Please show the answer only.

Logistic regression is a supervised learning algorithm used to estimate the model

parameters vec$()$ from a training dataset $D=\{(vec(x{)}^{(i)},y^{(i)}),i=1,2,$dots,$m,$yin$\{0,1\}\},$ such that

the resulting hypothesis function $h_{\frac{?}{\text{b}\text{a}\text{r}}()}(vec(x))=g(vec({)}^T*(vec(x)))$ can predict the probability that a new

input instance vec$(x)$ belongs to the positive class $($usually denoted as $1).$ To achieve this goal,

logistic regression employs a logistic or sigmoid activation function $g(z)=\frac{1}{1+e^{-z}}$ which

transforms the linear combination of input features and model parameters to a probability value

within the range $(0,1).$ The probability serves as the basis for classifying new data instances.

A common approach for logistic regression models is stochastic gradient ascent, which

involves iteratively updating the model parameters vec$()$ based on the gradients of log$-$likelihood

function. Stochastic gradient ascent aims to maximize the likelihood of observing the given

data and refine the model's parameters accordingly.

Please derive the following stochastic gradient ascent update rule for logistic regression

models,

${}_j^{(t+1)}={}_j^{(t)}+(y^{(i)}-h_{\frac{?}{\text{b}\text{a}\text{r}}()}(vec(x{)}^{(i)}))x_j^{(i)},j=0,1,$dots,$n$

where $$ is the learning rate, $y^{(i)}$ is the binary label of the $i$ th training instance vec$(x{)}^{(i)}($either $0$ or

$,h_{\frac{?}{\text{b}\text{a}\text{r}}()}(vec(x{)}^{(i)})$ is the predicted probability of labeling the $i$ th instance vec$(x{)}^{(i)}$ as positive $($class label

is $1)$ based on the current values of the model parameters vec$({)}^{(t)}*{}_j^{(t)}$ is the current value of the

parameter ${}_j*{}_j^{(t+1)}$ is the updated value of the parameter ${}_j.$

Given a neural network, its structure is shown below. $z_j^{[l]}$ is the output of the linear part of $j^{th}$

neuron in layer $l$;$a_j^{[l]}=g(z_j^{[l]})$ is the output of the activation part of $j^{th}$ neuron in layer $l$ and

$g(z)$ is the activation function.

I need show work about derive the following stochastic gradient ascent update rule for logistic regression

models please.