Backward Propagation

In $[3]$: 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.

$"$n$"$

# 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:

In $[]$: # Example usage

if $\_==""$:

# 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$)$

Original weights: $\left[\begin{array}{cccc}0\mathrm{.}2& 0\mathrm{.}5& 0\mathrm{.}3& 0\mathrm{.}2\end{array}\right]$

Updated weights: $\left[\begin{array}{cccc}0\mathrm{.}202& 0\mathrm{.}5002& 0\mathrm{.}3004& 0\mathrm{.}2006\end{array}\right]$