[DL Specialization] C2W1A3

Gradient Checking

Packages

import numpy as np
from testCases import *
from public_tests import *
from gc_utils import sigmoid, relu, dictionary_to_vector, vector_to_dictionary, gradients_to_vector

%load_ext autoreload
%autoreload 2

 

 

Dimensional Gradient Checking

forward propagation

def forward_propagation(x, theta):
    J = theta * x
    
    return J

x, theta = 2, 4
J = forward_propagation(x, theta)
print ("J = " + str(J))
forward_propagation_test(forward_propagation)

J = 8

 

backward propagation

def backward_propagation(x, theta):
    dtheta = x
    
    return dtheta

x, theta = 3, 4
dtheta = backward_propagation(x, theta)
print ("dtheta = " + str(dtheta))
backward_propagation_test(backward_propagation)

dtheta = 3

 

gradient check

def gradient_check(x, theta, epsilon=1e-7, print_msg=False):
    theta_plus = theta + epsilon                               
    theta_minus = theta - epsilon                              
    J_plus = forward_propagation(x, theta_plus)                
    J_minus = forward_propagation(x, theta_minus)              
    gradapprox = (J_plus - J_minus) / (2 * epsilon)
    grad = backward_propagation(x, theta)
    
    numerator = np.linalg.norm(grad - gradapprox)
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)
    difference = numerator / denominator
    
    if print_msg:
        if difference > 2e-7:
            print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
        else:
            print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")
    
    return difference

x, theta = 3, 4
difference = gradient_check(x, theta, print_msg=True)

difference = 7.814075313343006e-11

 

N-Dimensional Gradient Checking

def forward_propagation_n(X, Y, parameters):
    # retrieve parameters
    m = X.shape[1]
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]

    Z1 = np.dot(W1, X) + b1
    A1 = relu(Z1)
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)

    # Cost
    log_probs = np.multiply(-np.log(A3),Y) + np.multiply(-np.log(1 - A3), 1 - Y)
    cost = 1. / m * np.sum(log_probs)
    
    cache = (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3)
    
    return cost, cache

def backward_propagation_n(X, Y, cache):
    
    m = X.shape[1]
    (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3) = cache
    
    dZ3 = A3 - Y
    dW3 = 1. / m * np.dot(dZ3, A2.T)
    db3 = 1. / m * np.sum(dZ3, axis=1, keepdims=True)
    
    dA2 = np.dot(W3.T, dZ3)
    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    dW2 = 1. / m * np.dot(dZ2, A1.T) * 2
    db2 = 1. / m * np.sum(dZ2, axis=1, keepdims=True)
    
    dA1 = np.dot(W2.T, dZ2)
    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    dW1 = 1. / m * np.dot(dZ1, X.T)
    db1 = 4. / m * np.sum(dZ1, axis=1, keepdims=True)
    
    gradients = {"dZ3": dZ3, "dW3": dW3, "db3": db3,
                 "dA2": dA2, "dZ2": dZ2, "dW2": dW2, "db2": db2,
                 "dA1": dA1, "dZ1": dZ1, "dW1": dW1, "db1": db1}
    
    return gradients

 

gradient check n

def gradient_check_n(parameters, gradients, X, Y, epsilon=1e-7, print_msg=False)
    # Set-up variables
    parameters_values, _ = dictionary_to_vector(parameters)
    
    grad = gradients_to_vector(gradients)
    num_parameters = parameters_values.shape[0]
    J_plus = np.zeros((num_parameters, 1))
    J_minus = np.zeros((num_parameters, 1))
    gradapprox = np.zeros((num_parameters, 1))
    
    # Compute gradapprox
    for i in range(num_parameters):
        
        theta_plus = np.copy(parameters_values)
        theta_plus[i][0] = theta_plus[i][0] + epsilon
        J_plus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(theta_plus))
        
        theta_minus = np.copy(parameters_values)
        theta_minus[i][0] = theta_minus[i][0] - epsilon
        J_minus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(theta_minus))
        
        gradapprox[i] = (J_plus[i] - J_minus[i]) / (2 * epsilon)
        
    
    numerator = np.linalg.norm(grad - gradapprox)
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)
    difference = np.divide(numerator, denominator)
    
    if print_msg:
        if difference > 2e-7:
            print ("\033[93m" + "There is a mistake in the backward propagation! difference = " + str(difference) + "\033[0m")
        else:
            print ("\033[92m" + "Your backward propagation works perfectly fine! difference = " + str(difference) + "\033[0m")

    return difference

X, Y, parameters = gradient_check_n_test_case()

cost, cache = forward_propagation_n(X, Y, parameters)
gradients = backward_propagation_n(X, Y, cache)
difference = gradient_check_n(parameters, gradients, X, Y, 1e-7, True)
expected_values = [0.2850931567761623, 1.1890913024229996e-07]
assert not(type(difference) == np.ndarray), "You are not using np.linalg.norm for numerator or denominator"
assert np.any(np.isclose(difference, expected_values)), "Wrong value. It is not one of the expected values"

difference = 0.2850931567761623