Q$3(20$ points$)$

$5\mathrm{.}0\mathrm{.}1.$ Calculate the gradient of $f$ with respect to $x$ and $y$ using the backpropagation

algorithm

$5\mathrm{.}0\mathrm{.}2.$ Calculate the gradient of $f$ with respect to $x$ and $y$ using the computation graph in

tensorflow

$f(x,y)=\frac{x^3+y^4}{(x+y{)}^2}$

Declare intermediate variables, $q_1=x^3,q_2=y^4,q=q1+q2$ and so on$.$ Remember to use is needed when $x$ is used in

multiple paths of the computation$),$ similarly dfdy$.$

##code:

import sys

from packaging import version

import sklearn

import matplotlib.pyplot as plt

import numpy as np

import tensorflow as tf

from sklearn.preprocessing import add$\_$dummy$\_$feature

from sklearn.datasets import make$\_$blobs

from sklearn import metrics #Import scikit$-$learn metrics module for accuracy calculation

from sklearn.model$\_$selection import train$\_$test$\_$split

from sklearn.metrics import confusion$\_$matrix, ConfusionMatrixDisplay

from sklearn.datasets import make$\_$classification

print$("$Sklearn package",sys$.$version$\_$info$)$

print$("$Sklearn package",sklearn.$\_\_$version$\_\_)$

print$("$TensorFlow version:", tf$.\_\_$version$\_\_)$

assert sys$.$version$\_$info $>=(3,7)$

assert version.parse$($sklearn$.\_\_$version$\_\_)>=$ version.parse$("1\mathrm{.}0\mathrm{.}1")$

plt$.$rc$(\text{'}$font$\text{'},$ size$=12)$

plt$.$rc$(\text{'}$axes$\text{'},$ labelsize$=14,$ titlesize$=14)$

plt$.$rc$(\text{'}$legend$\text{'},$ fontsize$=14)$

plt$.$rc$(\text{'}$xtick$\text{'},$ labelsize$=10)$

plt$.$rc$(\text{'}$ytick$\text{'},$ labelsize$=10)$

##

x $=2$

y $=5$

#Forward pass

#Backward pass

#print$($f$)$

#print$($dfdx$,$dfdy$)$

#f $=12\mathrm{.}918367346938776$

#dfdx $=-3\mathrm{.}4460641399416914$

#dfdy $=6\mathrm{.}513119533527696$

##

#using computation graph from tensorflow

x $=$ tf$.$Variable$(2.)$

y $=$ tf$.$Variable$(5.)$

#print$($f$)$

#print$($grad$\_$x$,$grad$\_$y$)$