[此篇文章介绍关于SVM中的一些不懂的地方的公式推导，以及代码实现和一些SVM问题，通过做题检验掌握的效果。
]
一、代码实现
[调用sklearn包，进行SVM分类
]
#!/usr/bin/python
# -*- coding utf-8 -*-
import numpy as np
import matplotlib.pyplot as plt
import pandas as pd
import matplotlib as mpl
from sklearn import svm
from sklearn.model_selection import train_test_split
from sklearn.metrics import accuracy_score
def load_data():
path = 'E:数据挖掘Machine learning[小象学院]机器学习课件8.Regression代码8.Regressioniris.data'
# 读取文件路径
data = pd.read_csv(path, header = None)
# 从data 读取数据, x为前4列的所有数据, y为第5列数据
x, y = data[range(4)], data[4]
# 返回字符类别的位置索引, 因y数组包含三类, 对应返回下标值
y = pd.Categorical(y).codes
# 取x的前两列数据, 一般SVM只做二特征分类, 多特征的转化为多个二特征分类再bagging?
x = x[[0, 1]]
# x = x[[0 ,2]]
return x, y
def classifier(x,y):
# 鸢尾花包含四个特征属性, 包含三类标签, 山鸢尾(0), 变色鸢尾(1), 维吉尼亚鸢尾(2)
iris_feature = u'花萼长度', u'花萼宽度', u'花瓣长度', u'花瓣宽度'
# 按 0.6 的比例，test_data 占40%， train_data 占60%, random_state随机数的种子, 1为产生相同随机数, 产生不同随机数
x_train, x_test, y_train, y_test = train_test_split(x, y, random_state=1, train_size=0.6)
# 使用SVM进行分类训练, 包含关键字, C, gamma, kernel
# kernel='linear'时，为线性核，C越大分类效果越好, kernel= 'rbf' 时（default）, 为高斯核
# gamma值越小，分类界面越连续；gamma值越大，分类界面越“散”，分类效果越好
# decision_function_shape = 'ovr' 时，为one vs rest, 即一个类别与其他类别进行划分，decision_function_shape = 'ovo'
# 为one vs one,即将类别两两之间进行划分，用二分类的方法模拟多分类的结果
clf = svm.SVC(C=0.8, kernel='rbf', gamma=20, decision_function_shape='ovr')
clf.fit(x_train, y_train.ravel())
# score函数返回返回该次预测的系数R2, 在（0, 1)之间、accuracy_score指的是分类准确率，即分类正确占所有分类的百分比
# recall_score 召回率 = 提取出的正确信息条数 / 样本中的信息条数
print(clf.score(x_train, y_train))
print('训练集准确率：', accuracy_score(y_train, clf.predict(x_train)))
print(clf.score(x_test, y_test))
print('测试集准确率：', accuracy_score(y_test, clf.predict(x_test)))
# decision_function()的功能: 计算样本点到分割超平面的函数距离, 每一列的值代表距离各类别的距离
print('decision_function:
', clf.decision_function(x_train))
print('
predict:
', clf.predict(x_train))
# 画图
x1_min, x2_min = x.min() # 第0列的范围
x1_max, x2_max = x.max() # 第1列的范围
x1, x2 = np.mgrid[x1_min:x1_max:500j, x2_min:x2_max:500j] # 生成网格采样点
grid_test = np.stack((x1.flat, x2.flat), axis=1) # 测试点
# print 'grid_test =
', grid_test
# Z = clf.decision_function(grid_test) # 样本到决策面的距离
# print Z
grid_hat = clf.predict(grid_test) # 预测分类值
grid_hat = grid_hat.reshape(x1.shape) # 使之与输入的形状相同
mpl.rcParams['font.sans-serif'] = [u'SimHei']
mpl.rcParams['axes.unicode_minus'] = False
cm_light = mpl.colors.ListedColormap(['#A0FFA0', '#FFA0A0', '#A0A0FF'])
cm_dark = mpl.colors.ListedColormap(['g', 'r', 'b'])
plt.figure(facecolor='w')
plt.pcolormesh(x1, x2, grid_hat, cmap=cm_light)
plt.scatter(x[0], x[1], c=y, edgecolors='k', s=50, cmap=cm_dark) # 样本
plt.scatter(x_test[0], x_test[1], s=120, facecolors='none', zorder=10) # 圈中测试集样本
plt.xlabel(iris_feature[0], fontsize=13)
plt.ylabel(iris_feature[1], fontsize=13)
plt.xlim(x1_min, x1_max)
plt.ylim(x2_min, x2_max)
plt.title(u'鸢尾花SVM二特征分类', fontsize=16)
plt.grid(b=True, ls=':')
plt.tight_layout(pad=1.5)
plt.show()
if __name__ == "__main__":
x, y = load_data()
classifier(x, y)
[SMO算法
]
# -*- coding: utf-8 -*-
import numpy as np
import matplotlib.pyplot as plt
def loadDataSet(fileName):
# 数据矩阵
dataMat = []
# 标签向量
labelMat = []
# 打开文件
fr = open(fileName)
# 逐行读取
for line in fr.readlines():
# 去掉每一行首尾的空白符，例如'
','
',' ',' '
# 将每一行内容根据' '符进行切片
lineArr = line.strip().split(' ')
# 添加数据(100个元素排成一行)
dataMat.append([float(lineArr[0]), float(lineArr[1])])
# 添加标签(100个元素排成一行)
labelMat.append(float(lineArr[2]))
return dataMat, labelMat
def selectJrand(i, m):
# i为第一个alpha的下标，m是所有alpha的数目
j = i
while (j == i):
# uniform()方法将随机生成一个实数，它在[x, y)范围内
j = int(np.random.uniform(0, m))
return j
def clipAlpha(aj, H, L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def smoSimple(dataMatIn, classLabels, C, toler, maxIter):
# 转换为numpy的mat矩阵存储(100,2)
dataMatrix = np.mat(dataMatIn)
# 转换为numpy的mat矩阵存储并转置(100,1)
labelMat = np.mat(classLabels).transpose()
# 初始化b参数，统计dataMatrix的维度,m:行；n:列
b = 0
# 统计dataMatrix的维度,m:100行；n:2列
m, n = np.shape(dataMatrix)
# 初始化alpha参数，设为0
alphas = np.mat(np.zeros((m, 1)))
# 初始化迭代次数
iter_num = 0
# 最多迭代maxIter次
while (iter_num < maxIter):
alphaPairsChanged = 0
for i in range(m):
# 步骤1：计算误差Ei
# multiply(a,b)就是个乘法，如果a,b是两个数组，那么对应元素相乘
# .T为转置
fxi = float(np.multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[i, :].T)) + b
# 误差项计算公式
Ei = fxi - float(labelMat[i])
# 优化alpha，设定一定的容错率
if ((labelMat[i] * Ei < -toler) and (alphas[i] < C)) or ((labelMat[i] * Ei > toler) and (alphas[i] > 0)):
# 随机选择另一个alpha_i成对比优化的alpha_j
j = selectJrand(i, m)
# 步骤1，计算误差Ej
fxj = float(np.multiply(alphas, labelMat).T * (dataMatrix * dataMatrix[j, :].T)) + b
# 误差项计算公式
Ej = fxj - float(labelMat[j])
# 保存更新前的alpha值，使用深拷贝(完全拷贝)A深层拷贝为B，A和B是两个独立的个体
alphaIold = alphas[i].copy()
alphaJold = alphas[j].copy()
# 步骤2：计算上下界H和L
if (labelMat[i] != labelMat[j]):
L = max(0, alphas[j] - alphas[i])
H = min(C, C + alphas[j] - alphas[i])
else:
L = max(0, alphas[j] + alphas[i] - C)
H = min(C, alphas[j] + alphas[i])
if (L == H):
print("L == H")
continue
# 步骤3：计算eta
eta = 2.0 * dataMatrix[i, :] * dataMatrix[j, :].T - dataMatrix[i, :] * dataMatrix[i, :].T - dataMatrix[
j,
:] * dataMatrix[
j, :].T
if eta >= 0:
print("eta>=0")
continue
# 步骤4：更新alpha_j
alphas[j] -= labelMat[j] * (Ei - Ej) / eta
# 步骤5：修剪alpha_j
alphas[j] = clipAlpha(alphas[j], H, L)
if (abs(alphas[j] - alphaJold) < 0.00001):
print("alpha_j变化太小")
continue
# 步骤6：更新alpha_i
alphas[i] += labelMat[j] * labelMat[i] * (alphaJold - alphas[j])
# 步骤7：更新b_1和b_2
b1 = b - Ei - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[i, :].T - labelMat[
j] * (alphas[j] - alphaJold) * dataMatrix[j, :] * dataMatrix[i, :].T
b2 = b - Ej - labelMat[i] * (alphas[i] - alphaIold) * dataMatrix[i, :] * dataMatrix[j, :].T - labelMat[
j] * (alphas[j] - alphaJold) * dataMatrix[j, :] * dataMatrix[j, :].T
# 步骤8：根据b_1和b_2更新b
if (0 < alphas[i] < C):
b = b1
elif (0 < alphas[j] < C):
b = b2
else:
b = (b1 + b2) / 2.0
# 统计优化次数
alphaPairsChanged += 1
# 打印统计信息
print("第%d次迭代 样本：%d， alpha优化次数：%d" % (iter_num, i, alphaPairsChanged))
# 更新迭代次数
if (alphaPairsChanged == 0):
iter_num += 1
else:
iter_num = 0
print("迭代次数：%d" % iter_num)
return b, alphas
def get_w(dataMat, labelMat, alphas):
alphas, dataMat, labelMat = np.array(alphas), np.array(dataMat), np.array(labelMat)
# 我们不知道labelMat的shape属性是多少，
# 但是想让labelMat变成只有一列，行数不知道多少，
# 通过labelMat.reshape(1, -1)，Numpy自动计算出有100行，
# 新的数组shape属性为(100, 1)
# np.tile(labelMat.reshape(1, -1).T, (1, 2))将labelMat扩展为两列(将第1列复制得到第2列)
# dot()函数是矩阵乘，而*则表示逐个元素相乘
# w = sum(alpha_i * yi * xi)
w = np.dot((np.tile(labelMat.reshape(1, -1).T, (1, 2)) * dataMat).T, alphas)
return w.tolist()
def showClassifer(dataMat, w, b):
# 正样本
data_plus = []
# 负样本
data_minus = []
for i in range(len(dataMat)):
if labelMat[i] > 0:
data_plus.append(dataMat[i])
else:
data_minus.append(dataMat[i])
# 转换为numpy矩阵
data_plus_np = np.array(data_plus)
# 转换为numpy矩阵
data_minus_np = np.array(data_minus)
# 正样本散点图（scatter）
# transpose转置
plt.scatter(np.transpose(data_plus_np)[0], np.transpose(data_plus_np)[1], s=30, alpha=0.7)
# 负样本散点图（scatter）
plt.scatter(np.transpose(data_minus_np)[0], np.transpose(data_minus_np)[1], s=30, alpha=0.7)
# 绘制直线
x1 = max(dataMat)[0]
x2 = min(dataMat)[0]
a1, a2 = w
b = float(b)
a1 = float(a1[0])
a2 = float(a2[0])
y1, y2 = (-b - a1 * x1) / a2, (-b - a1 * x2) / a2
plt.plot([x1, x2], [y1, y2])
# 找出支持向量点
# enumerate在字典上是枚举、列举的意思
for i, alpha in enumerate(alphas):
# 支持向量机的点
if (abs(alpha) > 0):
x, y = dataMat[i]
plt.scatter([x], [y], s=150, c='none', alpha=0.7, linewidth=1.5, edgecolors='red')
plt.show()
if __name__ == '__main__':
dataMat, labelMat = loadDataSet('E:\数据挖掘\Machine learning\代码\SVM_Project1\testSet.txt')
b, alphas = smoSimple(dataMat, labelMat, 0.6, 0.001, 40)
w = get_w(dataMat, labelMat, alphas)
showClassifer(dataMat, w, b)
[核函数测试
]
# -*- coding: utf-8 -*-
import matplotlib.pyplot as plt
import numpy as np
import random
class optStruct:
def __init__(self, dataMatIn, classLabels, C, toler, kTup):
# 数据矩阵
self.X = dataMatIn
# 数据标签
self.labelMat = classLabels
# 松弛变量
self.C = C
# 容错率
self.tol = toler
# 矩阵的行数
self.m = np.shape(dataMatIn)[0]
# 根据矩阵行数初始化alphas矩阵，一个m行1列的全零列向量
self.alphas = np.mat(np.zeros((self.m, 1)))
# 初始化b参数为0
self.b = 0
# 根据矩阵行数初始化误差缓存矩阵，第一列为是否有效标志位，第二列为实际的误差E的值
self.eCache = np.mat(np.zeros((self.m, 2)))
# 初始化核K
self.K = np.mat(np.zeros((self.m, self.m)))
# 计算所有数据的核K
for i in range(self.m):
self.K[:, i] = kernelTrans(self.X, self.X[i, :], kTup)
def kernelTrans(X, A, kTup):
# 读取X的行列数
m, n = np.shape(X)
# K初始化为m行1列的零向量
K = np.mat(np.zeros((m, 1)))
# 线性核函数只进行内积
if kTup[0] == 'lin':
K = X * A.T
# 高斯核函数，根据高斯核函数公式计算
elif kTup[0] == 'rbf':
for j in range(m):
deltaRow = X[j, :] - A
K[j] = deltaRow * deltaRow.T
K = np.exp(K / (-1 * kTup[1] ** 2))
else:
raise NameError('核函数无法识别')
return K
def loadDataSet(fileName):
# 数据矩阵
dataMat = []
# 标签向量
labelMat = []
# 打开文件
fr = open(fileName)
# 逐行读取
for line in fr.readlines():
# 去掉每一行首尾的空白符，例如'
','
',' ',' '
# 将每一行内容根据' '符进行切片
lineArr = line.strip().split(' ')
# 添加数据(100个元素排成一行)
dataMat.append([float(lineArr[0]), float(lineArr[1])])
# 添加标签(100个元素排成一行)
labelMat.append(float(lineArr[2]))
return dataMat, labelMat
def calcEk(oS, k):
# multiply(a,b)就是个乘法，如果a,b是两个数组，那么对应元素相乘
# .T为转置
fXk = float(np.multiply(oS.alphas, oS.labelMat).T * oS.K[:, k] + oS.b)
# 计算误差项
Ek = fXk - float(oS.labelMat[k])
# 返回误差项
return Ek
def selectJrand(i, m):
j = i
while (j == i):
# uniform()方法将随机生成一个实数，它在[x, y)范围内
j = int(random.uniform(0, m))
return j
def selectJ(i, oS, Ei):
# 初始化
maxK = -1
maxDeltaE = 0
Ej = 0
# 根据Ei更新误差缓存
oS.eCache[i] = [1, Ei]
# 对一个矩阵.A转换为Array类型
# 返回误差不为0的数据的索引值
validEcacheList = np.nonzero(oS.eCache[:, 0].A)[0]
# 有不为0的误差
if (len(validEcacheList) > 1):
# 遍历，找到最大的Ek
for k in validEcacheList:
# 不计算k==i节省时间
if k == i:
continue
# 计算Ek
Ek = calcEk(oS, k)
# 计算|Ei - Ek|
deltaE = abs(Ei - Ek)
# 找到maxDeltaE
if (deltaE > maxDeltaE):
maxK = k
maxDeltaE = deltaE
Ej = Ek
# 返回maxK，Ej
return maxK, Ej
# 没有不为0的误差
else:
# 随机选择alpha_j的索引值
j = selectJrand(i, oS.m)
# 计算Ej
Ej = calcEk(oS, j)
# 返回j，Ej
return j, Ej
def updateEk(oS, k):
# 计算Ek
Ek = calcEk(oS, k)
# 更新误差缓存
oS.eCache[k] = [1, Ek]
def clipAlpha(aj, H, L):
if aj > H:
aj = H
if L > aj:
aj = L
return aj
def innerL(i, oS):
# 步骤1：计算误差Ei
Ei = calcEk(oS, i)
# 优化alpha,设定一定的容错率
if ((oS.labelMat[i] * Ei < -oS.tol) and (oS.alphas[i] < oS.C)) or (
(oS.labelMat[i] * Ei > oS.tol) and (oS.alphas[i] > 0)):
# 使用内循环启发方式2选择alpha_j,并计算Ej
j, Ej = selectJ(i, oS, Ei)
# 保存更新前的alpha值，使用深层拷贝
alphaIold = oS.alphas[i].copy()
alphaJold = oS.alphas[j].copy()
# 步骤2：计算上界H和下界L
if (oS.labelMat[i] != oS.labelMat[j]):
L = max(0, oS.alphas[j] - oS.alphas[i])
H = min(oS.C, oS.C + oS.alphas[j] - oS.alphas[i])
else:
L = max(0, oS.alphas[j] + oS.alphas[i] - oS.C)
H = min(oS.C, oS.alphas[j] + oS.alphas[i])
if L == H:
print("L == H")
return 0
# 步骤3：计算eta
eta = 2.0 * oS.K[i, j] - oS.K[i, i] - oS.K[j, j]
if eta >= 0:
print("eta >= 0")
return 0
# 步骤4：更新alpha_j
oS.alphas[j] -= oS.labelMat[j] * (Ei - Ej) / eta
# 步骤5：修剪alpha_j
oS.alphas[j] = clipAlpha(oS.alphas[j], H, L)
# 更新Ej至误差缓存
updateEk(oS, j)
if (abs(oS.alphas[j] - alphaJold) < 0.00001):
print("alpha_j变化太小")
return 0
# 步骤6：更新alpha_i
oS.alphas[i] += oS.labelMat[i] * oS.labelMat[j] * (alphaJold - oS.alphas[j])
# 更新Ei至误差缓存
updateEk(oS, i)
# 步骤7：更新b_1和b_2:
b1 = oS.b - Ei - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, i] - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.K[j, i]
b2 = oS.b - Ej - oS.labelMat[i] * (oS.alphas[i] - alphaIold) * oS.K[i, j] - oS.labelMat[j] * (
oS.alphas[j] - alphaJold) * oS.K[j, j]
# 步骤8：根据b_1和b_2更新b
if (0 < oS.alphas[i] < oS.C):
oS.b = b1
elif (0 < oS.alphas[j] < oS.C):
oS.b = b2
else:
oS.b = (b1 + b2) / 2.0
return 1
else:
return 0
def smoP(dataMatIn, classLabels, C, toler, maxIter, kTup=('lin', 0)):
# 初始化数据结构
oS = optStruct(np.mat(dataMatIn), np.mat(classLabels).transpose(), C, toler, kTup)
# 初始化当前迭代次数
iter = 0
entrieSet = True
alphaPairsChanged = 0
# 遍历整个数据集alpha都没有更新或者超过最大迭代次数，则退出循环
while (iter < maxIter) and ((alphaPairsChanged > 0) or (entrieSet)):
alphaPairsChanged = 0
# 遍历整个数据集
if entrieSet:
for i in range(oS.m):
# 使用优化的SMO算法
alphaPairsChanged += innerL(i, oS)
print("全样本遍历:第%d次迭代 样本:%d, alpha优化次数:%d" % (iter, i, alphaPairsChanged))
iter += 1
# 遍历非边界值
else:
# 遍历不在边界0和C的alpha
nonBoundIs = np.nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
for i in nonBoundIs:
alphaPairsChanged += innerL(i, oS)
print("非边界遍历:第%d次迭代 样本:%d, alpha优化次数:%d" % (iter, i, alphaPairsChanged))
iter += 1
# 遍历一次后改为非边界遍历
if entrieSet:
entrieSet = False
# 如果alpha没有更新，计算全样本遍历
elif (alphaPairsChanged == 0):
entrieSet = True
print("迭代次数:%d" % iter)
# 返回SMO算法计算的b和alphas
return oS.b, oS.alphas
def testRbf(k1=1.3):
# 加载训练集
dataArr, labelArr = loadDataSet('E:\数据挖掘\Machine learning\代码\SVM_Project3\testSetRBF.txt')
# 根据训练集计算b, alphas
b, alphas = smoP(dataArr, labelArr, 200, 0.0001, 100, ('rbf', k1))
datMat = np.mat(dataArr)
labelMat = np.mat(labelArr).transpose()
# 获得支持向量
svInd = np.nonzero(alphas.A > 0)[0]
sVs = datMat[svInd]
labelSV = labelMat[svInd]
print("支持向量个数:%d" % np.shape(sVs)[0])
m, n = np.shape(datMat)
errorCount = 0
for i in range(m):
# 计算各个点的核
kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', k1))
# 根据支持向量的点计算超平面，返回预测结果
predict = kernelEval.T * np.multiply(labelSV, alphas[svInd]) + b
# 返回数组中各元素的正负号，用1和-1表示，并统计错误个数
if np.sign(predict) != np.sign(labelArr[i]):
errorCount += 1
# 打印错误率
print('训练集错误率:%.2f%%' % ((float(errorCount) / m) * 100))
# 加载测试集
dataArr, labelArr = loadDataSet('E:\数据挖掘\Machine learning\代码\SVM_Project3\testSetRBF2.txt')
errorCount = 0
datMat = np.mat(dataArr)
labelMat = np.mat(labelArr).transpose()
m, n = np.shape(datMat)
for i in range(m):
# 计算各个点的核
kernelEval = kernelTrans(sVs, datMat[i, :], ('rbf', k1))
# 根据支持向量的点计算超平面，返回预测结果
predict = kernelEval.T * np.multiply(labelSV, alphas[svInd]) + b
# 返回数组中各元素的正负号，用1和-1表示，并统计错误个数
if np.sign(predict) != np.sign(labelArr[i]):
errorCount += 1
# 打印错误率
print('训练集错误率:%.2f%%' % ((float(errorCount) / m) * 100))
def showDataSet(dataMat, labelMat):
# 正样本
data_plus = []
# 负样本
data_minus = []
for i in range(len(dataMat)):
if labelMat[i] > 0:
data_plus.append(dataMat[i])
else:
data_minus.append(dataMat[i])
# 转换为numpy矩阵
data_plus_np = np.array(data_plus)
# 转换为numpy矩阵
data_minus_np = np.array(data_minus)
# 正样本散点图（scatter）
# transpose转置
plt.scatter(np.transpose(data_plus_np)[0], np.transpose(data_plus_np)[1])
# 负样本散点图（scatter）
plt.scatter(np.transpose(data_minus_np)[0], np.transpose(data_minus_np)[1])
# 显示
plt.show()
if __name__ == '__main__':
testRbf()
二、公式推导
(在样本空间中任意点x到超平面(w,b)的距离可写为：)
[r = frac{|w^Tx + b|}{||w||}
]
[推导如下：\
取x_0为任意点x在超平面y= w^Tx + b的投影\
wx_0 +b = 0 Longrightarrow |wvec {xx_0}| = |wvec r|= ||w||r \
另一方面:|wvec{xx_0}| = |w(x_0 -x)|=|-b-wx|=|b+wx|\
herefore r = frac{|w^Tx + b|}{||w||}]
[hat r=yf(x)=y(w^Tx + b)\
ilde r = ry = yfrac{|w^Tx + b|}{||w||}=frac {hat r}{||w||}\
\定义hat r为函数间隔， ilde r为几何间隔
]
[L(oldsymbol{w}, b, oldsymbol{alpha})=frac{1}{2}|oldsymbol{w}|^{2}+sum_{i=1}^{m} alpha_{i}left(1-y_{i}left(oldsymbol{w}^{ op} oldsymbol{x}_{i}+b
ight)
ight)
]
[原问题为极小极大问题min_{oldsymbol{w,b}}quad max_{oldsymbol{alpha}}quad L(w,b,alpha)\
转化为极大极小问题max_{oldsymbol{alpha}}quad min_{oldsymbol{w,b}}quad L(w,b,alpha)]
[推导如下：\
目标函数：minfrac{1}{2}||w||^2\
约束条件：y_i(w^Tx_i + b) geq 1\
herefore 对每个在y_i(w^Tx_i+b)-1的i乘以alpha_i\
herefore L(oldsymbol{w}, b, oldsymbol{alpha})=frac{1}{2}|oldsymbol{w}|^{2}+sum_{i=1}^{m} alpha_{i}left(1-y_{i}left(oldsymbol{w}^{ op} oldsymbol{x}_{i}+b
ight)
ight)]
[在其他的机器学习上述公式是L(oldsymbol{w}, b, oldsymbol{alpha})=frac{1}{2}|oldsymbol{w}|^{2}-sum_{i=1}^{m} alpha_{i}left(y_{i}left(oldsymbol{w}^{ op} oldsymbol{x}_{i}+b
ight)-1
ight)，两者等价
]
[egin{aligned}
w &= sum_{i=1}^malpha_iy_ioldsymbol{x}_i \
0 &=sum_{i=1}^malpha_iy_i
end{aligned}
]
[推导如下：\
egin{aligned}
L(oldsymbol{w},b,oldsymbol{alpha}) &= frac{1}{2}||oldsymbol{w}||^2+sum_{i=1}^malpha_i(1-y_i(oldsymbol{w}^Toldsymbol{x}_i+b)) \
& = frac{1}{2}||oldsymbol{w}||^2+sum_{i=1}^m(alpha_i-alpha_iy_i oldsymbol{w}^Toldsymbol{x}_i-alpha_iy_ib)\
& =frac{1}{2}oldsymbol{w}^Toldsymbol{w}+sum_{i=1}^malpha_i -sum_{i=1}^malpha_iy_ioldsymbol{w}^Toldsymbol{x}_i-sum_{i=1}^malpha_iy_ib
end{aligned}]
[frac {partial L}{partial oldsymbol{w}}=frac{1}{2} imes2 imesoldsymbol{w} + 0 - sum_{i=1}^{m}alpha_iy_i oldsymbol{x}_i-0= 0 Longrightarrow oldsymbol{w}=sum_{i=1}^{m}alpha_iy_i oldsymbol{x}_i
]
[frac {partial L}{partial b}=0+0-0-sum_{i=1}^{m}alpha_iy_i=0 Longrightarrow sum_{i=1}^{m}alpha_iy_i=0
]
[egin{aligned}
max_{oldsymbol{alpha}} & sum_{i=1}^malpha_i - frac{1}{2}sum_{i = 1}^msum_{j=1}^malpha_i alpha_j y_iy_joldsymbol{x}_i^Toldsymbol{x}_j \
s.t. & sum_{i=1}^m alpha_i y_i =0 \
& alpha_i geq 0 quad i=1,2,dots ,m
end{aligned}
]
(推导如下：\计算拉格朗日函数，即将求得的两个公式代入)
[egin{aligned}
min_{oldsymbol{w},b} L(oldsymbol{w},b,oldsymbol{alpha}) &=frac{1}{2}oldsymbol{w}^Toldsymbol{w}+sum_{i=1}^malpha_i -sum_{i=1}^malpha_iy_ioldsymbol{w}^Toldsymbol{x}_i-sum_{i=1}^malpha_iy_ib \
&=frac {1}{2}oldsymbol{w}^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i-oldsymbol{w}^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_
i -bsum _{i=1}^malpha_iy_i \
& = -frac {1}{2}oldsymbol{w}^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_i -bsum _{i=1}^malpha_iy_i
end{aligned}]
[egin{aligned}
min_{oldsymbol{w},b} L(oldsymbol{w},b,oldsymbol{alpha}) &= -frac {1}{2}oldsymbol{w}^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_i \
&=-frac {1}{2}(sum_{i=1}^{m}alpha_iy_ioldsymbol{x}_i)^T(sum _{i=1}^malpha_iy_ioldsymbol{x}_i)+sum _{i=1}^malpha_i \
&=-frac {1}{2}sum_{i=1}^{m}alpha_iy_ioldsymbol{x}_i^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_i \
&=sum _{i=1}^malpha_i-frac {1}{2}sum_{i=1 }^{m}sum_{j=1}^{m}alpha_ialpha_jy_iy_joldsymbol{x}_i^Toldsymbol{x}_j
end{aligned}]
[egin{aligned}
& min_{oldsymbol{alpha}}frac{1}{2}sum_{i = 1}^msum_{j=1}^malpha_i alpha_j y_iy_joldsymbol{x}_i^Toldsymbol{x}_j- sum_{i=1}^malpha_i\
& s.t. sum_{i=1}^m alpha_i y_i =0 \
& alpha_i geq 0 quad i=1,2,dots ,m
end{aligned}
]
[在原max_{oldsymbol{alpha}}quad min_{oldsymbol{w,b}}quad L(w,b,alpha)加负号，同样转化为约束最优化问题，为了求解最优解alpha^*
]
[计算得到\w^* = sum_{i =1}^m{alpha_i}^*y_ix_i\
b^* = y_i -sum_{i=1}^m{alpha_i}^*y_i（x_ix_j）\
分离得到超平面：\
w^*x+ b^* =0\
分类决策函数：\
f(x) =sign(w^*x+b^*)
]
[引入松弛因子xi_i的目标函数如下:\
]
[egin{aligned}
& min_{oldsymbol{w,b,xi}}frac{1}{2}||w||^2 +Csum_{i = 1}^mxi_i\
s.t. & y_i(w.x_i+b)geq1-xi_i, i=1,2,dots ,m \
& xi_i geq 0 quad i=1,2,dots ,m
end{aligned}
]
[同理如上式，构造拉格朗日函数L，再对w,b,xi分别求偏导，再代入L
]
[egin{aligned}
L(oldsymbol{w},b,oldsymbol{alpha},oldsymbol{xi},oldsymbol{mu}) &= frac{1}{2}||oldsymbol{w}||^2+Csum_{i=1}^m xi_i+sum_{i=1}^m alpha_i(1-xi_i-y_i(oldsymbol{w}^Toldsymbol{x}_i+b))-sum_{i=1}^mmu_i xi_i
end{aligned}
]
[对w,b,xi求偏导
]
[egin{aligned}
w &= sum_{i=1}^malpha_iy_ioldsymbol{x}_i \
0 &=sum_{i=1}^malpha_iy_i\
C & = a_i+mu_i
end{aligned}
]
[代入L
]
[egin{aligned}
min_{oldsymbol{w},b,oldsymbol{xi}}L(oldsymbol{w},b,oldsymbol{alpha},oldsymbol{xi},oldsymbol{mu}) &= frac{1}{2}||oldsymbol{w}||^2+Csum_{i=1}^m xi_i+sum_{i=1}^m alpha_i(1-xi_i-y_i(oldsymbol{w}^Toldsymbol{x}_i+b))-sum_{i=1}^mmu_i xi_i \
&=frac{1}{2}||oldsymbol{w}||^2+sum_{i=1}^malpha_i(1-y_i(oldsymbol{w}^Toldsymbol{x}_i+b))+Csum_{i=1}^m xi_i-sum_{i=1}^m alpha_i xi_i-sum_{i=1}^mmu_i xi_i \
& = -frac {1}{2}sum_{i=1}^{m}alpha_iy_ioldsymbol{x}_i^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_i +sum_{i=1}^m Cxi_i-sum_{i=1}^m alpha_i xi_i-sum_{i=1}^mmu_i xi_i \
& = -frac {1}{2}sum_{i=1}^{m}alpha_iy_ioldsymbol{x}_i^Tsum _{i=1}^malpha_iy_ioldsymbol{x}_i+sum _{i=1}^malpha_i +sum_{i=1}^m (C-alpha_i-mu_i)xi_i \
&=sum _{i=1}^malpha_i-frac {1}{2}sum_{i=1 }^{m}sum_{j=1}^{m}alpha_ialpha_jy_iy_joldsymbol{x}_i^Toldsymbol{x}_j
end{aligned}
]
[再求alpha极大max
]
[egin{aligned}
&max_{alpha}sum _{i=1}^malpha_i-frac {1}{2}sum_{i=1 }^{m}sum_{j=1}^{m}alpha_ialpha_jy_iy_joldsymbol{x}_i^Toldsymbol{x}_j\
转化为\
&min_{alpha}frac {1}{2}sum_{i=1 }^{m}sum_{j=1}^{m}alpha_ialpha_jy_iy_joldsymbol{x}_i^Toldsymbol{x}_j-sum _{i=1}^malpha_i\
&s.t.sum_{i=1}^m alpha_i y_i=0 \
&0 leqalpha_i leq C quad i=1,2,dots ,m
end{aligned}
]
[求最优解alpha^*
]
[计算得到\w^* = sum_{i =1}^m{alpha_i}^*y_ix_i\
b^* = (max_{i: y_i =1} w^*.x_i + min_{i: y_i =-1} w^x* +x_i)/2\
分离得到超平面：\
w^*x+ b^* =0\
分类决策函数：\
f(x) =sign(w^*x+b^*)
]
[left{egin{array}{l}
{alpha_{i}left(fleft(oldsymbol{x}_{i}
ight)-y_{i}-epsilon-xi_{i}
ight)=0} \ {hat{alpha}_{i}left(y_{i}-fleft(oldsymbol{x}_{i}
ight)-epsilon-hat{xi}_{i}
ight)=0} \ {alpha_{i} hat{alpha}_{i}=0, xi_{i} hat{xi}_{i}=0} \
{left(C-alpha_{i}
ight) xi_{i}=0,left(C-hat{alpha}_{i}
ight) hat{xi}_{i}=0}
end{array}
ight.
]
[推导如下：\
]
[left{egin{array}{l}2{fleft(oldsymbol{x}_{i}
ight)-y_{i}-epsilon-xi_{i} leq 0 } \ 3{y_{i}-fleft(oldsymbol{x}_{i}
ight)-epsilon-hat{xi}_{i} leq 0 } \ 4{-xi_{i} leq 0} \5{-hat{xi}_{i} leq 0}6end{array}
ight.
]
[left{egin{array}{l}
{alpha_ileft(fleft(oldsymbol{x}_{i}
ight)-y_{i}-epsilon-xi_{i}
ight) = 0 } \
{hat{alpha}_ileft(y_{i}-fleft(oldsymbol{x}_{i}
ight)-epsilon-hat{xi}_{i}
ight) = 0 } \
{-mu_ixi_{i} = 0 Rightarrow mu_ixi_{i} = 0 } \
{-hat{mu}_i hat{xi}_{i} = 0 Rightarrow hat{mu}_i hat{xi}_{i} = 0 }
end{array}
ight.
]
[ecauseegin{aligned}
mu_i=C-alpha_i \
hat{mu}_i=C-hat{alpha}_i
end{aligned}]
[left{egin{array}{l}
{alpha_ileft(fleft(oldsymbol{x}_{i}
ight)-y_{i}-epsilon-xi_{i}
ight) = 0 } \
{hat{alpha}_ileft(y_{i}-fleft(oldsymbol{x}_{i}
ight)-epsilon-hat{xi}_{i}
ight) = 0 } \
{(C-alpha_i)xi_{i} = 0 } \
{(C-hat{alpha}_i) hat{xi}_{i} = 0 }
end{array}
ight.
]
[前面硬间隔与软间隔均处理线性问题，而对非线性问题需要将低维空间映射到高维空间，引入核函数
]
[多项式核函数
]
[k(vec x,vec y)= (vec x,vec y +c)^2\
=(vec x, vec y)^2+2cvec xvec y+c^2\
=sum_{i =1}^n sum_{j=1}^m(x_ix_j)(y_iy_j)+sum_{i=1}^m(sqrt {2c}x_i sqrt{2cy_i})+c^2
]
(高斯核函数)
[k(vec x_1,vec x_2) = e^-frac{x_1^2+x_2^2}{2sigma^2}(1+frac {x_1 x_2}{sigma^2}+frac{x_1^2+x_2^2}{2sigma^2 sigma^2}+...+frac{x_1^n+x_2^n}{n!sigma^nsigma^n})
]
三、练习题目
3.1 给定三个数据点，正例点(x_1 = (3, 3)^T), (x_2 = (4, 3)^T),负例点(x_3 = (1, 1)^T)，求线性可分SVM
3.2 SVM可否用于多分类
3.3 SVM和Logistic回归的比较
3.4 核函数是什么？高斯核映射到无穷维是怎么回事？
3.5 如何理解SVM的损失函数？
3.6 使用高斯核函数，请描述SVM的参数C和 (sigma) 对分类器的影响
3.7 比较感知机的对偶性形式与线性可分支持向量机的对偶形式
3-8 证明内积的正整数幂函数：
[K(x,z) = (x,z)^p\
是正定核函数，此处p为正整数，x,z为R
]
3.9 线性支持向量机还可定义为以下形式：
[egin{aligned}
min_{oldsymbol{w,b,xi}}quad frac{1}{2}||w||^2+Csum_{i=1}^N{xi_i}^2\
s.t.quad y_i(oldsymbol w.{x_i}+b)geq 1-xi_i, i= 1,2,...,N\
{xi}_i geq 0, i=1, 2,...,N
end{aligned}\
求其对偶形式
]
3.10 给定数据点，正例点(x_1 = (1, 2)^T), (x_2 = (2, 3)^T), (x_3 = (1, 1)^T,)负例点(x_4 = (1, 1)^T),(x_5 = (1, 1)^T),求最大间隔分离超平面和分类决策函数，并在图上画出分离超平面，间隔边界及支持向量
3.11 分析SVM对噪声敏感的原因
3.12 使用核技巧推广对数几率回归，产生核对率回归
3.13 给出式（6.52）的KKT条件
3.13 讨论线性判别分析与线性核支持向量机在何种条件等价
四、参看文献
[1] 《机器学习》 邹博
[2] 《SVM的三重境界》 July
[3] 《pumpkin-book》 Datawhale
[4] 《机器学习》周志华
[5] 《机器学习实战》Peter
[6] 《统计学习方法》李航，清华大学出版社，2012
[7] 《机器学习算法精讲》 秦曾昌，2018