AIDeep Learning

DeepLearning.ai作业:(2-1)-- 深度学习的实践层面（Practical aspects of Deep Learning）

dl.ai, homework
阅读数量:次

  1. 不要抄作业！
  2. 我只是把思路整理了，供个人学习。
  3. 不要抄作业！

本周的作业分了3部分：

  • 初始化参数
  • 正则化（L2、dropout）
  • 梯度检验

part1：Initialization

主要说明的不同的初始化对迭代的影响。

首先，模型函数是这样的：

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def model(X, Y, learning_rate = 0.01, num_iterations = 15000, print_cost = True, initialization = "he"):
    """
    Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (2, number of examples)
    Y -- true "label" vector (containing 0 for red dots; 1 for blue dots), of shape (1, number of examples)
    learning_rate -- learning rate for gradient descent 
    num_iterations -- number of iterations to run gradient descent
    print_cost -- if True, print the cost every 1000 iterations
    initialization -- flag to choose which initialization to use ("zeros","random" or "he")
    
    Returns:
    parameters -- parameters learnt by the model
    """
        
    grads = {}
    costs = [] # to keep track of the loss
    m = X.shape[1] # number of examples
    layers_dims = [X.shape[0], 10, 5, 1]
    
    # Initialize parameters dictionary.
    if initialization == "zeros":
        parameters = initialize_parameters_zeros(layers_dims)
    elif initialization == "random":
        parameters = initialize_parameters_random(layers_dims)
    elif initialization == "he":
        parameters = initialize_parameters_he(layers_dims)

    # Loop (gradient descent)

    for i in range(0, num_iterations):

        # Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
        a3, cache = forward_propagation(X, parameters)
        
        # Loss
        cost = compute_loss(a3, Y)

        # Backward propagation.
        grads = backward_propagation(X, Y, cache)
        
        # Update parameters.
        parameters = update_parameters(parameters, grads, learning_rate)
        
        # Print the loss every 1000 iterations
        if print_cost and i % 1000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
            costs.append(cost)
            
    # plot the loss
    plt.plot(costs)
    plt.ylabel('cost')
    plt.xlabel('iterations (per hundreds)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

1. Zero Initialization

把参数全都置位0，结果是显而易见的，就是没有任何变化。

2. Random initialization

把W参数随机化了，但是乘以10倍系数，所以导致初始化的参数太大，收敛速度很慢

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def initialize_parameters_random(layers_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the size of each layer.
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
                    b1 -- bias vector of shape (layers_dims[1], 1)
                    ...
                    WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
                    bL -- bias vector of shape (layers_dims[L], 1)
    """
    
    np.random.seed(3)               # This seed makes sure your "random" numbers will be the as ours
    parameters = {}
    L = len(layers_dims)            # integer representing the number of layers
    
    for l in range(1, L):
        ### START CODE HERE ### (≈ 2 lines of code)
        parameters['W' + str(l)] = np.random.randn(layers_dims[l], layers_dims[l-1]) * 10
        parameters['b' + str(l)] = np.zeros((layers_dims[l], 1))
        ### END CODE HERE ###

    return parameters

结果一般般

3. He initialization

把W参数随机化，但是乘上系数 sqrt(2./layers_dims[l-1])

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def initialize_parameters_he(layers_dims):
    """
    Arguments:
    layer_dims -- python array (list) containing the size of each layer.
    
    Returns:
    parameters -- python dictionary containing your parameters "W1", "b1", ..., "WL", "bL":
                    W1 -- weight matrix of shape (layers_dims[1], layers_dims[0])
                    b1 -- bias vector of shape (layers_dims[1], 1)
                    ...
                    WL -- weight matrix of shape (layers_dims[L], layers_dims[L-1])
                    bL -- bias vector of shape (layers_dims[L], 1)
    """
    
    np.random.seed(3)
    parameters = {}
    L = len(layers_dims) - 1 # integer representing the number of layers
     
    for l in range(1, L + 1):
        ### START CODE HERE ### (≈ 2 lines of code)
        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))
        ### END CODE HERE ###
        
    return parameters

结果非常理想。

Part 2：Regularization

数据集：

模型函数：

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def model(X, Y, learning_rate = 0.3, num_iterations = 30000, print_cost = True, lambd = 0, keep_prob = 1):
    """
    Implements a three-layer neural network: LINEAR->RELU->LINEAR->RELU->LINEAR->SIGMOID.
    
    Arguments:
    X -- input data, of shape (input size, number of examples)
    Y -- true "label" vector (1 for blue dot / 0 for red dot), of shape (output size, number of examples)
    learning_rate -- learning rate of the optimization
    num_iterations -- number of iterations of the optimization loop
    print_cost -- If True, print the cost every 10000 iterations
    lambd -- regularization hyperparameter, scalar
    keep_prob - probability of keeping a neuron active during drop-out, scalar.
    
    Returns:
    parameters -- parameters learned by the model. They can then be used to predict.
    """
        
    grads = {}
    costs = []                            # to keep track of the cost
    m = X.shape[1]                        # number of examples
    layers_dims = [X.shape[0], 20, 3, 1]
    
    # Initialize parameters dictionary.
    parameters = initialize_parameters(layers_dims)

    # Loop (gradient descent)

    for i in range(0, num_iterations):

        # Forward propagation: LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID.
        if keep_prob == 1:
            a3, cache = forward_propagation(X, parameters)
        elif keep_prob < 1:
            a3, cache = forward_propagation_with_dropout(X, parameters, keep_prob)
        
        # Cost function
        if lambd == 0:
            cost = compute_cost(a3, Y)
        else:
            cost = compute_cost_with_regularization(a3, Y, parameters, lambd)
            
        # Backward propagation.
        assert(lambd==0 or keep_prob==1)    # it is possible to use both L2 regularization and dropout, 
                                            # but this assignment will only explore one at a time
        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)
        
        # Update parameters.
        parameters = update_parameters(parameters, grads, learning_rate)
        
        # Print the loss every 10000 iterations
        if print_cost and i % 10000 == 0:
            print("Cost after iteration {}: {}".format(i, cost))
        if print_cost and i % 1000 == 0:
            costs.append(cost)
    
    # plot the cost
    plt.plot(costs)
    plt.ylabel('cost')
    plt.xlabel('iterations (x1,000)')
    plt.title("Learning rate =" + str(learning_rate))
    plt.show()
    
    return parameters

没有使用正则化时，效果：

L2 正则

计算代价函数

$$J_{regularized} = \small \underbrace{-\frac{1}{m} \sum\limits_{i = 1}^{m} \large{(}\small y^{(i)}\log\left(a^{[L] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[L] (i)}\right) \large{)} }_\text{cross-entropy cost} + \underbrace{\frac{1}{m} \frac{\lambda}{2} \sum\limits_l\sum\limits_k\sum\limits_j W_{k,j}^{[l]2} }_\text{L2 regularization cost} $$

公式已经给了，只要加上后面那一项就可以了

使用np.sum(np.square(Wl))来计算$\sum\limits_k\sum\limits_j W_{k,j}^{[l]2}$

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# GRADED FUNCTION: compute_cost_with_regularization

def compute_cost_with_regularization(A3, Y, parameters, lambd):
    """
    Implement the cost function with L2 regularization. See formula (2) above.
    
    Arguments:
    A3 -- post-activation, output of forward propagation, of shape (output size, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    parameters -- python dictionary containing parameters of the model
    
    Returns:
    cost - value of the regularized loss function (formula (2))
    """
    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
    
    ### START CODE HERE ### (approx. 1 line)
    L2_regularization_cost = lambd / (m * 2) * (np.sum(np.square(W1)) + np.sum(np.square(W2)) + np.sum(np.square(W3)))
    ### END CODER HERE ###
    
    cost = cross_entropy_cost + L2_regularization_cost
    
    return cost

计算反向传播函数

在$dW$上加上了正则项$\frac{\lambda}{m} W$

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# GRADED FUNCTION: backward_propagation_with_regularization

def backward_propagation_with_regularization(X, Y, cache, lambd):
    """
    Implements the backward propagation of our baseline model to which we added an L2 regularization.
    
    Arguments:
    X -- input dataset, of shape (input size, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    cache -- cache output from forward_propagation()
    lambd -- regularization hyperparameter, scalar
    
    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 = A3 - Y
    
    ### START CODE HERE ### (approx. 1 line)
    dW3 = 1./m * np.dot(dZ3, A2.T) + lambd / m * W3
    ### END CODE HERE ###
    db3 = 1./m * np.sum(dZ3, axis=1, keepdims = True)
    
    dA2 = np.dot(W3.T, dZ3)
    dZ2 = np.multiply(dA2, np.int64(A2 > 0))
    ### START CODE HERE ### (approx. 1 line)
    dW2 = 1./m * np.dot(dZ2, A1.T) + lambd / m * W2
    ### END CODE HERE ###
    db2 = 1./m * np.sum(dZ2, axis=1, keepdims = True)
    
    dA1 = np.dot(W2.T, dZ2)
    dZ1 = np.multiply(dA1, np.int64(A1 > 0))
    ### START CODE HERE ### (approx. 1 line)
    dW1 = 1./m * np.dot(dZ1, X.T) + lambd / m * W1
    ### END CODE HERE ###
    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

加上L2正则项后，效果很明显：

dropout

在每一次迭代中，都随机删除一定概率的neurons。

1. Forward propagation with dropout

分4步:

  1. 每一层的$d^{[l]}$对应每一层的$a^{[l]}$,因为有m个样本，所以就有$D^{[1]} = [d^{1} d^{1} … d^{1}] $of the same dimension as $A^{[1]}$.使用np.random.rand(n,m)
  2. 将$D^{[l]}$布尔化， $ < keepprob$ 分为 1和0
  3. Set $A^{[1]}$ to $A^{[1]} * D^{[1]}$.
  4. Divide $A^{[1]}$ by keep_prob.

记得用cache把每一层的D都记录下来

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# GRADED FUNCTION: forward_propagation_with_dropout

def forward_propagation_with_dropout(X, parameters, keep_prob = 0.5):
    """
    Implements the forward propagation: LINEAR -> RELU + DROPOUT -> LINEAR -> RELU + DROPOUT -> LINEAR -> SIGMOID.
    
    Arguments:
    X -- input dataset, of shape (2, number of examples)
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape (20, 2)
                    b1 -- bias vector of shape (20, 1)
                    W2 -- weight matrix of shape (3, 20)
                    b2 -- bias vector of shape (3, 1)
                    W3 -- weight matrix of shape (1, 3)
                    b3 -- bias vector of shape (1, 1)
    keep_prob - probability of keeping a neuron active during drop-out, scalar
    
    Returns:
    A3 -- last activation value, output of the forward propagation, of shape (1,1)
    cache -- tuple, information stored for computing the backward propagation
    """
    
    np.random.seed(1)
    
    # 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)
    ### START CODE HERE ### (approx. 4 lines)         # Steps 1-4 below correspond to the Steps 1-4 described above. 
    D1 = np.random.rand(A1.shape[0], A1.shape[1])                                         # Step 1: initialize matrix D1 = np.random.rand(..., ...)
    D1 = D1 < keep_prob                                         # Step 2: convert entries of D1 to 0 or 1 (using keep_prob as the threshold)
    A1 = A1 * D1                                         # Step 3: shut down some neurons of A1
    A1 = A1 / keep_prob                                         # Step 4: scale the value of neurons that haven't been shut down
    ### END CODE HERE ###
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    ### START CODE HERE ### (approx. 4 lines)
    D2 = np.random.rand(A2.shape[0], A2.shape[1])                                              # Step 1: initialize matrix D2 = np.random.rand(..., ...)
    D2 = D2 < keep_prob                                         # Step 2: convert entries of D2 to 0 or 1 (using keep_prob as the threshold)
    A2 = A2 * D2                                               # Step 3: shut down some neurons of A2
    A2 = A2 / keep_prob                                         # Step 4: scale the value of neurons that haven't been shut down
    ### END CODE HERE ###
    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

2. Backward propagation with dropout

  1. reapplying the same mask $D^{[1]}$ to dA1.
  2. divide dA1 by keep_prob

反向传播的时候，让之前的删除的neurons依旧归0，然后也要除以keepprob，因为dA = np.dot(W.T, dZ)，并没有重复除以过系数。

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# GRADED FUNCTION: backward_propagation_with_dropout

def backward_propagation_with_dropout(X, Y, cache, keep_prob):
    """
    Implements the backward propagation of our baseline model to which we added dropout.
    
    Arguments:
    X -- input dataset, of shape (2, number of examples)
    Y -- "true" labels vector, of shape (output size, number of examples)
    cache -- cache output from forward_propagation_with_dropout()
    keep_prob - probability of keeping a neuron active during drop-out, scalar
    
    Returns:
    gradients -- A dictionary with the gradients with respect to each parameter, activation and pre-activation variables
    """
    
    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)
    ### START CODE HERE ### (≈ 2 lines of code)
    dA2 = dA2 * D2              # Step 1: Apply mask D2 to shut down the same neurons as during the forward propagation
    dA2 = dA2 / keep_prob             # Step 2: Scale the value of neurons that haven't been shut down
    ### END CODE HERE ###
    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)
    ### START CODE HERE ### (≈ 2 lines of code)
    dA1 = dA1 * D1              # Step 1: Apply mask D1 to shut down the same neurons as during the forward propagation
    dA1 = dA1 / keep_prob              # Step 2: Scale the value of neurons that haven't been shut down
    ### END CODE HERE ###
    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

最终结果,也还不错：

注意：

  • dropout也是正则化的一种
  • 训练的时候用，测试的时候不要用
  • 在正向传播和反向传播中都要用

Part3:Gradient Checking

首先写了一维的checking

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# GRADED FUNCTION: forward_propagation

def forward_propagation(x, theta):
    """
    Implement the linear forward propagation (compute J) presented in Figure 1 (J(theta) = theta * x)
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    
    Returns:
    J -- the value of function J, computed using the formula J(theta) = theta * x
    """
    
    ### START CODE HERE ### (approx. 1 line)
    J = np.dot(theta,x)
    ### END CODE HERE ###
    
    return J
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# GRADED FUNCTION: backward_propagation

def backward_propagation(x, theta):
    """
    Computes the derivative of J with respect to theta (see Figure 1).
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    
    Returns:
    dtheta -- the gradient of the cost with respect to theta
    """
    
    ### START CODE HERE ### (approx. 1 line)
    dtheta = x
    ### END CODE HERE ###
    
    return dtheta

根据公式：

$$ difference = \frac {\mid\mid grad - gradapprox \mid\mid_2}{\mid\mid grad \mid\mid_2 + \mid\mid gradapprox \mid\mid_2} $$

步骤是：

  1. $\theta^{+} = \theta + \varepsilon$
  2. $\theta^{-} = \theta - \varepsilon$
  3. $J^{+} = J(\theta^{+})$
  4. $J^{-} = J(\theta^{-})$
  5. $gradapprox = \frac{J^{+} - J^{-}}{2 \varepsilon}$
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def gradient_check(x, theta, epsilon = 1e-7):
    """
    Implement the backward propagation presented in Figure 1.
    
    Arguments:
    x -- a real-valued input
    theta -- our parameter, a real number as well
    epsilon -- tiny shift to the input to compute approximated gradient with formula(1)
    
    Returns:
    difference -- difference (2) between the approximated gradient and the backward propagation gradient
    """
    
    # Compute gradapprox using left side of formula (1). epsilon is small enough, you don't need to worry about the limit.
    ### START CODE HERE ### (approx. 5 lines)
    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
    ### END CODE HERE ###
    
    # Check if gradapprox is close enough to the output of backward_propagation()
    ### START CODE HERE ### (approx. 1 line)
    grad = backward_propagation(x, theta)
    ### END CODE HERE ###
    
    ### START CODE HERE ### (approx. 1 line)
    numerator = np.linalg.norm(grad - gradapprox)                               # Step 1'
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)                             # Step 2'
    difference = numerator / denominator                              # Step 3'
    ### END CODE HERE ###
    
    if difference < 1e-7:
        print ("The gradient is correct!")
    else:
        print ("The gradient is wrong!")
    
    return difference

在N维的空间中，

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def forward_propagation_n(X, Y, parameters):
    """
    Implements the forward propagation (and computes the cost) presented in Figure 3.
    
    Arguments:
    X -- training set for m examples
    Y -- labels for m examples 
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
                    W1 -- weight matrix of shape (5, 4)
                    b1 -- bias vector of shape (5, 1)
                    W2 -- weight matrix of shape (3, 5)
                    b2 -- bias vector of shape (3, 1)
                    W3 -- weight matrix of shape (1, 3)
                    b3 -- bias vector of shape (1, 1)
    
    Returns:
    cost -- the cost function (logistic cost for one example)
    """
    
    # retrieve parameters
    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 = relu(Z1)
    Z2 = np.dot(W2, A1) + b2
    A2 = relu(Z2)
    Z3 = np.dot(W3, A2) + b3
    A3 = sigmoid(Z3)

    # Cost
    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
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def backward_propagation_n(X, Y, cache):
    """
    Implement the backward propagation presented in figure 2.
    
    Arguments:
    X -- input datapoint, of shape (input size, 1)
    Y -- true "label"
    cache -- cache output from forward_propagation_n()
    
    Returns:
    gradients -- A dictionary with the gradients of the cost 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 = 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

这个时候，给了两个函数，可以在字典和向量结构相互转换，也就是要计算$\theta^{+}$时，把字典转为向量会比较好计算。

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dictionary_to_vector()
vector_to_dictionary()

J_plus[i]就是向量中的每一个元素，也就是W,b展开之后的每一项元素

  • To compute J_plus[i]:
    1. Set $\theta^{+}$ to np.copy(parameters_values)
    2. Set $\theta^{+}_i$ to $\theta^{+}_i + \varepsilon$
    3. Calculate $J^{+}_i$ using to forward_propagation_n(x, y, vector_to_dictionary($\theta^{+}$ )).
  • To compute J_minus[i]: do the same thing with $\theta^{-}$
  • Compute $gradapprox[i] = \frac{J^{+}_i - J^{-}_i}{2 \varepsilon}$

代码如下，记住 thetaplus是一个(n,1)的向量，循环计算每一个参数的gradapprox，再和原本的grad比较：

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# GRADED FUNCTION: gradient_check_n

def gradient_check_n(parameters, gradients, X, Y, epsilon = 1e-7):
    """
    Checks if backward_propagation_n computes correctly the gradient of the cost output by forward_propagation_n
    
    Arguments:
    parameters -- python dictionary containing your parameters "W1", "b1", "W2", "b2", "W3", "b3":
    grad -- output of backward_propagation_n, contains gradients of the cost with respect to the parameters. 
    x -- input datapoint, of shape (input size, 1)
    y -- true "label"
    epsilon -- tiny shift to the input to compute approximated gradient with formula(1)
    
    Returns:
    difference -- difference (2) between the approximated gradient and the backward propagation gradient
    """
    
    # 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):
        
        # Compute J_plus[i]. Inputs: "parameters_values, epsilon". Output = "J_plus[i]".
        # "_" is used because the function you have to outputs two parameters but we only care about the first one
        ### START CODE HERE ### (approx. 3 lines)
        thetaplus =  np.copy(parameters_values)                                      # Step 1
        thetaplus[i][0] = thetaplus[i][0] + epsilon                                # Step 2
        J_plus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaplus))                                   # Step 3
        ### END CODE HERE ###
        
        # Compute J_minus[i]. Inputs: "parameters_values, epsilon". Output = "J_minus[i]".
        ### START CODE HERE ### (approx. 3 lines)
        thetaminus = np.copy(parameters_values)                                        # Step 1
        thetaminus[i][0] = thetaminus[i][0] - epsilon                               # Step 2        
        J_minus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaminus))                                  # Step 3
        ### END CODE HERE ###
        
        # Compute gradapprox[i]
        ### START CODE HERE ### (approx. 1 line)
        gradapprox[i] = (J_plus[i] - J_minus[i]) / (2 * epsilon)
        ### END CODE HERE ###
    
    # Compare gradapprox to backward propagation gradients by computing difference.
    ### START CODE HERE ### (approx. 1 line)
    numerator = np.linalg.norm(grad - gradapprox)                                           # Step 1'
    denominator = np.linalg.norm(grad) + np.linalg.norm(gradapprox)                                         # Step 2'
    difference = numerator / denominator                                          # Step 3'
    ### END CODE HERE ###

    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

注意：

梯度检验太慢，不要在训练的时候运行，你运行只是为了保证你的算法是正确的。

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