前言：
文章以Andrew Ng 的 deeplearning.ai 视频课程为主线，记录Programming Assignments 的实现过程。相对于斯坦福的CS231n课程，Andrew的视频课程更加简单易懂，适合深度学习的入门者系统学习！
本次作业主要讲到Gradient Checking方法，使用这个方法能够较早的发现梯度计算问题，检验梯度计算是否正确，从而保证程序能够正确的执行。
其实梯度检查的主要思想就是高数中导数的定义，利用无线逼近的方法判断程序梯度计算是否存在问题：
1.1 1-dimensional gradient checking
梯度检查的主要步骤如下：
代码如下：
def forward_propagation(x, theta):
J=theta*x
return J
def backward_propagation(x, theta):
dtheta=x
return dtheta
def gradient_check(x, theta, epsilon = 1e-7):
thetaplus = theta+epsilon
thetaminus = theta-epsilon
J_plus = thetaplus*x
J_minus = thetaminus*x
gradapprox = (J_plus-J_minus)/(2*epsilon)
grad = x
numerator = np.linalg.norm(gradapprox-grad)
denominator = np.linalg.norm(gradapprox)+np.linalg.norm(grad)
difference = numerator/denominator
if difference < 1e-7:
print ("The gradient is correct!")
else:
print ("The gradient is wrong!")
return difference
1.2 N-dimensional gradient checking
def forward_propagation_n(X, Y, parameters):
m = X.shape[1]
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
W3 = parameters["W3"]
b3 = parameters["b3"]
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)
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):
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
def gradient_check_n(parameters, gradients, X, Y, epsilon = 1e-7):
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))
for i in range(num_parameters):
thetaplus = np.copy(parameters_values,True)
thetaplus[i,:] = thetaplus[i,:]+epsilon
J_plus[i], _ = forward_propagation_n(X, Y, vector_to_dictionary(thetaplus))
thetaminus = np.copy(parameters_values,True)
thetaminus[i,:] = thetaminus[i,:]-epsilon
J_minus[i], _ = forward_propagation_n(X,Y,vector_to_dictionary(thetaminus))
gradapprox[i] = (J_plus[i]-J_minus[i])/(2*epsilon)
numerator = np.linalg.norm(grad-gradapprox)
denominator = np.linalg.norm(grad)+np.linalg.norm(gradapprox)
difference = numerator/denominator
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
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