$1.$ Objective Function for Logistic Regression with L$2$ Regularization:

The objective function for logistic regression can be written as:

J$($$)=$i$=1$N$($yilog$($$($Txi$))+(1$yi$)$log$(1$$($Txi$)))$J$(\backslash$theta$)=-\backslash$sum$\_\{$i$=1\}^\{$N$\}\backslash$left$($ y$\_$i $\backslash$log$(\backslash$sigma$(\backslash$theta$^$T x$\_$i$))+(1-$ y$\_$i$)\backslash$log$(1-\backslash$sigma$(\backslash$theta$^$T x$\_$i$))\backslash$right$)$J$($$)=$i$=1$N$($yi$$log$($$($Txi$))+(1$yi$)$log$(1$$($Txi$)))$

Adding the L$2$ regularization term, the objective function becomes:

Jreg$($$)=$i$=1$N$($yilog$($$($Txi$))+(1$yi$)$log$(1$$($Txi$)))+$$2$$2$J$\_\{\backslash$text$\{$reg$\}\}(\backslash$theta$)=-\backslash$sum$\_\{$i$=1\}^\{$N$\}\backslash$left$($ y$\_$i $\backslash$log$(\backslash$sigma$(\backslash$theta$^$T x$\_$i$))+(1-$ y$\_$i$)\backslash$log$(1-\backslash$sigma$(\backslash$theta$^$T x$\_$i$))\backslash$right$)+\backslash$frac$\{\backslash$lambda$\}\{2\}\backslash |\backslash$theta$\backslash |^2$Jreg$($$)=$i$=1$N$($yi$$log$($$($Txi$))+(1$yi$)$log$(1$$($Txi$)))+2$$$$2$

The regularization term penalizes larger values of $\backslash$theta$,$ which helps prevent overfitting$.$

$2.$ Gradient Descent Update Rule for $\backslash$theta:

The gradient of the regularized objective function with respect to $\backslash$theta is derived as:

$$Jreg$($$)=$i$=1$N$($$($Txi$)$yi$)$xi$+$

abla$\_\backslash$theta J$\_\{\backslash$text$\{$reg$\}\}(\backslash$theta$)=\backslash$sum$\_\{$i$=1\}^\{$N$\}\backslash$left$(\backslash$sigma$(\backslash$theta$^$T x$\_$i$)-$ y$\_$i $\backslash$right$)$ x$\_$i $+\backslash$lambda $\backslash$theta$$$$Jreg$($$)=$i$=1$N$($$($Txi$)$yi$)$xi$+$

The update rule using gradient descent is:

:$=$$$$($i$=1$N$($$($Txi$)$yi$)$xi$+$$)\backslash$theta :$=\backslash$theta $-\backslash$eta $\backslash$left$(\backslash$sum$\_\{$i$=1\}^\{$N$\}(\backslash$sigma$(\backslash$theta$^$T x$\_$i$)-$ y$\_$i$)$ x$\_$i $+\backslash$lambda $\backslash$theta $\backslash$right$)$:$=$$$$($i$=1$N$($$($Txi$)$yi$)$xi$+$$)$

Where $\backslash$eta is the learning rate. This rule ensures that $\backslash$theta is updated iteratively while taking into account both the gradient of the loss function and the regularization term$.$