[AI教程]TensorFlow入门：通过二分类来比较神经网络模型和逻辑回归模型
介绍
本文主要是建立一个包含一个隐藏层的神经网络， 体会神经网络模型和逻辑回归模型的不同。
在本文中将涉及如下几个部分：
- 使用神经网络实现二分类；
- 使用非线性**函数tanh;
- 计算 cross entropy loss
- 实现正向传播和反向传播；
工具：TensorFlow 1.9.0 + Python 3.6.3
1、两个辅助.py文件
使用两个辅助,py文件写入一些经常使用的内容，通过导入包来进行调用。两个辅助文件如下：
1.1 planar_utils.py
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
1.2 test_Cases.py
import numpy as np
plot_decision_boundary
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)
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.]])}
return X_assess, parameters
def compute_cost_test_case():
np.random.seed(1)
Y_assess = np.random.randn(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': 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)
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)
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
2、主文件
2.1 Package imports
// Package imports
import numpy as np
import matplotlib.pyplot as plt
from test_Cases 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
print('import packages!')
2.2 可视化数据
X, Y = load_planar_dataset()
print('load dataset!')plt.scatter(X[0, :], X[1, :], c=Y[0, :], s=40, cmap=plt.cm.Spectral);
可视化数据集：
- 红色点 (label y=0)
- 蓝色点 (label y=1)
目标：建立模型，对不同颜色的样本点进行分类。可视化图片 - X 包含特征 (x1, x2)
- Y 包含标签 (红:0, 蓝:1).
2.3 初始化并输出相应数据
shape_X = X.shape
shape_Y = Y.shape
m = X.shape[1]
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))
输出结果如下：
The shape of X is: (2, 400)
The shape of Y is: (1, 400)
I have m = 400 training examples!
2.4 使用logistic回归进行分类
训练逻辑回归，绘出决策边界，输出准确率，代码如下:
// Train the logistic regression classifier
clf = sklearn.linear_model.LogisticRegressionCV();
clf.fit(X.T, Y[0, :].T);
// Plot the decision boundary for logistic regression
plot_decision_boundary(lambda x: clf.predict(x), X, Y[0, :])
plt.title("Logistic Regression")
// Print accuracy
LR_predictions = clf.predict(X.T) //预测结果 （0,1）
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)")
Logistic回归的准确率：47%,运行结果如下图所示：
可以看出逻辑回归不能很好的实现，接下来我们尝试使用一个简单的神经网络进行分类！
2.5 神经网络模型
2.5.1 定义输入大小并输出
def layer_sizes(X, Y):
"""
Arguments:
X -- 输入样本的shape(input size, number of examples)
Y -- 样本标签的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
"""
n_x = X.shape[0]
n_h = 4
n_y = Y.shape[0]
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))
2.5.2 初始化模型参数
# 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)
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))
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"]))
2.5.3 正向传播
# 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"
W1 = parameters["W1"]
b1 = parameters["b1"]
W2 = parameters["W2"]
b2 = parameters["b2"]
# Implement Forward Propagation to calculate A2 (probabilities)
Z1 = np.dot(W1,X) + b1
A1 = np.tanh(Z1)
Z2 = np.dot(W2,A1) + b2
A2 = sigmoid(Z2)
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)
print(np.mean(cache['Z1']) ,np.mean(cache['A1']),np.mean(cache['Z2']),np.mean(cache['A2']))
2.5.4 代价函数
# GRADED FUNCTION: compute_cost
def compute_cost(A2, Y, parameters):
m = Y.shape[1] # number of example
#由A2（Y^）预测值和Y实际值作为输入，计算出Cost
cost = - np.sum(Y * np.log(A2) + (1-Y) * np.log(1-A2)) / m # no need to use a for loop!
cost = np.squeeze(cost) # makes sure cost is the dimension we expect.
# E.g., turns [[17]] into 17
return cost
A2, Y_assess, parameters = compute_cost_test_case()
print("cost = " + str(compute_cost(A2, Y_assess, parameters)))
2.5.5 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]
W1 = parameters["W1"]
W2 = parameters["W2"]
A1 = cache["A1"]
A2 = cache["A2"]
# Backward propagation: calculate dW1, db1, dW2, db2.
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 - A1 * A1)
dW1 = 1 / m * np.dot(dZ1, X.T)
db1 = 1 / m * np.sum(dZ1, axis = 1, keepdims = True)
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"]))
2.5.6 梯度下降
# GRADED FUNCTION: update_parameters
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']
# 梯度下降，更新 W,b
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
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"]))
2.5.7 建立网络模型
# 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]
parameters = initialize_parameters(n_x, n_h, n_y)
W1 = parameters['W1']
b1 = parameters['b1']
W2 = parameters['W2']
b2 = parameters['b2']
# 迭代 (梯度下降)
for i in range(0, num_iterations):
# 正向传播 Forward propagation. Inputs: "X, parameters". Outputs: "A2, cache".
A2, cache = forward_propagation(X, parameters)
# 计算代价函数 Cost function. Inputs: "A2, Y, parameters". Outputs: "cost".
cost = compute_cost(A2, Y, parameters)
# 后向传播 Backpropagation. Inputs: "parameters, cache, X, Y". Outputs: "grads".
grads = backward_propagation(parameters, cache, X, Y)
# 梯度下降，更新参数 Gradient descent parameter update. Inputs: "parameters, grads". Outputs: "parameters".
parameters = update_parameters(parameters, grads)
# 输出代价值 Print the cost every 1000 iterations
if print_cost and i % 500 == 0:
print ("Cost after iteration %i: %f" %(i, cost))
return parameters
# GRADED FUNCTION: predict
def predict(parameters, 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.
A2, cache = forward_propagation(X, parameters)
predictions = np.array([0 if i <= 0.5 else 1 for i in np.squeeze(A2)])
return predictions
2.5.8使用神经网络训练planar dataset.
# Build a model with a n_h-dimensional hidden layer
parameters = nn_model(X, Y, n_h = 4, num_iterations = 10000, print_cost=True)
# 绘出决策边界
plot_decision_boundary(lambda x: predict(parameters, x.T), X, Y[0, :])
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) + '%')
训练结果如下：
Cost after iteration 0: 0.693048
Cost after iteration 500: 0.304722
Cost after iteration 1000: 0.288083
Cost after iteration 1500: 0.278875
Cost after iteration 2000: 0.254385
Cost after iteration 2500: 0.240464
Cost after iteration 3000: 0.233864
Cost after iteration 3500: 0.229744
Cost after iteration 4000: 0.226792
Cost after iteration 4500: 0.224505
Cost after iteration 5000: 0.222644
Cost after iteration 5500: 0.221079
Cost after iteration 6000: 0.219731
Cost after iteration 6500: 0.218550
Cost after iteration 7000: 0.217504
Cost after iteration 7500: 0.216570
Cost after iteration 8000: 0.219496
Cost after iteration 8500: 0.218996
Cost after iteration 9000: 0.218569
Cost after iteration 9500: 0.218181
Accuracy: 90%
可视化图片如下：
与Logistic回归相比，准确性得到了提高。模型已经学会了花的叶子图案！与Logistic回归不同，神经网络能够学习甚至高度非线性的决策边界。
本文内容编辑：邵光辉