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] denotes a quantity associated with the lth layer.
  • Example: a[L] is the Lth layer activation. W[L] and b[L] are the Lth layer parameters.
• Superscript (i) denotes a quantity associated with the ith example.
  • Example: x(i) is the ith training example.
• Lowerscript i denotes the ith entry of a vector.
  • Example: a[l]i denotes the ith entry of the lth layer‘s activations).

Let‘s get started!

1 - Packages

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

• numpy is the main package for scientific computing with Python.
• matplotlib is a library to plot graphs in Python.
• dnn_utils provides some necessary functions for this notebook.
• testCases provides some test cases to assess the correctness of your functions
• np.random.seed(1) is used to keep all the random function calls consistent. It will help us grade your work. Please don‘t change the seed.

  import numpy as np
  import h5py
  import matplotlib.pyplot as plt
  from testCases_v3 import *
  from dnn_utils_v2 import sigmoid, sigmoid_backward, relu, relu_backward
  
  %matplotlib inline
  plt.rcParams[‘figure.figsize‘] = (5.0, 4.0) # set default size of plots
  plt.rcParams[‘image.interpolation‘] = ‘nearest‘
  plt.rcParams[‘image.cmap‘] = ‘gray‘
  
  %load_ext autoreload
  %autoreload 2
  
  np.random.seed(1)

  2 - Outline of the Assignment

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

  • Initialize the parameters for a two-layer network and for an LL-layer neural network.
  • Implement the forward propagation module (shown in purple in the figure below).
    • Complete the LINEAR part of a layer‘s forward propagation step (resulting in Z^[l]).
    • 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((n_h,1))
      W2 = np.random.randn(n_y,n_h)*0.01
      b2 = np.zeros((n_y,1))
      ### END CODE HERE ###
      
      assert(W1.shape == (n_h, n_x))
      assert(b1.shape == (n_h, 1))
      assert(W2.shape == (n_y, n_h))
      assert(b2.shape == (n_y, 1))
      
      parameters = {"W1": W1,
                    "b1": b1,
                    "W2": W2,
                    "b2": b2}
      
      return parameters    

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

  W1 = [[ 0.01624345 -0.00611756 -0.00528172]
   [-0.01072969  0.00865408 -0.02301539]]
  b1 = [[ 0.]
   [ 0.]]
  W2 = [[ 0.01744812 -0.00761207]]
  b2 = [[ 0.]]

• Expected output:

  W1	[[ 0.01624345 -0.00611756 -0.00528172] [-0.01072969 0.00865408 -0.02301539]]
  b1	[[ 0.] [ 0.]]
  W2	[[ 0.01744812 -0.00761207]]
  b2	[[ 0.]]

3.2 - L-layer Neural Network

The initialization for a deeper L-layer neural network is more complicated because there are many more weight matrices and bias vectors. When completing the initialize_parameters_deep, you should make sure that your dimensions match between each layer. Recall that n[l]n[l] is the number of units in layer ll. Thus for example if the size of our input XX is (12288,209)(12288,209) (with m=209m=209 examples) then:

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.rand(shape) * 0.01.
• Use zeros initialization for the biases. Use np.zeros(shape).
• We will store n[l]n[l], the number of units in different layers, in a variable layer_dims. For example, the layer_dims for the "Planar Data classification model" from last week would have been [2,4,1]: There were two inputs, one hidden layer with 4 hidden units, and an output layer with 1 output unit. Thus means W1‘s shape was (4,2), b1 was (4,1), W2 was (1,4) and b2 was (1,1). Now you will generalize this to LL layers!
• Here is the implementation for L=1L=1 (one layer neural network). It should inspire you to implement the general case (L-layer neural network).

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

# 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): #my# range(1,3)  [1,2]   range(10) presents： range(0, 10) [0, 1, 2, 3, 4, 5, 6, 7, 8, 9]
        ### 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])
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))

W1 = [[ 0.01788628  0.0043651   0.00096497 -0.01863493 -0.00277388]
 [-0.00354759 -0.00082741 -0.00627001 -0.00043818 -0.00477218]
 [-0.01313865  0.00884622  0.00881318  0.01709573  0.00050034]
 [-0.00404677 -0.0054536  -0.01546477  0.00982367 -0.01101068]]
b1 = [[ 0.]
 [ 0.]
 [ 0.]
 [ 0.]]
W2 = [[-0.01185047 -0.0020565   0.01486148  0.00236716]
 [-0.01023785 -0.00712993  0.00625245 -0.00160513]
 [-0.00768836 -0.00230031  0.00745056  0.01976111]]
b2 = [[ 0.]
 [ 0.]
 [ 0.]]

Expected output:

4 - Forward propagation module

4.1 - Linear Forward

Now that you have initialized your parameters, you will do the forward propagation module. You will start by implementing some basic functions that you will use later when implementing the model. You will complete three functions in this order:

• LINEAR
• LINEAR -> ACTIVATION where ACTIVATION will be either ReLU or Sigmoid.
• [LINEAR -> RELU] ×× (L-1) -> LINEAR -> SIGMOID (whole model)

The linear forward module (vectorized over all the examples) computes the following equations:

where A[0]=XA[0]=X.

Exercise: Build the linear part of forward propagation.

Reminder: The mathematical representation of this unit is Z[l]=W[l]A[l?1]+b[l]Z[l]=W[l]A[l?1]+b[l]. You may also find np.dot() useful. If your dimensions don‘t match, printing W.shape may help.

# 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
    ### 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()

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

Expected output:

Building your Deep Neural Network: Step by Step¶