## $1$ Perceptron algorithm Classifier

The Perceptron algorithm, a fundamental concept in machine learning, serves as a building block for more complex models. It was initially proposed by Frank Rosenblatt in the late $1950$s and is widely used for binary classification tasks. The algorithm learns to classify input data points into two categories by adjusting its weights based on errors made during prediction. While relatively simple, the Perceptron algorithm forms the basis for neural networks and other sophisticated machine learning models. Its ability to learn from data and make decisions autonomously makes it a crucial tool in various fields, from pattern recognition to natural language processing.

Backward Propagation

$[]$ import numpy as $np$

def backward$\_$propagation$(x,y,$ predicted$\_$output, weights, learning$\_$rate$)$:

$"""$

Perform backward propagation for one perceptron to update weights.

Parameters:

x: Input data.

y: True label $($the actual output$).$

predicted$\_$output: Output of the perceptron $($from the forward pass$).$

weights: Weights matrix $($including bias$).$

learning$\_$rate: Learning rate $($alpha$).$

Returns:

updated$\_$weights: Updated weights matrix.

$"""$

# Write your code here

return updated$\_$weights

Use the following cell to test your solution in the previous cell. Here is a list of correct answers:

# Example usage

if $\_\_$name$\_\_=="\_\_$main$\_\_"$:

# Sample input data for one example

x $=$ np$.$array$([0\mathrm{.}1,0\mathrm{.}2,0\mathrm{.}3])$ # Input features

y $=1$ # True label

predicted$\_$output $=0\mathrm{.}8$ # Output of the perceptron $($from the forward pass$)$

weights $=$ np$.$array$([0\mathrm{.}2,0\mathrm{.}5,0\mathrm{.}3,0\mathrm{.}2])$ # Weights matrix $($including bias$)$

learning$\_$rate $=0\mathrm{.}01$ # Learning rate $($alpha$)$

# Perform backward propagation to update weights

updated$\_$weights $=$ backward$\_$propagation$($x$,$ y$,$ predicted$\_$output, weights, learning$\_$rate$)$

print$("$Original weights:", weights$)$

print$("$Updated weights: $",$ updated$\_$weights$)$