吴恩达 深度学习 第一课第三周 课程笔记和课后习题答案
1，课程小结
主要讲了只有一层隐层的浅层神经网络。
假设样本有三个特征，隐层有4个节点，则神经网络的结构如上图所示。正向传播计算有四个公式。
对于多样本，向量化计算如下：
常用的激活函数：
一般来说，tanh函数好过sigmod函数，因为其中心为0，除非是二元分类，一般不用sigmod函数；一般情况下，激活函数选用Relu函数。
反向传播计算梯度如下：
一般w初始化是采用随机数而不能采用全零处理，全零处理会导致每个节点演化相同，没有意义。
课后作业，建立一个神经网络：
导入需要的库：
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases import *
import sklearn
import sklearn.datasets
import sklearn.linear_model
from planar_utils import plot_decision_boundary, sigmoid, load_planar_dataset, load_extra_datasets
%matplotlib inline
np.random.seed(1) # set a seed so that the results are consistent
得到层数：
# 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)
初始化参数：
# 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
前向传播：
# 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
计算损失函数：
# 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(Y,np.log(A2))+np.multiply((1-Y),np.log(1-A2))
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
反向传播：
# 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 = None
# W2 = None
W1=parameters['W1']#n_h,n_x
W2=parameters['W2']#n_y,n_h
### END CODE HERE ###
# Retrieve also A1 and A2 from dictionary "cache".
### START CODE HERE ### (≈ 2 lines of code)
# A1 = None
# A2 = None
A1=cache['A1']#n_h,m
A2=cache['A2']#n_y,m
### 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 = None
# dW2 = None
# db2 = None
# dZ1 = None
# dW1 = None
# db1 = None
dZ2=A2-Y#n_y,m
dW2=np.dot(dZ2,A1.T)/m#n_y,n_h
db2=np.sum(dZ2,axis=1,keepdims=True)/m#n_y,1
dZ1=np.multiply(np.dot(W2.T,dZ2),(1-A1**2))#n_h,m
dW1=np.dot(dZ1,X.T)/m#n_h,n_x
db1=np.sum(dZ1,axis=1,keepdims=True)/m#n_h,1
### END CODE HERE ###
grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}
return grads
更新参数：
def update_parameters(parameters,grads,learning_rate=1.2):
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
dW1 = grads["dW1"]
db1 = grads["db1"]
dW2 = grads["dW2"]
db2 = grads["db2"]
###梯度更新，每迭代一次更新一次###
W1 -= learning_rate * dW1
b1 -= learning_rate * db1
W2 -= learning_rate * dW2
b2 -= learning_rate * db2
parameters = {"W1": W1,
"b1": b1,
"W2": W2,
"b2": b2}
return parameters
建立深度学习模型：
def nn_model(X,Y,n_h,num_iterations=10000,print_cost=False):
np.random.seed(3)
n_x = layer_sizes(X, Y)[0]
n_y = layer_sizes(X, Y)[2]
parameters=initialize_parameters(n_x,n_h,n_y)
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
for i in range(0,num_iterations):
A2,cache=forward_propagation(X,parameters)#前向传播节点
cost = compute_cost(A2, Y, parameters)#计算损失函数
grads=backward_propagation(parameters,cache,X,Y)#计算后向传播梯度
parameters=update_parameters(parameters,grads,learning_rate=1.2)#使用梯度更新W，b一次
if print_cost and i % 1000 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
return parameters
预测函数：
# 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
训练实例：
# 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))
# 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) + '%')