吴恩达深度学习笔记 第二章作业2

源码以及注释:

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
train_set_x_orig , train_set_y , test_set_x_orig , test_set_y , classes = load_dataset()
# Example of a picture
"""
index = 25
plt.imshow(train_set_x_orig[index])
print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")
"""

#Calculating the - m_train (number of training examples)  - m_test (number of test examples) - num_px (= height = width of a training image) 
m_train=train_set_x_orig.shape[0]
m_test=test_set_x_orig.shape[0]
num_px=train_set_x_orig.shape[1]
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))

# 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 ∗num_px ∗3, 1).
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
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[0:5,0]))

#standardize our data_set
train_set_x = train_set_x_flatten/255
test_set_x = test_set_x_flatten/255

#define sigmoid function
def sigmoid(z):
  s=1/(1+np.exp(-z))
  return s

#Initial function(Initial w and b)
def initialize_with_zeros(dim):
    w=np.zeros((dim,1)) #dim is the size we want
    b=0
    assert(w.shape == (dim, 1))
    assert(isinstance(b, float) or isinstance(b, int))
    return w, b
#Implement a function propagate() that computes the cost function and its gradient
def propagate(w, b, X, Y):
    # FORWARD PROPAGATION (FROM X TO COST)
    m = X.shape[1]
    A=sigmoid(np.dot(w.T,X)+b)
    cost=-1/m*np.sum(Y*np.log(A)+(1-Y)*np.log(1-A))
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    dw=1/m*np.dot(X,(A-Y).T)
    db=1/m*np.sum(A-Y)
    assert(dw.shape == w.shape)
    assert(db.dtype == float)
    cost = np.squeeze(cost)
    assert(cost.shape == ())    
    grads = {"dw": dw,
             "db": db}    
    return grads, cost
def optimize(w,b,X,Y,num_iterations,learning_rate,print_cost=False):
    costs = []   
    for i in range(num_iterations):
        grads,cost=propagate(w, b, X, Y)
        dw=grads["dw"]
        db=grads["db"]
        w=w-learning_rate*dw
        b=b-learning_rate*db
        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
def predict(w, b, X):
    m = X.shape[1]
    Y_prediction = np.zeros((1,m))
    w = w.reshape(X.shape[0], 1)
    A=sigmoid(np.dot(w.T,X)+b)
    for i in range(A.shape[1]):
        if(A[0,i]<=0.5):
            Y_prediction[0,i]=0
        else:
            Y_prediction[0,i]=1
    assert(Y_prediction.shape == (1, m))
    return Y_prediction
def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):
    # initialize parameters with zeros
    w,b=initialize_with_zeros(X_train.shape[0])
    params,grads,costs=optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)
    w=params["w"]
    b=params["b"]
    Y_prediction_test=predict(w, b, X_test)
    Y_prediction_train=predict(w, b, X_train)
    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
d = model(train_set_x, train_set_y, test_set_x, test_set_y, num_iterations = 2000, learning_rate = 0.005, print_cost = True)
#compare the learning curve of our model with several choices of learning rates.
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 center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
#plt.show()
#Test with your own image
my_image = "cat_in_iran.jpg"   # change this to the name of your image file 
fname =  my_image
image = np.array(ndimage.imread(fname, flatten=False))
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)
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.")



    
        
    
    

        
        
    
    
         
            
    
    
    
    
    
    

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.

---

欢迎来到您的第一个（必填）编程任务！你将建立一个逻辑回归分类器来识别猫。这个任务将指导你如何用神经网络的思维方式做到这一点，所以也将磨练你对深度学习的直觉。

说明：

不要在代码中使用循环（for / while），除非指令明确要求您这样做。

您将学习：

构建一个学习算法的总体架构，包括：

• 初始化参数
• 计算成本函数及其梯度
• 使用优化算法（梯度下降） 将上述三个函数按照正确的顺序收集到主模型函数中。

 

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.

In [14]:

 

 

 

 

 

1

import numpy as np

2

import matplotlib.pyplot as plt

3

import h5py

4

import scipy

5

from PIL import Image

6

from scipy import ndimage

7

from lr_utils import load_dataset

8

​

9

%matplotlib inline

 

 

 

 

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. 您将构建一个简单的图像识别算法，可以正确地将图片分类为猫或非猫

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

In [15]:

 

 

 

 

 

1

# Loading the data (cat/non-cat)

2

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.

train_set_x_orig和test_set_x_orig的每一行都是表示图像的数组。您可以通过运行以下代码来可视化示例。随意更改index值并重新运行以查看其他图像。

In [16]:

 

 

 

 

 

 

1

# Example of a picture

2

index = 14

3

plt.imshow(train_set_x_orig[index])

4

print ("y = " + str(train_set_y[:, index]) + ", it's a '" + classes[np.squeeze(train_set_y[:, index])].decode("utf-8") +  "' picture.")

 

 

 

 

y = [1], it's a 'cat' picture.

 

 

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. （深度学习中的许多编码错误来自不适合的矩阵/矢量维度。如果你能保证矩阵/向量维度的正确，可以消除许多错误）

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].

In [17]:

 

 

 

 

 

1

### START CODE HERE ### (≈ 3 lines of code)

2

m_train = train_set_x_orig.shape[0]

3

m_test = test_set_x_orig.shape[0]

4

num_px = train_set_x_orig.shape[1]

5

### END CODE HERE ###

6

​

7

print ("Number of training examples: m_train = " + str(m_train))

8

print ("Number of testing examples: m_test = " + str(m_test))

9

print ("Height/Width of each image: num_px = " + str(num_px))

10

print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")

11

print ("train_set_x shape: " + str(train_set_x_orig.shape))

12

print ("train_set_y shape: " + str(train_set_y.shape))

13

print ("test_set_x shape: " + str(test_set_x_orig.shape))

14

print ("test_set_y shape: " + str(test_set_y.shape))

 

 

 

 

Number of training examples: m_train = 209
Number of testing examples: m_test = 50
Height/Width of each image: num_px = 64
Each image is of size: (64, 64, 3)
train_set_x shape: (209, 64, 64, 3)
train_set_y shape: (1, 209)
test_set_x shape: (50, 64, 64, 3)
test_set_y shape: (1, 50)

 

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 ∗∗ num_px ∗∗ 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 ∗∗ num_px ∗∗ 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∗∗c∗∗d, a) is to use:

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

In [18]:

 

 

 

 

 

1

# Reshape the training and test examples

2

​

3

### START CODE HERE ### (≈ 2 lines of code)

4

train_set_x_flatten = train_set_x_orig.reshape(train_set_x_orig.shape[0], -1).T

5

test_set_x_flatten = test_set_x_orig.reshape(test_set_x_orig.shape[0], -1).T

6

### END CODE HERE ###

7

​

8

print ("train_set_x_flatten shape: " + str(train_set_x_flatten.shape))

9

print ("train_set_y shape: " + str(train_set_y.shape))

10

print ("test_set_x_flatten shape: " + str(test_set_x_flatten.shape))

11

print ("test_set_y shape: " + str(test_set_y.shape))

12

print ("sanity check after reshaping: " + str(train_set_x_flatten[0:5,0]))

 

 

 

 

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]

 

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.

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为了表示彩色图像，必须为每个像素指定红色，绿色和蓝色通道（RGB），因此像素值实际上是从0到255的三个数字的向量。

机器学习中一个常见的预处理步骤是对数据集进行中心化和标准化，这意味着您从每个示例中减去整个numpy数组的平均值，然后将每个示例除以整个numpy数组的标准偏差。但是对于图片数据集来说，它更简单，更方便，几乎可以将数据集的每一行除以255（像素通道的最大值）。

让我们标准化我们的数据集。

In [19]:

 

 

 

 

 

1

train_set_x = train_set_x_flatten/255.

2

test_set_x = test_set_x_flatten/255.

 

 

 

 

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

 

 

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)x(i):

z(i)=wTx(i)+b(1)(1)z(i)=wTx(i)+b

ŷ (i)=a(i)=sigmoid(z(i))(2)(2)y^(i)=a(i)=sigmoid(z(i))

(a(i),y(i))=−y(i)log(a(i))−(1−y(i))log(1−a(i))(3)(3)L(a(i),y(i))=−y(i)log⁡(a(i))−(1−y(i))log⁡(1−a(i))

 

The cost is then computed by summing over all training examples:

J=1m∑i=1m(a(i),y(i))(6)(6)J=1m∑i=1mL(a(i),y(i))

 

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

 

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().

---

建立神经网络的主要步骤是：

• 定义模型结构（如输入特征的个数）
• 初始化模型的参数
• 循环：
  • 计算当前损失（正向传播）
  • 计算当前梯度（反向传播）
  • 更新参数（梯度下降）

你经常分别建立1-3，并把它们整合到我们所说的一个函数中model()。

4.1 - Helper functions 辅助函数

Exercise: Using your code from "Python Basics", implement sigmoid(). As you've seen in the figure above, you need to compute sigmoid(wTx+b)=11+e−(wTx+b)sigmoid(wTx+b)=11+e−(wTx+b) to make predictions. Use np.exp().

In [20]:

 

 

 

 

 

1

# GRADED FUNCTION: sigmoid

2

​

3

def sigmoid(z):

4

    """

5

    Compute the sigmoid of z

6

​

7

    Arguments:

8

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

9

​

10

    Return:

11

    s -- sigmoid(z)

12

    """

13

​

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    ### START CODE HERE ### (≈ 1 line of code)

15

    s = 1 / (1 + np.exp(-z))

16

    ### END CODE HERE ###

17

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    return s

 

 

 

In [21]:

 

 

 

 

 

1

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

 

 

 

 

sigmoid([0, 2]) = [0.5        0.88079708]

 

Expected Output:

sigmoid([0, 2])

[ 0.5 0.88079708]

 

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.

In [22]:

 

 

 

 

 

1

# GRADED FUNCTION: initialize_with_zeros

2

​

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def initialize_with_zeros(dim):

4

    """

5

    This function creates a vector of zeros of shape (dim, 1) for w and initializes b to 0.

6

7

    Argument:

8

    dim -- size of the w vector we want (or number of parameters in this case)

9

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    Returns:

11

    w -- initialized vector of shape (dim, 1)

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    b -- initialized scalar (corresponds to the bias)

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    """

14

15

    ### START CODE HERE ### (≈ 1 line of code)

16

    w = np.zeros((dim, 1))

17

    b = 0

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    ### END CODE HERE ###

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​

20

    assert(w.shape == (dim, 1))

21

    assert(isinstance(b, float) or isinstance(b, int))

22

23

    return w, b

 

 

 

In [23]:

 

 

 

 

 

1

dim = 2

2

w, b = initialize_with_zeros(dim)

3

print ("w = " + str(w))

4

print ("b = " + str(b))

 

 

 

 

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

 

Expected Output:

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

For image inputs, w will be of shape (num_px ×× num_px ×× 3, 1).

 

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=σ(wTX+b)=(a(0),a(1),...,a(m−1),a(m))A=σ(wTX+b)=(a(0),a(1),...,a(m−1),a(m))
• You calculate the cost function: J=−1m∑mi=1y(i)log(a(i))+(1−y(i))log(1−a(i))J=−1m∑i=1my(i)log⁡(a(i))+(1−y(i))log⁡(1−a(i))

Here are the two formulas you will be using:

 

∂J∂w=1mX(A−Y)T(7)(7)∂J∂w=1mX(A−Y)T

∂J∂b=1m∑i=1m(a(i)−y(i))(8)(8)∂J∂b=1m∑i=1m(a(i)−y(i))

In [32]:

 

 

 

 

 

1

# GRADED FUNCTION: propagate

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​

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def propagate(w, b, X, Y):

4

    """

5

    Implement the cost function and its gradient for the propagation explained above

6

​

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    Arguments:

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    w -- weights, a numpy array of size (num_px * num_px * 3, 1)

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    b -- bias, a scalar

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    X -- data of size (num_px * num_px * 3, number of examples)

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    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) of size (1, number of examples)

12

​

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    Return:

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    cost -- negative log-likelihood cost for logistic regression

15

    dw -- gradient of the loss with respect to w, thus same shape as w

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    db -- gradient of the loss with respect to b, thus same shape as b

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    Tips:

19

    - Write your code step by step for the propagation. np.log(), np.dot()

20

    """

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    m = X.shape[1]

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    # FORWARD PROPAGATION (FROM X TO COST)

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    ### START CODE HERE ### (≈ 2 lines of code)

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    A = sigmoid(np.dot(w.T, X) + b)            # compute activation

27

    cost = -1 / m * np.sum(Y * np.log(A) + (1 - Y) * np.log(1 - A))         # compute cost

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    ### END CODE HERE ###

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    # BACKWARD PROPAGATION (TO FIND GRAD)

31

    ### START CODE HERE ### (≈ 2 lines of code)

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    dw = 1 / m * np.dot(X, (A - Y).T)

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    db = 1 / m * np.sum(A - Y)

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    ### END CODE HERE ###

35

    assert(dw.shape == w.shape)

36

    assert(db.dtype == float)

37

    cost = np.squeeze(cost)

38

    assert(cost.shape == ())

39

40

    grads = {"dw": dw,

41

             "db": db}

42

43

    return grads, cost

 

 

 

In [33]:

 

 

 

 

 

1

w, b, X, Y = np.array([[1],[2]]), 2, np.array([[1,2],[3,4]]), np.array([[1,0]])

2

grads, cost = propagate(w, b, X, Y)

3

print ("dw = " + str(grads["dw"]))

4

print ("db = " + str(grads["db"]))

5

print ("cost = " + str(cost))

 

 

 

 

dw = [[0.99993216]
 [1.99980262]]
db = 0.49993523062470574
cost = 6.000064773192205

 

Expected Output:

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

 

d) 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 ww and bb by minimizing the cost function JJ. For a parameter θθ, the update rule is θ=θ−α dθθ=θ−α dθ, where αα is the learning rate.

In [26]:

 

 

 

 

 

 

1

# GRADED FUNCTION: optimize

2

​

3

def optimize(w, b, X, Y, num_iterations, learning_rate, print_cost = False):

4

    """

5

    This function optimizes w and b by running a gradient descent algorithm

6

7

    Arguments:

8

    w -- weights, a numpy array of size (num_px * num_px * 3, 1)

9

    b -- bias, a scalar

10

    X -- data of shape (num_px * num_px * 3, number of examples)

11

    Y -- true "label" vector (containing 0 if non-cat, 1 if cat), of shape (1, number of examples)

12

    num_iterations -- number of iterations of the optimization loop

13

    learning_rate -- learning rate of the gradient descent update rule

14

    print_cost -- True to print the loss every 100 steps

15

16

    Returns:

17

    params -- dictionary containing the weights w and bias b

18

    grads -- dictionary containing the gradients of the weights and bias with respect to the cost function

19

    costs -- list of all the costs computed during the optimization, this will be used to plot the learning curve.

20

21

    Tips:

22

    You basically need to write down two steps and iterate through them:

23

        1) Calculate the cost and the gradient for the current parameters. Use propagate().

24

        2) Update the parameters using gradient descent rule for w and b.

25

    """

26

27

    costs = []

28

29

    for i in range(num_iterations):

30

31

32

        # Cost and gradient calculation (≈ 1-4 lines of code)

33

        ### START CODE HERE ### 

34

        grads, cost = propagate(w, b, X, Y)

35

        ### END CODE HERE ###

36

37

        # Retrieve derivatives from grads

38

        dw = grads["dw"]

39

        db = grads["db"]

40

41

        # update rule (≈ 2 lines of code)

42

        ### START CODE HERE ###

43

        w = w - learning_rate * dw

44

        b = b - learning_rate * db

45

        ### END CODE HERE ###

46

47

        # Record the costs

48

        if i % 100 == 0:

49

            costs.append(cost)

50

51

        # Print the cost every 100 training examples

52

        if print_cost and i % 100 == 0:

53

            print ("Cost after iteration %i: %f" %(i, cost))

54

55

    params = {"w": w,

56

              "b": b}

57

58

    grads = {"dw": dw,

59

             "db": db}

60

61

    return params, grads, costs

 

 

 

In [27]:

 

 

 

 

 

 

1

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

2

​

3

print ("w = " + str(params["w"]))

4

print ("b = " + str(params["b"]))

5

print ("dw = " + str(grads["dw"]))

6

print ("db = " + str(grads["db"]))

7

print(costs)

 

 

 

 

w = [[0.1124579 ]
 [0.23106775]]
b = 1.5593049248448891
dw = [[0.90158428]
 [1.76250842]]
db = 0.4304620716786828
[6.000064773192205]

 

Expected Output:

w	[[ 0.1124579 ] [ 0.23106775]]
b	1.55930492484
dw	[[ 0.90158428] [ 1.76250842]]
db	0.430462071679

 

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 Ŷ =A=σ(wTX+b)Y^=A=σ(wTX+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 forloop (though there is also a way to vectorize this).

In [36]:

 

 

 

 

 

1

# GRADED FUNCTION: predict

2

​

3

def predict(w, b, X):

4

    '''

5

    Predict whether the label is 0 or 1 using learned logistic regression parameters (w, b)

6

7

    Arguments:

8

    w -- weights, a numpy array of size (num_px * num_px * 3, 1)

9

    b -- bias, a scalar

10

    X -- data of size (num_px * num_px * 3, number of examples)

11

12

    Returns:

13

    Y_prediction -- a numpy array (vector) containing all predictions (0/1) for the examples in X

14

    '''

15

16

    m = X.shape[1]

17

    Y_prediction = np.zeros((1,m))

18

    w = w.reshape(X.shape[0], 1)

19

20

    # Compute vector "A" predicting the probabilities of a cat being present in the picture

21

    ### START CODE HERE ### (≈ 1 line of code)

22

    A = sigmoid(np.dot(w.T, X) + b)

23

    ### END CODE HERE ###

24

​

25

    for i in range(A.shape[1]):

26

27

        # Convert probabilities A[0,i] to actual predictions p[0,i]

28

        ### START CODE HERE ### (≈ 4 lines of code)

29

        if A[0, i] <= 0.5:

30

            Y_prediction[0, i] = 0

31

        else:

32

            Y_prediction[0, i] = 1

33

        ### END CODE HERE ###

34

35

    assert(Y_prediction.shape == (1, m))

36

37

    return Y_prediction

 

 

 

In [37]:

 

 

 

 

 

1

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

 

 

 

 

predictions = [[1. 1.]]

 

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

 

 

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()

In [38]:

 

 

 

 

 

 

1

# GRADED FUNCTION: model

2

​

3

def model(X_train, Y_train, X_test, Y_test, num_iterations = 2000, learning_rate = 0.5, print_cost = False):

4

    """

5

    Builds the logistic regression model by calling the function you've implemented previously

6

7

    Arguments:

8

    X_train -- training set represented by a numpy array of shape (num_px * num_px * 3, m_train)

9

    Y_train -- training labels represented by a numpy array (vector) of shape (1, m_train)

10

    X_test -- test set represented by a numpy array of shape (num_px * num_px * 3, m_test)

11

    Y_test -- test labels represented by a numpy array (vector) of shape (1, m_test)

12

    num_iterations -- hyperparameter representing the number of iterations to optimize the parameters

13

    learning_rate -- hyperparameter representing the learning rate used in the update rule of optimize()

14

    print_cost -- Set to true to print the cost every 100 iterations

15

16

    Returns:

17

    d -- dictionary containing information about the model.

18

    """

19

20

    ### START CODE HERE ###

21

22

    # initialize parameters with zeros (≈ 1 line of code)

23

    w, b = initialize_with_zeros(X_train.shape[0])

24

​

25

    # Gradient descent (≈ 1 line of code)

26

    parameters, grads, costs = optimize(w, b, X_train, Y_train, num_iterations, learning_rate, print_cost)

27

28

    # Retrieve parameters w and b from dictionary "parameters"

29

    w = parameters["w"]

30

    b = parameters["b"]

31

32

    # Predict test/train set examples (≈ 2 lines of code)

33

    Y_prediction_test = predict(w, b, X_test)

34

    Y_prediction_train = predict(w, b, X_train)

35

​

36

    ### END CODE HERE ###

37

​

38

    # Print train/test Errors

39

    print("train accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_train - Y_train)) * 100))

40

    print("test accuracy: {} %".format(100 - np.mean(np.abs(Y_prediction_test - Y_test)) * 100))

41

​

42

43

    d = {"costs": costs,

44

         "Y_prediction_test": Y_prediction_test, 

45

         "Y_prediction_train" : Y_prediction_train, 

46

         "w" : w, 

47

         "b" : b,

48

         "learning_rate" : learning_rate,

49

         "num_iterations": num_iterations}

50

51

    return d

 

 

 

 

Run the following cell to train your model.

In [39]:

 

 

 

 

 

 

1

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

 

 

 

 

Cost after iteration 0: 0.693147
Cost after iteration 100: 0.584508
Cost after iteration 200: 0.466949
Cost after iteration 300: 0.376007
Cost after iteration 400: 0.331463
Cost after iteration 500: 0.303273
Cost after iteration 600: 0.279880
Cost after iteration 700: 0.260042
Cost after iteration 800: 0.242941
Cost after iteration 900: 0.228004
Cost after iteration 1000: 0.214820
Cost after iteration 1100: 0.203078
Cost after iteration 1200: 0.192544
Cost after iteration 1300: 0.183033
Cost after iteration 1400: 0.174399
Cost after iteration 1500: 0.166521
Cost after iteration 1600: 0.159305
Cost after iteration 1700: 0.152667
Cost after iteration 1800: 0.146542
Cost after iteration 1900: 0.140872
train accuracy: 99.04306220095694 %
test accuracy: 70.0 %

 

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 error is 68%. 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.

In [40]:

 

 

 

 

 

 

1

# Example of a picture that was wrongly classified.

2

index = 1

3

plt.imshow(test_set_x[:,index].reshape((num_px, num_px, 3)))

4

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.")

 

 

 

 

y = 1, you predicted that it is a "cat" picture.

 

 

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

In [43]:

 

 

 

 

 

1

# Plot learning curve (with costs)

2

costs = np.squeeze(d['costs'])

3

plt.plot(costs)

4

plt.ylabel('cost')

5

plt.xlabel('iterations (per hundreds)')

6

plt.title("Learning rate =" + str(d["learning_rate"]))

7

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.

 

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 αα.

 

Choice of learning rate

Reminder: In order for Gradient Descent to work you must choose the learning rate wisely. The learning rate αα 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.

提醒：为了使梯度下降起作用，您必须明智地选择学习速度。学习率 α 决定了我们更新参数的速度。如果学习率太高，我们可能会“超过”最优值。同样，如果它太小，我们将需要太多迭代来收敛到最佳值。这就是为什么使用良好的学习速度至关​​重要。

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.

In [44]:

 

 

 

 

 

 

1

learning_rates = [0.01, 0.001, 0.0001]

2

models = {}

3

for i in learning_rates:

4

    print ("learning rate is: " + str(i))

5

    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)

6

    print ('\n' + "-------------------------------------------------------" + '\n')

7

​

8

for i in learning_rates:

9

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

10

​

11

plt.ylabel('cost')

12

plt.xlabel('iterations')

13

​

14

legend = plt.legend(loc='upper center', shadow=True)

15

frame = legend.get_frame()

16

frame.set_facecolor('0.90')

17

plt.show()

 

 

 

 

learning rate is: 0.01
train accuracy: 99.52153110047847 %
test accuracy: 68.0 %

-------------------------------------------------------

learning rate is: 0.001
train accuracy: 88.99521531100478 %
test accuracy: 64.0 %

-------------------------------------------------------

learning rate is: 0.0001
train accuracy: 68.42105263157895 %
test accuracy: 36.0 %

-------------------------------------------------------

 

 

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.)

 

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)!

In [45]:

 

 

 

 

 

 

1

## START CODE HERE ## (PUT YOUR IMAGE NAME) 

2

my_image = "cat_in_iran.jpg"   # change this to the name of your image file 

3

## END CODE HERE ##

4

​

5

# We preprocess the image to fit your algorithm.

6

fname =  my_image

7

image = np.array(ndimage.imread(fname, flatten=False))

8

my_image = scipy.misc.imresize(image, size=(num_px,num_px)).reshape((1, num_px*num_px*3)).T

9

my_predicted_image = predict(d["w"], d["b"], my_image)

10

​

11

plt.imshow(image)

12

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

 

 

 

 

D:\ANACONDA\lib\site-packages\ipykernel_launcher.py:7: DeprecationWarning: `imread` is deprecated!
`imread` is deprecated in SciPy 1.0.0.
Use ``matplotlib.pyplot.imread`` instead.
  import sys
D:\ANACONDA\lib\site-packages\ipykernel_launcher.py:8: DeprecationWarning: `imresize` is deprecated!
`imresize` is deprecated in SciPy 1.0.0, and will be removed in 1.2.0.
Use ``skimage.transform.resize`` instead.
  

 

y = 1.0, your algorithm predicts a "cat" picture.

 

 

What to remember from this assignment:

• Preprocessing the dataset is important.
• You implemented each function separately: initialize(), propagate(), optimize(). Then you built a model().
• 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