机器学习入门-支持向量机

基本定义

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给定一个训练样本集，分类学习最基本的想法就是基于训练集Ｄ在样本空间中找到一个划分超平面，将不同的类别的样本分开。这个划分超平面越鲁棒，对未见示例的泛化能力就越强。划分超平面可通过如下线性方程来描述。

$$W^{T}X+b=0$$ 其中$w=(w_{1};w_{2};...;w_{d})$是法向量，决定了超平面的方向。ｂ是位移项，决定了超平面与远点的距离。显然划分超平面可以被法向量ｗ和位移ｂ确定，将其记为$(w,b)$.样本空间中任一点ｘ到超平面$(w,b)$的距离可以写为 $$r=\frac{|W^{T}X+b|}{||w||}$$ 假设其能正确的分类，则对于$(x_{i},y_{i})\in D$,若$y_{i}=+1$,则有$w^{T}x_{i}+b>0$，若$y_{i}=－1$,则有$w^{T}x_{i}+b<0$，令 $$\begin{cases} & w^{T}x_{i}+b\geq +1 \text{ if } y_{i}=+1 \\ & w^{T}x_{i}+b\leq -1 \text{ if } y_{i}=-1 \end{cases}$$ 距离超平面最近的点使得等号成立，它们被称为“支持向量”，两个异类支持向量到超平面的距离之和为 $$\gamma =\frac{2}{||w||}$$ 它被称为间隔（ｍａｒｇｉｎ），优化目标则是找到具有最大间隔的划分超平面，因此ＳＶＭ也被称为最大间隔分类。因此优化目标可以具体化为 $$\underset{w,b}{min} \frac{1}{2}||w||^{2}$$ $$s.t. y_{i}(w^{T}x_{i}+b)\geq1,i=1,2,...m.$$ 这就是支持向量机的基本型。

对偶问题

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对于SVM的基本型，可以使用拉格朗日乘子法的到其对偶类型。
下面将对在多个约束的情况下的对偶问题进行一定的推倒。考虑具有ｍ个等式和ｎ个不等式的约束。

$$\underset{x}{min} f(x)$$ $$s.t. h_{i}(x)=0(i=1,...m)$$ $$s.t. g_{j}(x)\leq0(j=1,...m)$$ 引入拉格朗日乘子$\lambda=(\lambda_{1},\lambda_{2},...,\lambda_{m})^{T}$和$\mu=(\mu{1},\mu{2},...,\mu_{n})^{T}$,相应的拉格朗日函数为 $$L(x,\lambda,\mu)=f(x)+\sum_{i=1}^{m}\lambda_{i}h_{i}(x)+\sum_{j=1}^{n}\mu_{j}g_{j}(x)$$ 由不等式引入的ＫＫＴ条件为 $$\begin{cases} & g_{j}(x) \leq0 \text{ ; }\\ & \mu_{j}\geq0 \text{ ; } \\ & \mu_{j}g_{j}(x)=0\text{ ; } \end{cases}$$ 对于该主问题，设其对偶问题为$\tau(\lambda,\mu)=inf(f(x)+\sum_{i=1}^{m}\lambda_{i}h_{i}(x)+\sum_{j=1}^{n}\mu_{j}g_{j}(x))$
由上述条件显然$\sum_{i=1}^{m}\lambda_{i}h_{i}(x)+\sum_{j=1}^{n}\mu_{j}g_{j}(x)\leq0$
因此有 $$\tau(\lambda,\mu)\leq f(\tilde{x})$$ 因此对偶函数给出了主问题最优值的下界。很自然的一个问题是基于对偶函数能获得的最好下界是什么？这就引出了优化问题 $$\underset{\lambda,\mu}{max}\tau(\lambda,\mu) s.t. \mu\succeq0$$ 这就是主问题的对偶问题了。

下面考虑原问题。该问题的拉格朗日函数可以写为

$$L(w,b,\alpha)=\frac{1}{2}||w||^{2}+\sum_{i=1}^{m}\alpha_{i}(1-y_{i}(w^{T}x_{i}+b))$$ 对ｗ和ｂ偏导为０可以得到 $$w=\sum_{i=1}^{m}\alpha_{i}y_{i}x_{i}$$ $$\sum_{i=1}^{m}\alpha_{i}y_{i}=0$$ 从而得到对偶问题： $$\underset{\alpha}{max}\sum_{i=1}^{m}\alpha_{i}-\frac{1}{2}\sum_{i=1}^{m}\sum_{j=1}^{m}\alpha_{i}\alpha_{j}y_{i}y_{j}x_{i}^{T}x_{j}$$ $$s.t.\sum_{i=1}^{m}\alpha_{i}y_{i}=0,\alpha_{i}\geq0(i=1,2,...m)$$ 解出$\alpha$后，求出ｗ和ｂ即可得到模型 $$f(x)=\sum_{i=1}^{m}\alpha_{i}y_{i}x_{i}^{T}x+b$$
那么应该如何计算$\alpha$呢？

SMO算法

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ＳＭＯ的基本思路是先固定$\alpha_i$之外的所有参数，然后求$\alpha_i$上的极值。由于存在约束$\sum_{i=1}^{m}\alpha_{i}y_{i}=0$若固定$\alpha_i$之外的其他变量，则$\alpha_i$可由其它变量打出。于是每次选择两个变量$\alpha_i$和$\alpha_j$并固定其它参数。不断执行如下步骤直至收敛。

• 选取一对更新的变量$\alpha_i$、$\alpha_j$
• 固定$\alpha_i$和$\alpha_j$以外的参数，求解更新后的$\alpha_i$和$\alpha_j$。

SMO首先选取违背ＫＫＴ条件程度最大的变量。选取的第二个变量则应该使对应样本间的间隔最大。
对于ｂ则使用所有支持向量求解的平均值。

核函数

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对于某些原始样本空间内不存在一个能正确划分两类样本的超平面的问题，可将样本从原始空间映射到一个更高维的特征空间，使得样本在这个特征空间内线性可分。令$\phi(x)$表示将$x$映射后的特征向量，于是在特征空间内划分超平面所对应的模型可表示为

$$f(x)=w^{T}\phi(x)+b$$ 剩余推导都和上文类似，除了在计算维度上有一定难度上的提高。
对于映射有一些基本的经验，例如对于文本数据通常采用线性核，情况不明可以先采取高斯核。

• 线性核：　　$\kappa(x_{i},x_{j})=x_{i}^{T}x_j$
• 高斯核：　　$\kappa(x_{i},x_{j})=e^{-\frac{||x_{i}-x_{j}||^{2}}{2\sigma^{2}}}$

实战

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from numpy import *
from time import sleep

def loadDataSet(filename):
    dataMat=[]
    labelMat=[]
    with open(filename,'r') as fr:
        for line in fr.readlines():
            lineArr=line.strip().split('\t')
            dataMat.append([float(lineArr[0]),float(lineArr[1])])
            labelMat.append(float(lineArr[2]))
    return dataMat,labelMat

def selectJrand(i,m):
    j=i
    while(j==i):
        j=int(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):
    dataMatrix=mat(dataMatIn)
    labelMat=mat(classLabels).transpose()
    b=0;m,n=shape(dataMatrix)
    alphas=mat(zeros((m,1)))
    iter=0
    while(iter<maxIter):
        alphaPairsChanged=0
        for i in range(m):
            fXi=float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[i,:].T))+b
            Ei=fXi-float(labelMat[i])
            if(((labelMat[i]*Ei<-toler)and(alphas[i]<C))or((labelMat[i]*Ei>toler)and(alphas[i]>0))):
                j=selectJrand(i,m)
                fXj=float(multiply(alphas,labelMat).T*(dataMatrix*dataMatrix[j,:].T))+b
                Ej=fXj-float(labelMat[j])
                alphaIold=alphas[i].copy()
                alphaJold=alphas[j].copy()
                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
                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
                alphas[j]-=labelMat[j]*(Ei-Ej)/eta
                alphas[j]=clipAlpha(alphas[j],H,L)
                if(abs(alphas[j]-alphaJold)<0.00001):
                    print('j not moving enough')
                    continue
                alphas[i]+=labelMat[j]*labelMat[i]*(alphaJold-alphas[j])
                b1 = b - Ei- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[i,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[i,:]*dataMatrix[j,:].T
                b2 = b - Ej- labelMat[i]*(alphas[i]-alphaIold)*dataMatrix[i,:]*dataMatrix[j,:].T - labelMat[j]*(alphas[j]-alphaJold)*dataMatrix[j,:]*dataMatrix[j,:].T
                if (0 < alphas[i]) and (C > alphas[i]): b = b1
                elif (0 < alphas[j]) and (C > alphas[j]): b = b2
                else: b = (b1 + b2)/2.0
                alphaPairsChanged += 1
                print ("iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
        if (alphaPairsChanged == 0): iter += 1
        else: iter = 0
        print ("iteration number: %d" % iter)
    return b,alphas

def kernelTrans(X, A, kTup): #calc the kernel or transform data to a higher dimensional space
    m,n = shape(X)
    K = mat(zeros((m,1)))
    if kTup[0]=='lin': K = X * A.T   #linear kernel
    elif kTup[0]=='rbf':
        for j in range(m):
            deltaRow = X[j,:] - A
            K[j] = deltaRow*deltaRow.T
        K = exp(K/(-1*kTup[1]**2)) #divide in NumPy is element-wise not matrix like Matlab
    else: raise NameError('Houston We Have a Problem -- \
    That Kernel is not recognized')
    return K

class optStruct:
    def __init__(self,dataMatIn, classLabels, C, toler, kTup):  # Initialize the structure with the parameters 
        self.X = dataMatIn
        self.labelMat = classLabels
        self.C = C
        self.tol = toler
        self.m = shape(dataMatIn)[0]
        self.alphas = mat(zeros((self.m,1)))
        self.b = 0
        self.eCache = mat(zeros((self.m,2))) #first column is valid flag
        self.K = mat(zeros((self.m,self.m)))
        for i in range(self.m):
            self.K[:,i] = kernelTrans(self.X, self.X[i,:], kTup)

def calcEk(oS,k):
    fXk=float(multiply(oS.alphas,oS.labelMat).T*(oS.X*oS.X[k,:].T))+oS.b
    Ek=fXk-float(oS.labelMat[k])
    return Ek

def selectJ(i, oS, Ei):         #this is the second choice -heurstic, and calcs Ej
    maxK = -1; maxDeltaE = 0; Ej = 0
    oS.eCache[i] = [1,Ei]  #set valid #choose the alpha that gives the maximum delta E
    validEcacheList = nonzero(oS.eCache[:,0].A)[0]
    if (len(validEcacheList)) > 1:
        for k in validEcacheList:   #loop through valid Ecache values and find the one that maximizes delta E
            if k == i: continue #don't calc for i, waste of time
            Ek = calcEk(oS, k)
            deltaE = abs(Ei - Ek)
            if (deltaE > maxDeltaE):
                maxK = k; maxDeltaE = deltaE; Ej = Ek
        return maxK, Ej
    else:   #in this case (first time around) we don't have any valid eCache values
        j = selectJrand(i, oS.m)
        Ej = calcEk(oS, j)
    return j, Ej

def updateEk(oS, k):#after any alpha has changed update the new value in the cache
    Ek = calcEk(oS, k)
    oS.eCache[k] = [1,Ek]

def innerL(i, oS):
    Ei = calcEk(oS, i)
    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)):
        j,Ej = selectJ(i, oS, Ei) #this has been changed from selectJrand
        alphaIold = oS.alphas[i].copy(); alphaJold = oS.alphas[j].copy();
        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
        eta = 2.0 * oS.K[i,j] - oS.K[i,i] - oS.K[j,j] #changed for kernel
        if eta >= 0: print ("eta>=0"); return 0
        oS.alphas[j] -= oS.labelMat[j]*(Ei - Ej)/eta
        oS.alphas[j] = clipAlpha(oS.alphas[j],H,L)
        updateEk(oS, j) #added this for the Ecache
        if (abs(oS.alphas[j] - alphaJold) < 0.00001): print ("j not moving enough"); return 0
        oS.alphas[i] += oS.labelMat[j]*oS.labelMat[i]*(alphaJold - oS.alphas[j])#update i by the same amount as j
        updateEk(oS, i) #added this for the Ecache                    #the update is in the oppostie direction
        b1 = oS.b - Ei- oS.labelMat[i]*(oS.alphas[i]-alphaIold)*oS.K[i,i] - oS.labelMat[j]*(oS.alphas[j]-alphaJold)*oS.K[i,j]
        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]
        if (0 < oS.alphas[i]) and (oS.C > oS.alphas[i]): oS.b = b1
        elif (0 < oS.alphas[j]) and (oS.C > oS.alphas[j]): 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)):    #full Platt SMO
    oS = optStruct(mat(dataMatIn),mat(classLabels).transpose(),C,toler, kTup)
    iter = 0
    entireSet = True; alphaPairsChanged = 0
    while (iter < maxIter) and ((alphaPairsChanged > 0) or (entireSet)):
        alphaPairsChanged = 0
        if entireSet:   #go over all
            for i in range(oS.m):        
                alphaPairsChanged += innerL(i,oS)
                print ("fullSet, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
            iter += 1
        else:#go over non-bound (railed) alphas
            nonBoundIs = nonzero((oS.alphas.A > 0) * (oS.alphas.A < C))[0]
            for i in nonBoundIs:
                alphaPairsChanged += innerL(i,oS)
                print ("non-bound, iter: %d i:%d, pairs changed %d" % (iter,i,alphaPairsChanged))
            iter += 1
        if entireSet: entireSet = False #toggle entire set loop
        elif (alphaPairsChanged == 0): entireSet = True  
        print ("iteration number: %d" % iter)
    return oS.b,oS.alphas   
def calcWs(alphas,dataArr,classLabels):
    X = mat(dataArr); labelMat = mat(classLabels).transpose()
    m,n = shape(X)
    w = zeros((n,1))
    for i in range(m):
        w += multiply(alphas[i]*labelMat[i],X[i,:].T)
    return w

def testRbf(k1=1.3):
    dataArr,labelArr = loadDataSet('testSetRBF.txt')
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, ('rbf', k1)) #C=200 important
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]
    sVs=datMat[svInd] #get matrix of only support vectors
    labelSV = labelMat[svInd];
    print ("there are %d Support Vectors" % shape(sVs)[0])
    m,n = shape(datMat)
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the training error rate is: %f" % (float(errorCount)/m))
    dataArr,labelArr = loadDataSet('testSetRBF2.txt')
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],('rbf', k1))
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1    
    print ("the test error rate is: %f" % (float(errorCount)/m))

def img2vector(filename):
    returnVect = zeros((1,1024))
    fr = open(filename)
    for i in range(32):
        lineStr = fr.readline()
        for j in range(32):
            returnVect[0,32*i+j] = int(lineStr[j])
    return returnVect

def loadImages(dirName):
    from os import listdir
    hwLabels = []
    trainingFileList = listdir(dirName)           #load the training set
    m = len(trainingFileList)
    trainingMat = zeros((m,1024))
    for i in range(m):
        fileNameStr = trainingFileList[i]
        fileStr = fileNameStr.split('.')[0]     #take off .txt
        classNumStr = int(fileStr.split('_')[0])
        if classNumStr == 9: hwLabels.append(-1)
        else: hwLabels.append(1)
        trainingMat[i,:] = img2vector('%s/%s' % (dirName, fileNameStr))
    return trainingMat, hwLabels    

def testDigits(kTup=('rbf', 10)):
    dataArr,labelArr = loadImages('trainingDigits')
    b,alphas = smoP(dataArr, labelArr, 200, 0.0001, 10000, kTup)
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    svInd=nonzero(alphas.A>0)[0]
    sVs=datMat[svInd] 
    labelSV = labelMat[svInd]
    print ("there are %d Support Vectors" % shape(sVs)[0])
    m,n = shape(datMat)
    errorCount = 0
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1
    print ("the training error rate is: %f" % (float(errorCount)/m))
    dataArr,labelArr = loadImages('testDigits')
    errorCount = 0
    datMat=mat(dataArr); labelMat = mat(labelArr).transpose()
    m,n = shape(datMat)
    for i in range(m):
        kernelEval = kernelTrans(sVs,datMat[i,:],kTup)
        predict=kernelEval.T * multiply(labelSV,alphas[svInd]) + b
        if sign(predict)!=sign(labelArr[i]): errorCount += 1    
    print ("the test error rate is: %f" % (float(errorCount)/m))