SVM学习笔记5-SMO

首先拿出最后要求解的问题：$\underset{\alpha}{min}W(\alpha)=\frac{1}{2} \sum_{i,j=1}^{n}y^{(i)}y^{(j)}\alpha_{i}\alpha_{j}k_{ij}-\sum_{i=1}^{n}\alpha_{i}$,使得满足:
(1)$0 \leq \alpha_{i}\leq C,1 \leq i \leq n$
(2)$\sum_{i=1}^{n}\alpha_{i}y^{(i)}=0$

求解的策略是每次选出两个$\alpha$进行更新。比如选取的是$\alpha_{1},\alpha_{2}$。由于$\sum_{i=1}^{n}\alpha_{i}y^{(i)}=0$，所以$\alpha_{1}y^{(1)}+\alpha_{2}y^{(2)}=-\sum_{i=3}^{n}\alpha_{i}y^{(i)}$。等号右侧是一个常数，设为$\xi$。当$y^{(1)}$和$y^{(2)}$异号时，有$\alpha_{1}-\alpha_{2}=\xi$或者$\alpha_{2}-\alpha_{1}=\xi$。同时它们还要满足$0\leq \alpha \leq C$。

我们设$\alpha_{2}$的合法区间为[L,R],那么此时有$L=max(0,\alpha_{2}-\alpha_{1}),R=min(C,C+\alpha_{2}-\alpha_{1})$。同理当$y^{(1)}$和$y^{(2)}$同号时有$L=max(0,\alpha_{2}+\alpha_{1}-C),R=min(C,\alpha_{2}+\alpha_{1})$。

首先定义$u=w^{T}x+b$

将$\alpha_{1},\alpha_{2}$带入$W(\alpha)$中得到：
$W(\alpha)=\frac{1}{2}(k_{11}\alpha_{1}^{2}+k_{22}\alpha_{2}^{2})+sk_{12}\alpha_{1}\alpha_{2}+y^{(1)}\alpha_{1}v_{1}+y^{(2)}\alpha_{2}v_{2}-\alpha_{1}-\alpha_{2}+P$

其中:
(1)$s=y^{(1)}y^{(2)}$
(2)$v_{1}=\sum_{i=3}^{n}y^{(i)}\alpha_{i}k_{1i}=u_{1}-b-y^{(1)}\alpha_{1}^{old}k_{11}-y^{(2)}\alpha_{2}^{old}k_{12}$
(3)$v_{2}=\sum_{i=3}^{n}y^{(i)}\alpha_{i}k_{2i}=u_{2}-b-y^{(1)}\alpha_{1}^{old}k_{12}-y^{(2)}\alpha_{2}^{old}k_{22}$

由于$y^{(1)}\alpha_{1}+y^{(2)}\alpha_{2}=y^{(1)}\alpha_{1}^{old}+y^{(2)}\alpha_{2}^{old}=\xi$
两边同时乘以$y^{(1)}$,得到$\alpha_{1}+s\alpha_{2}=\alpha_{1}^{old}+s\alpha_{2}^{old}=y^{(1)}\xi=T$

所以$\alpha_{1}=T-s\alpha_{2}$，将其带入$W(\alpha)$，得到$W(\alpha)=\frac{1}{2}(k_{11}(T-s\alpha_{2})^{2}+k_{22}\alpha_{2}^{2})+sk_{12}(T-s\alpha_{2})\alpha_{2}+y^{(1)}(T-s\alpha_{2})v_{1}+y^{(2)}\alpha_{2}v_{2}-(T-s\alpha_{2})-\alpha_{2}+P$

其实这是一个关于$\alpha_{2}$的二次函数，在一阶导数等于0的地方取得最小值，一阶导数为：$\frac{\partial W}{\partial \alpha_{2}}=-sk_{11}(T-s\alpha_{2})+k_{22}\alpha_{2}+sk_{12}(T-s\alpha_{2})-k_{12}\alpha_{2}-y^{(2)}v_{1}+y^{(2)}v_{2}+s-1=0$

移项得：$\alpha_{2}(k_{11}+k_{22}-2k_{12})=s(k_{11}-k_{12})T+y^{(2)}(v_{1}-v_{2})+1-s$

将$v_{1},v_{2}$带入得：$\alpha_{2}(k_{11}+k_{22}-2k_{12})=\alpha_{2}^{old}(k_{11}+k_{22}-2k_{12})+y^{(2)}(u_{1}-u_{2}+y^{(2)}-y^{(1)})$

令:
(1)$\eta =k_{11}+k_{22}-2k_{12}$
(2)$E_{1}=u_{1}-y^{(1)}$
(3)$E_{2}=u_{2}-y^{(2)}$

那么有：
$\alpha_{2}^{new}=\alpha_{2}^{old}+\frac{y^{(2)}(E_{1}-E_{2})}{\eta }$

这里就求出了新的$\alpha_{2}$。需要注意的是如果$\alpha_{2}$不在上面求出的[L,R]区间，要做一下裁剪。

由$y^{(1)}\alpha_{1}+y^{(2)}\alpha_{2}=y^{(1)}\alpha_{1}^{old}+y^{(2)}\alpha_{2}^{old}$可得：
$\alpha_{1}^{new}=\alpha_{1}^{old}+y^{(1)}y^{(2)}(\alpha_{2}^{old}-\alpha_{2}^{new})$

最后更新b
$b=\left\{\begin{matrix} b_{1} & 0<\alpha_{1}<C\\
b_{2} & 0<\alpha_{2}<C\\
\frac{1}{2}(b_{1}+b_{2}) & other
\end{matrix}\right.$

其中
$b_{1}=b-E_{1}-y^{(1)}(\alpha_{1}^{new}-\alpha_{1}^{old})k_{11}-y^{(2)}(\alpha_{2}^{new}-\alpha_{2}^{old})k_{12}$
$b_{2}=b-E_{2}-y^{(1)}(\alpha_{1}^{new}-\alpha_{1}^{old})k_{12}-y^{(2)}(\alpha_{2}^{new}-\alpha_{2}^{old})k_{22}$

这样更新b会迫使输入$x_{1}$时输出$y^{(1)}$,输入$x_{2}$时输出$y^{(2)}$

from numpy import *
import operator
from time import sleep
import numpy as np;
from svmplatt import *;
import matplotlib.pyplot as plt 

class PlattSVM(object):
        def __init__(self):
                self.X = []
                self.labelMat = []
                self.C = 0.0
                self.tol = 0.0
                self.b = 0.0
                self.kValue=0.0
                self.maxIter=10000
                self.svIndx=[]
                self.sptVects=[]
                self.SVlabel=[] 

        def loadDataSet(self,fileName):
                fr = open(fileName)
                for line in fr.readlines():
                        lineArr = line.strip().split('\t')
                        self.X.append([float(lineArr[0]), float(lineArr[1])])
                        self.labelMat.append(float(lineArr[2]))
                self.initparam()        

        def kernels(self,dataMat,A):
                m,n=shape(dataMat)
                K=mat(zeros((m,1)))
                for j in range(m):
                        delta=dataMat[j,:]-A
                        K[j]=delta*delta.T
                K=exp(K/-1*self.kValue**2)
                return K

        def initparam(self):
                self.X = mat(self.X)
                self.labelMat = mat(self.labelMat).T
                self.m = shape(self.X)[0]
                self.lambdas = mat(zeros((self.m,1)))
                self.eCache = mat(zeros((self.m,2)))
                self.K = mat(zeros((self.m,self.m)))
                for i in range(self.m):
                        self.K[:,i] = self.kernels(self.X,self.X[i,:])

        def randJ(self,i):
                j=i
                while(j==i):
                        j = int(random.uniform(0,self.m))
                return j

        def clipLambda(self,aj,H,L):
                if aj > H: aj = H
                if L > aj: aj = L
                return aj

        def calcEk(self,k):
                return float(multiply(self.lambdas,self.labelMat).T*self.K[:,k] + self.b) - float(self.labelMat[k])

        def chooseJ(self,i,Ei):
                maxK = -1; maxDeltaE = 0; Ej = 0
                self.eCache[i] = [1,Ei]
                validEcacheList = nonzero(self.eCache[:,0].A)[0]
                if (len(validEcacheList)) > 1:
                        for k in validEcacheList:
                                if k == i: continue
                                Ek = self.calcEk(k)
                                deltaE = abs(Ei - Ek)
                                if (deltaE > maxDeltaE):
                                        maxK = k; maxDeltaE = deltaE; Ej = Ek
                        return maxK, Ej
                else:
                                j = self.randJ(i)
                                Ej = self.calcEk(j)
                return j, Ej

        def innerLoop(self,i):
                Ei = self.calcEk(i) 

                if ((self.labelMat[i]*Ei < -self.tol) and (self.lambdas[i] < self.C)) or ((self.labelMat[i]*Ei > self.tol) and (self.lambdas[i] > 0)):
                        j,Ej = self.chooseJ(i, Ei)
                        lambdaIold = self.lambdas[i].copy(); lambdaJold = self.lambdas[j].copy();
                        if (self.labelMat[i] != self.labelMat[j]):
                                L = max(0, self.lambdas[j] - self.lambdas[i])
                                H = min(self.C, self.C + self.lambdas[j] - self.lambdas[i])
                        else:
                                L = max(0, self.lambdas[j] + self.lambdas[i] - self.C)
                                H = min(self.C, self.lambdas[j] + self.lambdas[i])
                        if L==H:        return 0
                        eta = 2.0 * self.K[i,j] - self.K[i,i] - self.K[j,j]
                        if eta >= 0:    return 0
                        self.lambdas[j] -= self.labelMat[j]*(Ei - Ej)/eta
                        self.lambdas[j] = self.clipLambda(self.lambdas[j],H,L)
                        self.eCache[j] = [1,self.calcEk(j)]
                        if (abs(self.lambdas[j] - lambdaJold) < 0.00001):       return 0
                        self.lambdas[i] += self.labelMat[j]*self.labelMat[i]*(lambdaJold - self.lambdas[j])
                        self.eCache[i] = [1,self.calcEk(i)]    

                        b1 = self.b - Ei- self.labelMat[i]*(self.lambdas[i]-lambdaIold)*self.K[i,i] - self.labelMat[j]*(self.lambdas[j]-lambdaJold)*self.K[i,j]
                        b2 = self.b - Ej- self.labelMat[i]*(self.lambdas[i]-lambdaIold)*self.K[i,j] - self.labelMat[j]*(self.lambdas[j]-lambdaJold)*self.K[j,j]

                        if (0 < self.lambdas[i]) and (self.C > self.lambdas[i]): self.b = b1
                        elif (0 < self.lambdas[j]) and (self.C > self.lambdas[j]): self.b = b2
                        else: self.b = (b1 + b2)/2.0;
                        return 1
                else: return 0

        def train(self):    #full Platt SMO
                step = 0
                entireflag = True; lambdaPairsChanged = 0 

                while (step < self.maxIter) and ((lambdaPairsChanged > 0) or (entireflag)):
                        lambdaPairsChanged = 0
                        if entireflag:
                                for i in range(self.m):
                                        lambdaPairsChanged += self.innerLoop(i)
                                step += 1
                        else:
                                nonBoundIs = nonzero((self.lambdas.A > 0) * (self.lambdas.A < self.C))[0]                                  for i in nonBoundIs:                                         lambdaPairsChanged += self.innerLoop(i)                                 step += 1                         if entireflag: entireflag = False                          elif (lambdaPairsChanged == 0): entireflag = True                   self.svIndx = nonzero(self.lambdas.A>0)[0]
                self.sptVects = self.X[self.svIndx]
                self.SVlabel = self.labelMat[self.svIndx]      

        def scatterplot(self,plt):
                fig = plt.figure()
                ax = fig.add_subplot(111)
                for i in range(shape(self.X)[0]):
                        if self.lambdas[i] != 0:
                                ax.scatter(self.X[i,0],self.X[i,1],c='green',marker='s',s=50)
                        elif self.labelMat[i] == 1:
                                ax.scatter(self.X[i,0],self.X[i,1],c='blue',marker='o')
                        elif self.labelMat[i] == -1:
                                ax.scatter(self.X[i,0],self.X[i,1],c='red',marker='o')

svm = PlattSVM()
svm.C=70
svm.tol=0.001
svm.maxIter=2000
svm.kValue= 3.0
svm.loadDataSet('nolinear.txt')

svm.train()

print(svm.svIndx)
print(shape(svm.sptVects)[0])
print("b:",svm.b)

svm.scatterplot(plt)
plt.show()

　

实验结果