You are supposed to train a neural network that will guide auto$-$driving car. The training data consists of grayscale $6464$ pixel images. The training labels include the human driver's steering wheel angle in degrees and the human driver's speed in miles per hour in output layer. Your neural network consists of an input layer with $6464=4096$ units, a hidden layer with $2048$ units, and an output layer with $2$ units $($one for steering angle, one for speed$).$ You use the ReLU activation function for the hidden units and no activation function for the outputs $($or inputs$).$

a$)$ Calculate the number of parameters $($weights$)$ in this network. You can leave your answer as an expression. Make sure to account for the bias terms.

$(2\mathrm{.}5$ marks$)$

b$)$ You train your network with the cost function $J=0\mathrm{.}5*|y-z{|}^2.$ Use the following notation.

$x$ is a training image $($input$)$ vector with a $1$ component appended to the end, $y$ is a training label $($input$)$ vector, and $z$ is the output vector. All vectors are column vectors.

$r()=$max$\{0,\}$ is the ReLU activation function, $r^{\text{'}}()$ is its derivative if $Y>0,0$ otherwise$),$ and $r(v)$ is $r(*)$ applied component$-$wise to a vector.

$g$ is the vector of hidden unit values before the ReLU activation functions are applied, and $h=r(g)$ is the vector of hidden unit values after they are applied $($but we append a $1$ component to the end of $h).$

$V$ is the weight matrix mapping the input layer to the hidden layer; $g=Vx.$

$W$ is the weight matrix mapping the hidden layer to the output layer; $z=Wh.$

Derive del$\frac{J}{d}el{W}_{ij}.$

$(4$ marks$)$

c$)$ Write del$\frac{J}{d}$elW as an outer product of two vectors. del$\frac{J}{d}$elW is a matrix with the same dimensions as $W$; it's just like a gradient, except that $W$ and del$\frac{J}{d}$elW are matrices rather than vectors.

$(2$ marks$)$