支持向量机 (SVM)，序列最小优化算法 (SMO)

支持向量机(Support Vector Machine)由V.N. Vapnik，A.Y. Chervonenkis，C. Cortes 等在1964年提出。序列最小优化算法（Sequential minimal optimization）是一种用于解决支持向量机训练过程中所产生优化问题的算法。由John C. Platt于1998年提出。

支持向量机的推导在西瓜书，各大网站已经有详细的介绍。本文主要依据 John C. Platt 发表的文章《Sequential Minimal Optimization: A Fast Algorithm for Training Support Vector Machines》来实现 SVM 与 SMO 算法。

import numpy as np
from sklearn import datasets
import matplotlib.pyplot as plt

定义需要的数据，包含数据样本，数据标签，偏置 b，拉格朗日乘子 α，容忍系数 C 等。

class Par:
	def __init__(self,n,D,C,eps,tol):
		self.X=datasets.make_blobs(n_samples=n,n_features=D,centers=2,cluster_std=1.0,shuffle=True,random_state=None)
		self.point=self.X[0]
		self.target=self.X[1]
		self.target[np.nonzero(self.target==0)[0]]=-1
		self.w=np.zeros((1,D))[0]
		self.b=0
		self.E=-self.target
		self.alpha=np.zeros((1,n))[0]
		self.n=n
		self.C=C
		self.eps=eps
		self.tol=tol

定义核函数，预测公式。

def kernel(x,y):
	return np.dot(x,y.T)

def f(x):
	s=0
	for i in range(n):
		s+=P.alpha[i]*P.target[i]*kernel(P.point[i],x)
	return s-P.b

被选中的一对 α 更新细节：

def takeStep(i1,i2):
	if i1==i2:
		return 0
	alph2=P.alpha[i2]
	alph1=P.alpha[i1]
	y1=P.target[i1]
	y2=P.target[i2]
	s=y1*y2
	#Compute L,H via equations (13) and (14)
	if y1!=y2:
		L=max(0,alph2-alph1)
		H=min(P.C,P.C+alph2-alph1)
	else:
		L=max(0,alph2+alph1-P.C)
		H=min(P.C,alph2+alph1)
	if L==H:
		return 0
	k11=kernel(P.point[i1],P.point[i1])
	k12=kernel(P.point[i1],P.point[i2])
	k22=kernel(P.point[i2],P.point[i2])
	eta=k11+k22-2*k12
	if eta>0:
		a2=alph2+y2*(P.E[i1]-P.E[i2])/eta
	if a2<L:
		a2=L
	elif a2>H:
		a2=H
	else:
		f1=y1*(P.E[i1]+b)-alph1*k11-s*alph2*k12
		f2=y2*(P.E[i2]+b)-s*alph1*k12-alph2*k22
		L1=alph1+s*(alph2-L)
		H1=alph1+s*(alph2+H)
		psiL=L1*f1+L*f2+0.5*L1**2*k11+0.5*L**2*k22+s*L*L1*k12
		psiH=H1*f1+H*f2+0.5*H1**2*k11+0.5*H**2*k22+s*H*H1*k12
		Lobj = psiL
		Hobj = psiH
	if Lobj<Hobj-eps:
		a2=L
	elif Lobj>Hobj+eps:
		a2=H
	else:
		a2=alph2
	if abs(a2-alph2)<P.eps*(a2+alph2+P.eps):
		return 0
	a1=alph1+s*(alph2-a2)
	#Update threshold to reflect change in Lagrange multipliers
	b1=P.E[i1]+y1*(a1-alph1)*k11+y2*(a2-alph2)*k12+P.b
	b2=P.E[i2]+y1*(a1-alph1)*k12+y2*(a2-alph2)*k22+P.b
	if a1>0 and a1<P.C:
		P.b=b1
	elif a2>0 and a2<P.C:
		P.b=b2
	else:
		P.b=(b1+b2)/2
	#Update weight vector to reflect change in a1 & a2, if SVM is linear
	P.w=P.w+y1*(a1-alph1)*P.point[i1]+y2*(a2-alph2)*P.point[i2]
	#Store a1 in the alpha array
	P.alpha[i1]=a1
	#Store a2 in the alpha array
	P.alpha[i2]=a2
	#Update error cache using new Lagrange multipliers
	P.E[i1]=f(P.point[i1])-P.target[i1]
	P.E[i2]=f(P.point[i2])-P.target[i2]
	return 1

内循环选择第二个 α：

def examineExample(i2):
	global valid
	alph2=P.alpha[i2]
	y2=P.target[i2]
	r2=P.E[i2]*y2
	if (r2<-P.tol and alph2<P.C) or (r2>P.tol and alph2>0):
		valid=np.where((P.alpha!=0) & (P.alpha!=C))[0]
		Long=len(valid)
		if Long > 1:
			#i1 = result of second choice heuristic (section 2.2)
			best=-1
			if len(valid)>1:
				for k in valid:
					deltaE=abs(P.E[i2]-P.E[k])
					if deltaE>best:
						best=deltaE
						i1=k
					if takeStep(i1,i2):
						return 1
		#loop over all non-zero and non-C alpha, starting at a random point
		if Long>0:
			random_index=np.random.randint(0,Long)
			for i in np.hstack((valid[random_index:Long],valid[0:random_index])):
				i1=i
				if takeStep(i1,i2):
					return 1
		#loop over all possible i1, starting at a random point
		random_index=np.random.randint(0,n)
		for i in np.hstack((np.arange(random_index,n),np.arange(0,random_index))):
			#i1=loop variable
			i1=i
			if takeStep(i1,i2):
				return 1
	return 0

外循环选择第一个 α：

def SMO():
	global valid
	numChanged=0
	examineAll=1
	while numChanged>0 or examineAll:
		numChanged=0
		if examineAll:
			for i in range(n):
				numChanged+=examineExample(i)
		else:
			#loop I over examples where alpha is not 0 & not C
			for i in valid:
				numChanged+=examineExample(i)
		if examineAll==1:
			examineAll=0
		elif numChanged==0:
			examineAll=1

主函数入口：

if __name__ == '__main__':
	n=100       #样本个数
	C=10         
	eps=0.001  #停止精度
	tol=0.001   #分类容错率
	D=2            #样本维度
	P=Par(n,D,C,eps,tol) 
	SMO()
	#绘制图像
	plt.scatter(P.point[:,0],P.point[:,1],c=P.target)
	x=np.arange(-10,10,0.1)
	y=(P.b-P.w[0]*x)/P.w[1]
	plt.plot(x,y)
	plt.show()
	Y=kernel(P.point,P.w)-P.b
	count=0
	for i in range(n):
	if Y[i]*P.target[i]<0:
	count+=1
	print('Error Point num:',count)

单次测试结果：