吴恩达深度学习1.2练习_Neural Networks and Deep Learning

版权声明：本文为博主原创文章，未经博主允许不得转载。 https://blog.csdn.net/weixin_42432468

学习心得：
1、每周的视频课程看一到两遍
2、做笔记

3、做每周的作业练习，这个里面的含金量非常高。掌握后一定要自己敲一遍，这样以后用起来才能得心应手。

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二分类

• 预测输出为正类（+1）的概率：
  $$\hat{y}=P(y=1\mid x)\text{\hspace{1em}(1)}$$
  $$P(y\mid x)={\hat{y}}^y(1-\hat{y}{)}^{(1-\hat{y})}\text{\hspace{1em}(2)}$$
  $$sigmoid(z)=\sigma (z)=\frac{1}{1+e^{-z}}\text{\hspace{1em}(3)}$$

• 单样本正向传播：
  $$a=\hat{y}=\sigma (w^{T}x+b)\text{\hspace{1em}(1)}$$
  $$a\prime =a(1-a)\text{\hspace{1em}(2)}$$
  $$\mathcal{L}(a,y)=-y\log (a)-(1-y)\log (1-a)\text{\hspace{1em}(3)}$$

• 单样本反向传播：
  $$\mathrm{d}a=\frac{\partial L}{\partial a}=-\frac{y}{a}+\frac{1-y}{1-a}\text{\hspace{1em}(1)}$$
  $$\mathrm{d}z=\frac{\partial L}{\partial a}\times \frac{\partial a}{\partial z}=a-y\text{\hspace{1em}(2)}$$
  $$\mathrm{d}w=\mathrm{d}z\times \frac{\partial z}{\partial w}=(a-y)x\text{\hspace{1em}(3)}$$
  $$\mathrm{d}b=\mathrm{d}z\times \frac{\partial z}{\partial b}=(a-y)\text{\hspace{1em}(4)}$$

• 多样本的cost function和反向传播：
  $$J=\frac{1}{m}\sum _{i=1}^m\mathcal{L}(a^{(i)},y^{(i)})=-\frac{1}{m}\sum _{i=1}^m(y^{(i)}\log (a^{(i)})+(1-y^{(i)})\log (1-a^{(i)}))\text{\hspace{1em}(1)}$$

$$dw=\frac{\partial J}{\partial w}=\frac{1}{m}\sum _{i=1}^m[x^{(i)}\times (a^{(i)}-y^{(i)})]=\frac{1}{m}X(A-Y{)}^T\text{\hspace{1em}(2)}$$
$$db=\frac{\partial J}{\partial b}=\frac{1}{m}\sum _{i=1}^m(a^{(i)}-y^{(i)})\text{\hspace{1em}(3)}$$

• 梯度下降法：
  $$w=w-\alpha \times \mathrm{d}w\text{\hspace{1em}(1)}$$
  $$b=b-\alpha \times \mathrm{d}b\text{\hspace{1em}(2)}$$

Part 1：Python Basics with Numpy (optional assignment)

Building basic functions with numpy

Numpy is the main package for scientific computing in Python. It is maintained by a large community (www.numpy.org). In this exercise you will learn several key numpy functions such as np.exp, np.log, and np.reshape. You will need to know how to use these functions for future assignments.

1.1 - sigmoid function, np.exp()

Exercise: Build a function that returns the sigmoid of a real number x. Use math.exp(x) for the exponential function.

Reminder:

$sigmoid(x)=\frac{1}{1+e^{-x}}$ is sometimes also known as the logistic function. It is a non-linear function used not only in Machine Learning (Logistic Regression), but also in Deep Learning.

To refer to a function belonging to a specific package you could call it using package_name.function(). Run the code below to see an example with math.exp().

#GRADED FUNCTION:basic_sigmoid
import math

def basic_sigmoid(x):
    '''
    compute sigmoid of x
    
    Arguments:
    x --- A scalar
    
    returns:
    s --- sigmoid(x)
    '''
    
    s = 1.0/(1+1/math.exp(x))
    return s

basic_sigmoid(3)

Actually, we rarely use the “math” library in deep learning because the inputs of the functions are real numbers. In deep learning we mostly use matrices and vectors. This is why numpy is more useful.

One reason why we use “numpy” instead of “math” in Deep Learning

x = [1, 2, 3]
basic_sigmoid(x) # you will see this give an error when you run it, because x is a vector.

In fact, if $x=(x_1,x_2,...,x_n)$ is a row vector then $np.exp(x)$ will apply the exponential function to every element of x. The output will thus be: $np.exp(x)=(e^{x_1},e^{x_2},...,e^{x_n})$

import numpy as np
# example of np.exp()
x = np.array([1,2,3])
print (np.exp(x))

Furthermore, if x is a vector, then a Python operation such as $s=x+3$ or $s=\frac{1}{x}$ will output s as a vector of the same size as x.

# example of vector operation
x = np.array([1, 2, 3])
print (x + 3)

Exercise: Implement the sigmoid function using numpy.

Instructions: x could now be either a real number, a vector, or a matrix. The data structures we use in numpy to represent these shapes (vectors, matrices…) are called numpy arrays. You don’t need to know more for now.
$$\text{For }x\in {\mathbb{R}}^n\text{, }sigmoid(x)=sigmoid\left(\begin{array}{c}{x}_1\\ x_2\\ ...\\ x_n\end{array}\right)=\left(\begin{array}{c}\frac{1}{1+e^{-x_1}}\\ \frac{1}{1+e^{-x_2}}\\ ...\\ \frac{1}{1+e^{-x_n}}\end{array}\right)$$

#GRADED FUNCTION:basic_sigmoid
import numpy as np

def sigmoid(x):
    '''
    compute sigmoid of x
    
    Arguments:
    x --- A scalar
    
    returns:
    s --- sigmoid(x)
    '''
    
    s = 1.0/(1+1/np.exp(x))
    return s

x = np.array([1,2,3])
print (sigmoid(x))

1.2 - Sigmoid gradient

Exercise: Implement the function sigmoid_grad() to compute the gradient of the sigmoid function with respect to its input x. The formula is:
$$sigmoid\_derivative(x)=\sigma \prime (x)=\sigma (x)(1-\sigma (x))$$

You often code this function in two steps:

1. Set s to be the sigmoid of x. You might find your sigmoid(x) function useful.
2. Compute $\sigma \prime (x)=s(1-s)$

#GRADED FUNCTION :sigmoid_derivative
import numpy as np

def sigmoid_derivative(x):
    '''
    Compute the gradient (also called the slope or derivative) of the sigmoid function with respect to its input x.
    You can store the output of the sigmoid function into variables and then use it to calculate the gradient.

    Arguments:
    x -- A scalar or numpy array

    Return:
    ds -- Your computed gradient.
    '''

    s = 1.0/(1+1/np.exp(x))
    ds = s*(1-s)
    return ds

x = np.array([1,2,3])
print ('sigmoid_derivate(x)='+str(sigmoid_derivative(x)))

1.3 - Reshaping arrays

Two common numpy functions used in deep learning are np.shape and np.reshape().

• X.shape is used to get the shape (dimension) of a matrix/vector X.
• X.reshape(…) is used to reshape X into some other dimension.

For example, in computer science, an image is represented by a 3D array of shape $(length,height,depth=3)$. However, when you read an image as the input of an algorithm you convert it to a vector of shape $(length\ast height\ast 3,1)$. In other words, you “unroll”, or reshape, the 3D array into a 1D vector.

Exercise: Implement image2vector() that takes an input of shape (length, height, 3) and returns a vector of shape (lengthheight3, 1). For example, if you would like to reshape an array v of shape (a, b, c) into a vector of shape (a*b,c) you would do:

#v = v.reshape(v.shape[0]*v.shape[1],v.shape[2])# v.shape[0] = a ; v.shape[1] = b ; v.shape[2] = c

• Please don’t hardcode the dimensions of image as a constant. Instead look up the quantities you need with image.shape[0], etc.

#GRADED FUNCTION:image2vector
def image2vector(image):
    """
    Argument:
    image -- a numpy array of shape (length, height, depth)

    Returns:
    v -- a vector of shape (length*height*depth, 1)
    """
    v = image.reshape((image.shape[0]*image.shape[1]*image.shape[2],1))
    #v = image.reshape(image.shape[0]*image.shape[1]*image.shape[2],1)   #有区别吗？
    return v
    

# This is a 3 by 3 by 2 array, typically images will be (num_px_x, num_px_y,3) where 3 represents the RGB values
image = np.array([[[ 0.67826139,  0.29380381],
        [ 0.90714982,  0.52835647],
        [ 0.4215251 ,  0.45017551]],

       [[ 0.92814219,  0.96677647],
        [ 0.85304703,  0.52351845],
        [ 0.19981397,  0.27417313]],

       [[ 0.60659855,  0.00533165],
        [ 0.10820313,  0.49978937],
        [ 0.34144279,  0.94630077]]])
print ("image2vector(image) = " + str(image2vector(image)))

1.4 - Normalizing rows

Another common technique we use in Machine Learning and Deep Learning is to normalize our data. It often leads to a better performance because gradient descent converges faster after normalization. Here, by normalization we mean changing x to $$\frac{x}{\Vert x\Vert }$$ (dividing each row vector of x by its norm).

For example, if $$x=\left[\begin{array}{ccc}0& 3& 4\\ 2& 6& 4\end{array}\right]\text{\hspace{1em}(1)}$$ then $$\Vert x\Vert =np.linalg.norm(x,axis=1,keepdims=True)=\left[\begin{array}{c}5\\ \sqrt{56}\end{array}\right]\text{\hspace{1em}(2)}$$and $$x\_normalized=\frac{x}{\Vert x\Vert }=\left[\begin{array}{ccc}0& \frac{3}{5}& \frac{4}{5}\\ \frac{2}{\sqrt{56}}& \frac{6}{\sqrt{56}}& \frac{4}{\sqrt{56}}\end{array}\right]\text{\hspace{1em}(3)}$$ Note that you can divide matrices of different sizes and it works fine: this is called broadcasting and you’re going to learn about it in part 5.

Exercise: Implement normalizeRows() to normalize the rows of a matrix. After applying this function to an input matrix x, each row of x should be a vector of unit length (meaning length 1).

#GRADED FUNCTION:normalizeRows
def normalizeRows(x):
    x_norm = np.linalg.norm(x,axis=1,keepdims=True)  #计算每一行的长度，得到一个列向量
    x = x / x_norm  #利用numpy的广播，用矩阵与列向量相除。
    return x

# Graded Function : L1
def L1(yhat,y):
    '''
    Arguments:
    yhat -- vector of size m (predicted lables)
    y -- vector of size m (true lables)
    
    Returns:
    loss -- the value of the L1 loss function defined above
    
    '''
    
    loss = np.sum(np.abs(y-yhat))
    
    return loss

yhat = np.array([.9, 0.2, 0.1, .4, .9])
y = np.array([1, 0, 0, 1, 1])
print ('L1 = '+str(L1(yhat,y)))

# Graded Function : L2
def L2(yhat,y):
    '''
    Arguments:
    yhat -- vector of size m (predicted lables)
    y -- vector of size m (true lables)
    
    Returns:
    loss -- the value of the L2 loss function defined above
    
    '''
    
    #loss = np.sum(np.power((y-yhat),2))
    loss = np.dot((yhat-y),(yhat-y))
    return loss

yhat = np.array([.9, 0.2, 0.1, .4, .9])
y = np.array([1, 0, 0, 1, 1])
print ('L2 = '+str(L2(yhat,y)))

Part2 : Logistic Regression with a Neural Network mindset

Welcome to your first (required) programming assignment! You will build a logistic regression classifier to recognize cats. This assignment will step you through how to do this with a Neural Network mindset, and so will also hone your intuitions about deep learning.

Instructions:

• Do not use loops (for/while) in your code, unless the instructions explicitly ask you to do so.

You will learn to:

• Build the general architecture of a learning algorithm, including:
  • Initializing parameters
  • Calculating the cost function and its gradient
  • Using an optimization algorithm (gradient descent)
• Gather all three functions above into a main model function, in the right order.

2.1 - Packages

First, let’s run the cell below to import all the packages that you will need during this assignment.

• numpy is the fundamental package for scientific computing with Python.
• h5py is a common package to interact with a dataset that is stored on an H5 file.
• matplotlib is a famous library to plot graphs in Python.
• PIL and scipy are used here to test your model with your own picture at the end.

import numpy as np
import matplotlib.pyplot as plt
import h5py
import scipy
from PIL import Image
from scipy import ndimage
from lr_utils import load_dataset

%matplotlib inline
#加上这一句，就可以省去plt.show()

2.2 - Overview of the Problem set

Problem Statement: You are given a dataset (“data.h5”) containing:
- a training set of m_train images labeled as cat (y=1) or non-cat (y=0)
- a test set of m_test images labeled as cat or non-cat
- each image is of shape (num_px, num_px, 3) where 3 is for the 3 channels (RGB). Thus, each image is square (height = num_px) and (width = num_px).

You will build a simple image-recognition algorithm that can correctly classify pictures as cat or non-cat.

Let’s get more familiar with the dataset. Load the data by running the following code.

# Loading the data (cat/non-cat)
train_set_x_orig, train_set_y, test_set_x_orig, test_set_y, classes = load_dataset()

We added “_orig” at the end of image datasets (train and test) because we are going to preprocess them. After preprocessing, we will end up with train_set_x and test_set_x (the labels train_set_y and test_set_y don’t need any preprocessing).

Each line of your train_set_x_orig and test_set_x_orig is an array representing an image. You can visualize an example by running the following code. Feel free also to change the index value and re-run to see other images.

# Example of a picture
index = 24
plt.imshow(train_set_x_orig[index])
print (classes,type(classes))
print (train_set_x_orig.shape,test_set_x_orig.shape)
print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")

a = np.array([1,2,3,4,5,6,7,8,9,10,11,12])
print ('a:\n',a)
b = a.reshape((1,2,6))
print ('b:\n',b)
print ('b[0]:\n',b[0])     
'去掉最外面的中括号来计算，依次类推'
print ('b[0,1]:\n',b[0,1])
print ('b[0,1,1]:\n',b[0,1,1])
c = np.squeeze(b)  
'从数组的形状中删除单维度条目，即把shape中为1的维度去掉'
print ('c:\n',c)

Many software bugs in deep learning come from having matrix/vector dimensions that don’t fit. If you can keep your matrix/vector dimensions straight you will go a long way toward eliminating many bugs.

Exercise: Find the values for:
- m_train (number of training examples)
- m_test (number of test examples)
- num_px (= height = width of a training image)
Remember that train_set_x_orig is a numpy-array of shape (m_train, num_px, num_px, 3). For instance, you can access m_train by writing train_set_x_orig.shape[0].

#自己加一行
train_set_x_orig.shape

### START CODE HERE ### (≈ 3 lines of code)
m_train = train_set_x_orig.shape[0]
m_test = test_set_x_orig.shape[0]
num_px = train_set_x_orig.shape[1]
### END CODE HERE ###

print ("Number of training examples: m_train = " + str(m_train))
print ("Number of testing examples: m_test = " + str(m_test))
print ("Height/Width of each image: num_px = " + str(num_px))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_set_x shape: " + str(train_set_x_orig.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x shape: " + str(test_set_x_orig.shape))
print ("test_set_y shape: " + str(test_set_y.shape))

# 注：train和test的y都是行向量

Expected Output for m_train, m_test and num_px:

m_train：	209
m_test：	50
num_px：	64

For convenience, you should now reshape images of shape (num_px, num_px, 3) in a numpy-array of shape (num_px $\ast$ num_px $\ast$ 3, 1). After this, our training (and test) dataset is a numpy-array where each column represents a flattened image. There should be m_train (respectively m_test) columns.

Exercise: Reshape the training and test data sets so that images of size (num_px, num_px, 3) are flattened into single vectors of shape (num_px $\ast$ num_px $\ast$ 3, 1).

A trick when you want to flatten a matrix X of shape (a,b,c,d) to a matrix X_flatten of shape (b$\ast$c$\ast$d, a) is to use:

X_flatten = X.reshape(X.shape[0], -1).T      # X.T is the transpose of X

# Reshape the training and test examples

### START CODE HERE ### (≈ 2 lines of code)
train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T
test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T     
### END CODE HERE ###

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))
print ("train_set_y shape: " + str(train_set_y.shape))
print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))
print ("test_set_y shape: " + str(test_set_y.shape))
print ("sanity check after reshaping: " + str(train_set_x_flatten[0:6,0]))
#新增一条，查看reshape之前的数据
print ("sanity check before reshaping:\n " + str(train_set_x_orig[0,0,0:2,0:3]))

Expected Output:

train_set_x_flatten shape	(12288, 209)
train_set_y shape	(1, 209)
test_set_x_flatten shape	(12288, 50)
test_set_y shape	(1, 50)
sanity check after reshaping	[17 31 56 22 33]

To represent color images, the red, green and blue channels (RGB) must be specified for each pixel, and so the pixel value is actually a vector of three numbers ranging from 0 to 255.

One common preprocessing step in machine learning is to center and standardize your dataset, meaning that you substract the mean of the whole numpy array from each example, and then divide each example by the standard deviation of the whole numpy array. But for picture datasets, it is simpler and more convenient and works almost as well to just divide every row of the dataset by 255 (the maximum value of a pixel channel).

Let’s standardize our dataset.

train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.
print ("sanity check after standardize: \n" + str(train_set_x[0:6,0]))
print ("sanity check after standardize: \n" + str(train_set_x[0:6,0:1]))

What you need to remember:

Common steps for pre-processing a new dataset are:

• Figure out the dimensions and shapes of the problem (m_train, m_test, num_px, …)
• Reshape the datasets such that each example is now a vector of size (num_px * num_px * 3, 1)
• “Standardize” the data

2.3 - General Architecture of the learning algorithm

It’s time to design a simple algorithm to distinguish cat images from non-cat images.

You will build a Logistic Regression, using a Neural Network mindset. The following Figure explains why Logistic Regression is actually a very simple Neural Network!

Mathematical expression of the algorithm:

For one example $x^{(i)}$:
$$z^{(i)}=w^T{x}^{(i)}+b\text{\hspace{1em}(1)}$$
$${\hat{y}}^{(i)}=a^{(i)}=sigmoid(z^{(i)})\text{\hspace{1em}(2)}$$
$$\mathcal{L}(a^{(i)},y^{(i)})=-y^{(i)}\log (a^{(i)})-(1-y^{(i)})\log (1-a^{(i)})\text{\hspace{1em}(3)}$$

The cost is then computed by summing over all training examples:
$$J=\frac{1}{m}\sum _{i=1}^m\mathcal{L}(a^{(i)},y^{(i)})=-\frac{1}{m}\sum _{i=1}^m(y^{(i)}\log (a^{(i)})+(1-y^{(i)})\log (1-a^{(i)}))\text{\hspace{1em}(4)}$$

Key steps:
In this exercise, you will carry out the following steps:
- Initialize the parameters of the model
- Learn the parameters for the model by minimizing the cost
- Use the learned parameters to make predictions (on the test set)
- Analyse the results and conclude

2.4 - Building the parts of our algorithm

The main steps for building a Neural Network are:

1. Define the model structure (such as number of input features)
2. Initialize the model’s parameters
3. Loop:
   • Calculate current loss (forward propagation)
   • Calculate current gradient (backward propagation)
   • Update parameters (gradient descent)

You often build 1-3 separately and integrate them into one function we call model().

2.4.1 - Helper functions

Exercise: Using your code from “Python Basics”, implement sigmoid(). As you’ve seen in the figure above, you need to compute $$sigmoid(w^{T}x+b)=\frac{1}{1+e^{-(w^{T}x+b)}}$$to make predictions. Use np.exp().

# GRADED FUNCTION: sigmoid

def sigmoid(z):
    """
    Compute the sigmoid of z

    Arguments:
    z -- A scalar or numpy array of any size.

    Return:
    s -- sigmoid(z)
    """

    ### START CODE HERE ### (≈ 1 line of code)
    s = 1/(1+np.exp(-z))
    ### END CODE HERE ###
    
    return s

print ("sigmoid([0, 2]) = " + str(sigmoid(np.array([0,2]))))

Expected Output:

sigmoid([0, 2])

[ 0.5 0.88079708]

2.4.2 - Initializing parameters

Exercise: Implement parameter initialization in the cell below. You have to initialize w as a vector of zeros. If you don’t know what numpy function to use, look up np.zeros() in the Numpy library’s documentation.

# GRADED FUNCTION: initialize_with_zeros

def initialize_with_zeros(dim):
    """
    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.
    
    Argument:
    dim -- size of the w vector we want (or number of parameters in this case)
    
    Returns:
    w -- initialized vector of shape (dim, 1)
    b -- initialized scalar (corresponds to the bias)
    """
    
    ### START CODE HERE ### (≈ 1 line of code)
    w = np.zeros((dim,1))
    b = 0
    ### END CODE HERE ###

    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    
    return w, b

'''
assert expression [,arguments]
逻辑上等同于：
if not condition:
    raise AssertionError()

'''
#assert (len(train_set_x[1]) >=210),'x训练样本数小于210'

'''
isinstance()的用法是用来判断一个量是否是相应的类型，接受的参数一个是对象加一种类型
'''
a = 1
print(isinstance(a,int))
print(isinstance(a,float))

dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))

Expected Output:

w	[[ 0.] [ 0.]]
b	0

For image inputs, w will be of shape (num_px $\times$ num_px $\times$ 3, 1).

2.4.3 - Forward and Backward propagation

Now that your parameters are initialized, you can do the “forward” and “backward” propagation steps for learning the parameters.

Exercise: Implement a function propagate() that computes the cost function and its gradient.

Hints:

Forward Propagation:

• You get X
• You compute $A=\sigma (w^{T}X+b)=(a^{(0)},a^{(1)},...,a^{(m-1)},a^{(m)})$
• You calculate the cost function: $J=-\frac{1}{m}{\sum }_{i=1}^m[y^{(i)}\log (a^{(i)})+(1-y^{(i)})\log (1-a^{(i)})]$

Here are the two formulas you will be using:

$$dw=\frac{\partial J}{\partial w}=\frac{1}{m}\sum _{i=1}^m[x^{(i)}\times (a^{(i)}-y^{(i)})]=\frac{1}{m}X(A-Y{)}^T\text{\hspace{1em}(5)}$$
$$db=\frac{\partial J}{\partial b}=\frac{1}{m}\sum _{i=1}^m(a^{(i)}-y^{(i)})\text{\hspace{1em}(6)}$$

# GRADED FUNCTION: propagate

def propagate(w, b, X, Y):
    """
    Implement the cost function and its gradient for the propagation explained above

    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

    Return:
    cost -- negative log-likelihood cost for logistic regression
    dw -- gradient of the loss with respect to w, thus same shape as w
    db -- gradient of the loss with respect to b, thus same shape as b
    
    Tips:
    - Write your code step by step for the propagation. np.log(), np.dot()
    """
    
    m = X.shape[1]
    
    # FORWARD PROPAGATION (FROM X TO COST)
    ### START CODE HERE ### (≈ 2 lines of code)
    A = sigmoid(np.dot(w.T,X) + b)                                      # compute activation
    cost = (-1.0/m) * np.sum(np.multiply(Y, np.log(A)) + np.multiply((1-Y), np.log(1-A) ) )         # compute cost
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = (1.0/m)*np.dot(X, (A-Y).T)    
    db = (1.0/m)*np.sum(A-Y)
    ### END CODE HERE ###

    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())
    
    grads = {"dw": dw,
             "db": db}
    
    return grads, cost

w, b, X, Y = np.array([[1],[2]]), 2, np.array([[1,2],[3,4]]), np.array([[1,0]])
print (w.shape,type(w),Y.shape,type(Y))
grads, cost = propagate(w, b, X, Y)
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("cost = " + str(cost))

Expected Output:

dw	[[ 0.99993216] [ 1.99980262]]
db	0.499935230625
cost	6.000064773192205

2.4.4 - Optimization

• You have initialized your parameters.
• You are also able to compute a cost function and its gradient.
• Now, you want to update the parameters using gradient descent.

Exercise: Write down the optimization function. The goal is to learn $w$ and $b$ by minimizing the cost function $J$. For a parameter $\theta$, the update rule is $ \theta = \theta - \alpha \text{ } d\theta$, where $\alpha$ is the learning rate.

# GRADED FUNCTION: optimize

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):
    """
    This function optimizes w and b by running a gradient descent algorithm
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of shape (num_px * num_px * 3, number of examples)
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- True to print the loss every 100 steps
    
    Returns:
    params -- dictionary containing the weights w and bias b
    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function
    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.
    
    Tips:
    You basically need to write down two steps and iterate through them:
        1) Calculate the cost and the gradient for the current parameters. Use propagate().
        2) Update the parameters using gradient descent rule for w and b.
    """
    
    costs = []
    
    for i in range(num_iterations):
        
        
        # Cost and gradient calculation (≈ 1-4 lines of code)
        ### START CODE HERE ### 
        grads, cost = propagate(w, b, X, Y)
        ### END CODE HERE ###
        
        # Retrieve derivatives from grads
        dw = grads["dw"]
        db = grads["db"]
        
        # update rule (≈ 2 lines of code)
        ### START CODE HERE ###
        w = w - learning_rate * dw
        b = b - learning_rate * db
        ### END CODE HERE ###
        
        # Record the costs
        if i % 100 == 0:
            costs.append(cost)
        
        # Print the cost every 100 training examples
        if print_cost and i % 100 == 0:
            print ("Cost after iteration %i: %f" %(i, cost))
    
    params = {"w": w,
              "b": b}
    
    grads = {"dw": dw,
             "db": db}
    
    return params, grads, costs

params, grads, costs = optimize(w, b, X, Y, num_iterations= 100, learning_rate = 0.009, print_cost = False)

print ("w = " + str(params["w"]))
print ("b = " + str(params["b"]))
print ("dw = " + str(grads["dw"]))
print ("db = " + str(grads["db"]))
print ("costs = " + str(costs[-1]))

Expected Output:

b	1.55930492484
dw	[[ 0.90158428] [ 1.76250842]]

2.4.5 - Predict

Exercise: The previous function will output the learned w and b. We are able to use w and b to predict the labels for a dataset X. Implement the predict() function. There is two steps to computing predictions:

1. Calculate $\hat{Y}=A=\sigma (w^{T}X+b)$

2. Convert the entries of a into 0 (if activation <= 0.5) or 1 (if activation > 0.5), stores the predictions in a vector Y_prediction. If you wish, you can use an if/else statement in a for loop (though there is also a way to vectorize this).

# GRADED FUNCTION: predict

def predict(w, b, X):
    '''
    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)
    
    Arguments:
    w -- weights, a numpy array of size (num_px * num_px * 3, 1)
    b -- bias, a scalar
    X -- data of size (num_px * num_px * 3, number of examples)
    
    Returns:
    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X
    '''
    
    m = X.shape[1]
    Y_prediction = np.zeros((1,m))
    w = w.reshape(X.shape[0], 1)
    
    # Compute vector "A" predicting the probabilities of a cat being present in the picture
    ### START CODE HERE ### (≈ 1 line of code)
    A = sigmoid(np.dot(w.T, X) + b)
    ### END CODE HERE ###
    
    for i in range(A.shape[1]):
        
        # Convert probabilities A[0,i] to actual predictions p[0,i]
        ### START CODE HERE ### (≈ 4 lines of code)
        if A[0,i] > 0.5:
            Y_prediction[0, i] = 1
        else:
            Y_prediction[0, i] = 0
        ### END CODE HERE ###
    
    assert(Y_prediction.shape == (1, m))
    
    return Y_prediction

print ("predictions = " + str(predict(w, b, X)))

Expected Output:

predictions

[[ 1. 1.]]

What to remember:
You’ve implemented several functions that:

• Initialize (w,b)
• Optimize the loss iteratively to learn parameters (w,b):
  • computing the cost and its gradient
  • updating the parameters using gradient descent
• Use the learned (w,b) to predict the labels for a given set of examples

2.5 - Merge all functions into a model

You will now see how the overall model is structured by putting together all the building blocks (functions implemented in the previous parts) together, in the right order.

Exercise: Implement the model function. Use the following notation:
- Y_prediction for your predictions on the test set
- Y_prediction_train for your predictions on the train set
- w, costs, grads for the outputs of optimize()

# GRADED FUNCTION: model

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    """
    Builds the logistic regression model by calling the function you've implemented previously
    
    Arguments:
    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)
    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)
    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)
    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)
    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters
    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()
    print_cost -- Set to true to print the cost every 100 iterations
    
    Returns:
    d -- dictionary containing information about the model.
    """
    
    ### START CODE HERE ###
    
    # initialize parameters with zeros (≈ 1 line of code)
    w, b = initialize_with_zeros(X_train.shape[0])

    # Gradient descent (≈ 1 line of code)
    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost = False)
    
    # Retrieve parameters w and b from dictionary "parameters"
    w = parameters["w"]
    b = parameters["b"]
    
    # Predict test/train set examples (≈ 2 lines of code)
    Y_prediction_test = predict(w, b, X_test)
    Y_prediction_train = predict(w, b, X_train)

    ### END CODE HERE ###

    # Print train/test Errors
    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))
    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

    
    d = {"costs": costs,
         "Y_prediction_test": Y_prediction_test, 
         "Y_prediction_train" : Y_prediction_train, 
         "w" : w, 
         "b" : b,
         "learning_rate" : learning_rate,
         "num_iterations": num_iterations}
    
    return d

Run the following cell to train your model.

d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)

Expected Output:

Train Accuracy	99.04306220095694 %
Test Accuracy	70.0 %

Comment: Training accuracy is close to 100%. This is a good sanity check: your model is working and has high enough capacity to fit the training data. Test accuracy is 70%. It is actually not bad for this simple model, given the small dataset we used and that logistic regression is a linear classifier. But no worries, you’ll build an even better classifier next week!

Also, you see that the model is clearly overfitting the training data. Later in this specialization you will learn how to reduce overfitting, for example by using regularization. Using the code below (and changing the index variable) you can look at predictions on pictures of the test set.

# Example of a picture that was wrongly classified.
index = 1
plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))
print ("y = " + str(test_set_y[0,index]) + ", you predicted that it is a \"" + classes[int(d["Y_prediction_test"][0,index])].decode("utf-8") +  "\" picture.")

# classes[int()] 中的int函数是后加的，不然么会报错，跟新版本中，非数字索引被丢弃，成为一个error。

Let’s also plot the cost function and the gradients.

# Plot learning curve (with costs)
costs = np.squeeze(d['costs'])
#squeeze 函数：从数组的形状中删除单维度条目，即把shape中为1的维度去掉

plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

Interpretation:
You can see the cost decreasing. It shows that the parameters are being learned. However, you see that you could train the model even more on the training set. Try to increase the number of iterations in the cell above and rerun the cells. You might see that the training set accuracy goes up, but the test set accuracy goes down. This is called overfitting.

2.6 - Further analysis (optional/ungraded exercise)

Congratulations on building your first image classification model. Let’s analyze it further, and examine possible choices for the learning rate $\alpha$.

Choice of learning rate

Reminder:
In order for Gradient Descent to work you must choose the learning rate wisely. The learning rate $\alpha$ determines how rapidly we update the parameters. If the learning rate is too large we may “overshoot” the optimal value. Similarly, if it is too small we will need too many iterations to converge to the best values. That’s why it is crucial to use a well-tuned learning rate.

Let’s compare the learning curve of our model with several choices of learning rates. Run the cell below. This should take about 1 minute. Feel free also to try different values than the three we have initialized the learning_rates variable to contain, and see what happens.

learning_rates = [0.01, 0.001, 0.0001]
models = {}
for i in learning_rates:
    print ("learning rate is: " + str(i))
    models[str(i)] = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 1500, learning_rate = i, print_cost = False)
    print ('\n' + "-------------------------------------------------------" + '\n')

for i in learning_rates:
    plt.plot(np.squeeze(models[str(i)]["costs"]), label= str(models[str(i)]["learning_rate"]))

plt.ylabel('cost')
plt.xlabel('iterations')

legend = plt.legend(loc='upper right', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
plt.show()

Interpretation:

• Different learning rates give different costs and thus different predictions results.
• If the learning rate is too large (0.01), the cost may oscillate up and down. It may even diverge (though in this example, using 0.01 still eventually ends up at a good value for the cost).
• A lower cost doesn’t mean a better model. You have to check if there is possibly overfitting. It happens when the training accuracy is a lot higher than the test accuracy.
• In deep learning, we usually recommend that you:
  • Choose the learning rate that better minimizes the cost function.
  • If your model overfits, use other techniques to reduce overfitting. (We’ll talk about this in later videos.)

2.7 - Test with your own image (optional/ungraded exercise)

Congratulations on finishing this assignment. You can use your own image and see the output of your model. To do that:
1. Click on “File” in the upper bar of this notebook, then click “Open” to go on your Coursera Hub.
2. Add your image to this Jupyter Notebook’s directory, in the “images” folder
3. Change your image’s name in the following code
4. Run the code and check if the algorithm is right (1 = cat, 0 = non-cat)!

## START CODE HERE ## (PUT YOUR IMAGE NAME) 
# my_image = "Lion_waiting_in_Namibia.jpg"   # change this to the name of your image file 
my_image = "my_image2.jpg"
## END CODE HERE ##

# We preprocess the image to fit your algorithm.
fname = "images/" + my_image
image = np.array(plt.imread(fname))
my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T
my_predicted_image = predict(d["w"], d["b"], my_image)  
#此处为什么不需要my_image/255标准化处理，保证放入模型的数据与训练集的形式一样

plt.imshow(image)
print("y = " + str(np.squeeze(my_predicted_image)) + ", your algorithm predicts a \"" + classes[int(np.squeeze(my_predicted_image)),].decode("utf-8") +  "\" picture.")

What to remember from this assignment:

1. Preprocessing the dataset is important.
2. You implemented each function separately: initialize(), propagate(), optimize(). Then you built a model().
3. Tuning the learning rate (which is an example of a “hyperparameter”) can make a big difference to the algorithm. You will see more examples of this later in this course!

Finally, if you’d like, we invite you to try different things on this Notebook. Make sure you submit before trying anything. Once you submit, things you can play with include:
- Play with the learning rate and the number of iterations
- Try different initialization methods and compare the results
- Test other preprocessings (center the data, or divide each row by its standard deviation)

Bibliography:

• http://www.wildml.com/2015/09/implementing-a-neural-network-from-scratch/
• https://stats.stackexchange.com/questions/211436/why-do-we-normalize-images-by-subtracting-the-datasets-image-mean-and-not-the-c

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