[深度学习] 第一课 Building your Deep Neural Network: Step by Step：编程练习-学习笔记

Building your Deep Neural Network: Step by Step

Welcome to your week 4 assignment (part 1 of 2)! You have previously trained a 2-layer Neural Network (with a single hidden layer). This week, you will build a deep neural network, with as many layers as you want!

• In this notebook, you will implement all the functions required to build a deep neural network.
• In the next assignment, you will use these functions to build a deep neural network for image classification.

After this assignment you will be able to:

• Use non-linear units like ReLU to improve your model
• Build a deeper neural network (with more than 1 hidden layer)
• Implement an easy-to-use neural network class

Notation:

• Superscript [l][l] denotes a quantity associated with the lthlth layer.
  • Example: a[L]a[L] is the LthLth layer activation. W[L]W[L] and b[L]b[L] are the LthLth layer parameters.
• Superscript (i)(i) denotes a quantity associated with the ithith example.
  • Example: x(i)x(i) is the ithith training example.
• Lowerscript ii denotes the ithith entry of a vector.
  • Example: a[l]iai[l] denotes the ithith entry of the lthlth layer's activations).

1 - Packages

Let's first import all the packages that you will need during this assignment.

• numpy is the main package for scientific computing with Python.
• matplotlib is a library to plot graphs in Python.
• dnn_utils provides some necessary functions for this notebook.
• testCases provides some test cases to assess the correctness of your functions
• np.random.seed(1) is used to keep all the random function calls consistent. It will help us grade your work. Please don't change the seed.

import numpy as np
import h5py
import matplotlib.pyplot as plt
from testCases_v4 import *
from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward

%matplotlib inline
plt.rcParams['figure.figsize'] = (5.0, 4.0) # set default size of plots
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

%load_ext autoreload
%autoreload 2

np.random.seed(1)

2 - Outline of the Assignment

To build your neural network, you will be implementing several "helper functions". These helper functions will be used in the next assignment to build a two-layer neural network and an L-layer neural network. Each small helper function you will implement will have detailed instructions that will walk you through the necessary steps. Here is an outline of this assignment, you will:

• Initialize the parameters for a two-layer network and for an LL-layer neural network.
• Implement the forward propagation module (shown in purple in the figure below).
  • Complete the LINEAR part of a layer's forward propagation step (resulting in Z[l]Z[l]).
  • We give you the ACTIVATION function (relu/sigmoid).
  • Combine the previous two steps into a new [LINEAR->ACTIVATION] forward function.
  • Stack the [LINEAR->RELU] forward function L-1 time (for layers 1 through L-1) and add a [LINEAR->SIGMOID] at the end (for the final layer LL). This gives you a new L_model_forward function.
• Compute the loss.
• Implement the backward propagation module (denoted in red in the figure below).
  • Complete the LINEAR part of a layer's backward propagation step.
  • We give you the gradient of the ACTIVATE function (relu_backward/sigmoid_backward)
  • Combine the previous two steps into a new [LINEAR->ACTIVATION] backward function.
  • Stack [LINEAR->RELU] backward L-1 times and add [LINEAR->SIGMOID] backward in a new L_model_backward function
• Finally update the parameters.

Note that for every forward function, there is a corresponding backward function. That is why at every step of your forward module you will be storing some values in a cache. The cached values are useful for computing gradients. In the backpropagation module you will then use the cache to calculate the gradients. This assignment will show you exactly how to carry out each of these steps.

3 - Initialization

You will write two helper functions that will initialize the parameters for your model. The first function will be used to initialize parameters for a two layer model. The second one will generalize this initialization process to LL layers.

3.1 - 2-layer Neural Network

Exercise: Create and initialize the parameters of the 2-layer neural network.

Instructions:

• The model's structure is: LINEAR -> RELU -> LINEAR -> SIGMOID.
• Use random initialization for the weight matrices. Use np.random.randn(shape)*0.01 with the correct shape.
• Use zero initialization for the biases. Use np.zeros(shape).

# GRADED FUNCTION: initialize_parameters

def initialize_parameters(n_x, n_h, n_y):
    """
    Argument:
    n_x -- size of the input layer
    n_h -- size of the hidden layer
    n_y -- size of the output layer
    
    Returns:
    parameters -- python dictionary containing your parameters:
                    W1 -- weight matrix of shape (n_h, n_x)
                    b1 -- bias vector of shape (n_h, 1)
                    W2 -- weight matrix of shape (n_y, n_h)
                    b2 -- bias vector of shape (n_y, 1)
    """
    
    np.random.seed(1)
    
    ### START CODE HERE ### (≈ 4 lines of code)
    W1 = np.random.randn(n_h, n_x) * 0.01
    b1 = np.zeros(shape=(n_h, 1))
    W2 = np.random.randn(n_y, n_h) * 0.01
    b2 = np.zeros(shape=(n_y, 1))
    ### END CODE HERE ###
    
    assert(W1.shape == (n_h, n_x))
    assert(b1.shape == (n_h, 1))
    assert(W2.shape == (n_y, n_h))
    assert(b2.shape == (n_y, 1))
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters    

parameters = initialize_parameters(3,2,1)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

Exercise: Implement initialization for an L-layer Neural Network.

Instructions:

• The model's structure is [LINEAR -> RELU] ×× (L-1) -> LINEAR -> SIGMOID. I.e., it has L−1L−1 layers using a ReLU activation function followed by an output layer with a sigmoid activation function.
• Use random initialization for the weight matrices. Use np.random.randn(shape) * 0.01.
• Use zeros initialization for the biases. Use np.zeros(shape).
• We will store n[l]n[l], the number of units in different layers, in a variable layer_dims. For example, the layer_dims for the "Planar Data classification model" from last week would have been [2,4,1]: There were two inputs, one hidden layer with 4 hidden units, and an output layer with 1 output unit. Thus means W1's shape was (4,2), b1 was (4,1), W2 was (1,4) and b2 was (1,1). Now you will generalize this to LL layers!
• Here is the implementation for L=1L=1 (one layer neural network). It should inspire you to implement the general case (L-layer neural network).

    if L == 1:
        parameters["W" + str(L)] = np.random.randn(layer_dims[1], layer_dims[0]) * 0.01
        parameters["b" + str(L)] = np.zeros((layer_dims[1], 1))

# GRADED FUNCTION: initialize_parameters_deep

def initialize_parameters_deep(layer_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the dimensions of each layer in our network
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    Wl -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
                    bl -- bias vector of shape (layer_dims[l], 1)
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layer_dims)            # number of layers in the network

    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l - 1]) * 0.01
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
        ### END CODE HERE ###
        
        assert(parameters['W' + str(l)].shape == (layer_dims[l], layer_dims[l-1]))
        assert(parameters['b' + str(l)].shape == (layer_dims[l], 1))

        
    return parameters

parameters = initialize_parameters_deep([5,4,3]) # 第一层5个神经元，第二层4个，第三层3个
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

# GRADED FUNCTION: linear_forward

def linear_forward(A, W, b):
    """
    Implement the linear part of a layer's forward propagation.

    Arguments:
    A -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)

    Returns:
    Z -- the input of the activation function, also called pre-activation parameter 
    cache -- a python dictionary containing "A", "W" and "b" ; stored for computing the backward pass efficiently
    """
    
    ### START CODE HERE ### (≈ 1 line of code)
    Z = np.dot(W, A) + b
    print("Z:"+str(Z))
    ### END CODE HERE ###
    
    assert(Z.shape == (W.shape[0], A.shape[1]))
    cache = (A, W, b)
    
    return Z, cache

A, W, b = linear_forward_test_case()
print("A:"+str(A))     # A 输入是按一列一列排列。 此处是2个样本，每个样本3个特征、
print("W:"+str(W))     # W 是一行一行排列，所以直接W点积A。 此处为一个W有3个weight。（1,3）*（3,2） = （1,2） = Z的维度
print("b:"+str(b))

Z, linear_cache = linear_forward(A, W, b)
print("Z = " + str(Z))

def sigmoid(Z):
    """
    Implements the sigmoid activation in numpy
    
    Arguments:
    Z -- numpy array of any shape
    
    Returns:
    A -- output of sigmoid(z), same shape as Z
    cache -- returns Z as well, useful during backpropagation
    """
    
    A = 1/(1+np.exp(-Z))
    cache = Z
    
    return A, cache

def relu(Z):
    """
    Implement the RELU function.

    Arguments:
    Z -- Output of the linear layer, of any shape

    Returns:
    A -- Post-activation parameter, of the same shape as Z
    cache -- a python dictionary containing "A" ; stored for computing the backward pass efficiently
    """
    
    A = np.maximum(0,Z)
    
    assert(A.shape == Z.shape)
    
    cache = Z 
    return A, cache

# GRADED FUNCTION: linear_activation_forward

def linear_activation_forward(A_prev, W, b, activation):
    """
    Implement the forward propagation for the LINEAR->ACTIVATION layer

    Arguments:
    A_prev -- activations from previous layer (or input data): (size of previous layer, number of examples)
    W -- weights matrix: numpy array of shape (size of current layer, size of previous layer)
    b -- bias vector, numpy array of shape (size of the current layer, 1)
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"

    Returns:
    A -- the output of the activation function, also called the post-activation value 
    cache -- a python dictionary containing "linear_cache" and "activation_cache";
             stored for computing the backward pass efficiently
    """
    
    if activation == "sigmoid":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = sigmoid(Z)
        print('sigmoid A:'+str(A))
        print('sigmoid activation_cache:'+str(activation_cache))
        ### END CODE HERE ###
    
    elif activation == "relu":
        # Inputs: "A_prev, W, b". Outputs: "A, activation_cache".
        ### START CODE HERE ### (≈ 2 lines of code)
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = relu(Z)
        ### END CODE HERE ###
    
    assert (A.shape == (W.shape[0], A_prev.shape[1]))
    # activation_cache 就是Z
    cache = (linear_cache, activation_cache)

    return A, cache

A_prev, W, b = linear_activation_forward_test_case()

A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "sigmoid")
print("With sigmoid: A = " + str(A))

A, linear_activation_cache = linear_activation_forward(A_prev, W, b, activation = "relu")
print("With ReLU: A = " + str(A))

Exercise: Implement the forward propagation of the above model.

Instruction: In the code below, the variable AL will denote A[L]=σ(Z[L])=σ(W[L]A[L−1]+b[L])A[L]=σ(Z[L])=σ(W[L]A[L−1]+b[L]). (This is sometimes also called Yhat, i.e., this is Ŷ Y^.)

Tips:

• Use the functions you had previously written
• Use a for loop to replicate [LINEAR->RELU] (L-1) times
• Don't forget to keep track of the caches in the "caches" list. To add a new value c to a list, you can use list.append(c).

# GRADED FUNCTION: L_model_forward

def L_model_forward(X, parameters):
    """
    Implement forward propagation for the [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID computation
    
    Arguments:
    X -- data, numpy array of shape (input size, number of examples)
    parameters -- output of initialize_parameters_deep()
    
    Returns:
    AL -- last post-activation value
    caches -- list of caches containing:
                every cache of linear_activation_forward() (there are L-1 of them, indexed from 0 to L-1)
    """

    caches = []
    A = X
    L = len(parameters) // 2                  # number of layers in the neural network
    
    # Implement [LINEAR -> RELU]*(L-1). Add "cache" to the "caches" list.
    for l in range(1, L):
        A_prev = A 
        ### START CODE HERE ### (≈ 2 lines of code)
        A, cache = linear_activation_forward(A_prev, 
                                             parameters['W' + str(l)], 
                                             parameters['b' + str(l)], 
                                             activation='relu')
        caches.append(cache)
        ### END CODE HERE ###
    
    # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
    ### START CODE HERE ### (≈ 2 lines of code)
    AL, cache = linear_activation_forward(A, 
                                          parameters['W' + str(L)], 
                                          parameters['b' + str(L)], 
                                          activation='sigmoid')
    caches.append(cache)    
    ### END CODE HERE ###
    
    assert(AL.shape == (1,X.shape[1]))
            
    return AL, caches

X, parameters = L_model_forward_test_case_2hidden()
#print('X:'+str(X))    # 一列是一个样本。 此处样本数4个，特征5个。
#print('parameters:'+str(parameters)) 
#  W1 每一行是一套weight。 此处有5列，说明特征有5个。 有4行，说明此层有4个神经元。
#  Z1 每一行是一层的输出。 此处有4列，说明有此层有4个神经元， 此处有4行，说明有4个样本。
AL, caches = L_model_forward(X, parameters)  
print("AL = " + str(AL))
print("Length of caches list = " + str(len(caches)))

# GRADED FUNCTION: compute_cost

def compute_cost(AL, Y):
    """
    Implement the cost function defined by equation (7).

    Arguments:
    AL -- probability vector corresponding to your label predictions, shape (1, number of examples)
    Y -- true "label" vector (for example: containing 0 if non-cat, 1 if cat), shape (1, number of examples)

    Returns:
    cost -- cross-entropy cost
    """
    
    m = Y.shape[1]

    # Compute loss from aL and y.
    ### START CODE HERE ### (≈ 1 lines of code)
    cost = (-1 / m) * np.sum(np.multiply(Y, np.log(AL)) + np.multiply(1 - Y, np.log(1 - AL)))
    ### END CODE HERE ###
    cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).
    assert(cost.shape == ())
    
    return cost

Y, AL = compute_cost_test_case()
print("AL:"+str(AL))
print("Y:"+str(Y))
print("cost = " + str(compute_cost(AL, Y)))

Now, similar to forward propagation, you are going to build the backward propagation in three steps:

• LINEAR backward
• LINEAR -> ACTIVATION backward where ACTIVATION computes the derivative of either the ReLU or sigmoid activation
• [LINEAR -> RELU] ×× (L-1) -> LINEAR -> SIGMOID backward (whole model)

# GRADED FUNCTION: linear_backward

def linear_backward(dZ, cache):
    """
    Implement the linear portion of backward propagation for a single layer (layer l)

    Arguments:
    dZ -- Gradient of the cost with respect to the linear output (of current layer l)
    cache -- tuple of values (A_prev, W, b) coming from the forward propagation in the current layer

    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    A_prev, W, b = cache
    m = A_prev.shape[1]

    ### START CODE HERE ### (≈ 3 lines of code)    
    dW = np.dot(dZ, cache[0].T) / m
    #db = np.squeeze(np.sum(dZ, axis=1, keepdims=True)) / m  # 输出 db:0.506294475007
    db = np.sum(dZ, axis=1, keepdims=True) / m   # 如果不添加np.squeeze，则会输出 db:[[ 0.50629448]]
    dA_prev = np.dot(cache[1].T, dZ)
    ### END CODE HERE ###
    
    print("db:"+str(db))
    print("b:"+str(b))
    print("db.shape:"+str(db.shape))
    print("b.shape:"+str(b.shape))
    
    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)   #原文本错误，db.shape = (), b.shape = (1,1) 不可能相等
    #assert (isinstance(db, float))
    
    return dA_prev, dW, db

# Set up some test inputs
dZ, linear_cache = linear_backward_test_case()

dA_prev, dW, db = linear_backward(dZ, linear_cache)
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db))

def relu_backward(dA, cache):
    """
    Implement the backward propagation for a single RELU unit.

    Arguments:
    dA -- post-activation gradient, of any shape
    cache -- 'Z' where we store for computing backward propagation efficiently

    Returns:
    dZ -- Gradient of the cost with respect to Z
    """
    
    Z = cache
    dZ = np.array(dA, copy=True) # just converting dz to a correct object.
    
    # When z <= 0, you should set dz to 0 as well. 
    dZ[Z <= 0] = 0
    
    assert (dZ.shape == Z.shape)
    
    return dZ

def sigmoid_backward(dA, cache):
    """
    Implement the backward propagation for a single SIGMOID unit.

    Arguments:
    dA -- post-activation gradient, of any shape
    cache -- 'Z' where we store for computing backward propagation efficiently

    Returns:
    dZ -- Gradient of the cost with respect to Z
    """
    
    Z = cache
    
    s = 1/(1+np.exp(-Z))
    dZ = dA * s * (1-s)
    
    assert (dZ.shape == Z.shape)
    
    return dZ

# GRADED FUNCTION: linear_activation_backward

def linear_activation_backward(dA, cache, activation):
    """
    Implement the backward propagation for the LINEAR->ACTIVATION layer.
    
    Arguments:
    dA -- post-activation gradient for current layer l 
    cache -- tuple of values (linear_cache, activation_cache) we store for computing backward propagation efficiently
    activation -- the activation to be used in this layer, stored as a text string: "sigmoid" or "relu"
    
    Returns:
    dA_prev -- Gradient of the cost with respect to the activation (of the previous layer l-1), same shape as A_prev
    dW -- Gradient of the cost with respect to W (current layer l), same shape as W
    db -- Gradient of the cost with respect to b (current layer l), same shape as b
    """
    linear_cache, activation_cache = cache
    
    if activation == "relu":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = relu_backward(dA, activation_cache)
        ### END CODE HERE ###
        
    elif activation == "sigmoid":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = sigmoid_backward(dA, activation_cache)
        ### END CODE HERE ###
        
    # Shorten the code
    dA_prev, dW, db = linear_backward(dZ, linear_cache)
    return dA_prev, dW, db

dAL, linear_activation_cache = linear_activation_backward_test_case()
print ("linear_activation_cache:"+str(linear_activation_cache))

dA_prev, dW, db = linear_activation_backward(dAL, linear_activation_cache, activation = "sigmoid")
print ("sigmoid:")
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db) + "\n")

dA_prev, dW, db = linear_activation_backward(dAL, linear_activation_cache, activation = "relu")
print ("relu:")
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db))

# GRADED FUNCTION: L_model_backward  
  
def L_model_backward(AL, Y, caches):  
    """ 
    Implement the backward propagation for the [LINEAR->RELU] * (L-1) -> LINEAR -> SIGMOID group 
     
    Arguments: 
    AL -- probability vector, output of the forward propagation (L_model_forward()) 
    Y -- true "label" vector (containing 0 if non-cat, 1 if cat) 
    caches -- list of caches containing: 
                every cache of linear_activation_forward() with "relu" (it's caches[l], for l in range(L-1) i.e l = 0...L-2) 
                the cache of linear_activation_forward() with "sigmoid" (it's caches[L-1]) 
     
    Returns: 
    grads -- A dictionary with the gradients 
             grads["dA" + str(l)] = ...  
             grads["dW" + str(l)] = ... 
             grads["db" + str(l)] = ...  
    """  
    grads = {}  
    L = len(caches) # the number of layers  
    m = AL.shape[1]  
    Y = Y.reshape(AL.shape) # after this line, Y is the same shape as AL  
      
    # Initializing the backpropagation   # cost funtion 对AL求偏导
    ### START CODE HERE ### (1 line of code)  
    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))  
    ### END CODE HERE ###  
      
    # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "AL, Y, caches". Outputs: "grads["dAL"], grads["dWL"], grads["dbL"]  
    ### START CODE HERE ### (approx. 2 lines)  
    current_cache = caches[L-1]
    grads["dA" + str(L)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, 
                                                                                                  current_cache, 
                                                                                                  activation = "sigmoid")
      ### END CODE HERE ###  
      # 此处L=2。 l=0.当l=0，求 dA(l+1)=dA1 需要 dA(l+2)=dA2
    for l in reversed(range(L-1)):  
        # lth layer: (RELU -> LINEAR) gradients.  
        # Inputs: "grads["dA" + str(l + 2)], caches". Outputs: "grads["dA" + str(l + 1)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)]   
        ### START CODE HERE ### (approx. 5 lines)  
        current_cache = caches[l]  
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 2)], 
                                                                    current_cache, 
                                                                    activation = "relu")
        grads["dA" + str(l + 1)] = dA_prev_temp  
        grads["dW" + str(l + 1)] = dW_temp  
        grads["db" + str(l + 1)] = db_temp  
        ### END CODE HERE ###  
  
    return grads  

AL, Y_assess, caches = L_model_backward_test_case()
#print("caches:"+str(caches))
grads = L_model_backward(AL, Y_assess, caches)
print_grads(grads)

# GRADED FUNCTION: update_parameters

def update_parameters(parameters, grads, learning_rate):
    """
    Update parameters using gradient descent
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    grads -- python dictionary containing your gradients, output of L_model_backward
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
                  parameters["W" + str(l)] = ... 
                  parameters["b" + str(l)] = ...
    """
    
    L = len(parameters) // 2 # number of layers in the neural network

    # Update rule for each parameter. Use a for loop.
    ### START CODE HERE ### (≈ 3 lines of code)
    for l in range(L):
        parameters["W" + str(l + 1)] = parameters["W" + str(l + 1)] - learning_rate * grads["dW" + str(l + 1)]
        parameters["b" + str(l + 1)] = parameters["b" + str(l + 1)] - learning_rate * grads["db" + str(l + 1)]
    ### END CODE HERE ###
    return parameters

parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads, 0.1)

print ("W1 = "+ str(parameters["W1"]))
print ("b1 = "+ str(parameters["b1"]))
print ("W2 = "+ str(parameters["W2"]))
print ("b2 = "+ str(parameters["b2"]))

7 - Conclusion

Congrats on implementing all the functions required for building a deep neural network!

We know it was a long assignment but going forward it will only get better. The next part of the assignment is easier.

In the next assignment you will put all these together to build two models:

• A two-layer neural network
• An L-layer neural network