$($b$)$ Consider a sample of $N=125$ observations $(x_i,y_i),$ where, for each obser$-$

vation $i,x_i$ denotes the input variable and $y_i$ the numerical target variable

$($all quantities are uni$-$dimensional$).$ From the data, the following summary

quantities can be computed:

${\displaystyle \underset{i=1}{\overset{N}{}}}{x}_i{y}_i=112,{\displaystyle \underset{i=1}{\overset{N}{}}}{e}^{x_i}=207$

A neural network is trained on these data using the following average loss

function and weight decay penalization function:

$E(w)=-\frac{1}{N}{\displaystyle \underset{i=1}{\overset{N}{}}}(y_i{x}_{i}w-e^{w{x}_i}),Q(w)=\frac{1}{2}{w}^2.$

The network is trained using a gradient descent algorithm with learning rate

equal to $0\mathrm{.}05$ and penalty parameter equal to $0\mathrm{.}5.$ At iteration $(t)$ the current

estimate of the weight $w$ is $0\mathrm{.}75.$ Using the information available, compute

the updated estimate of $w$ at iteration $(t+1).$