[DL Specialization] C2W1A2

Regularization

Packages

# import packages
import numpy as np
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets
import scipy.io
from reg_utils import sigmoid, relu, plot_decision_boundary, initialize_parameters, load_2D_dataset, predict_dec
from reg_utils import compute_cost, predict, forward_propagation, backward_propagation, update_parameters
from testCases import *
from public_tests import *

%matplotlib inline
plt.rcParams['figure.figsize'] = (7.0, 4.0) 
plt.rcParams['image.interpolation'] = 'nearest'
plt.rcParams['image.cmap'] = 'gray'

%load_ext autoreload
%autoreload 2

 

Problem Statement

프랑스 골키퍼가 공을 차서 프랑스 팀 선수들이 헤딩할 수 있는 위치 추천

 

Loading the Dataset

train_X, train_Y, test_X, test_Y = load_2D_dataset()

Blue: 프랑스 선수가 헤딩함

Red: 상대팀 선수가 헤딩함

 

Non-Regularized Model

def model(X, Y, learning_rate = 0.3, num_iterations = 30000, print_cost = True, lambd = 0, keep_prob = 1):
 
    grads = {}
    costs = []                           
    m = X.shape[1]                        
    layers_dims = [X.shape[0], 20, 3, 1]
    
    parameters = initialize_parameters(layers_dims)


    for i in range(0, num_iterations):

        if keep_prob == 1:
            a3, cache = forward_propagation(X, parameters)
        elif keep_prob < 1:
            a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)
        
        if lambd == 0:
            cost = compute_cost(a3, Y)
        else:
            cost = compute_cost_with_regularization(a3, Y, parameters, lambd)
            
        assert (lambd == 0 or keep_prob == 1)   
        if lambd == 0 and keep_prob == 1:
            grads = backward_propagation(X, Y, cache)
        elif lambd != 0:
            grads = backward_propagation_with_regularization(X, Y, cache, lambd)
        elif keep_prob < 1:
            grads = backward_propagation_with_dropout(X, Y, cache, keep_prob)
        
        parameters = update_parameters(parameters, grads, learning_rate)
        
        if print_cost and i % 10000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
        if print_cost and i % 1000 == 0:
            costs.append(cost)
    
    plt.plot(costs)
    plt.ylabel('cost')
    plt.xlabel('iterations (x1,000)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

parameters = model(train_X, train_Y)
print ("On the training set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)

Cost after iteration 0: 0.6557412523481002
Cost after iteration 10000: 0.16329987525724204
Cost after iteration 20000: 0.13851642423234922

On the training set:
Accuracy: 0.9478672985781991
On the test set:
Accuracy: 0.915

plt.title("Model without regularization")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

 

L2 Regularization

compute cost

def compute_cost_with_regularization(A3, Y, parameters, lambd):
    m = Y.shape[1]
    W1 = parameters["W1"]
    W2 = parameters["W2"]
    W3 = parameters["W3"]
    
    cross_entropy_cost = compute_cost(A3, Y) # This gives you the cross-entropy part of the cost
    
    L2_regularization_cost = (lambd / (2 * m)) * (np.sum(np.square(W1)) + np.sum(np.square(W2)) + np.sum(np.square(W3)))
    
    cost = cross_entropy_cost + L2_regularization_cost
    
    return cost

A3, t_Y, parameters = compute_cost_with_regularization_test_case()
cost = compute_cost_with_regularization(A3, t_Y, parameters, lambd=0.1)
print("cost = " + str(cost))

compute_cost_with_regularization_test(compute_cost_with_regularization)

cost = 1.7864859451590758

 

backward propagation

def backward_propagation_with_regularization(X, Y, cache, lambd):
    
    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) + (lambd / m) * W3
    
    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) + (lambd / m) * W2
    
    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) + (lambd / m) * W1
    
    db1 = 1. / 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

t_X, t_Y, cache = backward_propagation_with_regularization_test_case()

grads = backward_propagation_with_regularization(t_X, t_Y, cache, lambd = 0.7)
print ("dW1 = \n"+ str(grads["dW1"]))
print ("dW2 = \n"+ str(grads["dW2"]))
print ("dW3 = \n"+ str(grads["dW3"]))
backward_propagation_with_regularization_test(backward_propagation_with_regularization)

dW1 = 
[[-0.25604646  0.12298827 -0.28297129]
 [-0.17706303  0.34536094 -0.4410571 ]]
dW2 = 
[[ 0.79276486  0.85133918]
 [-0.0957219  -0.01720463]
 [-0.13100772 -0.03750433]]
dW3 = 
[[-1.77691347 -0.11832879 -0.09397446]]

parameters = model(train_X, train_Y, lambd = 0.7)
print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)

Cost after iteration 0: 0.6974484493131264
Cost after iteration 10000: 0.2684918873282238
Cost after iteration 20000: 0.26809163371273004

On the train set:
Accuracy: 0.9383886255924171
On the test set:
Accuracy: 0.93

plt.title("Model with L2-regularization")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)

 

Dropout

forward propagation

def forward_propagation_with_dropout(X, parameters, keep_prob = 0.5):
    
    np.random.seed(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)
    D1 = np.random.rand(A1.shape[0], A1.shape[1])
    D1 = D1 < keep_prob
    A1 = np.multiply(A1, D1)
    A1 = A1 / keep_prob
    
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    D2 = np.random.rand(A2.shape[0], A2.shape[1])
    D2 = D2 < keep_prob
    A2 = np.multiply(A2, D2)
    A2 = A2 / keep_prob
    
    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)
    
    cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)
    
    return A3, cache

t_X, parameters = forward_propagation_with_dropout_test_case()

A3, cache = forward_propagation_with_dropout(t_X, parameters, keep_prob=0.7)
print ("A3 = " + str(A3))

forward_propagation_with_dropout_test(forward_propagation_with_dropout)

A3 = [[0.36974721 0.00305176 0.04565099 0.49683389 0.36974721]]

 

backward propagation

def backward_propagation_with_dropout(X, Y, cache, keep_prob):
    
    m = X.shape[1]
    (Z1, D1, A1, W1, b1, Z2, D2, 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)
    dA2 = np.multiply(dA2, D2)
    dA2 = dA2 / keep_prob 
    
    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    dW2 = 1./m * np.dot(dZ2, A1.T)
    db2 = 1./m * np.sum(dZ2, axis=1, keepdims=True)
    
    dA1 = np.dot(W2.T, dZ2)
    dA1 = np.multiply(dA1, D1)  
    dA1 = dA1 / keep_prob
    
    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    dW1 = 1./m * np.dot(dZ1, X.T)
    db1 = 1./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

t_X, t_Y, cache = backward_propagation_with_dropout_test_case()

gradients = backward_propagation_with_dropout(t_X, t_Y, cache, keep_prob=0.8)

print ("dA1 = \n" + str(gradients["dA1"]))
print ("dA2 = \n" + str(gradients["dA2"]))

backward_propagation_with_dropout_test(backward_propagation_with_dropout)

dA1 = 
[[ 0.36544439  0.         -0.00188233  0.         -0.17408748]
 [ 0.65515713  0.         -0.00337459  0.         -0.        ]]
dA2 = 
[[ 0.58180856  0.         -0.00299679  0.         -0.27715731]
 [ 0.          0.53159854 -0.          0.53159854 -0.34089673]
 [ 0.          0.         -0.00292733  0.         -0.        ]]

parameters = model(train_X, train_Y, keep_prob = 0.86, learning_rate = 0.3)

print ("On the train set:")
predictions_train = predict(train_X, train_Y, parameters)
print ("On the test set:")
predictions_test = predict(test_X, test_Y, parameters)

Cost after iteration 0: 0.6543912405149825
Cost after iteration 10000: 0.0610169865749056
Cost after iteration 20000: 0.060582435798513114

On the train set:
Accuracy: 0.9289099526066351
On the test set:
Accuracy: 0.95

plt.title("Model with dropout")
axes = plt.gca()
axes.set_xlim([-0.75,0.40])
axes.set_ylim([-0.75,0.65])
plot_decision_boundary(lambda x: predict_dec(parameters, x.T), train_X, train_Y)