DeepLearning入门03
具有一个隐藏层的平面数据分类
1.依赖包
首先安装需要的包，然后用import导入：
- numpy is the fundamental package for scientific computing with Python.
- sklearn provides simple and efficient tools for data mining and data analysis.
- matplotlib is a library for plotting graphs in Python.
- testCases_v2 provides some test examples to assess the correctness of your functions
- planar_utils provide various useful functions used in this assignment
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases_v2 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
其中，testCases_v2.py文件内容如下：
import numpy as np
def layer_sizes_test_case():
np.random.seed(1)
X_assess = np.random.randn(5, 3)
Y_assess = np.random.randn(2, 3)
return X_assess, Y_assess
def initialize_parameters_test_case():
n_x, n_h, n_y = 2, 4, 1
return n_x, n_h, n_y
def forward_propagation_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
b1 = np.random.randn(4,1)
b2 = np.array([[ -1.3]])
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': b1,
'b2': b2}
return X_assess, parameters
def compute_cost_test_case():
np.random.seed(1)
Y_assess = (np.random.randn(1, 3) > 0)
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': np.array([[ 0.],
[ 0.],
[ 0.],
[ 0.]]),
'b2': np.array([[ 0.]])}
a2 = (np.array([[ 0.5002307 , 0.49985831, 0.50023963]]))
return a2, Y_assess, parameters
def backward_propagation_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
Y_assess = (np.random.randn(1, 3) > 0)
parameters = {'W1': np.array([[-0.00416758, -0.00056267],
[-0.02136196, 0.01640271],
[-0.01793436, -0.00841747],
[ 0.00502881, -0.01245288]]),
'W2': np.array([[-0.01057952, -0.00909008, 0.00551454, 0.02292208]]),
'b1': np.array([[ 0.],
[ 0.],
[ 0.],
[ 0.]]),
'b2': np.array([[ 0.]])}
cache = {'A1': np.array([[-0.00616578, 0.0020626 , 0.00349619],
[-0.05225116, 0.02725659, -0.02646251],
[-0.02009721, 0.0036869 , 0.02883756],
[ 0.02152675, -0.01385234, 0.02599885]]),
'A2': np.array([[ 0.5002307 , 0.49985831, 0.50023963]]),
'Z1': np.array([[-0.00616586, 0.0020626 , 0.0034962 ],
[-0.05229879, 0.02726335, -0.02646869],
[-0.02009991, 0.00368692, 0.02884556],
[ 0.02153007, -0.01385322, 0.02600471]]),
'Z2': np.array([[ 0.00092281, -0.00056678, 0.00095853]])}
return parameters, cache, X_assess, Y_assess
def update_parameters_test_case():
parameters = {'W1': np.array([[-0.00615039, 0.0169021 ],
[-0.02311792, 0.03137121],
[-0.0169217 , -0.01752545],
[ 0.00935436, -0.05018221]]),
'W2': np.array([[-0.0104319 , -0.04019007, 0.01607211, 0.04440255]]),
'b1': np.array([[ -8.97523455e-07],
[ 8.15562092e-06],
[ 6.04810633e-07],
[ -2.54560700e-06]]),
'b2': np.array([[ 9.14954378e-05]])}
grads = {'dW1': np.array([[ 0.00023322, -0.00205423],
[ 0.00082222, -0.00700776],
[-0.00031831, 0.0028636 ],
[-0.00092857, 0.00809933]]),
'dW2': np.array([[ -1.75740039e-05, 3.70231337e-03, -1.25683095e-03,
-2.55715317e-03]]),
'db1': np.array([[ 1.05570087e-07],
[ -3.81814487e-06],
[ -1.90155145e-07],
[ 5.46467802e-07]]),
'db2': np.array([[ -1.08923140e-05]])}
return parameters, grads
def nn_model_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
Y_assess = (np.random.randn(1, 3) > 0)
return X_assess, Y_assess
def predict_test_case():
np.random.seed(1)
X_assess = np.random.randn(2, 3)
parameters = {'W1': np.array([[-0.00615039, 0.0169021 ],
[-0.02311792, 0.03137121],
[-0.0169217 , -0.01752545],
[ 0.00935436, -0.05018221]]),
'W2': np.array([[-0.0104319 , -0.04019007, 0.01607211, 0.04440255]]),
'b1': np.array([[ -8.97523455e-07],
[ 8.15562092e-06],
[ 6.04810633e-07],
[ -2.54560700e-06]]),
'b2': np.array([[ 9.14954378e-05]])}
return parameters, X_assess
其中，planar_utils.py内容如下：
import matplotlib.pyplot as plt
import numpy as np
import sklearn
import sklearn.datasets
import sklearn.linear_model
def plot_decision_boundary(model, X, y):
# Set min and max values and give it some padding
x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
h = 0.01
# Generate a grid of points with distance h between them
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
# Predict the function value for the whole grid
Z = model(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
# Plot the contour and training examples
plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
plt.ylabel('x2')
plt.xlabel('x1')
plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)
def sigmoid(x):
"""
Compute the sigmoid of x
Arguments:
x -- A scalar or numpy array of any size.
Return:
s -- sigmoid(x)
"""
s = 1/(1+np.exp(-x))
return s
def load_planar_dataset():
np.random.seed(1)
m = 400 # number of examples
N = int(m/2) # number of points per class
D = 2 # dimensionality
X = np.zeros((m,D)) # data matrix where each row is a single example
Y = np.zeros((m,1), dtype='uint8') # labels vector (0 for red, 1 for blue)
a = 4 # maximum ray of the flower
for j in range(2):
ix = range(N*j,N*(j+1))
t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta
r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
Y[ix] = j
X = X.T
Y = Y.T
return X, Y
def load_extra_datasets():
N = 200
noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)
noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)
blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)
gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)
no_structure = np.random.rand(N, 2), np.random.rand(N, 2)
return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure
也可以到链接地址下载： https://pan.baidu.com/s/1Zog4gUubAzvtN7rQe7MUbQ
2.数据集
- 首先，让我们获取您将要处理的数据集。以下代码将“花”类数据集加载到变量X和Y中。
本模型将实现生成花的模型，并使用神经网络进行模拟：
# Package imports
import numpy as np
import matplotlib.pyplot as plt
from testCases_v2 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
def load_planar_dataset():
np.random.seed(1)
m = 400 # 样本数量
N = int(m/2) # 每个类别的样本量
D = 2 # 维度数
X = np.zeros((m,D)) # 初始化X
Y = np.zeros((m,1), dtype='uint8') # 初始化Y
a = 4 # 花儿的最大长度
for j in range(2):
ix = range(N*j,N*(j+1))
t = np.linspace(j*3.12,(j+1)*3.12,N) + np.random.randn(N)*0.2 # theta
r = a*np.sin(4*t) + np.random.randn(N)*0.2 # radius
X[ix] = np.c_[r*np.sin(t), r*np.cos(t)]
Y[ix] = j
X = X.T
Y = Y.T
return X, Y
X, Y = load_planar_dataset()
# The data looks like a “flower” with some red (label y=0) and some blue (y=1) points.
# Your goal is to build a model to fit this data.
# Visualize the data:
#plt.scatter(X[0, :], X[1, :], c=Y, s=40, cmap=plt.cm.Spectral);
plt.scatter(X[0, :], X[1, :], c=Y.flatten(), s=40, cmap=plt.cm.Spectral);
这样你可以得到:
- 一个包含特征(x1, x2) 的 numpy-array (matrix) X
- 一个包含标签(red:0, blue:1)的numpy-array (vector) Y
我们可以用代码看一下数据的样子：
### START CODE HERE ### (≈ 3 lines of code)
shape_X = X.shape
shape_Y = Y.shape
m = X.shape[1] # training set size
### END CODE HERE ###
print ('The shape of X is: ' + str(shape_X))
print ('The shape of Y is: ' + str(shape_Y))
print ('I have m = %d training examples!' % (m))
The shape of X is: (2, 400)
The shape of Y is: (1, 400)
I have m = 400 training examples!
3.简单的逻辑回归
构建一个完整的神经网络之前，让我们先看看如何回归逻辑在这个问题上执行。您可以使用sklearn的内置功能来做到这一点。
运行下面的代码来训练数据集上logistic回归分类。
# Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y.T);
# Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y.flatten())
plt.title("Logistic Regression")
# Print accuracy
LR_predictions = clf.predict(X.T)
print ('Accuracy of logistic regression: %d ' % float((np.dot(Y,LR_predictions) + np.dot(1-Y,1-LR_predictions))/float(Y.size)*100) +
'% ' + "(percentage of correctly labelled datapoints)")
Accuracy of logistic regression: 47 % (percentage of correctly labelled datapoints)
def plot_decision_boundary(model, X, y):
# Set min and max values and give it some padding
x_min, x_max = X[0, :].min() - 1, X[0, :].max() + 1
y_min, y_max = X[1, :].min() - 1, X[1, :].max() + 1
h = 0.01
# Generate a grid of points with distance h between them
xx, yy = np.meshgrid(np.arange(x_min, x_max, h), np.arange(y_min, y_max, h))
# Predict the function value for the whole grid
Z = model(np.c_[xx.ravel(), yy.ravel()])
Z = Z.reshape(xx.shape)
# Plot the contour and training examples
plt.contourf(xx, yy, Z, cmap=plt.cm.Spectral)
plt.ylabel('x2')
plt.xlabel('x1')
plt.scatter(X[0, :], X[1, :], c=y, cmap=plt.cm.Spectral)
该数据集不是线性可分的，所以回归效果并不是很好，希望神经网络会做的更好。现在，让我们试试这个！
4.神经网络模型
Logistic回归没有对“花集”很好地工作。你要训练神经网络具有单隐层。
模型如下:
需要的数学知识：
对任意 x ( i ) x^{(i)} x(i)
(1) z [ 1 ] ( i ) = W [ 1 ] x ( i ) + b [ 1 ] ( i ) z^{[1] (i)} = W^{[1]} x^{(i)} + b^{[1] (i)}\tag{1} z[1](i)=W[1]x(i)+b[1](i)(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 ] ( i ) z^{[2] (i)} = W^{[2]} a^{[1] (i)} + b^{[2] (i)}\tag{3} z[2](i)=W[2]a[1](i)+b[2](i)(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)
KaTeX parse error: Expected & or \\ or \cr or \end at position 43: …gin{cases} 1 & \̲m̲b̲o̲x̲{if } a^{[2](i)…
这里我们依然使用交叉熵损失函数：
(6) J = − 1 m ∑ i = 0 m ( y ( i ) log ( a [ 2 ] ( i ) ) + ( 1 − y ( i ) ) log ( 1 − a [ 2 ] ( i ) ) ) J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large\left(\small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large \right) \small \tag{6} J=−m1i=0∑m(y(i)log(a[2](i))+(1−y(i))log(1−a[2](i)))(6)
构建神经网络的一般方法是：
1.定义神经网络结构（输入单元数，隐藏单元数等）。
2.初始化模型的参数
3.循环：
- 实现正向传播
- 计算损失
- 实施反向传播来获得梯度
- 更新参数（梯度下降）
你经常构建辅助函数计算步骤1-3，然后将它们合并成一个功能我们称之为nn_model（）。一旦你已经建立nn_model（），并学会了正确的参数，你可以对新数据的预测。
4.1 定义神经网络结构
练习：定义三个变量：
- n_x：输入图层的大小
- n_h：隐藏层的大小（设置为4）
- n_y：输出图层的大小
提示：使用X和Y的形状来查找n_x和n_y。此外，将隐藏层大小硬编码为4。
# 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)
X_assess, Y_assess = layer_sizes_test_case()
(n_x, n_h, n_y) = layer_sizes(X_assess, Y_assess)
print("The size of the input layer is: n_x = " + str(n_x))
print("The size of the hidden layer is: n_h = " + str(n_h))
print("The size of the output layer is: n_y = " + str(n_y))
The size of the input layer is: n_x = 5
The size of the hidden layer is: n_h = 4
The size of the output layer is: n_y = 2
4.2 - 初始化模型的参数
练习：实现函数initialize_parameters()。
说明：
- 确保您参数的大小合适的参考神经网络上图中如果需要的话。
- 您将初始化随机值的权重矩阵。
- 使用：
np.random.randn（A，B）* 0.01随机初始化形状的矩阵（a，b）。 - 您将偏置向量初始化为零。
- 使用：
np.zeros((a，b))用零初始化形状（a，b）的矩阵。
# 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)
b1 = np.zeros((n_h, 1))
W2 = np.random.randn(n_y, n_h)
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
n_x, n_h, n_y = initialize_parameters_test_case()
parameters = initialize_parameters(n_x, n_h, n_y)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
W1 = [[-0.41675785 -0.05626683]
[-2.1361961 1.64027081]
[-1.79343559 -0.84174737]
[ 0.50288142 -1.24528809]]
b1 = [[0.]
[0.]
[0.]
[0.]]
W2 = [[-1.05795222 -0.90900761 0.55145404 2.29220801]]
b2 = [[0.]]
4.3 循环
4.3.1实现前向传播
-
根据数学公式我们知道这是一个分类问题，我们可以使用下面激活函数：
-
sigmoid()激活函数
-
np.tanh()激活函数
-
-
还需要实现下面步骤：
-
1.使用
parameters[".."]从字典“parameters”（这是initialize_parameters()的输出）中检索每个参数。 -
2.实现前向传播。计算 Z [ 1 ] , A [ 1 ] , Z [ 2 ] 和 A [ 2 ] Z^{[1]}, A^{[1]}, Z^{[2]} 和 A^{[2]} Z[1],A[1],Z[2]和A[2]（所有对训练集中所有例子的预测的向量矩阵）。
- 反向传播中所需的值存储在“缓存”中。缓存将作为反向传播函数的输入给出。
-
# 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
X_assess, parameters = forward_propagation_test_case()
A2, cache = forward_propagation(X_assess, parameters)
# Note: we use the mean here just to make sure that your output matches ours.
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))
0.26281864019752443 0.09199904522700113 -1.3076660128732143 0.21287768171914198
实现交叉熵的计算
交叉熵数学公式：
J = − 1 m ∑ i = 0 m ( y ( i ) log ( a [ 2 ] ( i ) ) + ( 1 − y ( i ) ) log ( 1 − a [ 2 ] ( i ) ) ) J = - \frac{1}{m} \sum\limits_{i = 0}^{m} \large\left(\small y^{(i)}\log\left(a^{[2] (i)}\right) + (1-y^{(i)})\log\left(1- a^{[2] (i)}\right) \large \right) \small J=−m1i=0∑m(y(i)log(a[2](i))+(1−y(i))log(1−a[2](i)))
具体实现如下：
#logprobs = np.multiply(np.log(A2),Y)
#cost = - np.sum(logprobs) # no need to use a for loop!
# 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 = -(1.0/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
A2, Y_assess, parameters = compute_cost_test_case()
print("cost = " + str(compute_cost(A2, Y_assess, parameters)))
cost = 0.6930587610394646
4.3.2 实现反向传播：backward_propagation()
反向传播是深度学习中很困难的问题，包含很多数学公式：
具体代码如下：
# 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 = 1.0/m*np.dot(dZ2, A1.T)
db2 = 1.0/m*np.sum(dZ2, axis=1, keepdims=True)
dZ1 = np.dot(W2.T, dZ2)*(1-np.power(A1, 2))
dW1 = 1.0/m*np.dot(dZ1, X.T)
db1 = 1.0/m*np.sum(dZ1, axis=1, keepdims=True)
### END CODE HERE ###
grads = {"dW1": dW1,
"db1": db1,
"dW2": dW2,
"db2": db2}
return grads
parameters, cache, X_assess, Y_assess = backward_propagation_test_case()
grads = backward_propagation(parameters, cache, X_assess, Y_assess)
print ("dW1 = "+ str(grads["dW1"]))
print ("db1 = "+ str(grads["db1"]))
print ("dW2 = "+ str(grads["dW2"]))
print ("db2 = "+ str(grads["db2"]))
dW1 = [[ 0.00301023 -0.00747267]
[ 0.00257968 -0.00641288]
[-0.00156892 0.003893 ]
[-0.00652037 0.01618243]]
db1 = [[ 0.00176201]
[ 0.00150995]
[-0.00091736]
[-0.00381422]]
dW2 = [[ 0.00078841 0.01765429 -0.00084166 -0.01022527]]
db2 = [[-0.16655712]]
4.3.3 更新参数
为了更新(W1, b1, W2, b2)，我们必须使用 (dW1, db1, dW2, db2)来更新.
梯度下降规则是 θ = θ − α ∂ J ∂ θ \theta = \theta - \alpha \frac{\partial J }{ \partial \theta } θ=θ−α∂θ∂J,其中 α \alpha α是学习率， θ \theta θ是代表需要更新的参数。不同的学习率对梯度下降影响很大，下图演示了合适的学习率 和不合适学习率的效果。
更新参数实现算法如下：
# 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
parameters, grads = update_parameters_test_case()
parameters = update_parameters(parameters, grads)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
W1 = [[-0.00643025 0.01936718]
[-0.02410458 0.03978052]
[-0.01653973 -0.02096177]
[ 0.01046864 -0.05990141]]
b1 = [[-1.02420756e-06]
[ 1.27373948e-05]
[ 8.32996807e-07]
[-3.20136836e-06]]
W2 = [[-0.01041081 -0.04463285 0.01758031 0.04747113]]
b2 = [[0.00010457]]
4.4整合来实现nn_model()
具体代码如下：
# 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, learning_rate = 1.2)
### 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
X_assess, Y_assess = nn_model_test_case()
parameters = nn_model(X_assess, Y_assess, 4, num_iterations=10000, print_cost=True)
print("W1 = " + str(parameters["W1"]))
print("b1 = " + str(parameters["b1"]))
print("W2 = " + str(parameters["W2"]))
print("b2 = " + str(parameters["b2"]))
Cost after iteration 0: 0.041960
Cost after iteration 1000: 0.000266
Cost after iteration 2000: 0.000134
Cost after iteration 3000: 0.000090
Cost after iteration 4000: 0.000068
Cost after iteration 5000: 0.000054
Cost after iteration 6000: 0.000045
Cost after iteration 7000: 0.000039
Cost after iteration 8000: 0.000034
Cost after iteration 9000: 0.000030
W1 = [[-0.89587042 1.18044635]
[-2.14783312 1.70666862]
[-1.50260821 -1.21347886]
[ 0.80826745 -1.65434514]]
b1 = [[ 0.19050922]
[ 0.01614166]
[-0.34103273]
[-0.25208981]]
W2 = [[-2.90757381 -3.18177289 0.36186225 4.50758023]]
b2 = [[0.24451252]]
4.5 预测
利用上述代码实现 predict()函数.
使用前向传播实现预测结果。
其中
p r e d i c t i o n s = y p r e d i c t i o n = { 1 , i f a c t i v a t i o n > 0.5 0 , o t h e r w i s e predictions = y_{prediction} = \begin{cases} 1, {if } activation { > 0.5} \\ 0, {otherwise } \end{cases} predictions=yprediction={1,ifactivation>0.50,otherwise
# 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
parameters, X_assess = predict_test_case()
predictions = predict(parameters, X_assess)
print("predictions mean = " + str(np.mean(predictions)))
predictions mean = 0.6666666666666666
# 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.flatten())
plt.title("Decision Boundary for hidden layer size " + str(4))
Cost after iteration 0: 1.127380
Cost after iteration 1000: 0.288553
Cost after iteration 2000: 0.276386
Cost after iteration 3000: 0.268077
Cost after iteration 4000: 0.263069
Cost after iteration 5000: 0.259617
Cost after iteration 6000: 0.257070
Cost after iteration 7000: 0.255105
Cost after iteration 8000: 0.253534
Cost after iteration 9000: 0.252245
Text(0.5, 1.0, 'Decision Boundary for hidden layer size 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) + '%')
Accuracy: 91%
该模型已经模拟了花的叶子模式！与逻辑回归不同，神经网络能够进一步学习高度非线性决策边界。
现在，让我们试试几个隐藏的图层大小。
4.6 调整隐藏的图层大小
代码如下：
# 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.flatten())
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))
Accuracy for 1 hidden units: 61.5 %
Accuracy for 2 hidden units: 70.5 %
Accuracy for 3 hidden units: 66.25 %
Accuracy for 4 hidden units: 90.75 %
Accuracy for 5 hidden units: 91.0 %
Accuracy for 20 hidden units: 91.5 %
Accuracy for 50 hidden units: 90.75 %
结论：
- 较大的模型（具有更多隐藏单位）能够更好地适应训练集，直到最终最大的模型过度拟合数据。
- 最好的隐藏层大小似乎在n_h = 5左右。实际上，这里的值似乎很好地适合数据而不会引起明显的过度拟合。
- 您稍后还将学习正则化，它可以让您使用非常大的模型（例如n_h = 50）而不会过度拟合。
你学会了：
- 使用隐藏层构建完整的神经网络
- 充分利用非线性单元
- 实现前向传播和反向传播，并训练神经网络
- 查看改变隐藏层大小的影响，包括过度拟合。
4.7其他数据集的性能
如果需要，可以为以下每个数据集重新运行整个笔记本（减去数据集部分）。
def load_extra_datasets():
N = 200
noisy_circles = sklearn.datasets.make_circles(n_samples=N, factor=.5, noise=.3)
noisy_moons = sklearn.datasets.make_moons(n_samples=N, noise=.2)
blobs = sklearn.datasets.make_blobs(n_samples=N, random_state=5, n_features=2, centers=6)
gaussian_quantiles = sklearn.datasets.make_gaussian_quantiles(mean=None, cov=0.5, n_samples=N, n_features=2, n_classes=2, shuffle=True, random_state=None)
no_structure = np.random.rand(N, 2), np.random.rand(N, 2)
return noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure
# Datasets
noisy_circles, noisy_moons, blobs, gaussian_quantiles, no_structure = load_extra_datasets()
datasets = {"noisy_circles": noisy_circles,
"noisy_moons": noisy_moons,
"blobs": blobs,
"gaussian_quantiles": gaussian_quantiles}
### START CODE HERE ### (choose your dataset)
dataset = "noisy_moons"
### END CODE HERE ###
X, Y = datasets[dataset]
X, Y = X.T, Y.reshape(1, Y.shape[0])
# make blobs binary
if dataset == "blobs":
Y = Y%2
# Visualize the data
plt.scatter(X[0, :], X[1, :], c=Y.flatten(), s=40, cmap=plt.cm.Spectral);
