logT$

$T$

$T^2$

$\sqrt[\phantom{2}]{T}$

Q$3$

Let's now dilate slightly differently. All temporal convolutions are of size $2$ and dilated with $k+1$ for layers

$k=0$dots. We increase the number of layers until the receptive field is larger than $1000.$ How many matrix

multiplications does the gradient go through from $l_T$ to $x_0?$ Consider the most efficient implementation.

$logT$

$T$

$T^2$

$\sqrt[\phantom{2}]{T}$