改进深度神经网络之三：梯度校验

1 导入所需的包

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

2 梯度校验是如何实现的

反向传播中计算梯度 $\frac{\partial J}{\partial \theta}$ ,此处的 $\theta$ 代表模型的参数， $J$ 是成本函数。

导数（或者说梯度）的定义如下：

\[\frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon}\]

通过上述的公式，使用较小的 $\varepsilon$ 值来确保你的 $\frac{\partial J}{\partial \theta}$ 计算正确。

3 一维的梯度校验

假设一维线性函数为： $J(\theta)=\theta x$ 。一维线性模型如下图所示。

3.1 前向传播与反向传播过程

# 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
    """
    J = x * theta 
    
    return J

# 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
    """
    
    dtheta = x
    
    return dtheta

3.2 梯度校验

步骤如下：

计算 “gradapprox” ：

$ \theta^{+} = \theta + \varepsilon $

$ \theta^{-} = \theta - \varepsilon $

$ J^{+} = J(\theta^{+}) $

$ J^{-} = J(\theta^{-}) $

$ gradapprox = \frac{J^{+} - J^{-}}{2 \varepsilon} $

使用反向传播计算梯度，将计算的梯度存在变量 “grad” 中。

使用以下公式计算 “gradapprox” 和 “grad” 之间的相对差异：

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

如果这个差异很小（比如小于 $10^{-7}$ ），即代表梯度计算无误。否则，梯度计算中可能存在错误。

# 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.
    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
    
    # Check if gradapprox is close enough to the output of backward_propagation()
    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 ("The gradient is correct!")
    else:
        print ("The gradient is wrong!")
    
    return difference

使用以下代码进行测试：

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

运行结果如下：
The gradient is correct!
difference = 2.919335883291695e-10

4 N维的梯度校验

N维线性模型如下图所示（ LINEAR -> RELU -> LINEAR -> RELU -> LINEAR -> SIGMOID ）：

4.1 前向传播与反向传播过程

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

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

4.2 梯度校验

同样的，还是需要比较 “gradapprox” 和 “grad” 之间的相对差异。计算 “gradapprox” 的公式如下：

\[\frac{\partial J}{\partial \theta} = \lim_{\varepsilon \to 0} \frac{J(\theta + \varepsilon) - J(\theta - \varepsilon)}{2 \varepsilon}\]

不同的是，此时的 $ \theta $ 不再是一个标量，而是一个 dictionary 类型的变量 parameters 。使用函数 dictionary_to_vector() 将变量 parameters 转换为一个向量 values ，反函数 vector_to_dictionary ，可以将 values 转换回 parameters 。如下图所示。

梯度计算的步骤如下：

For each i in num_parameters:

计算 J_plus[i]:
1. 使用 np.copy(parameters_values) 初始化 $ \theta^{+} $
2. 令 $ \theta^{+}_i $ 为 $ \theta^{+}_i + \varepsilon $
3. 使用 forward_propagation_n(x, y, vector_to_dictionary($ \theta^{+} $ )) 计算 $ J^{+}_i $
计算 J_minus[i]: 使用 $\theta^{-}$ 做与上面步骤相似的操作
计算 $ gradapprox[i] = \frac{J^{+}_i - J^{-}_i}{2 \varepsilon} $

接下来便可以使用以下公式来计算 “gradapprox” 和 “grad” 之间的相对差异：
$difference = \frac {\| grad - gradapprox \|_2}{\| grad \|_2 + \| gradapprox \|_2 }$

# 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
        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
        
        # Compute J_minus[i]. Inputs: "parameters_values, epsilon". Output = "J_minus[i]".
        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
        
        # Compute gradapprox[i]
        gradapprox[i] = (J_plus[i] - J_minus[i]) / (2 * epsilon)
    
    # Compare gradapprox to backward propagation gradients by computing difference.
    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 ("\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

使用以下代码进行测试：

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)

运行结果如下：
There is a mistake in the backward propagation! difference = 0.2850931566540251

4.3 反向传播中的bug修复

通过上面的梯度校验，发现反向传播过程中有计算错误，修改其错误代码如下：

# dW2 = 1./m * np.dot(dZ2, A1.T) * 2 有误的
dW2 = 1./m * np.dot(dZ2, A1.T) # 修改的

# db1 = 4./m * np.sum(dZ1, axis=1, keepdims = True) 有误的
db1 = 1./m * np.sum(dZ1, axis=1, keepdims = True) # 修改的

5 总结

梯度校验是验证 grads （由反向传播计算得出） 与 gradapprox （由前向传播计算得出） 之间的接近程度。

梯度校验很慢，因此不可每次训练迭代中都运行它。通常只使用它以确保代码是的正确性，然后将其关闭。

梯度校验不适用于Dropout。通常在没有Dropout的情况下运行梯度校验算法，以确保反向传播过程的计算无误，然后再添加Dropout技术。

Improving Deep Neural Networks Part Three

Gradient Checking