$(1)$ Recall that supervised gradient descent algorithms are made up of a few key components:

an activation function which describes how the input features $(vec(x))$ and the weights $(vec(w))$ are related to the predicted output feature $(hat(y)).$

a loss function, $L(vec(w)),$ which describes how we measure the accuracy of predictions $($ hat$(y)$ vs$.y)$ as a function of the weights.

the gradient of the loss function, $\frac{\text{d}\text{e}\text{l}\text{L}((vec(w)))}{\text{d}\text{e}\text{l}(vec(w))}.$

the update of the weights using the gradient, $\{$:$(vec(w))=(vec(w))-\frac{\text{d}\text{e}\text{l}\text{L}((vec(w)))}{\text{d}\text{e}\text{l}(vec(w))}).$

Below are three of each type of the first three components $($the update is the same process in all gradient descent algorithms$),$ written in mathematical notation, for three different gradient descent algorithms we discussed in class. Connect the three components to the corresponding model. For example $($this is NOT the exact answer$),$ model $($A$)$ may be made up of components: activation function $(1),$ loss function $($c$),$ and gradient $($ii$).$

$\backslash$table$[[$Model$,$Activation Fxn: hat$(y),$Loss Fxn: $L(vec(w)),$Gradient: $\frac{\text{d}\text{e}\text{l}\text{L}((vec(w)))}{\text{d}\text{e}\text{l}(vec(w))}$