吴恩达2021年版深度学习作业(编程作业)第一章Week2 assigment2_2_2021-1-9 吴恩达- cnn课后编程

这是第一个编程任务，在本任务你会构造一个逻辑回归分类器来识别猫.本任务会一步一步教你如何用神经网络的思想来构建逻辑回归分类器,同时也会提高你关于深度学习的直觉.

说明:

不要使用for/while循环来实现你的代码,除非练习主动要求你这么去做

在本任务,你会学习到:

1. 构建一个典型的深度学习算法框架,包括:

1. 初始化参数

1. 计算cost函数以及它的梯度

1. 使用优化算法(梯度下降)

1. 按照合理的顺序,将上述三个函数集合到一个逻辑回归主模型内

 
# 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.

1. 导入函数包

 
## 1 - Packages ##
 
First, let's run the cell below to import all the packages that you will need during this assignment. 
- [numpy](www.numpy.org) is the fundamental package for scientific computing with Python.
- [h5py](http://www.h5py.org) is a common package to interact with a dataset that is stored on an H5 file.
- [matplotlib](http://matplotlib.org) is a famous library to plot graphs in Python.
- [PIL](http://www.pythonware.com/products/pil/) and [scipy](https://www.scipy.org/) 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

2.了解习题集

问题概述:在本练习内有一个数据集("data.h5"),其中包括:

a.一个包含m_train个训练样本的训练集,其中照片被标记为猫(y = 1)和非猫(y = 0)

b.一个包含m_test个训练样本的训练集,其中照片被标记为猫和非猫

c.每张照片的维度为(num_px,num_px,3),其中3是3个RGB通道,并且每张照片都是正方形的(高度 =num_px,宽度 =num_px)

 
## 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.

加载数据集

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

我们加上了"orig"的标识符是因为后面我们需要对数据进行预处理,在预处理后,我们才能够得到训练集train_set_x,以及测试机test_set_x

每一个train_set_x_orig以及test_set_x_orig都代表了一张图片,你可以运行下面的代码来查看他们具体的图像,你也可以更换index来查看其他图片.

 
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. 

查看index=5的具体图像

 
 
# Example of a picture
index = 5
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.")
 
"""
Expected Output:y = [0], it's a 'non-cat' picture.
"""

image: index = 5

在深度学习中,很多软件层面的错误来自于矩阵/向量维度不匹配,如果你能保证你的矩阵/向量维度正确,你就能够顺利地消除很多bug

练习:查看下面这些参数的值:

1. m_train(训练样本的数量)

1. m_test(测试样本的数量)

1. num_px(训练图片的高度以及宽度)

 
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]`.

 
 
### 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))
 
"""
Expected Output:
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)
"""

为了方便,你应该将每张图片的矩阵维度重新变换(将(num_px, num_px, 3)变换为一个numpy数组的维度(num_px * num_px* 3, 1)).之后,我们的训练集和测试集中,每张图片就会变成一个只有一列的numpy数组,反映到在整个数据集中,就是(num_px * num_px* 3, m_train)/(num_px * num_px* 3, m_test)

练习:将训练集和测试集的图片维度从(num_px, num_px, 3)转换为(num_px * num_px* 3, 1)

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

 
#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
 
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]))
"""
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]
"""

为了表示图片的颜色,图片中的每个像素都会拥有红\绿\蓝三个颜色通道,并且每个像素中这个通道的值都在0~255之间.

在机器学习中,一个最常见的有u处理步骤就是使你的数据集中心化以及归一化,这意味着你需要先计算整个numpy数组的平均值,并将数组中每一个数都除以平均值.但对于图片数据集而言,这一步更加简单方便,你只需要将整个数据集都除以255就可以了(每个像素通道的最大值)

 
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).
 
<!-- During the training of your model, you're going to multiply weights and add biases to some initial inputs in order to observe neuron activations. Then you backpropogate with the gradients to train the model. But, it is extremely important for each feature to have a similar range such that our gradients don't explode. You will see that more in detail later in the lectures. !--> 
 
Let's standardize our dataset.

 
 
train_set_x = train_set_x_flatten/255.
test_set_x = test_set_x_flatten/255.

3.学习算法的总体框架

1. 你会使用逻辑回归来进行一个简单的神经网络搭建

1. 关键步骤

1. 初始化模型参数

1. 通过最小化损失函数来进行模型参数的学习

1. 使用训练好的参数来进行预测

1. 分析预测结果以及得出结论

 
## 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!**
 
<img src="images/LogReg_kiank.png" style="width:650px;height:400px;">
 
**Mathematical expression of the algorithm**:
 
For one example $x^{(i)}$:
$$z^{(i)} = w^T x^{(i)} + b \tag{1}$$
$$\hat{y}^{(i)} = a^{(i)} = sigmoid(z^{(i)})\tag{2}$$ 
$$ \mathcal{L}(a^{(i)}, y^{(i)}) =  - y^{(i)}  \log(a^{(i)}) - (1-y^{(i)} )  \log(1-a^{(i)})\tag{3}$$
 
The cost is then computed by summing over all training examples:
$$ J = \frac{1}{m} \sum_{i=1}^m \mathcal{L}(a^{(i)}, y^{(i)})\tag{6}$$
**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.构建算法的组成部分

1. 构建神经网络的步骤包括

1. 明确模型结构(里插入输入特征的数量)

1. 初始化模型参数

1. 循环训练

1. 计算损失函数(正向传播)

1. 计算梯度(反向传播)

1. 更新参数(梯度下降)

练习:使用Python中的基础函数来实现sigmoid函数

 
## 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()`.
 
### 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]
"""

练习:初始化参数.你必须将w初始化为值全部为0的向量.如果你不知道怎么使用numpy函数库,在numpy的说明文档里面查看np.zeros()的使用说明

 
### 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
 
dim = 2
w, b = initialize_with_zeros(dim)
print ("w = " + str(w))
print ("b = " + str(b))
 
"""
**Expected Output**: 
w = [[0.]
 [0.]]
b = 0
"""

现在你已经将所有的参数都初始化了,你可以进行正向传播以及反向传播来进行参数的学习

练习:实现propagate()函数来进行cost函数算以及梯度的计算

 
### 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: 
$$ \frac{\partial J}{\partial w} = \frac{1}{m}X(A-Y)^T\tag{7}$$
$$ \frac{\partial J}{\partial b} = \frac{1}{m} \sum_{i=1}^m (a^{(i)}-y^{(i)})\tag{8}$$

 
 
# 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)
    cost = - np.sum(np.multiply(Y ,np.log(A))
                    +np.multiply((1 - Y),np.log(1- A))) / m   
    ### END CODE HERE ###
    
    # BACKWARD PROPAGATION (TO FIND GRAD)
    ### START CODE HERE ### (≈ 2 lines of code)
    dw = ( np.dot(X, (A - Y).T) ) / m
    db = np.sum(A - Y) / m
    
    ### 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]])
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 = 0.49993523062470574
cost = 6.000064773192205
"""

你已经初始化参数,并且计算了损失函数以及梯度,现在你可以使用梯度下降来更新你的参数了

练习:实现优化函数,其目标是学习w以及b的值来最小化损失函数J.对于一个参数θ来说,更新规则是

θ = θ - αdθ,其中α是学习率

 
### 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 $w$ 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)
 
"""
**Expected Output**: 
w = [[0.1124579 ]
 [0.23106775]]
b = 1.5593049248448891
dw = [[0.90158428]
 [1.76250842]]
db = 0.4304620716786828
[6.000064773192205]
"""

练习:之前的函数已经输出了学习后的w和b,我们现在能利用他们来预测训练集中的x标签了,实现predict()函数.其中分为两步:

a.计算yhat

b.使yhat收敛为0或1(如果 激活函数的值<=0.5则为0,其余为1)

 
**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.]]
"""

你需要记住的是:

a.实现上面的函数你需要做到:

1. 初始化w和b

1. 每次迭代优化损失函数并且学习参数w,b:

1. 计算损失函数以及梯度

1. 利用梯度下降更新参数

1. 使用学习后的参数w和b来预测labels

 
<font color='blue'>
**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.将函数整合到模型内

在这一步骤中,你将会看到如何将之前实现的所有函数按照正确的顺序整合到一个模型当中\

练习:应用下面的提示实现model函数.

1. 利用Y_prediction函数来预测测试机

1. 利用Y_prediction_train来预测训练集

1. optimize()函数的返回值是w,costs,grads

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

训练集准确率接近100%.现在明确的选择应该是检查你的模型过拟合训练集的可能性.测试集的准确率是70%,实际上这对于一个简单模型来说已经算是不差的结果了,毕竟数据集很小,并且你的逻辑回归只是一个线性的分类器,不过别担心,下周你会构建一个更好的分类器.

现在让我们来看一下损失函数和梯度下降

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

 
 
# Plot learning curve (with costs)
costs = np.squeeze(d['costs'])
plt.plot(costs)
plt.ylabel('cost')
plt.xlabel('iterations (per hundreds)')
plt.title("Learning rate =" + str(d["learning_rate"]))
plt.show()

你hi看到cost是一个不断下降的曲线,这表明参数是有被学习的,你会发现对于你的训练集而言,你还可以继续训练下去.尝试增加训练轮次,并重新运行上面的代码,你可能会发现训练集的精度继续上升,但是测试集的精度却开始下降,这就是过拟合.

 
**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. 

选择不同的学习率

 
## 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 center', shadow=True)
frame = legend.get_frame()
frame.set_facecolor('0.90')
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 %
-------------------------------------------------------
"""

解释:

1. 不同的学习率导致不同的costs进而影响预测结果

1. 如果学习率过大(0.01),cost可能会上下震荡,甚至最后可能会发散(虽然在本例子中是收敛的)

1. cost更低并不意味着你有一个更好的模型,你必须检查一下看看是否过拟合了.当训练精度远高于测试精度的时候,就说明过拟合了

1. 在深度学习中,建议你:

1. 选择不同的学习率进行测试,选择最佳的学习率来减少损失函数

1. 如果你的模型过拟合了,使用其他手段来降低过拟合的程度(后面的视频会讲到)

 
**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.) 
 

测试你自己的图像

 
## 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 = "" #这里填上自己的图片名字,记得加后缀
## END CODE HERE ##
 
# We preprocess the image to fit your algorithm.
fname = "images/" + my_image    #记得把你图片放到images文件夹下面,不然报错找不到
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.")

我没有测,因为懒