Implement the function Logistic$\_$Regression$\_$via$\_$GD$($P$,$y$,$lr$)$:

Input: an $np$ array $\text{'}P\text{'}$ of $\text{'}n\text{'}$ rows and $\text{'}d\text{'}$ columns, a label vector $\text{'}y\text{'}$ of $\text{'}n\text{'}$ entries and learning rate parameter $\text{'}$Ir$\text{'}.$

Output: The function computes the output vector $\text{'}w\text{'}($and $\text{'}b\text{'})$ which minimzes the logistic regression cost function on $\text{'}P\text{'}$ and $\text{'}y\text{'}.$

The implementation should be fully yours. Don't use library implementation!

It should be done by implementing Gradient descent $($with $\text{'}$Ir$\text{'}$ as the learning rate$)$ to solve logistic regression.

Tip: The gradients may be large, you can use $\frac{1}{n}$gradL $($which is the true empirical loss' gradient$)$

$[]$ def sigmoid$($z$)$:

return $\frac{1}{1+np.\text{e}\text{x}\text{p}(-z)}$

# For now, ignore the lambda, you will need it later

def Logistic$\_$Regression$\_$via$\_$GD $(P,y,lr,$ lamda $=0$ :

# Implement here

Implement the function Predict$($w$,$b$,$p$)$:

$$ Input: an input vector $x$ which represents a sample, a vector $($numpy$)\text{'}w\text{'}$ and a number $\text{'}b\text{'}.$

Output: the class prediction for $\text{'}p\text{'}$ of the logistic regression model defined by $\text{'}w\text{'}$ and $\text{'}b\text{'}.$

Note: It is your choice if use predictions of $\frac{1}{-}1$ or $\frac{1}{0}.$ Make sure to adjust your choice to the true labels $($for comparison$).$

$[]$ def predict $(x,w,b)$ :

# Implement here

Call 'Logistic$\_$Regression$\_$via$\_$GD$($Py$,$lr$)\text{'},$ where $\text{'}P\text{'}$ and $\text{'}y\text{'}$ are the training data and the corresponding labels.

Try to find the best Ir for the learning task.

$[]$ # Implement here

Run the model on the test data, print the accuracy and plot hyperplane on the test data using 'plot' as defined above.

$[]$ # Implement here