trash.py

# Jupyter
# Notebook
# Building
# your
# Deep
# Neural
# Network - Step
# by
# Step
# v5
# Last
# Checkpoint: 2
# hours
# ago
# Autosave
# Failed!
# Python
# 3
# File
# Edit
# View
# Insert
# Cell
# Kernel
# Widgets
# Help
#
# Building
# your
# Deep
# Neural
# Network: Step
# by
# Step
# Welcome
# to
# your
# week
# 4
# assignment(part
# 1
# of
# 2)! You
# have
# previously
# trained
# a
# 2 - layer
# Neural
# Network(
# with a single hidden layer).This week, you will build a deep neural network, with as many layers as you want!
#
# In
# this
# notebook, you
# will
# implement
# all
# the
# functions
# required
# to
# build
# a
# deep
# neural
# network.
# In
# the
# next
# assignment, you
# will
# use
# these
# functions
# to
# build
# a
# deep
# neural
# network
# for image classification.
#     After
#     this
#     assignment
#     you
#     will
#     be
#     able
#     to:
#
# Use
# non - linear
# units
# like
# ReLU
# to
# improve
# your
# model
# Build
# a
# deeper
# neural
# network(
# with more than 1 hidden layer)
# Implement
# an
# easy - to - use
# neural
# network
#
#
# class
#     Notation:
#
#
# Superscript[l][l]
# denotes
# a
# quantity
# associated
# with the  lthlth  layer.
# Example: a[L]
# a[L] is the
# LthLth
# layer
# activation.W[L]
# W[L] and b[L]
# b[L]
# are
# the
# LthLth
# layer
# parameters.
# Superscript(i)(i)
# denotes
# a
# quantity
# associated
# with the  ithith  example.
# Example: x(i)
# x(i) is the
# ithith
# training
# example.
# Lowerscript
# ii
# denotes
# the
# ithith
# entry
# of
# a
# vector.
# Example: a[l]
# iai[l]
# denotes
# the
# ithith
# entry
# of
# the
# lthlth
# layer
# 's activations).
# Let
# 's get started!
#
# 1 - Packages
# Let
# 's first import all the packages that you will need during this assignment.
#
# numpy is the
# main
# package
# for scientific computing with Python.
# matplotlib is a
# library
# to
# plot
# graphs in Python.
# dnn_utils
# provides
# some
# necessary
# functions
# for this notebook.
#     testCases
#     provides
#     some
#     test
#     cases
#     to
#     assess
#     the
#     correctness
#     of
#     your
#     functions
# np.random.seed(1) is used
# to
# keep
# all
# the
# random
# function
# calls
# consistent.It
# will
# help
# us
# grade
# your
# work.Please
# don
# 't change the seed.
#
# import numpy as np
# import h5py
# import matplotlib.pyplot as plt
# from testCases_v3 import *
# from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward
# ​
# % matplotlib
# inline
# plt.rcParams['figure.figsize'] = (5.0, 4.0)  # set default size of plots
# plt.rcParams['image.interpolation'] = 'nearest'
# plt.rcParams['image.cmap'] = 'gray'
# ​
# % load_ext
# autoreload
# % autoreload
# 2
# ​
# np.random.seed(1)
# 2 - Outline
# of
# the
# Assignment
# To
# build
# your
# neural
# network, you
# will
# be
# implementing
# several
# "helper functions".These
# helper
# functions
# will
# be
# used in the
# next
# assignment
# to
# build
# a
# two - layer
# neural
# network and an
# L - layer
# neural
# network.Each
# small
# helper
# function
# you
# will
# implement
# will
# have
# detailed
# instructions
# that
# will
# walk
# you
# through
# the
# necessary
# steps.Here is an
# outline
# of
# this
# assignment, you
# will:
#
# Initialize
# the
# parameters
# for a two - layer network and for an  LL -layer neural network.
# Implement
# the
# forward
# propagation
# module(shown in purple in the
# figure
# below).
# Complete
# the
# LINEAR
# part
# of
# a
# layer
# 's forward propagation step (resulting in  Z[l]Z[l] ).
# We
# give
# you
# the
# ACTIVATION
# function(relu / sigmoid).
# Combine
# the
# previous
# two
# steps
# into
# a
# new[LINEAR->ACTIVATION] forward
# function.
# Stack
# the[LINEAR->RELU] forward
# function
# L - 1
# time(
# for layers 1 through L-1) and add a[LINEAR->SIGMOID] at the end ( for the final layer  LL ).This gives you a new L_model_forward function.
# Compute
# the
# loss.
# Implement
# the
# backward
# propagation
# module(denoted in red in the
# figure
# below).
# Complete
# the
# LINEAR
# part
# of
# a
# layer
# 's backward propagation step.
# We
# give
# you
# the
# gradient
# of
# the
# ACTIVATE
# function(relu_backward / sigmoid_backward)
# Combine
# the
# previous
# two
# steps
# into
# a
# new[LINEAR->ACTIVATION] backward
# function.
# Stack[LINEAR->RELU] backward
# L - 1
# times and add[LINEAR->SIGMOID] backward in a
# new
# L_model_backward
# function
# Finally
# update
# the
# parameters.
#
# Figure
# 1
#
# Note
# that
# for every forward function, there is a corresponding backward function.That is why at every step of your forward module you will be storing some values in a cache.The cached values are useful for computing gradients.In the backpropagation module you will then use the cache to calculate the gradients.This assignment will show you exactly how to carry out each of these steps.
#
# 3 - Initialization
# You
# will
# write
# two
# helper
# functions
# that
# will
# initialize
# the
# parameters
# for your model.The first function will be used to initialize parameters for a two layer model.The second one will generalize this initialization process to  LL  layers.
#
# 3.1 - 2 - layer
# Neural
# Network
# Exercise: Create and initialize
# the
# parameters
# of
# the
# 2 - layer
# neural
# network.
#
# Instructions:
#
# The
# model
# 's structure is: LINEAR -> RELU -> LINEAR -> SIGMOID.
# Use
# random
# initialization
# for the weight matrices.Use np.random.randn(shape) * 0.01 with the correct shape.
# Use
# zero
# initialization
# for the biases.Use np.zeros(shape).
#
# (
#     # GRADED FUNCTION: initialize_parameters
#     ​
#
#
# def initialize_parameters(n_x, n_h, n_y):
#     """
#     Argument:
#     n_x -- size of the input layer
#     n_h -- size of the hidden layer
#     n_y -- size of the output layer
#
#     Returns:
#     parameters -- python dictionary containing your parameters:
#                     W1 -- weight matrix of shape (n_h, n_x)
#                     b1 -- bias vector of shape (n_h, 1)
#                     W2 -- weight matrix of shape (n_y, n_h)
#                     b2 -- bias vector of shape (n_y, 1)
#     """
#
#     np.random.seed(1)
#
#     ### START CODE HERE ### (≈ 4 lines of code)
#     W1 = np.random.randn(n_h, n_x) * 0.01
#     b1 = np.zeros((n_h, 1))
#     W2 = np.random.randn(n_y, n_h) * 0.01
#     b2 = np.zeros((n_y, 1))
#     ### END CODE HERE ###
#
#     assert (W1.shape == (n_h, n_x))
#     assert (b1.shape == (n_h, 1))
#     assert (W2.shape == (n_y, n_h))
#     assert (b2.shape == (n_y, 1))
#
#     parameters = {"W1": W1,
#                   "b1": b1,
#                   "W2": W2,
#                   "b2": b2}
#
#     return parameters
#
#
# parameters = initialize_parameters(3, 2, 1)
# print("W1 = " + str(parameters["W1"]))
# print("b1 = " + str(parameters["b1"]))
# print("W2 = " + str(parameters["W2"]))
# print("b2 = " + str(parameters["b2"]))
# W1 = [[0.01624345 - 0.00611756 - 0.00528172]
#       [-0.01072969  0.00865408 - 0.02301539]]
# b1 = [[0.]
#       [0.]]
# W2 = [[0.01744812 - 0.00761207]]
# b2 = [[0.]]
# Expected
# output:
#
# W1[[0.01624345 - 0.00611756 - 0.00528172][-0.01072969
# 0.00865408 - 0.02301539]]
# b1[[0.][0.]]
# W2[[0.01744812 - 0.00761207]]
# b2[[0.]]
# 3.2 - L - layer
# Neural
# Network
# The
# initialization
# for a deeper L-layer neural network is more complicated because there are many more weight matrices and bias vectors.When completing the  initialize_parameters_deep, you should make sure that your dimensions match between each layer.Recall that  n[l]n[l] is the number of units in layer  ll.Thus for example if the size of our input  XX is (12288, 209)(12288, 209)  (with  m=209m=209  examples) then:
#
# Shape
# of
# W
# Shape
# of
# b
# Activation
# Shape
# of
# Activation
# Layer
# 1	(n[1] ,12288)(n[1] ,12288) 	 (n[1] ,1)(n[1] ,1) 	 Z[1 ] =W[1
# ] X +b[1
# ]Z[1 ] =W[1
# ] X +b[1] 	 (n[1] ,209)(n[1] ,209)
# Layer 2	 (n[2] ,n[1])(n[2] ,n[1]) 	 (n[2] ,1)(n[2] ,1) 	 Z[2 ] =W[2
# ]A[1 ] +b[2
# ]Z[2 ] =W[2
# ]A[1 ] +b[2] 	 (n[2] ,209)(n[2] ,209)
# ⋮⋮ 	 ⋮⋮ 	 ⋮⋮ 	 ⋮⋮ 	 ⋮⋮
# Layer L- 1	(n[L−1] ,n[L−2])(n[L−1] ,n[L−2]) 	 (n[L−1] ,1)(n[L−1] ,1) 	 Z[L−1]=W[L−1]A[L−2]+b[L−1]Z[L−1]=W[L−1]A[L−2]+b[L−1] 	 (n[L−1] ,209)(n[L−1] ,209)
# Layer L	 (n[L] ,n[L−1])(n[L] ,n[L−1]) 	 (n[L] ,1)(n[L] ,1) 	 Z[L ] =W[L
# ]A[L−1]+b[L
# ]Z[L ] =W[L
# ]A[L−1]+b[L] 	 (n[L] ,209)(n[L] ,209)
# Remember that when we compute  W X +bW X +b  in python, it carries out broadcasting. For example, if: W=jmpknqlorX= adgbehcfib= stu(2)
# (2)W
# = [ jklmnopqr]X
# = [ abcdefghi]b \
# = [ stu]
#
# Then
# WX+ b WX+ b
# will
# be:
#
# WX+ b = (ja+ k d+ l g)+ s (ma+ n d+ o g)+ t (pa+ q d+ r g)+ u (jb+ k e+ l h)+ s (mb+ n e+ o h)+ t (p
#     b+ q e+ r h)+ u (jc+ k f+ l i)+ s (mc+ n f+ o i)+ t (pc+ q f+ r i)+ u (3)
# (3)W
# X+ b = [ (ja+ k d+ l g)+ s (jb+ k e+ l h)+ s (jc+ k f+ l i)+ s (ma+ n d+ o g)+ t (mb+ n e+ o h)+ t (mc+ n f+ o i)+ t (p
#     a+ q d+ r g)+ u (pb+ q e+ r h)+ u (pc+ q f+ r i)+ u ]
#
# Exercise: Implement
# initialization
# for an L- l ayer Neural Network.
#
# Instructions:
#
# The
# model'
# s structure is [LINEAR -> RELU]  ××  (L-1) -> LINEAR -> SIGMOID. I.e., it has  L−1L−1  layers using a ReLU activation function followed by an output layer with a sigmoid activation function.
# Use
# random
# initialization
# for the weight matrices.Use np.random.rand(shape) * 0.01.
# Use
# zeros
# initialization
# for the biases.Use np.zeros(shape).
# We
# will
# store  n
# [l
# ]n[l] , the number of units in different layers, in a variable layer_dims. For example, the layer_dims for the "Planar Data classification model" from last week would have been [2 ,4 ,1]: There were two inputs, one hidden layer with 4 hidden units, and an output layer with 1 output unit. Thus means W1's shape was (4,2), b1 was (4,1), W2 was (1,4) and b2 was (1,1). Now you will generalize this to  LL  layers!
# Here is the implementation for  L=1L=1  (one layer neural network). It should inspire you to implement the general case (L-layer neural network).
# if L == 1:
#     parameters["W" + str(L)] = np.random.randn(layer_dims[1], layer_dims[0]) * 0.01
#     parameters["b" + str(L)] = np.zeros((layer_dims[1], 1))
#
# l
# # GRADED FUNCTION: initialize_parameters_deep
# ​
# def initialize_parameters_deep(layer_dims):
#     """
#     Arguments:
#     layer_dims -- python array (list) containing the dimensions of each layer in our network
#
#     Returns:
#     parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
#                     Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
#                     bl -- bias vector of shape (layer_dims[l], 1)
#     """
#
#     np.random.seed(3)
#     parameters = {}
#     L = len(layer_dims)            # number of layers in the network
# ​
# for l in range(1, L):
#     ### START CODE HERE ### (≈ 2 lines of code)
#     parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[ l -1]) * 0.01
#     parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
#     ### END CODE HERE ###
#
#     asser t(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[ l -1]))
#     asser t(parameters['b' + str(l)].shape == (layer_dims[l], 1))
# ​
#
# return parameters
#
# parameters = initialize_parameters_deep([5 ,4 ,3])
# print("W1 = " + str(parameters["W1"]))
# print("b1 = " + str(parameters["b1"]))
# print("W2 = " + str(parameters["W2"]))
# print("b2 = " + str(parameters["b2"]))
# W1 = [[ 0.01788628  0.0043651   0.00096497 -0.01863493 - 0.00277388]
#       [-0.00354759 - 0.00082741 - 0.00627001 - 0.00043818 - 0.00477218]
#       [-0.01313865  0.00884622  0.00881318  0.01709573  0.00050034]
# [-0.00404677 - 0.0054536 - 0.01546477
# 0.00982367 - 0.01101068]]
# b1 = [[0.]
#       [0.]
#       [0.]
#       [0.]]
# W2 = [[-0.01185047 - 0.0020565   0.01486148  0.00236716]
#       [-0.01023785 - 0.00712993  0.00625245 - 0.00160513]
# [-0.00768836 - 0.00230031
# 0.00745056
# 0.01976111]]
# b2 = [[0.]
#       [0.]
#       [0.]]
# Expected
# output:
#
# W1[[0.01788628 0.0043651 0.00096497 - 0.01863493 - 0.00277388][
#     -0.00354759 - 0.00082741 - 0.00627001 - 0.00043818 - 0.00477218][-0.01313865
# 0.00884622
# 0.00881318
# 0.01709573
# 0.00050034] [-0.00404677 - 0.0054536 - 0.01546477 0.00982367 - 0.01101068]]
# b1[[0.][0.][0.][0.]]
# W2[[-0.01185047 - 0.0020565 0.01486148 0.00236716][-0.01023785 - 0.00712993
# 0.00625245 - 0.00160513] [-0.00768836 - 0.00230031 0.00745056 0.01976111]]
# b2[[0.][0.][0.]]
# 4 - Forward
# propagation
# module
# 4.1 - Linear
# Forward
# Now
# that
# you
# have
# initialized
# your
# parameters, you
# will
# do
# the
# forward
# propagation
# module.You
# will
# start
# by
# implementing
# some
# basic
# functions
# that
# you
# will
# use
# later
# when
# implementing
# the
# model.You
# will
# complete
# three
# functions in this
# order:
#
# LINEAR
# LINEAR -> ACTIVATION
# where
# ACTIVATION
# will
# be
# either
# ReLU or Sigmoid.
# [LINEAR -> RELU]  ××  (L - 1) -> LINEAR -> SIGMOID(whole
# model)
# The
# linear
# forward
# module(vectorized
# over
# all
# the
# examples) computes
# the
# following
# equations:
#
# Z[l] = W[l]
# A[l−1]+b[l](4)
# (4)
# Z[l] = W[l]
# A[l−1]+b[l]
#
# where
# A[0] = XA[0] = X.
#
# Exercise: Build
# the
# linear
# part
# of
# forward
# propagation.
#
# Reminder: The
# mathematical
# representation
# of
# this
# unit is Z[l] = W[l]
# A[l−1]+b[l]
# Z[l] = W[l]
# A[l−1]+b[l].You
# may
# also
# find
# np.dot()
# useful.If
# your
# dimensions
# don
# 't match, printing W.shape may help.
#
# + b
# # GRADED FUNCTION: linear_forward
# ​
#
# def linear_forward(A, W, b):
#     """
#     Implement the linear part of a layer's forward propagation.
# ​
#     Arguments:
#     A -- activations from previous layer (or input data): (size of previous layer, number of examples)
#     W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
#     b -- bias vector, numpy array of shape (size of the current layer, 1)
# ​
#     Returns:
#     Z -- the input of the activation function, also called pre-activation parameter
#     cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
#     """
#
#     ### START CODE HERE ### (≈ 1 line of code)
#     Z = np.matmul(W, A) + b
#     ### END CODE HERE ###
#
#     assert (Z.shape == (W.shape[0], A.shape[1]))
#     cache = (A, W, b)
#
#     return Z, cache
#
#
# linear_forward(A, W, b)
# A, W, b = linear_forward_test_case()
# ​
# Z, linear_cache = linear_forward(A, W, b)
# print("Z = " + str(Z))
# Z = [[3.26295337 - 1.23429987]]
# Expected
# output:
#
# Z[[3.26295337 - 1.23429987]]
# 4.2 - Linear - Activation
# Forward
# In
# this
# notebook, you
# will
# use
# two
# activation
# functions:
#
# Sigmoid: σ(Z) = σ(WA + b) = 11 + e−(WA + b)
# σ(Z) = σ(WA + b) = 11 + e−(WA + b).We
# have
# provided
# you
# with the sigmoid function.This function returns two items: the
# activation
# value
# "a" and a
# "cache"
# that
# contains
# "Z"(it
# 's what we will feed in to the corresponding backward function). To use it you could just call:
#
# A, activation_cache = sigmoid(Z)
# ReLU: The
# mathematical
# formula
# for ReLu is A=RELU(Z)=max(0, Z)A=RELU(Z)=max(0, Z).We have provided you with the relu function.This function returns two items: the
# activation
# value
# "A" and a
# "cache"
# that
# contains
# "Z"(it
# 's what we will feed in to the corresponding backward function). To use it you could just call:
#
# A, activation_cache = relu(Z)
# For
# more
# convenience, you
# are
# going
# to
# group
# two
# functions(Linear and Activation)
# into
# one
# function(LINEAR->ACTIVATION).Hence, you
# will
# implement
# a
# function
# that
# does
# the
# LINEAR
# forward
# step
# followed
# by
# an
# ACTIVATION
# forward
# step.
#
# Exercise: Implement
# the
# forward
# propagation
# of
# the
# LINEAR->ACTIVATION
# layer.Mathematical
# relation is: A[l] = g(Z[l]) = g(W[l]
# A[l−1]+b[l])A[l] = g(Z[l]) = g(W[l]
# A[l−1]+b[l])  where
# the
# activation
# "g"
# can
# be
# sigmoid() or relu().Use
# linear_forward() and the
# correct
# activation
# function.
#
#     linear_activation_forward(A_prev, W, b, activation)
# # GRADED FUNCTION: linear_activation_forward
# ​
#
# def linear_activation_forward(A_prev, W, b, activation):
#     """
#     Implement the forward propagation for the LINEAR->ACTIVATION layer
# ​
#     Arguments:
#     A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
#     W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
#     b -- bias vector, numpy array of shape (size of the current layer, 1)
#     activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
# ​
#     Returns:
#     A -- the output of the activation function, also called the post-activation value
#     cache -- a python dictionary containing "linear_cache" and "activation_cache";
#              stored for computing the backward pass efficiently
#     """
#
#     if activation == "sigmoid":
#         # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
#         ### START CODE HERE ### (≈ 2 lines of code)
#         Z, linear_cache = linear_forward(A_prev, W, b)
#         A, activation_cache = sigmoid(Z)
#         ### END CODE HERE ###
#
#     elif activation == "relu":
#         # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
#         ### START CODE HERE ### (≈ 2 lines of code)
#         Z, linear_cache = linear_forward(A_prev, W, b)
#         A, activation_cache = relu(Z)
#         ### END CODE HERE ###
#
#     assert (A.shape == (W.shape[0], A_prev.shape[1]))
#     cache = (linear_cache, activation_cache)
#
# ​
# return A, cache
#
# A_prev, W, b = linear_activation_forward_test_case()
# ​
# A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation="sigmoid")
# print("With sigmoid: A = " + str(A))
# ​
# A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation="relu")
# print("With ReLU: A = " + str(A))
# With
# sigmoid: A = [[0.96890023  0.11013289]]
# With
# ReLU: A = [[3.43896131  0.]]
# Expected
# output:
#
# With
# sigmoid: A[[0.96890023 0.11013289]]
# With
# ReLU: A[[3.43896131 0.]]
# Note: In
# deep
# learning, the
# "[LINEAR->ACTIVATION]"
# computation is counted as a
# single
# layer in the
# neural
# network, not two
# layers.
#
# d) L - Layer
# Model
# For
# even
# more
# convenience
# when
# implementing
# the
# LL - layer
# Neural
# Net, you
# will
# need
# a
# function
# that
# replicates
# the
# previous
# one(linear_activation_forward
# with RELU)  L−1L−1  times, then follows that with one linear_activation_forward with SIGMOID.
#
# Figure 2: [LINEAR -> RELU]  ××  (L - 1) -> LINEAR -> SIGMOID
# model
#
# Exercise: Implement
# the
# forward
# propagation
# of
# the
# above
# model.
#
# Instruction: In
# the
# code
# below, the
# variable
# AL
# will
# denote
# A[L] = σ(Z[L]) = σ(W[L]
# A[L−1]+b[L])A[L] = σ(Z[L]) = σ(W[L]
# A[L−1]+b[L]).(This is sometimes also called Yhat, i.e., this is Ŷ Y ^.)
#
# Tips:
#
# Use
# the
# functions
# you
# had
# previously
# written
# Use
# a
# for loop to replicate[LINEAR->RELU] (L-1) times
# Don't forget to keep track of the caches in the "caches" list. To add a new value c to a list, you can use list.append(c).
#
# ache
# # GRADED FUNCTION: L_model_forward
# ​
#
#
# def L_model_forward(X, parameters):
#     """
#     Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation
#
#     Arguments:
#     X -- data, numpy array of shape (input size, number of examples)
#     parameters -- output of initialize_parameters_deep()
#
#     Returns:
#     AL -- last post-activation value
#     caches -- list of caches containing:
#                 every cache of linear_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
#                 the cache of linear_sigmoid_forward() (there is one, indexed L-1)
#     """
#
# ​
# caches = []
# A = X
# L = len(parameters) // 2  # number of layers in the neural network
#
# # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
# for l in range(1, L):
#     A_prev = A
#     ### START CODE HERE ### (≈ 2 lines of code)
#     A, cache = linear_activation_forward(A_prev, parameters['W' + str(l)], parameters['b' + str(l)], activation="relu")
#     caches.append(cache)
#     ### END CODE HERE ###
#
# # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
# ### START CODE HERE ### (≈ 2 lines of code)
# AL, cache = linear_activation_forward(A, parameters['W' + str(L)], parameters['b' + str(L)], activation="sigmoid")
# caches.append(cache)
# ### END CODE HERE ###
#
# assert (AL.shape == (1, X.shape[1]))
#
# return AL, caches
#
# X, parameters = L_model_forward_test_case_2hidden()
# AL, caches = L_model_forward(X, parameters)
# print("AL = " + str(AL))
# print("Length of caches list = " + str(len(caches)))
# AL = [[0.03921668  0.70498921  0.19734387  0.04728177]]
# Length
# of
# caches
# list = 3
# AL[[0.03921668 0.70498921 0.19734387 0.04728177]]
# Length
# of
# caches
# list
# 3
# Great! Now
# you
# have
# a
# full
# forward
# propagation
# that
# takes
# the
# input
# X and outputs
# a
# row
# vector
# A[L]
# A[L]
# containing
# your
# predictions.It
# also
# records
# all
# intermediate
# values in "caches".Using
# A[L]
# A[L], you
# can
# compute
# the
# cost
# of
# your
# predictions.
#
# 5 - Cost
# function
# Now
# you
# will
# implement
# forward and backward
# propagation.You
# need
# to
# compute
# the
# cost, because
# you
# want
# to
# check if your
# model is actually
# learning.
#
# Exercise: Compute
# the
# cross - entropy
# cost
# JJ, using
# the
# following
# formula:
# −1
# m∑i = 1
# m(y(i)
# log(a[L](i)) + (1−y(i))log(1−a[L](i)))(7)
# (7)−1
# m∑i = 1
# m(y(i)
# log⁡(a[L](i)) + (1−y(i))log⁡(1−a[L](i)))
#
#
# # GRADED FUNCTION: compute_cost
# ​
#
# def compute_cost(AL, Y):
#     """
#     Implement the cost function defined by equation (7).
# ​
#     Arguments:
#     AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
#     Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)
# ​
#     Returns:
#     cost -- cross-entropy cost
#     """
#
#     m = Y.shape[1]
#
# ​
# # Compute loss from aL and y.
# ### START CODE HERE ### (≈ 1 lines of code)
#
# cost = - np.sum(1 / m * (np.multiply(Y, np.log(AL)) + np.multiply((1 - Y), np.log(1 - AL))))
#
# ### END CODE HERE ###
#
# cost = np.squeeze(cost)  # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
# assert (cost.shape == ())
#
# return cost
#
# Y, AL = compute_cost_test_case()
# ​
# print("cost = " + str(compute_cost(AL, Y)))
# cost = 0.414931599615
# Expected
# Output:
#
# cost
# 0.41493159961539694
# 6 - Backward
# propagation
# module
# Just
# like
# with forward propagation, you will implement helper functions for backpropagation.Remember that back propagation is used to calculate the gradient of the loss function with respect to the parameters.
#
# Reminder:
#
# Figure
# 3: Forward and Backward
# propagation
# for LINEAR->RELU->LINEAR->SIGMOID
# The
# purple
# blocks
# represent
# the
# forward
# propagation, and the
# red
# blocks
# represent
# the
# backward
# propagation.
# Now, similar
# to
# forward
# propagation, you
# are
# going
# to
# build
# the
# backward
# propagation in three
# steps:
#
# LINEAR
# backward
# LINEAR -> ACTIVATION
# backward
# where
# ACTIVATION
# computes
# the
# derivative
# of
# either
# the
# ReLU or sigmoid
# activation
# [LINEAR -> RELU]  ××  (L - 1) -> LINEAR -> SIGMOID
# backward(whole
# model)
# 6.1 - Linear
# backward
# For
# layer
# ll, the
# linear
# part is: Z[l] = W[l]
# A[l−1]+b[l]
# Z[l] = W[l]
# A[l−1]+b[l](followed
# by
# an
# activation).
#
# Suppose
# you
# have
# already
# calculated
# the
# derivative
# dZ[l] =∂∂Z[l]
# dZ[l] =∂L∂Z[l].You
# want
# to
# get(dW[l], db[l]
# dA[l−1])(dW[l], db[l]dA[l−1]).
#
# Figure
# 4
# The
# three
# outputs(dW[l], db[l], dA[l])(dW[l], db[l], dA[l])
# are
# computed
# using
# the
# input
# dZ[l]
# dZ[l].Here
# are
# the
# formulas
# you
# need:
# dW[l] =∂∂W[l] = 1
# mdZ[l]
# A[l−1]T(8)
# (8)
# dW[l] =∂L∂W[l] = 1
# mdZ[l]
# A[l−1]T
#
# db[l] =∂∂b[l] = 1
# m∑i = 1
# mdZ[l](i)(9)
# (9)
# db[l] =∂L∂b[l] = 1
# m∑i = 1
# mdZ[l](i)
#
# dA[l−1]=∂∂A[l−1]=W[l]
# TdZ[l](10)
# (10)
# dA[l−1]=∂L∂A[l−1]=W[l]
# TdZ[l]
#
# Exercise: Use
# the
# 3
# formulas
# above
# to
# implement
# linear_backward().
#
# rue
# # GRADED FUNCTION: linear_backward
# ​
#
# def linear_backward(dZ, cache):
#     """
#     Implement the linear portion of backward propagation for a single layer (layer l)
# ​
#     Arguments:
#     dZ -- Gradient of the cost with respect to the linear output (of current layer l)
#     cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer
# ​
#     Returns:
#     dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
#     dW -- Gradient of the cost with respect to W (current layer l), same shape as W
#     db -- Gradient of the cost with respect to b (current layer l), same shape as b
#     """
#     A_prev, W, b = cache
#     m = A_prev.shape[1]
#
# ​
# ### START CODE HERE ### (≈ 3 lines of code)
# dW = (1 / m) * np.matmul(dZ, np.transpose(A_prev))
# db = (1 / m) * np.sum(dZ, axis=1, keepdims=True)
# dA_prev = np.matmul(np.transpose(W), dZ)
# ### END CODE HERE ###
#
# assert (dA_prev.shape == A_prev.shape)
# assert (dW.shape == W.shape)
# assert (db.shape == b.shape)
#
# return dA_prev, dW, db
#
# dZ, linear_cache)
# # Set up some test inputs
# dZ, linear_cache = linear_backward_test_case()
# ​
# dA_prev, dW, db = linear_backward(dZ, linear_cache)
# print("dA_prev = " + str(dA_prev))
# print("dW = " + str(dW))
# print("db = " + str(db))
# dA_prev = [[0.51822968 - 0.19517421]
# [-0.40506361  0.15255393]
# [2.37496825 - 0.89445391]]
# dW = [[-0.10076895  1.40685096  1.64992505]]
# db = [[0.50629448]]
# Expected
# Output:
#
# dA_prev[[0.51822968 - 0.19517421][-0.40506361
# 0.15255393] [2.37496825 - 0.89445391]]
# dW[[-0.10076895 1.40685096 1.64992505]]
# db[[0.50629448]]
# 6.2 - Linear - Activation
# backward
# Next, you
# will
# create
# a
# function
# that
# merges
# the
# two
# helper
# functions: linear_backward and the
# backward
# step
# for the activation linear_activation_backward.
#
# To help you implement linear_activation_backward, we provided two backward functions:
#
#     sigmoid_backward: Implements
# the
# backward
# propagation
# for SIGMOID unit.You can call it as follows:
#     dZ = sigmoid_backward(dA, activation_cache)
# relu_backward: Implements
# the
# backward
# propagation
# for RELU unit.You can call it as follows:
#     dZ = relu_backward(dA, activation_cache)
# If
# g(.)g(.) is the
# activation
# function, sigmoid_backward and relu_backward
# compute
# dZ[l] = dA[l]∗g′(Z[l])(11)
# (11)
# dZ[l] = dA[l]∗g′(Z[l])
#
# .
#
# Exercise: Implement
# the
# backpropagation
# for the LINEAR->ACTIVATION layer.
#
# linear_activation_backward(dA, cache, activation)
# # GRADED FUNCTION: linear_activation_backward
# ​
#
#
# def linear_activation_backward(dA, cache, activation):
#     """
#     Implement the backward propagation for the LINEAR->ACTIVATION layer.
#
#     Arguments:
#     dA -- post-activation gradient for current layer l
#     cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
#     activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
#
#     Returns:
#     dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
#     dW -- Gradient of the cost with respect to W (current layer l), same shape as W
#     db -- Gradient of the cost with respect to b (current layer l), same shape as b
#     """
#     linear_cache, activation_cache = cache
#
#     if activation == "relu":
#         ### START CODE HERE ### (≈ 2 lines of code)
#         dZ = relu_backward(dA, activation_cache)
#         dA_prev, dW, db = linear_backward(dZ, linear_cache)
#         ### END CODE HERE ###
#
#     elif activation == "sigmoid":
#         ### START CODE HERE ### (≈ 2 lines of code)
#         dZ = sigmoid_backward(dA, activation_cache)
#         dA_prev, dW, db = linear_backward(dZ, linear_cache)
#         ### END CODE HERE ###
#
#     return dA_prev, dW, db
#
#
# AL, linear_activation_cache = linear_activation_backward_test_case()
# ​
# dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation="sigmoid")
# print("sigmoid:")
# print("dA_prev = " + str(dA_prev))
# print("dW = " + str(dW))
# print("db = " + str(db) + "\n")
# ​
# dA_prev, dW, db = linear_activation_backward(AL, linear_activation_cache, activation="relu")
# print("relu:")
# print("dA_prev = " + str(dA_prev))
# print("dW = " + str(dW))
# print("db = " + str(db))
# sigmoid:
# dA_prev = [[0.11017994  0.01105339]
#            [0.09466817  0.00949723]
# [-0.05743092 - 0.00576154]]
# dW = [[0.10266786  0.09778551 - 0.01968084]]
# db = [[-0.05729622]]
#
# relu:
# dA_prev = [[0.44090989  0.]
#            [0.37883606  0.]
# [-0.2298228
# 0.]]
# dW = [[0.44513824  0.37371418 - 0.10478989]]
# db = [[-0.20837892]]
# Expected
# output
# with sigmoid:
#
# dA_prev[[0.11017994 0.01105339][0.09466817
# 0.00949723] [-0.05743092 - 0.00576154]]
# dW[[0.10266786 0.09778551 - 0.01968084]]
# db[[-0.05729622]]
# Expected
# output
# with relu:
#
# dA_prev[[0.44090989 0.][0.37883606
# 0.] [-0.2298228 0.]]
# dW[[0.44513824 0.37371418 - 0.10478989]]
# db[[-0.20837892]]
# 6.3 - L - Model
# Backward
# Now
# you
# will
# implement
# the
# backward
# function
# for the whole network.Recall that when you implemented the L_model_forward function, at each iteration, you stored a cache which contains (X, W, b, and z).In the back propagation module, you will use those variables to compute the gradients.Therefore, in the L_model_backward function, you will iterate through all the hidden layers backward, starting from layer  LL.On each step, you will use the cached values for layer  ll  to backpropagate through layer  ll.Figure 5 below shows the backward pass.
#
# Figure
# 5: Backward
# pass
# Initializing
# backpropagation: To
# backpropagate
# through
# this
# network, we
# know
# that
# the
# output is, A[L] = σ(Z[L])
# A[L] = σ(Z[L]).Your
# code
# thus
# needs
# to
# compute
# dAL =∂∂A[L] =∂L∂A[L].To
# do
# so, use
# this
# formula(derived
# using
# calculus
# which
# you
# don
# 't need in-depth knowledge of):
#
# dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))  # derivative of cost with respect to AL
# You
# can
# then
# use
# this
# post - activation
# gradient
# dAL
# to
# keep
# going
# backward.As
# seen in Figure
# 5, you
# can
# now
# feed in dAL
# into
# the
# LINEAR->SIGMOID
# backward
# function
# you
# implemented(which
# will
# use
# the
# cached
# values
# stored
# by
# the
# L_model_forward
# function).After
# that, you
# will
# have
# to
# use
# a
# for loop to iterate through all the other layers using the LINEAR->RELU backward function.You should store each dA, dW, and db in the grads dictionary.To do so, use this formula:
#
#     grads["dW" + str(l)] = dW[l](15)
#     (15)
# grads["dW" + str(l)] = dW[l]
#
# For
# example,
# for l=3l=3  this would store  dW[l]dW[l] in grads["dW3"].
#
# Exercise: Implement
# backpropagation
# for the[LINEAR->RELU]  ××  (L-1) -> LINEAR -> SIGMOID model.
#
# 2
# # GRADED FUNCTION: L_model_backward
# ​
#
#
# def L_model_backward(AL, Y, caches):
#     """
#     Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group
#
#     Arguments:
#     AL -- probability vector, output of the forward propagation (L_model_forward())
#     Y -- true "label" vector (containing 0 if non-cat, 1 if cat)
#     caches -- list of caches containing:
#                 every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2)
#                 the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1])
#
#     Returns:
#     grads -- A dictionary with the gradients
#              grads["dA" + str(l)] = ...
#              grads["dW" + str(l)] = ...
#              grads["db" + str(l)] = ...
#     """
#     grads = {}
#     L = len(caches)  # the number of layers
#     m = AL.shape[1]
#     Y = Y.reshape(AL.shape)  # after this line, Y is the same shape as AL
#
#     # Initializing the backpropagation
#     ### START CODE HERE ### (1 line of code)
#     dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
#     ### END CODE HERE ###
#
#     # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]
#     ### START CODE HERE ### (approx. 2 lines)
#     current_cache = caches[L - 1]
#     grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache,
#                                                                                                   activation="sigmoid")
#     ### END CODE HERE ###
#
#     for l in reversed(range(L - 1)):
#         # lth layer: (RELU -> LINEAR) gradients.
#         # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]
#         ### START CODE HERE ### (approx. 5 lines)
#         current_cache = caches[l]
#         dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 2)], current_cache,
#                                                                     activation="relu")
#         grads["dA" + str(l + 1)] = grads["dA" + str(l + 2)]
#         grads["dW" + str(l + 1)] = dW_temp
#         grads["db" + str(l + 1)] = db_temp
#         ### END CODE HERE ###
#
# ​
# return grads
#
# AL, Y_assess, caches = L_model_backward_test_case()
# grads = L_model_backward(AL, Y_assess, caches)
# print_grads(grads)
# dW1 = [[0.41010002  0.07807203  0.13798444  0.10502167]
#        [0.          0.          0.          0.]
# [0.05283652
# 0.01005865
# 0.01777766
# 0.0135308]]
# db1 = [[-0.22007063]
#        [0.]
#        [-0.02835349]]
# dA1 = [[0.12913162 - 0.44014127]
#        [-0.14175655  0.48317296]
# [0.01663708 - 0.05670698]]
# Expected
# Output
#
# dW1[[0.41010002 0.07807203 0.13798444 0.10502167][0.
# 0.
# 0.
# 0.] [0.05283652 0.01005865 0.01777766 0.0135308]]
# db1[[-0.22007063][0.][-0.02835349]]
# dA1[[0.12913162 - 0.44014127][-0.14175655
# 0.48317296] [0.01663708 - 0.05670698]]
# 6.4 - Update
# Parameters
# In
# this
# section
# you
# will
# update
# the
# parameters
# of
# the
# model, using
# gradient
# descent:
#
# W[l] = W[l]−α
# dW[l](16)
# (16)
# W[l] = W[l]−α
# dW[l]
#
# b[l] = b[l]−α
# db[l](17)
# (17)
# b[l] = b[l]−α
# db[l]
#
# where
# αα is the
# learning
# rate.After
# computing
# the
# updated
# parameters, store
# them in the
# parameters
# dictionary.
#
# Exercise: Implement
# update_parameters()
# to
# update
# your
# parameters
# using
# gradient
# descent.
#
# Instructions: Update
# parameters
# using
# gradient
# descent
# on
# every
# W[l]
# W[l] and b[l]
# b[l]
# for l=1, 2, ..., Ll=1, 2, ..., L.
#
#
# # GRADED FUNCTION: update_parameters
# ​
#
# def update_parameters(parameters, grads, learning_rate):
#     """
#     Update parameters using gradient descent
#
#     Arguments:
#     parameters -- python dictionary containing your parameters
#     grads -- python dictionary containing your gradients, output of L_model_backward
#
#     Returns:
#     parameters -- python dictionary containing your updated parameters
#                   parameters["W" + str(l)] = ...
#                   parameters["b" + str(l)] = ...
#     """
#
#     L = len(parameters) // 2  # number of layers in the neural network
#
# ​
# # Update rule for each parameter. Use a for loop.
# ### START CODE HERE ### (≈ 3 lines of code)
# for l in range(1, L + 1):
#     parameters["W" + str(l)] += - learning_rate * grads['dW' + str(l)]
#     parameters["b" + str(l)] += - learning_rate * grads['db' + str(l)]
# ​
# ### END CODE HERE ###
# return parameters
#
# parameters, grads = update_parameters_test_case()
# parameters = update_parameters(parameters, grads, 0.1)
# ​
# print("W1 = " + str(parameters["W1"]))
# print("b1 = " + str(parameters["b1"]))
# print("W2 = " + str(parameters["W2"]))
# print("b2 = " + str(parameters["b2"]))
# W1 = [[-0.59562069 - 0.09991781 - 2.14584584  1.82662008]
#       [-1.76569676 - 0.80627147  0.51115557 - 1.18258802]
# [-1.0535704 - 0.86128581
# 0.68284052
# 2.20374577]]
# b1 = [[-0.04659241]
#       [-1.28888275]
#       [0.53405496]]
# W2 = [[-0.55569196  0.0354055   1.32964895]]
# b2 = [[-0.84610769]]
# Expected
# Output:
#
# W1[[-0.59562069 - 0.09991781 - 2.14584584 1.82662008][-1.76569676 - 0.80627147
# 0.51115557 - 1.18258802] [-1.0535704 - 0.86128581 0.68284052 2.20374577]]
# b1[[-0.04659241][-1.28888275][0.53405496]]
# W2[[-0.55569196 0.0354055 1.32964895]]
# b2[[-0.84610769]]
# 7 - Conclusion
# Congrats
# on
# implementing
# all
# the
# functions
# required
# for building a deep neural network!
#
# We
# know
# it
# was
# a
# long
# assignment
# but
# going
# forward
# it
# will
# only
# get
# better.The
# next
# part
# of
# the
# assignment is easier.
#
# In
# the
# next
# assignment
# you
# will
# put
# all
# these
# together
# to
# build
# two
# models:
#
# A
# two - layer
# neural
# network
# An
# L - layer
# neural
# network
# You
# will in fact
# use
# these
# models
# to
# classify
# cat
# vs
# non - cat
# images!
#
#
# ​