纵观SMO算法，其核心是怎么选择每轮优化的两个拉格朗日乘子，标准的SMO算法是通过判断乘子是否违反原问题的KKT条件来选择待优化乘子的，这里可能有一个问题，回顾原问题的KKT条件：
是否违反它，与这几个因素相关：拉格朗日乘子、样本标记
、偏置
。
的更新依赖于两个优化拉格朗日乘子，这就可能出现这种情况：拉格朗日乘子
已经能使目标函数达到最优，而SMO算法本身并不能确定当前由于两个优化拉格朗日乘子计算得到的
是否就是使目标函数达到最优的那个
，换句话说，对一些本来不违反KKT条件的点，由于上次迭代选择了不合适的
，使得它们出现违反KKT条件的情况，导致后续出现一些耗时而无用的搜索，针对标准SMO的缺点，出现了以下几种改进方法，它们同样是通过KKT条件来选择乘子，不同之处为使用对偶问题的KKT条件。
1、通过Maximal Violating Pair选择优化乘子
原问题的对偶问题指的是：
它的拉格朗日方程可以写为：
定义，则对偶问题的KKT条件为：
总结以下可分为3种情况：
分类标记可以取1或-1，按照正类和负类区分指标集，
、
、
、
、
，如图：
图一
整理一下就得到KKT条件的新形式：
从图上也可以看到，当分类器对全部样本都分类正确的时候，必有：
像标准SMO一样，要精确达到最优值显然也是没必要的，因此这里也需要加个容忍值。
引入容忍值后的KKT条件为：
根据前面的说明可以定义“违反对”：
那么我们到底要优化哪两个拉格朗日乘子呢？由“违反对”的定义可知，和
差距最大的那两个点最违反KKT条件的，描述为：
于是就是一个“Maximal Violating Pair，一个示意图如下，其中
和
就是一对MVP：
2、通过Second Order Information选择优化乘子
不论是通过一阶信息还是二阶信息来选择优化乘子，都是基于以下这个定理：
定理一：如果矩阵为半正定矩阵，当且仅当待优化乘子为“违反对”时，对于SMO类型的算法，其目标函数严格递减(如：
)。
First Order Information选择方法我就不介绍了，直接看Second Order Information选择方法：
当选定一个乘子后，另一个乘子选择的条件是使得当前“乘子对”为“违反对”且能使目标函数值最小，在最优化迭代类型方法中非常重要的一个工具就是函数的taylor展开式，优化的过程可以概括为一系列选搜索方向、选搜索步长的过程，对目标函数w展开得：
，这里d分别为两个待选择乘子的优化方向。
于是最小化就变成了：
由优化条件可知：
这样选择乘子的过程就变成了以下优化问题：
总结通过Second Order Information选择优化乘子的方法如下：
1、首先选择一个乘子，条件如下：
2、选择另外一个乘子的过程就是求解的过程，设
为第二个乘子，则有：
将条件：
带入有：
于是目标函数在处达到最小值：
用式子表示就是：
当这种情况出现时可以用一个很小的值来代替它，具体可见《A study on SMO-type decomposition methods for support vector machines》一文中“Non-Positive Definite Kernel Matrices”小节。
4、算法实现
将代码加入到了LeftNotEasy的pymining项目中了，可以从http://code.google.com/p/python-data-mining-platform/下载。
几点说明：
1、训练和测试数据格式为：value1 value2 value3 …… valuen，label=+1，数据集放在***.data中，对应的label放在***.labels中，也可以使用libsvm的数据集；
2、分类器目前支持的核为RBF、Linear、Polynomial、Sigmoid、将来支持String Kernel；
3、训练集和测试集的输入支持dense matrix 和 sparse matrix，其中sparse matrix采用CSR表示法；
4、对于不平衡数据的处理一般来说从三个方面入手：
1)、对正例和负例赋予不同的C值，例如正例远少于负例，则正例的C值取得较大，这种方法的缺点是可能会偏离原始数据的概率分布；
2)、对训练集的数据进行预处理即对数量少的样本以某种策略进行采样，增加其数量或者减少数量多的样本，典型的方法如：随机插入法，缺点是可能出现
overfitting，较好的是：Synthetic Minority Over-sampling TEchnique(SMOTE)，其缺点是只能应用在具体的特征空间中，不适合处理那些无法用
特征向量表示的问题，当然增加样本也意味着训练时间可能增加；
3)、基于核函数的不平衡数据处理。
本文就以容易实现为原则，采用第一种方式，配置文件中的节点<c>中的值代表的是正例的代价值，模型中的npRatio用来计算负例的代价值，例如正例远多于负例，则这个比率就大于一，反之亦然，原理如下图所示：
和
异号且
的情形(
的情况类似)
和
同号的情形
5、算法最耗时的地方是优化乘子的选择和更新一阶导数信息，这两个地方都需要去计算核函数值，而核函数值的计算最终都需要去做内积运算，这就意味着原始空间的维度很高会增加内积运算的时间；对于dense matrix我就直接用numpy的dot了，而sparse matrix采用的是CSR表示法，求它的内积我实验过的方法有三种，第一种不需要额外空间，但时间复杂度为O(nlgn)，第二种需要一个hash表(用dictionary代替了)，时间复杂度为线性，第三种需要一个bitmap(使用BitVector)，时间复杂度也为线性，实际使用中第一种速度最快，我就暂时用它了，应该还有更快的方法，希望高人们能指点一下；另外由于使用dictionary缓存核矩阵，遇到训练数据很大的数据集很容易挂掉，所以在程序中，当dictionary的内存占用达到配置文件的阈值时会将其中相对次要的元素删掉，保留对角线上的内积值。
6、使用psyco
相对于c，python在进行高维度数值计算时表现的会比较差，为了加快数值计算的速度，我使用了psyco来进行加速，它的使用很简单，加两句话就行，用程序测试一下使用psyco之前和之后的表现：
定义一个1164维的向量，代码如下：
2: import psyco
3: psyco.full()
4:
5: def dot():
6: vec 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7: sum = 0.0
for i in range(len(vec)):
9: sum += vec[i] * vec[i]
return sum
11:
:
13: start = time.clock()
14:
for i in range(100000):
16: dot()
17:
'\n'
19:
20: end = time.clock()
'seconds.'