DeepLearing学习笔记-改善深层神经网络(第一周作业-3-梯度校验)

1-背景:

在神经网络计算过程中,对后向传播的梯度进行校验,确保其计算无误。至于,前向传播,由于相对简单,所以,一般不会出错,在前向传播的基础上利用计算出来的代价 J 我们可以进行后向梯度的校验。

公式原理如下:

Jθ=limε0J(θ+ε)J(θε)2ε(1)

在前向传播的基础上,我们可以获取到 J ,也就可以上公式1的计算,将计算的结果值和反向传播的梯度值按照一定的方式做比较即可。

2- 一维梯度校验

J(θ)=θx ,线性模型如下:
DeepLearing学习笔记-改善深层神经网络(第一周作业-3-梯度校验)_第1张图片
该模型的前向传播和后向传播的代码如下:

# 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 = theta * x
    ### END CODE HERE ###

    return J

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

输出结果:
J = 8

由于 J(θ)=θx 所以 dtheta=Jθ=x

# 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

x, theta = 2, 4
dtheta = backward_propagation(x, theta)
print ("dtheta = " + str(dtheta))

输出结果:
dtheta = 2

2-1 一维梯度校验

步骤如下:

  • 计算梯度近似值 “gradapprox”:

    1. θ+=θ+ε
    2. θ=θε
    3. J+=J(θ+)
    4. J=J(θ)
    5. gradapprox=J+J2ε
  • 进行反向传播,获取梯度值 “grad”

  • 按照如下公式计算两者difference :
    difference=gradgradapprox2grad2+gradapprox2(2)

范数的计算可以用np.linalg.norm(...)
当difference 足够小(< 107 ),则可以视为梯度校验通过。

2-2 梯度校验代码:

# GRADED FUNCTION: gradient_check

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

测试:

x, theta = 2, 4
difference = gradient_check(x, theta)
print("difference = " + str(difference))

运行结果如下:

The gradient is correct!
difference = 2.91933588329e-10

difference 明显小于阈值 107 ,校验通过。

3- N微梯度校验

本文采用3层神经网络做说明,模型:LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID
DeepLearing学习笔记-改善深层神经网络(第一周作业-3-梯度校验)_第2张图片

3-1 前向传播和后向传播:

前向传播代码:

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

这里的后向传播,故意在dW2和db1这里写错:

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)#不用乘以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 = 1./m * np.sum(dZ1, axis=1, keepdims = True)#分子是1,不是4

    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

3-2 梯度校验

对每个参数执行如下操作:

  • 计算J_plus[i]:
    1. θ+ = np.copy(parameters_values)
    2. θ+i = θ+i+ε
    3. 采用forward_propagation_n(x, y, vector_to_dictionary( θ+ ))计算 J+i
  • 对于 θ 同理计算 J_minus[i]
  • 计算 gradapprox[i]=J+iJi2ε

gradapprox中的 gradapprox[i] 对应的是参数parameter_values[i]的梯度近似值 。gradapprox 向量和后向传播的梯度的相似按照如下公式估算.:

difference=gradgradapprox2grad2+gradapprox2(3)

3-2 梯度校验

# 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)#将字典转为向量形式
    num_parameters = parameters_values.shape[0]#参数个数
    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 ###
    #注意这里epsilon值的问题,如果是1e-6是可以的,使得梯度校验通过,epsilon值越小,反而和导数越不一致
    if difference > 1e-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

测试代码:

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)

测试结果:
There is a mistake in the backward propagation! difference = 0.285093156654
明显存在梯度计算问题。将反向梯度计算的dW2和db1进行修改,重新运行:
There is a mistake in the backward propagation! difference = 1.18855520355e-07
虽然现实出现问题,但是different值已经很接近阈值了,此时,我们单独修改计算双边梯度之后的epsilon=1e-6,不修改判断的阈值。
输出:
Your backward propagation works perfectly fine! difference = 8.26588225515e-09
所以,要注意看difference值是否和阈值相距很大。本文为何就差那么一些,导致需要修改epsilon,可能是由于代价函数在局部存在毛刺,导致估算值和后向梯度计算结果,存在超于阈值的偏差。另外,relu的导数在0处有歧义,也可能导致此处的不够准确。

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