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：
∂J∂θ=limε→0J(θ+ε)−J(θ−ε)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
当 J(θ)=θx ，线性模型如下:
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=∂J∂θ=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”:
- θ+=θ+ε
- θ−=θ−ε
- J+=J(θ+)
- J−=J(θ−)
- gradapprox=J+−J−2ε
进行反向传播，获取梯度值 “grad”
- 按照如下公式计算两者difference :
difference=∣∣grad−gradapprox∣∣2∣∣grad∣∣2+∣∣gradapprox∣∣2(2)
范数的计算可以用np.linalg.norm(...)
当difference 足够小(< 10−7 )，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-10difference 明显小于阈值 10−7 ，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
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 gradients3-2 梯度校验
对每个参数执行如下操作:
- 计算
J_plus[i]:
- θ+ =
np.copy(parameters_values) - θ+i = θ+i+ε
- 采用
forward_propagation_n(x, y, vector_to_dictionary(θ+))计算 J+i
- θ+ =
- 对于 θ− 同理计算
J_minus[i] - 计算 gradapprox[i]=J+i−J−i2ε
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=∥grad−gradapprox∥2∥grad∥2+∥gradapprox∥2(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。