这篇文章我们主要学习一下如何改善神经网络，并给出相关代码，如有不足之处，还请大家指正，希望与大家共同进步！
这节中我们要做的有：

初始化参数

读取并绘制数据
初始化为零
随机初始化
抑梯度异常初始化

正则化模型

读取并绘制数据集
不使用正则化
使用正则化
   L2正则化
   随机删除节点

梯度校验

高维

初始化
1.初始化为0
笔记：
1.通常来说，零初始化都会导致神经网络无法打破对称性，最后导致的结果是无论网络有多少层，最终只能得到和Logistic函数相同的效果
2.权重矩阵W[L]应该被随机初始化，以打破对称性
3.在随机初始化后，每个神经元就可以开始学习输入的不同功能

def initialize_parameters_zeros(layers_dims):
    """
    将模型的参数全部设置为0
    
    参数：
        layers_dims - 列表，模型的层数和对应每一层的节点的数量
    返回
        parameters - 包含了所有W和b的字典
            W1 - 权重矩阵，维度为（layers_dims[1], layers_dims[0]）
            b1 - 偏置向量，维度为（layers_dims[1],1）
            ···
            WL - 权重矩阵，维度为（layers_dims[L], layers_dims[L -1]）
            bL - 偏置向量，维度为（layers_dims[L],1）
    """
    parameters = {}
    
    L = len(layers_dims) #网络层数
    
    for l in range(1,L):
        parameters["W" + str(l)] = np.zeros((layers_dims[l],layers_dims[l-1]))
        parameters["b" + str(l)] = np.zeros((layers_dims[l],1))
        
        #使用断言确保我的数据格式是正确的
        assert(parameters["W" + str(l)].shape == (layers_dims[l],layers_dims[l-1]))
        assert(parameters["b" + str(l)].shape == (layers_dims[l],1))
        
    return parameters

2.随机初始化

def initialize_parameters_random(layers_dims):
    """
    参数：
        layers_dims - 列表，模型的层数和对应每一层的节点的数量
    返回
        parameters - 包含了所有W和b的字典
            W1 - 权重矩阵，维度为（layers_dims[1], layers_dims[0]）
            b1 - 偏置向量，维度为（layers_dims[1],1）
            ···
            WL - 权重矩阵，维度为（layers_dims[L], layers_dims[L -1]）
            b1 - 偏置向量，维度为（layers_dims[L],1）
    """
    
    np.random.seed(3)               # 指定随机种子
    parameters = {}
    L = len(layers_dims)            # 层数
    
    for l in range(1, L):
        parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l - 1]) * 10 #使用10倍缩放
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        
        #使用断言确保我的数据格式是正确的
        assert(parameters["W" + str(l)].shape == (layers_dims[l],layers_dims[l-1]))
        assert(parameters["b" + str(l)].shape == (layers_dims[l],1))
        
    return parameters

3.抑制梯度异常初始化

def initialize_parameters_he(layers_dims):
    """
    参数：
        layers_dims - 列表，模型的层数和对应每一层的节点的数量
    返回
        parameters - 包含了所有W和b的字典
            W1 - 权重矩阵，维度为（layers_dims[1], layers_dims[0]）
            b1 - 偏置向量，维度为（layers_dims[1],1）
            ···
            WL - 权重矩阵，维度为（layers_dims[L], layers_dims[L -1]）
            b1 - 偏置向量，维度为（layers_dims[L],1）
    """
    
    np.random.seed(3)               # 指定随机种子
    parameters = {}
    L = len(layers_dims)            # 层数
    
    for l in range(1, L):
        parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l - 1]) * np.sqrt(2 / layers_dims[l - 1])
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        
        #使用断言确保我的数据格式是正确的
        assert(parameters["W" + str(l)].shape == (layers_dims[l],layers_dims[l-1]))
        assert(parameters["b" + str(l)].shape == (layers_dims[l],1))
        
    return parameters

正则化模型

1.L2正则化

def compute_cost_with_regularization(A3,Y,parameters,lambd):
    """
    实现公式2的L2正则化计算成本
    
    参数：
        A3 - 正向传播的输出结果，维度为（输出节点数量，训练/测试的数量）
        Y - 标签向量，与数据一一对应，维度为(输出节点数量，训练/测试的数量)
        parameters - 包含模型学习后的参数的字典
    返回：
        cost - 使用公式2计算出来的正则化损失的值
    
    """
    m = Y.shape[1]
    W1 = parameters["W1"]
    W2 = parameters["W2"]
    W3 = parameters["W3"]
    
    cross_entropy_cost = reg_utils.compute_cost(A3,Y)
    
    L2_regularization_cost = lambd * (np.sum(np.square(W1)) + np.sum(np.square(W2))  + np.sum(np.square(W3))) / (2 * m)
    
    cost = cross_entropy_cost + L2_regularization_cost
    
    return cost

#当然，因为改变了成本函数，我们也必须改变向后传播的函数， 所有的梯度都必须根据这个新的成本值来计算。

def backward_propagation_with_regularization(X, Y, cache, lambd):
    """
    实现我们添加了L2正则化的模型的后向传播。
    
    参数：
        X - 输入数据集，维度为（输入节点数量，数据集里面的数量）
        Y - 标签，维度为（输出节点数量，数据集里面的数量）
        cache - 来自forward_propagation（）的cache输出
        lambda - regularization超参数，实数
    
    返回：
        gradients - 一个包含了每个参数、激活值和预激活值变量的梯度的字典
    """
    
    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 * W3) / m )
    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 * W2) / m)
    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 * W1) / m)
    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
    

2.随机删除节点

def forward_propagation_with_dropout(X,parameters,keep_prob=0.5):
    """
    实现具有随机舍弃节点的前向传播。
    LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.
    
    参数：
        X  - 输入数据集，维度为（2，示例数）
        parameters - 包含参数“W1”，“b1”，“W2”，“b2”，“W3”，“b3”的python字典：
            W1  - 权重矩阵，维度为（20,2）
            b1  - 偏向量，维度为（20,1）
            W2  - 权重矩阵，维度为（3,20）
            b2  - 偏向量，维度为（3,1）
            W3  - 权重矩阵，维度为（1,3）
            b3  - 偏向量，维度为（1,1）
        keep_prob  - 随机删除的概率，实数
    返回：
        A3  - 最后的激活值，维度为（1,1），正向传播的输出
        cache - 存储了一些用于计算反向传播的数值的元组
    """
    np.random.seed(1)
    
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]
    
    #LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
    Z1 = np.dot(W1,X) + b1
    A1 = reg_utils.relu(Z1)
    
    #下面的步骤1-4对应于上述的步骤1-4。
    D1 = np.random.rand(A1.shape[0],A1.shape[1])    #步骤1：初始化矩阵D1 = np.random.rand(..., ...)
    D1 = D1 < keep_prob                             #步骤2：将D1的值转换为0或1（使​​用keep_prob作为阈值）
    A1 = A1 * D1                                    #步骤3：舍弃A1的一些节点（将它的值变为0或False）
    A1 = A1 / keep_prob                             #步骤4：缩放未舍弃的节点(不为0)的值
    """
    #不理解的同学运行一下下面代码就知道了。
    import numpy as np
    np.random.seed(1)
    A1 = np.random.randn(1,3)
    
    D1 = np.random.rand(A1.shape[0],A1.shape[1])
    keep_prob=0.5
    D1 = D1 < keep_prob
    print(D1)
    
    A1 = 0.01
    A1 = A1 * D1
    A1 = A1 / keep_prob
    print(A1)
    """
    
    Z2 = np.dot(W2,A1) + b2
    A2 = reg_utils.relu(Z2)
    
    #下面的步骤1-4对应于上述的步骤1-4。
    D2 = np.random.rand(A2.shape[0],A2.shape[1])    #步骤1：初始化矩阵D2 = np.random.rand(..., ...)
    D2 = D2 < keep_prob                             #步骤2：将D2的值转换为0或1（使​​用keep_prob作为阈值）
    A2 = A2 * D2                                    #步骤3：舍弃A1的一些节点（将它的值变为0或False）
    A2 = A2 / keep_prob                             #步骤4：缩放未舍弃的节点(不为0)的值
    
    Z3 = np.dot(W3, A2) + b3
    A3 = reg_utils.sigmoid(Z3)
    
    cache = (Z1, D1, A1, W1, b1, Z2, D2, A2, W2, b2, Z3, A3, W3, b3)
    
    return A3, cache

梯度校验

梯度校验就是为了验证我们所得到的的反向传播的函数是否正确
1.一维线性

def forward_propagation(x,theta):
    """
    
    实现图中呈现的线性前向传播（计算J）（J（theta）= theta * x）
    
    参数：
    x  - 一个实值输入
    theta  - 参数，也是一个实数
    
    返回：
    J  - 函数J的值，用公式J（theta）= theta * x计算
    """
    J = np.dot(theta,x)
    
    return J
    def backward_propagation(x,theta):
    """
    计算J相对于θ的导数。
    
    参数：
        x  - 一个实值输入
        theta  - 参数，也是一个实数
    
    返回：
        dtheta  - 相对于θ的成本梯度
    """
    dtheta = x
    
    return dtheta

梯度检验

def gradient_check(x,theta,epsilon=1e-7):
    """
    
    实现图中的反向传播。
    
    参数：
        x  - 一个实值输入
        theta  - 参数，也是一个实数
        epsilon  - 使用公式（3）计算输入的微小偏移以计算近似梯度
    
    返回：
        近似梯度和后向传播梯度之间的差异
    """
    
    #使用公式（3）的左侧计算gradapprox。
    thetaplus = theta + epsilon                               # Step 1
    thetaminus = theta - epsilon                              # Step 2
    J_plus = forward_propagation(x, thetaplus)                # Step 3
    J_minus = forward_propagation(x, thetaminus)              # Step 4
    gradapprox = (J_plus - J_minus) / (2 * epsilon)           # Step 5
    
    
    #检查gradapprox是否足够接近backward_propagation（）的输出
    grad = backward_propagation(x, theta)
    
    numerator = np.linalg.norm(grad - gradapprox)                      # Step 1'
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)    # Step 2'
    difference = numerator / denominator                               # Step 3'
    
    if difference < 1e-7:
        print("梯度检查：梯度正常!")
    else:
        print("梯度检查：梯度超出阈值!")
    
    return difference

2.高维

def forward_propagation_n(X,Y,parameters):
    """
    实现图中的前向传播（并计算成本）。
    
    参数：
        X - 训练集为m个例子
        Y -  m个示例的标签
        parameters - 包含参数“W1”，“b1”，“W2”，“b2”，“W3”，“b3”的python字典：
            W1  - 权重矩阵，维度为（5,4）
            b1  - 偏向量，维度为（5,1）
            W2  - 权重矩阵，维度为（3,5）
            b2  - 偏向量，维度为（3,1）
            W3  - 权重矩阵，维度为（1,3）
            b3  - 偏向量，维度为（1,1）
   
    返回：
        cost - 成本函数（logistic）
    """
    m = X.shape[1]
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]
    
    # LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
    Z1 = np.dot(W1,X) + b1
    A1 = gc_utils.relu(Z1)
    
    Z2 = np.dot(W2,A1) + b2
    A2 = gc_utils.relu(Z2)
    
    Z3 = np.dot(W3,A2) + b3
    A3 = gc_utils.sigmoid(Z3)
    
    #计算成本
    logprobs = np.multiply(-np.log(A3), Y) + np.multiply(-np.log(1 - A3), 1 - Y)
    cost = (1 / m) * np.sum(logprobs)
    
    cache = (Z1, A1, W1, b1, Z2, A2, W2, b2, Z3, A3, W3, b3)

    return cost, cache

def backward_propagation_n(X,Y,cache):
    """
    实现图中所示的反向传播。
    
    参数：
        X - 输入数据点（输入节点数量，1）
        Y - 标签
        cache - 来自forward_propagation_n（）的cache输出
    
    返回：
        gradients - 一个字典，其中包含与每个参数、激活和激活前变量相关的成本梯度。
    """
    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)
    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  # Should not multiply by 2
    dW2 = 1. / m * np.dot(dZ2, A1.T)
    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) # Should not multiply by 4
    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

梯度检查

def gradient_check_n(parameters,gradients,X,Y,epsilon=1e-7):
    """
    检查backward_propagation_n是否正确计算forward_propagation_n输出的成本梯度
    
    参数：
        parameters - 包含参数“W1”，“b1”，“W2”，“b2”，“W3”，“b3”的python字典：
        grad_output_propagation_n的输出包含与参数相关的成本梯度。
        x  - 输入数据点，维度为（输入节点数量，1）
        y  - 标签
        epsilon  - 计算输入的微小偏移以计算近似梯度
    
    返回：
        difference - 近似梯度和后向传播梯度之间的差异
    """
    #初始化参数
    parameters_values , keys = gc_utils.dictionary_to_vector(parameters) #keys用不到
    grad = gc_utils.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))
    
    #计算gradapprox
    for i in range(num_parameters):
        #计算J_plus [i]。输入：“parameters_values，epsilon”。输出=“J_plus [i]”
        thetaplus = np.copy(parameters_values)                                                  # Step 1
        thetaplus[i][0] = thetaplus[i][0] + epsilon                                             # Step 2
        J_plus[i], cache = forward_propagation_n(X,Y,gc_utils.vector_to_dictionary(thetaplus))  # Step 3 ，cache用不到
        
        #计算J_minus [i]。输入：“parameters_values，epsilon”。输出=“J_minus [i]”。
        thetaminus = np.copy(parameters_values)                                                 # Step 1
        thetaminus[i][0] = thetaminus[i][0] - epsilon                                           # Step 2        
        J_minus[i], cache = forward_propagation_n(X,Y,gc_utils.vector_to_dictionary(thetaminus))# Step 3 ，cache用不到
        
        #计算gradapprox[i]
        gradapprox[i] = (J_plus[i] - J_minus[i]) / (2 * epsilon)
        
    #通过计算差异比较gradapprox和后向传播梯度。
    numerator = np.linalg.norm(grad - gradapprox)                                     # Step 1'
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)                   # Step 2'
    difference = numerator / denominator                                              # Step 3'
    
    if difference < 1e-7:
        print("梯度检查：梯度正常!")
    else:
        print("梯度检查：梯度超出阈值!")
    
    return difference

相关库代码

testCases.py

import numpy as np

def compute_cost_with_regularization_test_case():
    np.random.seed(1)
    Y_assess = np.array([[1, 1, 0, 1, 0]])
    W1 = np.random.randn(2, 3)
    b1 = np.random.randn(2, 1)
    W2 = np.random.randn(3, 2)
    b2 = np.random.randn(3, 1)
    W3 = np.random.randn(1, 3)
    b3 = np.random.randn(1, 1)
    parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2, "W3": W3, "b3": b3}
    a3 = np.array([[ 0.40682402,  0.01629284,  0.16722898,  0.10118111,  0.40682402]])
    return a3, Y_assess, parameters

def backward_propagation_with_regularization_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(3, 5)
    Y_assess = np.array([[1, 1, 0, 1, 0]])
    cache = (np.array([[-1.52855314,  3.32524635,  2.13994541,  2.60700654, -0.75942115],
         [-1.98043538,  4.1600994 ,  0.79051021,  1.46493512, -0.45506242]]),
  np.array([[ 0.        ,  3.32524635,  2.13994541,  2.60700654,  0.        ],
         [ 0.        ,  4.1600994 ,  0.79051021,  1.46493512,  0.        ]]),
  np.array([[-1.09989127, -0.17242821, -0.87785842],
         [ 0.04221375,  0.58281521, -1.10061918]]),
  np.array([[ 1.14472371],
         [ 0.90159072]]),
  np.array([[ 0.53035547,  5.94892323,  2.31780174,  3.16005701,  0.53035547],
         [-0.69166075, -3.47645987, -2.25194702, -2.65416996, -0.69166075],
         [-0.39675353, -4.62285846, -2.61101729, -3.22874921, -0.39675353]]),
  np.array([[ 0.53035547,  5.94892323,  2.31780174,  3.16005701,  0.53035547],
         [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
         [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ]]),
  np.array([[ 0.50249434,  0.90085595],
         [-0.68372786, -0.12289023],
         [-0.93576943, -0.26788808]]),
  np.array([[ 0.53035547],
         [-0.69166075],
         [-0.39675353]]),
  np.array([[-0.3771104 , -4.10060224, -1.60539468, -2.18416951, -0.3771104 ]]),
  np.array([[ 0.40682402,  0.01629284,  0.16722898,  0.10118111,  0.40682402]]),
  np.array([[-0.6871727 , -0.84520564, -0.67124613]]),
  np.array([[-0.0126646]]))
    return X_assess, Y_assess, cache

def forward_propagation_with_dropout_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(3, 5)
    W1 = np.random.randn(2, 3)
    b1 = np.random.randn(2, 1)
    W2 = np.random.randn(3, 2)
    b2 = np.random.randn(3, 1)
    W3 = np.random.randn(1, 3)
    b3 = np.random.randn(1, 1)
    parameters = {"W1": W1, "b1": b1, "W2": W2, "b2": b2, "W3": W3, "b3": b3}
    
    return X_assess, parameters

def backward_propagation_with_dropout_test_case():
    np.random.seed(1)
    X_assess = np.random.randn(3, 5)
    Y_assess = np.array([[1, 1, 0, 1, 0]])
    cache = (np.array([[-1.52855314,  3.32524635,  2.13994541,  2.60700654, -0.75942115],
           [-1.98043538,  4.1600994 ,  0.79051021,  1.46493512, -0.45506242]]), np.array([[ True, False,  True,  True,  True],
           [ True,  True,  True,  True, False]], dtype=bool), np.array([[ 0.        ,  0.        ,  4.27989081,  5.21401307,  0.        ],
           [ 0.        ,  8.32019881,  1.58102041,  2.92987024,  0.        ]]), np.array([[-1.09989127, -0.17242821, -0.87785842],
           [ 0.04221375,  0.58281521, -1.10061918]]), np.array([[ 1.14472371],
           [ 0.90159072]]), np.array([[ 0.53035547,  8.02565606,  4.10524802,  5.78975856,  0.53035547],
           [-0.69166075, -1.71413186, -3.81223329, -4.61667916, -0.69166075],
           [-0.39675353, -2.62563561, -4.82528105, -6.0607449 , -0.39675353]]), np.array([[ True, False,  True, False,  True],
           [False,  True, False,  True,  True],
           [False, False,  True, False, False]], dtype=bool), np.array([[ 1.06071093,  0.        ,  8.21049603,  0.        ,  1.06071093],
           [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ],
           [ 0.        ,  0.        ,  0.        ,  0.        ,  0.        ]]), np.array([[ 0.50249434,  0.90085595],
           [-0.68372786, -0.12289023],
           [-0.93576943, -0.26788808]]), np.array([[ 0.53035547],
           [-0.69166075],
           [-0.39675353]]), np.array([[-0.7415562 , -0.0126646 , -5.65469333, -0.0126646 , -0.7415562 ]]), np.array([[ 0.32266394,  0.49683389,  0.00348883,  0.49683389,  0.32266394]]), np.array([[-0.6871727 , -0.84520564, -0.67124613]]), np.array([[-0.0126646]]))


    return X_assess, Y_assess, cache

def gradient_check_n_test_case(): 
    np.random.seed(1)
    x = np.random.randn(4,3)
    y = np.array([1, 1, 0])
    W1 = np.random.randn(5,4) 
    b1 = np.random.randn(5,1) 
    W2 = np.random.randn(3,5) 
    b2 = np.random.randn(3,1) 
    W3 = np.random.randn(1,3) 
    b3 = np.random.randn(1,1) 
    parameters = {"W1": W1,
                  "b1": b1,
                  "W2": W2,
                  "b2": b2,
                  "W3": W3,
                  "b3": b3}

    
    return x, y, parameters

init_utils.py

# -*- coding: utf-8 -*-

#init_utils.py

import numpy as np
import matplotlib.pyplot as plt
import sklearn
import sklearn.datasets


def sigmoid(x):
    """
    Compute the sigmoid of x
 
    Arguments:
    x -- A scalar or numpy array of any size.
 
    Return:
    s -- sigmoid(x)
    """
    s = 1/(1+np.exp(-x))
    return s
 
def relu(x):
    """
    Compute the relu of x
 
    Arguments:
    x -- A scalar or numpy array of any size.
 
    Return:
    s -- relu(x)
    """
    s = np.maximum(0,x)
    
    return s
    
def compute_loss(a3, Y):
    
    """
    Implement the loss function
    
    Arguments:
    a3 -- post-activation, output of forward propagation
    Y -- "true" labels vector, same shape as a3
    
    Returns:
    loss - value of the loss function
    """
    
    m = Y.shape[1]
    logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
    loss = 1./m * np.nansum(logprobs)
    
    return loss
    
def forward_propagation(X, parameters):
    """
    Implements the forward propagation (and computes the loss) presented in Figure 2.
    
    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat)
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape ()
                    b1 -- bias vector of shape ()
                    W2 -- weight matrix of shape ()
                    b2 -- bias vector of shape ()
                    W3 -- weight matrix of shape ()
                    b3 -- bias vector of shape ()
    
    Returns:
    loss -- the loss function (vanilla logistic loss)
    """
        
    # retrieve parameters
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]
    
    # LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
    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)
    
    cache = (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3)
    
    return a3, cache
 
def backward_propagation(X, Y, cache):
    """
    Implement the backward propagation presented in figure 2.
    
    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat)
    cache -- cache output from forward_propagation()
    
    Returns:
    gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
    """
    m = X.shape[1]
    (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3) = cache
    
    dz3 = 1./m * (a3 - Y)
    dW3 = np.dot(dz3, a2.T)
    db3 = np.sum(dz3, axis=1, keepdims = True)
    
    da2 = np.dot(W3.T, dz3)
    dz2 = np.multiply(da2, np.int64(a2 > 0))
    dW2 = np.dot(dz2, a1.T)
    db2 = np.sum(dz2, axis=1, keepdims = True)
    
    da1 = np.dot(W2.T, dz2)
    dz1 = np.multiply(da1, np.int64(a1 > 0))
    dW1 = np.dot(dz1, X.T)
    db1 = 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
 
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 n_model_backward
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
                  parameters['W' + str(i)] = ... 
                  parameters['b' + str(i)] = ...
    """
    
    L = len(parameters) // 2 # number of layers in the neural networks
 
    # Update rule for each parameter
    for k in range(L):
        parameters["W" + str(k+1)] = parameters["W" + str(k+1)] - learning_rate * grads["dW" + str(k+1)]
        parameters["b" + str(k+1)] = parameters["b" + str(k+1)] - learning_rate * grads["db" + str(k+1)]
        
    return parameters
    
def predict(X, y, parameters):
    """
    This function is used to predict the results of a  n-layer neural network.
    
    Arguments:
    X -- data set of examples you would like to label
    parameters -- parameters of the trained model
    
    Returns:
    p -- predictions for the given dataset X
    """
    
    m = X.shape[1]
    p = np.zeros((1,m), dtype = np.int)
    
    # Forward propagation
    a3, caches = forward_propagation(X, parameters)
    
    # convert probas to 0/1 predictions
    for i in range(0, a3.shape[1]):
        if a3[0,i] > 0.5:
            p[0,i] = 1
        else:
            p[0,i] = 0
 
    # print results
    print("Accuracy: "  + str(np.mean((p[0,:] == y[0,:]))))
    
    return p
    
def load_dataset(is_plot=True):
    np.random.seed(1)
    train_X, train_Y = sklearn.datasets.make_circles(n_samples=300, noise=.05)
    np.random.seed(2)
    test_X, test_Y = sklearn.datasets.make_circles(n_samples=100, noise=.05)
    # Visualize the data
    if is_plot:
        plt.scatter(train_X[:, 0], train_X[:, 1], c=train_Y, s=40, cmap=plt.cm.Spectral);
    train_X = train_X.T
    train_Y = train_Y.reshape((1, train_Y.shape[0]))
    test_X = test_X.T
    test_Y = test_Y.reshape((1, test_Y.shape[0]))
    return train_X, train_Y, test_X, test_Y
 
def plot_decision_boundary(model, X, y):
    # Set min and max values and give it some padding
    x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
    y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
    h = 0.01
    # Generate a grid of points with distance h between them
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
    # Predict the function value for the whole grid
    Z = model(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    # Plot the contour and training examples
    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    plt.ylabel('x2')
    plt.xlabel('x1')
    plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)
    plt.show()
 
def predict_dec(parameters, X):
    """
    Used for plotting decision boundary.
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    X -- input data of size (m, K)
    
    Returns
    predictions -- vector of predictions of our model (red: 0 / blue: 1)
    """
    
    # Predict using forward propagation and a classification threshold of 0.5
    a3, cache = forward_propagation(X, parameters)
    predictions = (a3>0.5)
    return predictions

reg_utils.py

# -*- coding: utf-8 -*-

#reg_utils.py

import numpy as np
import matplotlib.pyplot as plt
import scipy.io as sio

def sigmoid(x):
    """
    Compute the sigmoid of x
 
    Arguments:
    x -- A scalar or numpy array of any size.
 
    Return:
    s -- sigmoid(x)
    """
    s = 1/(1+np.exp(-x))
    return s
 
def relu(x):
    """
    Compute the relu of x
 
    Arguments:
    x -- A scalar or numpy array of any size.
 
    Return:
    s -- relu(x)
    """
    s = np.maximum(0,x)
    
    return s


def initialize_parameters(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":
                    W1 -- weight matrix of shape (layer_dims[l], layer_dims[l-1])
                    b1 -- bias vector of shape (layer_dims[l], 1)
                    Wl -- weight matrix of shape (layer_dims[l-1], layer_dims[l])
                    bl -- bias vector of shape (1, layer_dims[l])
                    
    Tips:
    - For example: the layer_dims for the "Planar Data classification model" would have been [2,2,1]. 
    This means W1's shape was (2,2), b1 was (1,2), W2 was (2,1) and b2 was (1,1). Now you have to generalize it!
    - In the for loop, use parameters['W' + str(l)] to access Wl, where l is the iterative integer.
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layer_dims) # number of layers in the network
 
    for l in range(1, L):
        parameters['W' + str(l)] = np.random.randn(layer_dims[l], layer_dims[l-1]) / np.sqrt(layer_dims[l-1])
        parameters['b' + str(l)] = np.zeros((layer_dims[l], 1))
        
        assert(parameters['W' + str(l)].shape == layer_dims[l], layer_dims[l-1])
        assert(parameters['W' + str(l)].shape == layer_dims[l], 1)
 
        
    return parameters

def forward_propagation(X, parameters):
    """
    Implements the forward propagation (and computes the loss) presented in Figure 2.
    
    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat)
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape ()
                    b1 -- bias vector of shape ()
                    W2 -- weight matrix of shape ()
                    b2 -- bias vector of shape ()
                    W3 -- weight matrix of shape ()
                    b3 -- bias vector of shape ()
    
    Returns:
    loss -- the loss function (vanilla logistic loss)
    """
        
    # retrieve parameters
    W1 = parameters["W1"]
    b1 = parameters["b1"]
    W2 = parameters["W2"]
    b2 = parameters["b2"]
    W3 = parameters["W3"]
    b3 = parameters["b3"]
    
    # LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
    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)
    
    cache = (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3)
    
    return a3, cache


 
def compute_cost(a3, Y):
    """
    Implement the cost function
    
    Arguments:
    a3 -- post-activation, output of forward propagation
    Y -- "true" labels vector, same shape as a3
    
    Returns:
    cost - value of the cost function
    """
    m = Y.shape[1]
    
    logprobs = np.multiply(-np.log(a3),Y) + np.multiply(-np.log(1 - a3), 1 - Y)
    cost = 1./m * np.nansum(logprobs)
    
    return cost

def backward_propagation(X, Y, cache):
    """
    Implement the backward propagation presented in figure 2.
    
    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- true "label" vector (containing 0 if cat, 1 if non-cat)
    cache -- cache output from forward_propagation()
    
    Returns:
    gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
    """
    m = X.shape[1]
    (z1, a1, W1, b1, z2, a2, W2, b2, z3, a3, W3, b3) = cache
    
    dz3 = 1./m * (a3 - Y)
    dW3 = np.dot(dz3, a2.T)
    db3 = np.sum(dz3, axis=1, keepdims = True)
    
    da2 = np.dot(W3.T, dz3)
    dz2 = np.multiply(da2, np.int64(a2 > 0))
    dW2 = np.dot(dz2, a1.T)
    db2 = np.sum(dz2, axis=1, keepdims = True)
    
    da1 = np.dot(W2.T, dz2)
    dz1 = np.multiply(da1, np.int64(a1 > 0))
    dW1 = np.dot(dz1, X.T)
    db1 = 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

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 n_model_backward
    
    Returns:
    parameters -- python dictionary containing your updated parameters 
                  parameters['W' + str(i)] = ... 
                  parameters['b' + str(i)] = ...
    """
    
    L = len(parameters) // 2 # number of layers in the neural networks
 
    # Update rule for each parameter
    for k in range(L):
        parameters["W" + str(k+1)] = parameters["W" + str(k+1)] - learning_rate * grads["dW" + str(k+1)]
        parameters["b" + str(k+1)] = parameters["b" + str(k+1)] - learning_rate * grads["db" + str(k+1)]
        
    return parameters



    
def load_2D_dataset(is_plot=True):
    data = sio.loadmat('datasets/data.mat')
    train_X = data['X'].T
    train_Y = data['y'].T
    test_X = data['Xval'].T
    test_Y = data['yval'].T
    if is_plot:
        plt.scatter(train_X[0, :], train_X[1, :], c=train_Y, s=40, cmap=plt.cm.Spectral);
    
    return train_X, train_Y, test_X, test_Y

def predict(X, y, parameters):
    """
    This function is used to predict the results of a  n-layer neural network.
    
    Arguments:
    X -- data set of examples you would like to label
    parameters -- parameters of the trained model
    
    Returns:
    p -- predictions for the given dataset X
    """
    
    m = X.shape[1]
    p = np.zeros((1,m), dtype = np.int)
    
    # Forward propagation
    a3, caches = forward_propagation(X, parameters)
    
    # convert probas to 0/1 predictions
    for i in range(0, a3.shape[1]):
        if a3[0,i] > 0.5:
            p[0,i] = 1
        else:
            p[0,i] = 0
 
    # print results
    print("Accuracy: "  + str(np.mean((p[0,:] == y[0,:]))))
    
    return p

def plot_decision_boundary(model, X, y):
    # Set min and max values and give it some padding
    x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
    y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
    h = 0.01
    # Generate a grid of points with distance h between them
    xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
    # Predict the function value for the whole grid
    Z = model(np.c_[xx.ravel(), yy.ravel()])
    Z = Z.reshape(xx.shape)
    # Plot the contour and training examples
    plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
    plt.ylabel('x2')
    plt.xlabel('x1')
    plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)
    plt.show()
 
def predict_dec(parameters, X):
    """
    Used for plotting decision boundary.
    
    Arguments:
    parameters -- python dictionary containing your parameters 
    X -- input data of size (m, K)
    
    Returns
    predictions -- vector of predictions of our model (red: 0 / blue: 1)
    """
    
    # Predict using forward propagation and a classification threshold of 0.5
    a3, cache = forward_propagation(X, parameters)
    predictions = (a3>0.5)
    return predictions