Neural Networks and Deep Learning

Certificate

Logistic Regression

$z = w^Tx+b$

$\hat{y} = a = \sigma(z)$

$L(a,y) = -(y \log(a) + (1-y)\log(1-a))$

$\frac{dL}{dz} = \frac{dL}{da} \times \frac{da}{dz}$

$\frac{dL}{da} = \frac{a-y}{a(1-a)}$

$\frac{da}{dz} = \frac{d}{dz}\sigma(z) = a(1-a),a = \sigma(z)$

$\frac{dL}{dz} = a - y$

Example: Build logistic regression from scratch to classify cats

$z^{(i)} = w^Tx^{(i)}+b$

$\hat{y}^{(i)} = a^{(i)} = \text{sigmoid}(z^{(i)})$

$L(a^{(i)}, y^{(i)}) = -y\log(a) - (1-y)\log(1-a)$

The cost function $J = \frac{1}{m}\sum^{m}_{i=1}L(a^{(i)}, y^{(i)})$

Backward Propagation

$\frac{\partial J}{\partial w} = \frac{1}{m}X(A-Y)^T$

$\frac{\partial J}{\partial b} = \frac{1}{m}(a^i - y^i)$

Results

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 %

Code

import numpy as np

Shallow Neural Network

Activation functions:

• Sigmoid $a = \sigma(z) = \frac{1}{1+e^{-z}}$

• Tanh $a = \text{tanh}(z) = \frac{e^z - e^{-z}}{e^z + e^{-z}}$

• Relu $a = max(0, z)$

• Leak Relu $a = max(0.01z, z)$

• Softmax $a = \frac{e^z_i}{\sum^{N}_{j=1}e^{z_i}}$

Derivatives of Activation functions:

• Sigmoid $g'(z) = a(1-a)$

• Tanh $g'(z) = 1 - a^2$

• Relu $g'(z) = 0 \text{ if } z<0 \text{ or }1 \text{ if } z>0$

Even for a basic Neural Network, there are many design decisions to make:

1. # of hidden layers (depth)

2. # of units per hidden layer (width)

3. Type of activation function (nonlinearity)

4. Form of objective function

5. 

Deep Neural Network

Notation

• Superscript $[𝑙]$denotes a quantity associated with the $𝑙𝑡ℎ$layer.

• Superscript $(𝑖)$denotes a quantity associated with the $𝑖𝑡ℎ$example.

• Lowerscript $𝑖$denotes the $𝑖𝑡ℎ$entry of a vector.

Forward Propagation

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

$𝑍^{[𝑙]}=𝑊^{[𝑙]}𝐴^{[𝑙−1]}+𝑏^𝑙$ where $𝐴^{[0]}=𝑋$

Linear-Activation Forward

Two activation functions are used:

• Sigmoid $A = \sigma(Z) = \sigma(WA+b) = \frac{1}{1+e^{-(WA+b)}}$

• ReLU $A = \max(0, Z)$

L-Layer Model

Cost Function $J = -\frac{1}{m}\sum_{i=1}^{m}(y^{(i)}\log(a^{[L](i) })+(1-y^{(i)})\log(1-a^{[L](i)}))$

Backward Propagation

Linear Backward

For layer $𝑙$, the linear part is: $𝑍^{[𝑙]}=𝑊^{[𝑙]}𝐴^{[𝑙−1]}+𝑏^{[𝑙]}$ (followed by an activation).

Suppose you have already calculated the derivative $dZ^{[l]} = \frac{\partial L}{\partial Z^{[l]}}$. You want to get $(𝑑𝑊^{[𝑙]},𝑑𝑏^{[𝑙]},𝑑𝐴^{[𝑙−1]})$.

The three outputs $(𝑑𝑊^{[𝑙]},𝑑𝑏^{[𝑙]},𝑑𝐴^{[𝑙−1]})$are computed using the input $𝑑𝑍[𝑙]$.

Here are the formulas you need:

$dW^{[l]} = \frac{\partial J}{\partial W^{[l]}} = \frac{1}{m}dZ^{[l]}A^{[l-1]T}$

$db^{[l]} = \frac{\partial J}{\partial b^{[l]}} = \frac{1}{m}\sum_{i=1}^{m}dZ^{[l](i)}$

$dA^{[i-1]} = \frac{\partial L}{\partial A^{[l-1]}} = W^{[l]T}dZ^{[l]}$

Linear-Activation Backward

$dZ^{[l]} = dA^{[l]} * g'(Z^{[l]})$

Update Parameters

$W^{[l]} = W^{[l]}-\alpha dW^{[l]}$

$b^{[l]} = b^{[l]} - \alpha db^{[l]}$ where $\alpha$ is the learning rate

Example: Implement Neural Network from Scratch

import numpy as np
import h5py
import matplotlib.pyplot as plt
from testCases import *
from dnn_utils import sigmoid, sigmoid_backward, relu, relu_backward
from public_tests import *

import copy
%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)

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)
    
    #(≈ 4 lines of code)
    # W1 = ...
    # b1 = ...
    # W2 = ...
    # b2 = ...
    # YOUR CODE STARTS HERE
    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))
    
    # YOUR CODE ENDS HERE
    
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2}
    
    return parameters    

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):
        #(≈ 2 lines of code)
        # parameters['W' + str(l)] = ...
        # parameters['b' + str(l)] = ...
        # YOUR CODE STARTS HERE
        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))
        
        # YOUR CODE ENDS 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
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 tuple containing "A", "W" and "b" ; stored for computing the backward pass efficiently
    """
    
    #(≈ 1 line of code)
    # Z = ...
    # YOUR CODE STARTS HERE
    Z = np.dot(W,A) + b
    
    # YOUR CODE ENDS HERE
    cache = (A, W, b)
    
    return Z, cache

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 tuple containing "linear_cache" and "activation_cache";
             stored for computing the backward pass efficiently
    """
    
    if activation == "sigmoid":
        #(≈ 2 lines of code)
        # Z, linear_cache = ...
        # A, activation_cache = ...
        # YOUR CODE STARTS HERE
        Z, linear_cache = linear_forward(A_prev, W, b)
        A, activation_cache = sigmoid(Z)
        # YOUR CODE ENDS HERE
    
    elif activation == "relu":
        #(≈ 2 lines of code)
        # Z, linear_cache = ...
        # A, activation_cache = ...
        # YOUR CODE STARTS HERE
        Z, linear_cache = linear_forward(A_prev, W, b) 
        A, activation_cache = relu(Z)
        # YOUR CODE ENDS HERE
    cache = (linear_cache, activation_cache)

    return A, cache

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 -- activation value from the output (last) layer
    caches -- list of caches containing:
                every cache of linear_activation_forward() (there are L 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.
    # The for loop starts at 1 because layer 0 is the input
    for l in range(1, L):
        A_prev = A 
        #(≈ 2 lines of code)
        # A, cache = ...
        # caches ...
        # YOUR CODE STARTS HERE
        A, cache = linear_activation_forward(A_prev, parameters['W' + str(l)], parameters['b' + str(l)], "relu")
        caches.append(cache)
        
        # YOUR CODE ENDS HERE
    
    # Implement LINEAR -> SIGMOID. Add "cache" to the "caches" list.
    #(≈ 2 lines of code)
    # AL, cache = ...
    # caches ...
    # YOUR CODE STARTS HERE
    AL, cache = linear_activation_forward(A, parameters['W' + str(L)], parameters['b' + str(L)], "sigmoid")
    caches.append(cache)
    # YOUR CODE ENDS HERE
          
    return AL, caches
    
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.
    # (≈ 1 lines of code)
    # cost = ...
    # YOUR CODE STARTS HERE
    
    cost = (-1/m) * (np.dot(Y, np.log(AL).T) + np.dot((1-Y), np.log(1-AL).T))
    # YOUR CODE ENDS HERE
    
    cost = np.squeeze(cost)      # To make sure your cost's shape is what we expect (e.g. this turns [[17]] into 17).

    
    return cost
    
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 = ...
    # db = ... sum by the rows of dZ with keepdims=True
    # dA_prev = ...
    # YOUR CODE STARTS HERE
    dW = (1/m) * np.dot(dZ, A_prev.T)
    db = (1/m) * np.sum(dZ, axis=1, keepdims=True)
    dA_prev = np.dot(W.T,dZ)
    
    # YOUR CODE ENDS HERE
    
    return dA_prev, dW, db
    
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":
        #(≈ 2 lines of code)
        # dZ =  ...
        # dA_prev, dW, db =  ...
        # YOUR CODE STARTS HERE
        dZ = relu_backward(dA, activation_cache)
        
        # YOUR CODE ENDS HERE
        
    elif activation == "sigmoid":
        #(≈ 2 lines of code)
        # dZ =  ...
        # dA_prev, dW, db =  ...
        # YOUR CODE STARTS HERE
        
        dZ = sigmoid_backward(dA, activation_cache)
        # YOUR CODE ENDS HERE
        
    dA_prev, dW, db = linear_backward(dZ, linear_cache)
    
    return dA_prev, dW, db

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
    #(1 line of code)
    # dAL = ...
    # YOUR CODE STARTS HERE
    dAL = - (np.divide(Y, AL) - np.divide(1 - Y, 1 - AL))
    
    # YOUR CODE ENDS HERE
    
    # Lth layer (SIGMOID -> LINEAR) gradients. Inputs: "dAL, current_cache". Outputs: "grads["dAL-1"], grads["dWL"], grads["dbL"]
    #(approx. 5 lines)
    # current_cache = ...
    # dA_prev_temp, dW_temp, db_temp = ...
    # grads["dA" + str(L-1)] = ...
    # grads["dW" + str(L)] = ...
    # grads["db" + str(L)] = ...
    # YOUR CODE STARTS HERE
    
    current_cache = caches[L-1] # Last Layer
    grads["dA" + str(L-1)], grads["dW" + str(L)], grads["db" + str(L)] = linear_activation_backward(dAL, current_cache, "sigmoid")
    # YOUR CODE ENDS HERE
    
    # Loop from l=L-2 to l=0
    for l in reversed(range(L-1)):
        # lth layer: (RELU -> LINEAR) gradients.
        # Inputs: "grads["dA" + str(l + 1)], current_cache". Outputs: "grads["dA" + str(l)] , grads["dW" + str(l + 1)] , grads["db" + str(l + 1)] 
        #(approx. 5 lines)
        # current_cache = ...
        # dA_prev_temp, dW_temp, db_temp = ...
        # grads["dA" + str(l)] = ...
        # grads["dW" + str(l + 1)] = ...
        # grads["db" + str(l + 1)] = ...
        # YOUR CODE STARTS HERE
        current_cache = caches[l]
        dA_prev_temp, dW_temp, db_temp = linear_activation_backward(grads["dA" + str(l + 1)], current_cache, activation = "relu")
        grads["dA" + str(l)] = dA_prev_temp
        grads["dW" + str(l + 1)] = dW_temp
        grads["db" + str(l + 1)] = db_temp
        
        # YOUR CODE ENDS HERE

    return grads

def update_parameters(params, grads, learning_rate):
    """
    Update parameters using gradient descent
    
    Arguments:
    params -- 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)] = ...
    """
    parameters = copy.deepcopy(params)
    L = len(parameters) // 2 # number of layers in the neural network

    # Update rule for each parameter. Use a for loop.
    #(≈ 2 lines of code)
    for l in range(L):
        # parameters["W" + str(l+1)] = ...
        # parameters["b" + str(l+1)] = ...
        # YOUR CODE STARTS HERE
        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)]
        
        # YOUR CODE ENDS HERE
    return parameters

Deep Neural Network - Application

2-Layer Neural Network to Classify Cat Images

Output

Training Accuracy: 0.9999999999999998

Prediction Accuracy: 0.72

Cost after iteration 0: 0.693049735659989
Cost after iteration 100: 0.6464320953428849
Cost after iteration 200: 0.6325140647912677
Cost after iteration 300: 0.6015024920354665
Cost after iteration 400: 0.5601966311605747
Cost after iteration 500: 0.5158304772764729
Cost after iteration 600: 0.4754901313943325
Cost after iteration 700: 0.43391631512257495
Cost after iteration 800: 0.4007977536203886
Cost after iteration 900: 0.3580705011323798
Cost after iteration 1000: 0.3394281538366413
Cost after iteration 1100: 0.30527536361962654
Cost after iteration 1200: 0.2749137728213015
Cost after iteration 1300: 0.2468176821061484
Cost after iteration 1400: 0.19850735037466102
Cost after iteration 1500: 0.17448318112556638
Cost after iteration 1600: 0.1708076297809692
Cost after iteration 1700: 0.11306524562164715
Cost after iteration 1800: 0.09629426845937156
Cost after iteration 1900: 0.0834261795972687
Cost after iteration 2000: 0.07439078704319085
Cost after iteration 2100: 0.06630748132267933
Cost after iteration 2200: 0.05919329501038172
Cost after iteration 2300: 0.053361403485605606
Cost after iteration 2400: 0.04855478562877019
Cost after iteration 2499: 0.04421498215868956

Code

import time
import numpy as np
import h5py
import matplotlib.pyplot as plt
import scipy
from PIL import Image
from scipy import ndimage
from dnn_app_utils_v3 import *
from public_tests import *

%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)

train_x_orig, train_y, test_x_orig, test_y, classes = load_data()

index = 10
plt.imshow(train_x_orig[index])
print ("y = " + str(train_y[0,index]) + ". It's a " + classes[train_y[0,index]].decode("utf-8") +  " picture.")

# Explore your dataset 
m_train = train_x_orig.shape[0]
num_px = train_x_orig.shape[1]
m_test = test_x_orig.shape[0]

print ("Number of training examples: " + str(m_train))
print ("Number of testing examples: " + str(m_test))
print ("Each image is of size: (" + str(num_px) + ", " + str(num_px) + ", 3)")
print ("train_x_orig shape: " + str(train_x_orig.shape))
print ("train_y shape: " + str(train_y.shape))
print ("test_x_orig shape: " + str(test_x_orig.shape))
print ("test_y shape: " + str(test_y.shape))

# Reshape the training and test examples 
train_x_flatten = train_x_orig.reshape(train_x_orig.shape[0], -1).T   # The "-1" makes reshape flatten the remaining dimensions
test_x_flatten = test_x_orig.reshape(test_x_orig.shape[0], -1).T

# Standardize data to have feature values between 0 and 1.
train_x = train_x_flatten/255.
test_x = test_x_flatten/255.

print ("train_x's shape: " + str(train_x.shape))
print ("test_x's shape: " + str(test_x.shape))

### CONSTANTS DEFINING THE MODEL ####
n_x = 12288     # num_px * num_px * 3
n_h = 7
n_y = 1
layers_dims = (n_x, n_h, n_y)
learning_rate = 0.0075

def two_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
    """
    Implements a two-layer neural network: LINEAR->RELU->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (n_x, number of examples)
    Y -- true "label" vector (containing 1 if cat, 0 if non-cat), of shape (1, number of examples)
    layers_dims -- dimensions of the layers (n_x, n_h, n_y)
    num_iterations -- number of iterations of the optimization loop
    learning_rate -- learning rate of the gradient descent update rule
    print_cost -- If set to True, this will print the cost every 100 iterations 
    
    Returns:
    parameters -- a dictionary containing W1, W2, b1, and b2
    """
    
    np.random.seed(1)
    grads = {}
    costs = []                              # to keep track of the cost
    m = X.shape[1]                           # number of examples
    (n_x, n_h, n_y) = layers_dims
    
    # Initialize parameters dictionary, by calling one of the functions you'd previously implemented
    #(≈ 1 line of code)
    # parameters = ...
    # YOUR CODE STARTS HERE
    parameters = initialize_parameters(n_x, n_h, n_y)
    
    # YOUR CODE ENDS HERE
    
    # Get W1, b1, W2 and b2 from the dictionary parameters.
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    
    # Loop (gradient descent)

    for i in range(0, num_iterations):

        # Forward propagation: LINEAR -> RELU -> LINEAR -> SIGMOID. Inputs: "X, W1, b1, W2, b2". Output: "A1, cache1, A2, cache2".
        #(≈ 2 lines of code)
        # A1, cache1 = ...
        # A2, cache2 = ...
        # YOUR CODE STARTS HERE
        A1, cache1 = linear_activation_forward(X, W1, b1, activation="relu")
        A2, cache2 = linear_activation_forward(A1, W2, b2, activation="sigmoid")
        
        # YOUR CODE ENDS HERE
        
        # Compute cost
        #(≈ 1 line of code)
        # cost = ...
        # YOUR CODE STARTS HERE
        cost = compute_cost(A2, Y)
        
        # YOUR CODE ENDS HERE
        
        # Initializing backward propagation
        dA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))
        
        # Backward propagation. Inputs: "dA2, cache2, cache1". Outputs: "dA1, dW2, db2; also dA0 (not used), dW1, db1".
        #(≈ 2 lines of code)
        # dA1, dW2, db2 = ...
        # dA0, dW1, db1 = ...
        # YOUR CODE STARTS HERE
        dA2 = - (np.divide(Y, A2) - np.divide(1 - Y, 1 - A2))
        
        dA1, dW2, db2 = linear_activation_backward(dA2, cache2, activation="sigmoid")
        dA0, dW1, db1 = linear_activation_backward(dA1, cache1, activation="relu")
        
        # YOUR CODE ENDS HERE
        
        # Set grads['dWl'] to dW1, grads['db1'] to db1, grads['dW2'] to dW2, grads['db2'] to db2
        grads['dW1'] = dW1
        grads['db1'] = db1
        grads['dW2'] = dW2
        grads['db2'] = db2
        
        # Update parameters.
        #(approx. 1 line of code)
        # parameters = ...
        # YOUR CODE STARTS HERE
        parameters = update_parameters(parameters, grads, learning_rate)
        
        # YOUR CODE ENDS HERE

        # Retrieve W1, b1, W2, b2 from parameters
        W1 = parameters["W1"]
        b1 = parameters["b1"]
        W2 = parameters["W2"]
        b2 = parameters["b2"]
        
        # Print the cost every 100 iterations
        if print_cost and i % 100 == 0 or i == num_iterations - 1:
            print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))
        if i % 100 == 0 or i == num_iterations:
            costs.append(cost)

    return parameters, costs

def plot_costs(costs, learning_rate=0.0075):
    plt.plot(np.squeeze(costs))
    plt.ylabel('cost')
    plt.xlabel('iterations (per hundreds)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()

parameters, costs = two_layer_model(train_x, train_y, layers_dims = (n_x, n_h, n_y), num_iterations = 2500, print_cost=True)
plot_costs(costs, learning_rate)

4-Layer Neural Network

Output

Train Accuracy: 0.9999999999999998

Prediction Accuracy: 0.74

Cost after iteration 0: 0.6950464961800915
Cost after iteration 100: 0.5892596054583805
Cost after iteration 200: 0.5232609173622991
Cost after iteration 300: 0.4497686396221906
Cost after iteration 400: 0.4209002161883899
Cost after iteration 500: 0.37246403061745953
Cost after iteration 600: 0.3474205187020191
Cost after iteration 700: 0.31719191987370265
Cost after iteration 800: 0.2664377434774658
Cost after iteration 900: 0.21991432807842573
Cost after iteration 1000: 0.1435789889362377
Cost after iteration 1100: 0.4530921262322132
Cost after iteration 1200: 0.09499357670093511
Cost after iteration 1300: 0.08014128076781366
Cost after iteration 1400: 0.0694023400553646
Cost after iteration 1500: 0.060216640231745895
Cost after iteration 1600: 0.05327415758001879
Cost after iteration 1700: 0.04762903262098432
Cost after iteration 1800: 0.04297588879436867
Cost after iteration 1900: 0.03903607436513823
Cost after iteration 2000: 0.03568313638049028
Cost after iteration 2100: 0.032915263730546776
Cost after iteration 2200: 0.030472193059120623
Cost after iteration 2300: 0.028387859212946117
Cost after iteration 2400: 0.026615212372776077
Cost after iteration 2499: 0.024821292218353375

Code

### CONSTANTS ###
layers_dims = [12288, 20, 7, 5, 1] #  4-layer model

def L_layer_model(X, Y, layers_dims, learning_rate = 0.0075, num_iterations = 3000, print_cost=False):
    """
    Implements a L-layer neural network: [LINEAR->RELU]*(L-1)->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (n_x, number of examples)
    Y -- true "label" vector (containing 1 if cat, 0 if non-cat), of shape (1, number of examples)
    layers_dims -- list containing the input size and each layer size, of length (number of layers + 1).
    learning_rate -- learning rate of the gradient descent update rule
    num_iterations -- number of iterations of the optimization loop
    print_cost -- if True, it prints the cost every 100 steps
    
    Returns:
    parameters -- parameters learnt by the model. They can then be used to predict.
    """

    np.random.seed(1)
    costs = []                         # keep track of cost
    
    # Parameters initialization.
    #(≈ 1 line of code)
    # parameters = ...
    # YOUR CODE STARTS HERE
    parameters = initialize_parameters_deep(layers_dims)
    
    # YOUR CODE ENDS HERE
    
    # Loop (gradient descent)
    for i in range(0, num_iterations):

        # Forward propagation: [LINEAR -> RELU]*(L-1) -> LINEAR -> SIGMOID.
        #(≈ 1 line of code)
        # AL, caches = ...
        # YOUR CODE STARTS HERE
        AL, caches = L_model_forward(X, parameters)
        
        # YOUR CODE ENDS HERE
        
        # Compute cost.
        #(≈ 1 line of code)
        # cost = ...
        # YOUR CODE STARTS HERE
        cost = compute_cost(AL, Y)
        
        # YOUR CODE ENDS HERE
    
        # Backward propagation.
        #(≈ 1 line of code)
        # grads = ...    
        # YOUR CODE STARTS HERE
        grads = L_model_backward(AL, Y, caches)
        
        # YOUR CODE ENDS HERE
 
        # Update parameters.
        #(≈ 1 line of code)
        # parameters = ...
        # YOUR CODE STARTS HERE
        parameters = update_parameters(parameters, grads, learning_rate)
        
        # YOUR CODE ENDS HERE
                
        # Print the cost every 100 iterations
        if print_cost and i % 100 == 0 or i == num_iterations - 1:
            print("Cost after iteration {}: {}".format(i, np.squeeze(cost)))
        if i % 100 == 0 or i == num_iterations:
            costs.append(cost)
    
    return parameters, costs

parameters, costs = L_layer_model(train_x, train_y, layers_dims, num_iterations = 2500, print_cost = True)

pred_train = predict(train_x, train_y, parameters)

pred_test = predict(test_x, test_y, parameters)