https://www.deeplearning.ai/ 上第一课第三周的编程作业介绍了如何实现一个隐藏层的神经网络。
https://github.com/stormstone/deeplearning.ai 这里可以获得编程作业
整体框架
这里做的是一个二分分类问题，X是input的特征，Y是分类标记，n_h是样本数。
这里把训练一个模型分为五步：
- 初始化参数W1、b1、W2、b2
- 正向传播，根据W1、b1、W2、b2计算当前输入A2，并保存其中的Z1、A1、Z2以便之后做反向传播
- 根据预测值A2及实际值Y，用损失函数计算所有样本的误差，即cost值（实时观察cost值变化情况）
- 反向传播，根据使用的激活函数、W1、W2、A1、A2、X、Y，链式法则求导，得出dW1、db1、dW2、db2
- 根据得出的dW1、db1、dW2、db2，去更新W1、b1、W2、b2。然后重复第2~5步操作，直到达到设定的循环次数
def nn_model(X, Y, n_h, num_iterations = 10000, print_cost=False):
"""
Arguments:
X -- dataset of shape (2, number of examples)
Y -- labels of shape (1, number of examples)
n_h -- size of the hidden layer
num_iterations -- Number of iterations in gradient descent loop
print_cost -- if True, print the cost every 1000 iterations
Returns:
parameters -- parameters learnt by the model. They can then be used to predict.
"""
np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]
# Initialize parameters, then retrieve W1, b1, W2, b2. Inputs: "n_x, n_h, n_y". Outputs = "W1, b1, W2, b2, parameters".
### START CODE HERE ### (≈ 5 lines of code)
parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###
# Loop (gradient descent)
for i in range(0, num_iterations):
### START CODE HERE ### (≈ 4 lines of code)
# Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)
# Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)
# Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)
# Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
parameters = update_parameters(parameters, grads,learning_rate = 0.1)
### END CODE HERE ###
# Print the cost every 1000 iterations
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
return parameterstips
- 激活函数的作用，当没有激活函数，每个神经元做的操作都是y=kx+b，计算一下会发现，不管有多少层、每层多少个神经元，最后实际上都只是y=kx+b。而实际生活中，很多分类问题都不是线性就可以解决的，所以需要激活函数让这条分类的线弯起来。
- 而不同激活函数如sigmoid、tanh、ReLU的区别在于sigmoid的输出在(0,1)，所以适合二分分类问题，表示是否一类的概率。tanh相比sigmoid的优势据说是数据对称？这个没弄懂。ReLU的优势在于，在输入过小或过大时，斜率不会太小，提高效率。（sigmoid和tanh的曲线在输入过小或过大时斜率都趋向于零）
Q
- tanh相比sigmoid的优势是？
- 损失函数使用 L = y*log(a) + (1-y) * log(1-a)，a为预测值，是否会出现斜率为零的情况，就是dW1、dW2、db1、db2皆为零，但这是可能不是最优值？对于不同的损失函数，应该如何得知这种情况呢？
附录
以下是函数的实现，这里隐藏层的激活函数用tanh，输出层激活函数用sigmoid，所以反向传播这样求导。
# GRADED FUNCTION: initialize_parameters
def initialize_parameters(n_x, n_h, n_y):
"""
Argument:
n_x -- size of the input layer
n_h -- size of the hidden layer
n_y -- size of the output layer
Returns:
params -- python dictionary containing your parameters:
W1 -- weight matrix of shape (n_h, n_x)
b1 -- bias vector of shape (n_h, 1)
W2 -- weight matrix of shape (n_y, n_h)
b2 -- bias vector of shape (n_y, 1)
"""
np.random.seed(2) # we set up a seed so that your output matches ours although the initialization is random.
### START CODE HERE ### (≈ 4 lines of code)
W1 = np.random.rand(n_h, n_x) * 0.01
b1 = np.zeros((n_h,1))
W2 = np.random.rand(n_y, n_h) * 0.01
b2 = np.zeros((n_y,1))
### END CODE HERE ###
assert (W1.shape == (n_h, n_x))
assert (b1.shape == (n_h, 1))
assert (W2.shape == (n_y, n_h))
assert (b2.shape == (n_y, 1))
parameters = {
"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}
return parameters
# GRADED FUNCTION: forward_propagation
def forward_propagation(X, parameters):
"""
Argument:
X -- input data of size (n_x, m)
parameters -- python dictionary containing your parameters (output of initialization function)
Returns:
A2 -- The sigmoid output of the second activation
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2"
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###
# Implement Forward Propagation to calculate A2 (probabilities)
### START CODE HERE ### (≈ 4 lines of code)
Z1 = np.dot(W1,X)+b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2,A1)+b2
A2 = sigmoid(Z2)
### END CODE HERE ###
assert(A2.shape == (1, X.shape[1]))
cache = {
"Z1": Z1,
"A1": A1,
"Z2": Z2,
"A2": A2}
return A2, cache
# GRADED FUNCTION: compute_cost
def compute_cost(A2, Y, parameters):
"""
Computes the cross-entropy cost given in equation (13)
Arguments:
A2 -- The sigmoid output of the second activation, of shape (1, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
parameters -- python dictionary containing your parameters W1, b1, W2 and b2
Returns:
cost -- cross-entropy cost given equation (13)
"""
m = Y.shape[1] # number of example
# Compute the cross-entropy cost
### START CODE HERE ### (≈ 2 lines of code)
logprobs = np.log(A2) * Y + (1-Y)* np.log(1-A2)
cost = -np.sum(logprobs)/m
### END CODE HERE ###
cost = np.squeeze(cost) # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
assert(isinstance(cost, float))
return cost
# GRADED FUNCTION: backward_propagation
def backward_propagation(parameters, cache, X, Y):
"""
Implement the backward propagation using the instructions above.
Arguments:
parameters -- python dictionary containing our parameters
cache -- a dictionary containing "Z1", "A1", "Z2" and "A2".
X -- input data of shape (2, number of examples)
Y -- "true" labels vector of shape (1, number of examples)
Returns:
grads -- python dictionary containing your gradients with respect to different parameters
"""
m = X.shape[1]
# First, retrieve W1 and W2 from the dictionary "parameters".
### START CODE HERE ### (≈ 2 lines of code)
W1 = parameters["W1"]
W2 = parameters["W2"]
### END CODE HERE ###
# Retrieve also A1 and A2 from dictionary "cache".
### START CODE HERE ### (≈ 2 lines of code)
A1 = cache["A1"]
A2 = cache["A2"]
### END CODE HERE ###
# Backward propagation: calculate dW1, db1, dW2, db2.
### START CODE HERE ### (≈ 6 lines of code, corresponding to 6 equations on slide above)
dZ2 = A2 - Y
dW2 = np.dot(dZ2,A1.T) / m
db2 = np.sum(dZ2,axis = 1, keepdims = True) / m
dZ1 = np.dot(W2.T,dZ2) * (1 - np.power(A1, 2))
dW1 = np.dot(dZ1,X.T) / m
db1 = np.sum(dZ1,axis = 1, keepdims = True) / m
### END CODE HERE ###
grads = {
"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}
return grads
# GRADED FUNCTION: update_parameters
def update_parameters(parameters, grads, learning_rate = 1.2):
"""
Updates parameters using the gradient descent update rule given above
Arguments:
parameters -- python dictionary containing your parameters
grads -- python dictionary containing your gradients
Returns:
parameters -- python dictionary containing your updated parameters
"""
# Retrieve each parameter from the dictionary "parameters"
### START CODE HERE ### (≈ 4 lines of code)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
### END CODE HERE ###
# Retrieve each gradient from the dictionary "grads"
### START CODE HERE ### (≈ 4 lines of code)
dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]
## END CODE HERE ###
# Update rule for each parameter
### START CODE HERE ### (≈ 4 lines of code)
W1 -= learning_rate * dW1
b1 -= learning_rate * db1
W2 -= learning_rate * dW2
b2 -= learning_rate * db2
### END CODE HERE ###
parameters = {
"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}
return parameters