title: DeepLearning.ai作业:(1-3)-- 浅层神经网络（Shallow neural networks）
tags:

• dl.ai
• homework
  categories:
• AI
• Deep Learning
  date: 2018-09-12 15:49:22
  id: 2018091216

首发于个人博客:fangzh.top，欢迎来访

1. 不要抄作业！
2. 我只是把思路整理了，供个人学习。
3. 不要抄作业！

数据集

数据集是一个类似花的数据集。

而如果用传统的logistic regression，做出来的就是一个二分类问题，简单粗暴的划出了一条线，

可以看见，准确率只有47%。

所以就需要构建神经网络模型了。

神经网络模型

Reminder: The general methodology to build a Neural Network is to:

1. Define the neural network structure ( # of input units,  # of hidden units, etc).
2. Initialize the model's parameters
3. Loop:
- Implement forward propagation
- Compute loss
- Implement backward propagation to get the gradients
- Update parameters (gradient descent)

已经给出思路了：

1. 定义神经网络的结构
2. 初始化模型参数
3. 循环： 计算正向传播
4. 计算损失函数
5. 计算反向传播来得到grad
6. 更新参数

1. 定义神经网络结构

# GRADED FUNCTION: layer_sizes

def layer_sizes(X, Y):
"""
Arguments:
X -- input dataset of shape (input size, number of examples)
Y -- labels of shape (output size, number of examples)

Returns:
n_x -- the size of the input layer
n_h -- the size of the hidden layer
n_y -- the size of the output layer
"""
### START CODE HERE ### (≈ 3 lines of code)
n_x = X.shape[0] # size of input layer
n_h = 4
n_y = Y.shape[0] # size of output layer
### END CODE HERE ###
return (n_x, n_h, n_y)

[/code]

2. 初始化参数

来初始化w和b的参数

w:

np.random.rand(a,b) * 0.01

b:

np.zeros((a,b))

# 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.randn(n_h, n_x) * 0.01
b1 = np.zeros((n_h, 1))
W2 = np.random.randn(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

[/code]

3. loop

在这里可以使用sigmoid()来做输出层的函数，np.tanh()来做hidden layer的激活函数。

3.1 forward propagation

在这个函数中，输入的是X，和parameters，然后就可以根据

(1)z[1](i)=W[1]x(i)+b[1]z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1]}\tag{1}z[1](i)=W[1]x(i)+b[1](1)
(2)a[1](i)=tanh⁡(z[1](i))a^{[1] (i)} = \tanh(z^{[1] (i)})\tag{2}a[1](i)=tanh(z[1](i))(2)
(3)z[2](i)=W[2]a[1](i)+b[2]z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2]}\tag{3}z[2](i)=W[2]a[1](i)+b[2](3)
(4)y^(i)=a[2](i)=σ(z[2](i))\hat{y}^{(i)} = a^{[2] (i)} = \sigma(z^{ [2] (i)})\tag{4}y^​(i)=a[2](i)=σ(z[2](i))(4)

得到每一层的Z和A了。

# 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

[/code]

3.2 cost

接下来，在得到A2的值后，就可以根据公式来计算损失函数了。

J=−1m∑i=0m(y(i)log⁡(a[2](i))+(1−y(i))log⁡(1−a[2](i)))J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large{(} \small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large{)} \smallJ=−m1​i=0∑m​(y(i)log(a[2](i))+(1−y(i))log(1−a[2](i)))

在这里需要注意的是交叉熵的计算，交叉熵使用np.multiply()来计算，然后用np.sum()，求和。

而单单计算

logprobs = np.multiply(np.log(A2),Y)

是不够的，因为这个只得到了公式的前一半的部分，Y=0的部分在元素相乘中就相当于没有了，所以还要再后面加一项

np.multiply(np.log(1-A2),1-Y)

# 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.multiply(np.log(A2),Y)  + np.multiply(np.log(1-A2),1-Y)
cost =  -1 / m *  np.sum(logprobs)
### 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

[/code]

3.3 backworad propagation

NG说神经网络中最难理解的是这个，但是现在公式已经帮我们推倒好了。

其中， g[1]′(Z[1])g^{[1]'}(Z^{[1]})g[1]′(Z[1]) using

(1 - np.power(A1, 2))

可以看到，公式中需要的变量有X,Y,A,W,然后输出一个字典结构的grads

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 = 1 / m * np.dot(dZ2, A1.T)
db2 = 1 / m * np.sum(dZ2, axis=1, keepdims=True)
dZ1 = np.dot(W2.T, dZ2) * (1 - np.power(A1, 2))
dW1 = 1 / m * np.dot(dZ1, X.T)
db1 = 1 / m * np.sum(dZ1, axis=1, keepdims=True)
### END CODE HERE ###

grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}

return grads

[/code]

3.4 update parameters

最后根据得到的grads，乘上学习速率，就可以更新参数了。

# 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 = W1 - learning_rate * dW1
b1 = b1 - learning_rate * db1
W2 = W2 - learning_rate * dW2
b2 = b2 - learning_rate * db2
### END CODE HERE ###

parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}

return parameters

[/code]

然后把更新完的参数再传入前面的循环中，不断循环，直到达到循环的次数。

nn_model

把前面的函数都调用过来。

模型中传入的参数是，X,Y，和迭代次数

1. 首先需要得到你要设计的神经网络结构，调用

   layer_sizes()

   得到了n_x,n_y，也就是输入层和输出层。
2. 初始化参数

   initialize_parameters(n_x, n_h, n_y)

   ,得到初始化的 W1, b1, W2, b2
3. 然后开始循环 使用

   forward_propagation(X, parameters)

   ,先得到各个神经元的计算值。
4. 然后

   compute_cost(A2, Y, parameters)

   ,得到cost

5. backward_propagation(parameters, cache, X, Y)

   计算出每一步的梯度

6. update_parameters(parameters, grads)

   更新一下参数

返回训练完的parameters

# GRADED FUNCTION: nn_model

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)

### 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 parameters

[/code]

预测

得到训练后的parameters，再用

forward_propagation(X, parameters)

计算出输出层最终的值A2，以0.5为分界，分为0和1。

# GRADED FUNCTION: predict

def predict(parameters, X):
"""
Using the learned parameters, predicts a class for each example in X

Arguments:
parameters -- python dictionary containing your parameters
X -- input data of size (n_x, m)

Returns
predictions -- vector of predictions of our model (red: 0 / blue: 1)
"""

# Computes probabilities using forward propagation, and classifies to 0/1 using 0.5 as the threshold.
### START CODE HERE ### (≈ 2 lines of code)
A2, cache = forward_propagation(X, parameters)
predictions = (A2 > 0.5)
### END CODE HERE ###

return predictions

[/code]

# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)

# Plot the decision boundary
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
plt.title("Decision Boundary for hidden layer size " + str(4))

[/code]

可以看到，训练后神经网络得到的分界线更为合理。

# Print accuracy
predictions = predict(parameters, X)
print ('Accuracy: %d' % float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100) + '%')

[/code]

准确率高达90%

优化参数

这个时候就可以设置不同的hidden_layer的维度大小[1, 2, 3, 4, 5, 20, 50]

# This may take about 2 minutes to run

plt.figure(figsize=(16, 32))
hidden_layer_sizes = [1, 2, 3, 4, 5, 20, 50]
for i, n_h in enumerate(hidden_layer_sizes):
plt.subplot(5, 2, i+1)
plt.title('Hidden Layer of size %d' % n_h)
parameters = nn_model(X, Y, n_h, num_iterations = 5000)
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y)
predictions = predict(parameters, X)
accuracy = float((np.dot(Y,predictions.T) + np.dot(1-Y,1-predictions.T))/float(Y.size)*100)
print ("Accuracy for {} hidden units: {} %".format(n_h, accuracy))

[/code]

Accuracy for 1 hidden units: 67.5 %
Accuracy for 2 hidden units: 67.25 %
Accuracy for 3 hidden units: 90.75 %
Accuracy for 4 hidden units: 90.5 %
Accuracy for 5 hidden units: 91.25 %
Accuracy for 20 hidden units: 90.0 %
Accuracy for 50 hidden units: 90.25 %

得到的结果在n_h = 5时有最大值。

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