基于matlab的SMO实现

这是我在机器学习课程上的作业，用matlab实现的SMO，记录一下体会。

我实现了简化版SMO代码，网络上流传的大部分也都是这种思路的代码，主要参考了Peter《机器学习实战》中关于SMO算法的部分。感谢yqx老师。

我自己编写的简化版代码：

function [alpha,bias] = my_seqminoptSimple(training,groupIndex,C,maxIter,tol)

% init
[sampleNum,featuerNum]=size(training);
alpha=zeros(sampleNum,1);
bias=0;
iteratorTimes=0;

K=training*training';
while iteratorTimes<maxIter
%iteratorTimes=iteratorTimes+1;
alphaPairsChanged=0;
% calculate predict value
%K=training*training';
%g=(alpha.*groupIndex)'*K+repmat(bias,1,sampleNum);
%g=g';

% calculate error
%E=g-groupIndex;

% find alpha1
for i=1:sampleNum
g1=(alpha.*groupIndex)'*(training*training(i,:)')+bias;
E1=g1-groupIndex(i,1);
% choose i: avoid KKT conditions
if(((E1*groupIndex(i,1)<-tol)&&alpha(i,1)<C)||((E1*groupIndex(i,1)>tol)&&alpha(i,1)>0))
% choose j: different from i
j=i;
while j==i
j=randi(sampleNum);
end

alpha1=i;
alpha2=j;

% updata alpha1 and alpha2
alpha1Old=alpha(alpha1,1);
alpha2Old=alpha(alpha2,1);
y1=groupIndex(alpha1,1);
y2=groupIndex(alpha2,1);

g2=(alpha.*groupIndex)'*(training*training(j,:)')+bias;
E2=g2-groupIndex(j,1);

if y1~=y2
L=max(0,alpha2Old-alpha1Old);
H=min(C,C+alpha2Old-alpha1Old);
else
L=max(0,alpha2Old+alpha1Old-C);
H=min(C,alpha2Old+alpha1Old);
end

if L==H
fprintf('H==L\n');
continue;
end

parameter=K(alpha1,alpha1)+K(alpha2,alpha2)-2*K(alpha1,alpha2);

if parameter<=0
fprintf('parameter<=0\n');
continue;
end

alpha2New=alpha2Old+y2*(E1-E2)/parameter;

if alpha2New>H
alpha2New=H;
end

if alpha2New<L
alpha2New=L;
end

if abs(alpha2New-alpha2Old)<=0.0001
fprintf('change small\n');
continue;
end

alpha1New=alpha1Old+y1*y2*(alpha2Old-alpha2New);

% updata bias
bias1=-E1-y1*K(alpha1,alpha1)*(alpha1New-alpha1Old)-y2*K(alpha2,alpha1)*(alpha2New-alpha2Old)+bias;
bias2=-E2-y1*K(alpha1,alpha2)*(alpha1New-alpha1Old)-y2*K(alpha2,alpha2)*(alpha2New-alpha2Old)+bias;

if alpha1New>0&&alpha1New<C
bias=bias1;
elseif alpha2New>0&&alpha2New<C
bias=bias2;
else
bias=(bias2+bias1)/2;
end

alpha(alpha1,1)=alpha1New;
alpha(alpha2,1)=alpha2New;
alphaPairsChanged=alphaPairsChanged+1;
end
end

if alphaPairsChanged==0
iteratorTimes=iteratorTimes+1;
else
iteratorTimes=0;
end
fprintf('iteratorTimes=%d\n',iteratorTimes);

end

《机器学习实战》部分简化版SMO代码，感谢Peter的代码。

'''
Created on Nov 4, 2010
Chapter 5 source file for Machine Learing in Action
@author: Peter
'''
from numpy import *
from time import sleep

def loadDataSet(fileName):
dataMat = []; labelMat = []
fr = open(fileName)
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 #we want to select any J not equal to 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 checks if an example violates KKT conditions
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])#update i by the same amount as j
#the update is in the oppostie direction
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

dataArr,labelArr=loadDataSet('testSet.txt')
b,alphas=smoSimple(dataArr, labelArr, 0.6, 0.001, 40)

SMO算法的基本思想就是对于还有大量变量的优化问题不好求解，我们就采用比较简单的思路：每次只更新两个变量的值，找到一个较好的解。

SMO理论不再重复，需要的可以参考传送门，这是一个去年选了这门课的一个师兄总结的，感谢。

我这里主要讨论一下我对SMO的体会吧。简化版SMO和原始SMO主要的区别在于两个更新变量选取。

思路一

选取违反KKT最大的变量α1，再选择更新值|E1-E2|最大的α2。但是简单的这么操作更新几次后就卡死了，看来SMO并没有如此简单。

思路二

选取违反KKT最严重的α1，再随机选一个α2。

思路三

遍历的选一个α1，再随机选一个α2。

这种简化的SMO就是选择思路三，而且必须加上很多的限制项，防止陷入一对选取的值然后就卡死不动了。具体的限制请参考SMO代码。

对于高级用户来说，想要实现完整版的SMO，请参考《机器学习实战》第6章的内容。