$(4$ points$)$ Derive $\frac{\text{d}\text{e}\text{l}\text{J}}{del{W}_{ij}}.$

$(2$ points$)$ Write $\frac{\text{d}\text{e}\text{l}\text{J}}{\text{d}\text{e}\text{l}\text{W}}$ as an outer product of two vectors. $\frac{\text{d}\text{e}\text{l}\text{J}}{\text{d}\text{e}\text{l}\text{W}}$ is a matrix with the same dimen$-$

sions as $W$; it is just like a gradient, except that $W$ and $\frac{\text{d}\text{e}\text{l}\text{J}}{\text{d}\text{e}\text{l}\text{W}}$ are matrices rather than vectors.

$(7$ points$)$ Derive $\frac{\text{d}\text{e}\text{l}\text{J}}{del{V}_{ij}}.$ Problem $5$

You want to train a neural network to recognize the age and sex based on the face image. Your

training data consists of grayscale $3232$ pixel images. The training labels include the age and sex

of each person. Your neural network consists of an input layer with $3232=1,024$ units, a hidden

layer with $2,048$ units, and an output layer with $2$ units. You use the ReLU activation function for

the hidden units and no activation function for the outputs $($or inputs$).$ You train your network

with the cost function $J=\frac{1}{2}||y-z|{|}^2.$ Please use the following notations to answer questions:

$x$ is a training image $($input$)$ vector with a $1$ component appended to the end $($as a bias$),y$ 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 $(1$ if $>0,0$ other$-$

wise$),$ 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 $1$ com$-$

ponent to the end of $h$ as the bias term$).$

$V$ 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; $z=Wh.$

Questions:

$(2$ points$)$ Calculate the number of parameters $($weights$)$ in this network. You can leave your

answer as an expression.