DEEP LEARNING

# Notice that you don't need any other packages.

import numpy as np

import pandas as pd

import random

from matplotlib import pyplot as plt

random.seed$(42)$ # NEVER change this line; this is for grading

# Reading the dataset

data $=$ pd$.$read$\_$csv$(\text{'}./$fashion$\_$data.csv$\text{'})$

# The data pre$-$processing is done for you. Please do NOT edit the cell

# However, you should understand what these codes are doing

data $=$ np$.$array$($data$)$

m$,$ n $=$ data.shape

np$.$random.shuffle$($data$)$ # shuffle before splitting into dev and training sets

data$\_$dev $=$ data$[0$:$400].$T

Y$\_$dev $=$ data$\_$dev$[-1]$

X$\_$dev $=$ data$\_$dev$[0$:n$-1]$

X$\_$dev $=$ X$\_$dev $/255.$

data$\_$train $=$ data$[400$:m$].$T

Y$\_$train $=$ data$\_$train$[-1]$

X$\_$train $=$ data$\_$train$[0$:n$-1]$

X$\_$train $=$ X$\_$train $/255.$

$\_,$m$\_$train $=$ X$\_$train.shape

Part $1$: Building your own neural network

# define a global variable specifying the number of hidden neurons after the first layer

# not the best practice, but we will do it for this mid$-$term project

num$\_$hidden$\_$neurons $=20$

This is the main part of the mid$-$term. You must not change the definition of the function. In fact, the comments are going to help you go through the implementation and they are all you need

$1\mathrm{.}1$ Initialize the parameter in the neural network

# Initialize the parameters in the neural network

# Based on the figure above, we need the weight and bias matrices.

# W$1,$ b$1$ are the matrices for the first layer

# W$2,$ b$2$ are the matrices for the second layer

# You should think about the sizes of the matrices

# then initialize elements in the matrix to be random numbers between $-0\mathrm{.}5$ to $+0\mathrm{.}5$

def init$\_$params$()$:

W$1=$ # Your code here

b$1=$ # Your code here

W$2=$ # Your code here

b$2=$ # Your code here

return W$1,$ b$1,$ W$2,$ b$2$

$1\mathrm{.}2$ Implement the non$-$linearity functions and its derivatives

# As a starting point, you only need a ReLu function, its derivative, and the softmax function

def ReLU$($Z$)$:

# Your code here

def ReLU$\_$deriv$($Z$)$:

# Your code here

def softmax$($Z$)$:

# Your code here

return A

$1\mathrm{.}3$ Implement the forward propagation function

# In the forward propagation function, X is the inputs $($the image in vector form$),$ and we pass all the weights and biases

def forward$\_$prop$($W$1,$ b$1,$ W$2,$ b$2,$ X$)$:

Z$1=$ # Your code here

A$1=$ # Your code here

Z$2=$ # Your code here

A$2=$ # Your code here

return Z$1,$ A$1,$ Z$2,$ A$2$

$1\mathrm{.}4$ Implement the backward propagation function

# This one hot function is to convert a numeric number into a one$-$hot vector

def one$\_$hot$($Y$)$:

# Your code here

return one$\_$hot$\_$Y

# Now performing the backward propagation

# Each function is only one line, but lots of Calculus behind

def backward$\_$prop$($Z$1,$ A$1,$ Z$2,$ A$2,$ W$1,$ W$2,$ X$,$ Y$)$:

one$\_$hot$\_$Y $=$ one$\_$hot$($Y$)$

dZ$2=$ # Your code here

dW$2=$ # Your code here

db$2=$ # Your code here

dZ$1=$ # Your code here

dW$1=$ # Your code here

db$1=$ # Your code here

return dW$1,$ db$1,$ dW$2,$ db$2$

# Finally, we are ready to update the parameters

def update$\_$params$($W$1,$ b$1,$ W$2,$ b$2,$ dW$1,$ db$1,$ dW$2,$ db$2,$ alpha$)$:

W$1=$ # Your code here

b$1=$ # Your code here

W$2=$ # Your code here

b$2=$ # Your code here

return W$1,$ b$1,$ W$2,$ b$2$

$1\mathrm{.}5$ Performing the gradient descent

# Implement the helper function. We need to convert the softmax output into a numeric label

# This is done through get$\_$predictions function

def get$\_$predictions$($A$2)$:

# Your code here

# We also want to have a simple function to compute the accuracy. Notice that "predictions" and $"$Y$"$ are the same shape

def get$\_$accuracy$($predictions$,$ Y$)$:

return # Your code here

# Finally, we are ready to implement gradient descent

def gradient$\_$descent$($X$,$ Y$,$ alpha, iterations$)$:

W$1,$ b$1,$ W$2,$ b$2=$ # Your code here $-$ using the function you have implemented

for i in range$($iterations$)$:

Z$1,$ A$1,$ Z$2,$ A$2=$ # Your code here $-$ using the function you have implemented

dW$1,$ db$1,$ dW$2,$ db$2=$ # Your code here $-$ using the function you have implemented

W$1,$ b$1,$ W$2,$ b$2=$ # Your code here $-$ using the function you have implemented

if i $\%10==0$:

print$("$Iteration: $",$ i$)$

predictions $=$ get$\_$predictions$($A$2)$

print$($get$\_$accuracy$($predictions$,$ Y$))$

return W$1,$ b$1,$ W$2,$ b$2$

W$1,$ b$1,$ W$2,$ b$2=$ gradient$\_$descent$($X$\_$train, Y$\_$train, $0\mathrm{.}10,500)$

$1\mathrm{.}6$ Validation Set Performance

def make$\_$predictions$($X$,$ W$1,$ b$1,$ W$2,$ b$2)$:

$\_,\_,\_,$ A$2=$ forward$\_$prop$($W$1,$ b$1,$ W$2,$ b$2,$ X$)$

predictions $=$ get$\_$predictions$($A$2)$

return predictions

dev$\_$predictions $=$ make$\_$predictions$($X$\_$dev, W$1,$ b$1,$ W$2,$ b$2)$

get$\_$accuracy$($dev$\_$predictions, Y$\_$dev$)$

Part $2$: Error Analysis and Performance Improvements

Based on the neural network, you should recommend some next steps in this part. Some ideas include investigating where the model fails to predict and$/$or trying to improve the model performance through, for example, different activation functions, expanding the network complexity.

It is crucial to provide reasoning