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

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 import *
from dnn_utils 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:

  • Implement the forward propagation module (shown in purple in the figure below).

    • We give you the ACTIVATION function (relu/sigmoid).

    • Combine the previous two steps into a new [LINEAR->ACTIVATION] 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

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)*.01
    b1 = np.zeros((n_h, 1))
    W2 = np.random.randn(n_y, n_h)*.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.]]

3.2 - L-layer Neural Network

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

Instructions:

  • Use random initialization for the weight matrices. Use np.random.rand(shape) * 0.01.

  • Use zeros initialization for the biases. Use np.zeros(shape).

  •   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])*.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.]]

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.

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

Exercise: Build the linear part of forward propagation.

# 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))
"""
Z = [[ 3.26295337 -1.23429987]]
"""

4.2 - Linear-Activation Forward

In this notebook, you will use two activation functions:

  • A, activation_cache = sigmoid(Z)
  • A, activation_cache = relu(Z)

    For more convenience, you are going to group two functions (Linear and Activation) into one function (LINEAR->ACTIVATION). Hence, you will implement a function that does the LINEAR forward step followed by an ACTIVATION forward step.

# 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)
        ### 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]))
    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))
With sigmoid: A = [[ 0.96890023  0.11013289]]
With ReLU: A = [[ 3.43896131  0.        ]]

Note: In deep learning, the "[LINEAR->ACTIVATION]" computation is counted as a single layer in the neural network, not two layers.

d) L-Layer Model

Exercise: Implement the forward propagation of the above model.

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_relu_forward() (there are L-1 of them, indexed from 0 to L-2)
                the cache of linear_sigmoid_forward() (there is one, indexed 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()
AL, caches = L_model_forward(X, parameters)
print("AL = " + str(AL))
print("Length of caches list = " + str(len(caches)))
AL = [[ 0.03921668  0.70498921  0.19734387  0.04728177]]
Length of caches list = 3

5 - Cost function

Now you will implement forward and backward propagation. You need to compute the cost, because you want to check if your model is actually learning.

# 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 = -np.sum(Y*np.log(AL) + (1 - Y)*np.log(1 - AL))/m
    ### 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("cost = " + str(compute_cost(AL, Y)))
"""
cost = 0.414931599615
"""

6 - Backward propagation module

Just like with forward propagation, you will implement helper functions for backpropagation. Remember that back propagation is used to calculate the gradient of the loss function with respect to the parameters.

This is why we talk about backpropagation.

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

6.1 - Linear backward

Exercise: Use the 3 formulas above to implement linear_backward().

# 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, A_prev.T)/m
    db = dZ.sum(axis = 1, keepdims = True)/m
    dA_prev = np.dot(W.T, dZ)
    ### END CODE HERE ###

    assert (dA_prev.shape == A_prev.shape)
    assert (dW.shape == W.shape)
    assert (db.shape == b.shape)

    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))
dA_prev = [[ 0.51822968 -0.19517421]
           [-0.40506361 0.15255393]
           [ 2.37496825 -0.89445391]]

dW = [[-0.10076895 1.40685096 1.64992505]]

db = [[ 0.50629448]]

6.2 - Linear-Activation backward

Next, you will create a function that merges the two helper functions: linear_backward and the backward step for the activation linear_activation_backward.

To help you implement linear_activation_backward, we provided two backward functions:

  • sigmoid_backward: Implements the backward propagation for SIGMOID unit. You can call it as follows:

dZ = sigmoid_backward(dA, activation_cache)
  • relu_backward: Implements the backward propagation for RELU unit. You can call it as follows:

dZ = relu_backward(dA, activation_cache)

Exercise: Implement the backpropagation for the LINEAR->ACTIVATION layer.

# 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)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###

    elif activation == "sigmoid":
        ### START CODE HERE ### (≈ 2 lines of code)
        dZ = sigmoid_backward(dA, activation_cache)
        dA_prev, dW, db = linear_backward(dZ, linear_cache)
        ### END CODE HERE ###

    return dA_prev, dW, db
AL, linear_activation_cache = linear_activation_backward_test_case()


dA_prev, dW, db = linear_activation_backward(AL, 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(AL, linear_activation_cache, activation = "relu")
print ("relu:")
print ("dA_prev = "+ str(dA_prev))
print ("dW = " + str(dW))
print ("db = " + str(db))
sigmoid:
dA_prev = [[ 0.11017994 0.01105339]
           [ 0.09466817 0.00949723]
           [-0.05743092 -0.00576154]]

dW = [[ 0.10266786 0.09778551 -0.01968084]]

db = [[-0.05729622]]


relu:
dA_prev = [[ 0.44090989 0. ]
           [ 0.37883606 0. ]
           [-0.2298228 0. ]]

dW = [[ 0.44513824 0.37371418 -0.10478989]]

db = [[-0.20837892]]

6.3 - L-Model Backward

dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL)) # derivative of cost with respect to AL

You can then use this post-activation gradient dAL to keep going backward. As seen in Figure 5, you can now feed in dAL into the LINEAR->SIGMOID backward function you implemented (which will use the cached values stored by the L_model_forward function). After that, you will have to use a for loop to iterate through all the other layers using the LINEAR->RELU backward function. You should store each dA, dW, and db in the grads dictionary. To do so, use this formula :

# 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
    ### 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 ###

    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()
grads = L_model_backward(AL, Y_assess, caches)
print_grads(grads)
dW1 = [[ 0.41010002 0.07807203 0.13798444 0.10502167]
       [ 0. 0. 0. 0. ]
       [ 0.05283652 0.01005865 0.01777766 0.0135308 ]]
db1 = [[-0.22007063]
       [ 0. ]
       [-0.02835349]]
dA1 = [[ 0.12913162 -0.44014127]
       [-0.14175655 0.48317296]
       [ 0.01663708 -0.05670698]]

6.4 - Update Parameters

In this section you will update the parameters of the model, using gradient descent:

Exercise: Implement update_parameters() to update your parameters using gradient descent.

# 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(1,L + 1):
        parameters['W' + str(l)] -= learning_rate*grads['dW' + str(l)]
        parameters['b' + str(l)] -= learning_rate*grads['db' + str(l)]
    ### 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"]))
W1 = [[-0.59562069 -0.09991781 -2.14584584 1.82662008]
      [-1.76569676 -0.80627147 0.51115557 -1.18258802]
      [-1.0535704 -0.86128581 0.68284052 2.20374577]]

b1 = [[-0.04659241]
      [-1.28888275]
      [ 0.53405496]]

W2 = [[-0.55569196 0.0354055 1.32964895]]

b2 = [[-0.84610769]]

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

You will in fact use these models to classify cat vs non-cat images!

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