logistic回归详解（四）：梯度下降训练逻辑回归python实现
1.逻辑回归梯度下降的迭代公式
在参考文献1中，我们推导出了逻辑回归的参数迭代公式为:
接下来我们按照这个思路，用python来手动实现以下逻辑回归算法。
2.准备数据
-0.017612 14.053064 0
-1.395634 4.662541 1
-0.752157 6.538620 0
-1.322371 7.152853 0
0.423363 11.054677 0
0.406704 7.067335 1
0.667394 12.741452 0
-2.460150 6.866805 1
0.569411 9.548755 0
-0.026632 10.427743 0
0.850433 6.920334 1
1.347183 13.175500 0
1.176813 3.167020 1
-1.781871 9.097953 0
-0.566606 5.749003 1
0.931635 1.589505 1
-0.024205 6.151823 1
-0.036453 2.690988 1
-0.196949 0.444165 1
1.014459 5.754399 1
1.985298 3.230619 1
-1.693453 -0.557540 1
-0.576525 11.778922 0
-0.346811 -1.678730 1
-2.124484 2.672471 1
1.217916 9.597015 0
-0.733928 9.098687 0
-3.642001 -1.618087 1
0.315985 3.523953 1
1.416614 9.619232 0
-0.386323 3.989286 1
0.556921 8.294984 1
1.224863 11.587360 0
-1.347803 -2.406051 1
1.196604 4.951851 1
0.275221 9.543647 0
0.470575 9.332488 0
-1.889567 9.542662 0
-1.527893 12.150579 0
-1.185247 11.309318 0
-0.445678 3.297303 1
1.042222 6.105155 1
-0.618787 10.320986 0
1.152083 0.548467 1
0.828534 2.676045 1
-1.237728 10.549033 0
-0.683565 -2.166125 1
0.229456 5.921938 1
-0.959885 11.555336 0
0.492911 10.993324 0
0.184992 8.721488 0
-0.355715 10.325976 0
-0.397822 8.058397 0
0.824839 13.730343 0
1.507278 5.027866 1
0.099671 6.835839 1
-0.344008 10.717485 0
1.785928 7.718645 1
-0.918801 11.560217 0
-0.364009 4.747300 1
-0.841722 4.119083 1
0.490426 1.960539 1
-0.007194 9.075792 0
0.356107 12.447863 0
0.342578 12.281162 0
-0.810823 -1.466018 1
2.530777 6.476801 1
1.296683 11.607559 0
0.475487 12.040035 0
-0.783277 11.009725 0
0.074798 11.023650 0
-1.337472 0.468339 1
-0.102781 13.763651 0
-0.147324 2.874846 1
0.518389 9.887035 0
1.015399 7.571882 0
-1.658086 -0.027255 1
1.319944 2.171228 1
2.056216 5.019981 1
-0.851633 4.375691 1
-1.510047 6.061992 0
-1.076637 -3.181888 1
1.821096 10.283990 0
3.010150 8.401766 1
-1.099458 1.688274 1
-0.834872 -1.733869 1
-0.846637 3.849075 1
1.400102 12.628781 0
1.752842 5.468166 1
0.078557 0.059736 1
0.089392 -0.715300 1
1.825662 12.693808 0
0.197445 9.744638 0
0.126117 0.922311 1
-0.679797 1.220530 1
0.677983 2.556666 1
0.761349 10.693862 0
-2.168791 0.143632 1
1.388610 9.341997 0
0.317029 14.739025 0
原始数据为文本，共三列，其中第一列第二列分别为两个特征，第三列为类别。
3.处理原始数据
首先从文本中将数据读入，变成我们想要的训练集
def parse_data():
data = np.loadtxt('data.csv')
dataMat = data[:, 0:-1]
classLabels = data[:, -1]
dataMat = np.insert(dataMat, 0, 1, axis=1)
return dataMat, classLabels
注意dataMat = np.insert(dataMat, 0, 1, axis=1)这一行的作用是插入偏置项。原始数据集中是二维特征，加入偏置项以后变成了三维。
4.定义sigmoid函数与loss function
接下来定义sigmoid 函数与loss function
def sigmoid(x):
return 1 / (1 + np.exp(-x))
def loss_funtion(dataMat, classLabels, weights):
m, n = np.shape(dataMat)
loss = 0.0
for i in range(m):
sum_theta_x = 0.0
for j in range(n):
sum_theta_x += dataMat[i, j] * weights.T[0, j]
propability = sigmoid(sum_theta_x)
loss += -classLabels[i, 0] * np.log(propability) - (1 - classLabels[i, 0]) * np.log(1 - propability)
return loss
注意loss function的公式，可以从参考文献2中查找详细推导过程。
5.定义梯度下降过程
定义梯度下降过程，并且画出迭代过程loss function的变化曲线。完整代码如下
import numpy as np
import matplotlib.pyplot as plt
def sigmoid(x):
return 1 / (1 + np.exp(-x))
def parse_data():
data = np.loadtxt('data.csv')
dataMat = data[:, 0:-1]
classLabels = data[:, -1]
dataMat = np.insert(dataMat, 0, 1, axis=1)
return dataMat, classLabels
def loss_funtion(dataMat, classLabels, weights):
m, n = np.shape(dataMat)
loss = 0.0
for i in range(m):
sum_theta_x = 0.0
for j in range(n):
sum_theta_x += dataMat[i, j] * weights.T[0, j]
propability = sigmoid(sum_theta_x)
loss += -classLabels[i, 0] * np.log(propability) - (1 - classLabels[i, 0]) * np.log(1 - propability)
return loss
def grad_descent(dataMatIn, classLabels):
dataMatrix = np.mat(dataMatIn) #(m,n)
labelMat = np.mat(classLabels).T
m, n = np.shape(dataMatrix)
weights = np.ones((n, 1))
alpha = 0.01
maxstep = 10000
eps = 0.0001
count = 0
loss_array = []
for i in range(maxstep):
loss = loss_funtion(dataMatrix, labelMat, weights)
h_theta_x = sigmoid(dataMatrix * weights)
e = h_theta_x - labelMat
new_weights = weights - alpha * dataMatrix.T * e
new_loss = loss_funtion(dataMatrix, labelMat, new_weights)
loss_array.append(new_loss)
if abs(new_loss - loss) < eps:
break
else:
weights = new_weights
count += 1
print "count is: ", count
print "loss is: ", loss
print "weights is: ", weights
return weights, loss_array
def plotloss(loss_array):
n = len(loss_array)
plt.xlabel("iteration num")
plt.ylabel("loss")
plt.scatter(range(1, n+1), loss_array)
plt.show()
data, labels = parse_data()
r, loss_array = grad_descent(data, labels)
r = np.mat(r).transpose()
plotloss(loss_array)
将代码run起来以后，输出如下：
count is: 797
loss is: 9.393884167889501
weights is: [[13.11270842]
[ 1.1410689 ]
[-1.78438114]]
得到的图像如下:
可以看出在迭代过程中，loss function还是比较稳定地变小的。
参考文献
1.https://blog.****.net/bitcarmanlee/article/details/51473567
2.https://blog.****.net/bitcarmanlee/article/details/51165444