支持向量机1

男娘i 2022-06-08 11:19 192阅读 0赞

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##### 1 简介 #####

支持向量机基本上是最好的有监督学习算法了。最开始接触SVM是去年暑假的时候，老师要求交《统计学习理论》的报告，那时去网上下了一份入门教程，里面讲的很通俗，当时只是大致了解了一些相关概念。这次斯坦福提供的学习材料，让我重新学习了一些SVM知识。我看很多正统的讲法都是从VC 维理论和结构风险最小原理出发，然后引出SVM什么的，还有些资料上来就讲分类超平面什么的。这份材料从前几节讲的logistic回归出发，引出了SVM，既揭示了模型间的联系，也让人觉得过渡更自然。

##### 2 重新审视logistic回归 #####

Logistic回归目的是从特征学习出一个0/1分类模型，而这个模型是将特性的线性组合作为自变量，由于自变量的取值范围是负无穷到正无穷。因此，使用logistic函数（或称作sigmoid函数）将自变量映射到(0,1)上，映射后的值被认为是属于y=1的概率。

形式化表示就是

假设函数

[![clip\_image001][clip_image001]][clip_image001_clip_image001]

其中x是n维特征向量，函数g就是logistic函数。

[![clip\_image002][clip_image002]][clip_image002_clip_image002]的图像是

[![clip\_image003][clip_image003]][clip_image003_clip_image003]

可以看到，将无穷映射到了(0,1)。

而假设函数就是特征属于y=1的概率。

[![clip\_image004][clip_image004]][clip_image004_clip_image004]

当我们要判别一个新来的特征属于哪个类时，只需求[![clip\_image006][clip_image006]][clip_image006_clip_image006]，若大于0.5就是y=1的类，反之属于y=0类。

再审视一下[![clip\_image006\[1\]][clip_image006_1]][clip_image006_1_clip_image006_1]，发现[![clip\_image006\[2\]][clip_image006_2]][clip_image006_2_clip_image006_2]只和[![clip\_image008][clip_image008]][clip_image008_clip_image008]有关，[![clip\_image008\[1\]][clip_image008_1]][clip_image008_1_clip_image008_1]>0，那么[![clip\_image010][clip_image010]][clip_image010_clip_image010]，g(z)只不过是用来映射，真实的类别决定权还在[![clip\_image008\[2\]][clip_image008_2]][clip_image008_2_clip_image008_2]。还有当[![clip\_image012][clip_image012]][clip_image012_clip_image012]时，[![clip\_image006\[3\]][clip_image006_3]][clip_image006_3_clip_image006_3]=1，反之[![clip\_image006\[4\]][clip_image006_4]][clip_image006_4_clip_image006_4]=0。如果我们只从[![clip\_image008\[3\]][clip_image008_3]][clip_image008_3_clip_image008_3]出发，希望模型达到的目标无非就是让训练数据中y=1的特征[![clip\_image012\[1\]][clip_image012_1]][clip_image012_1_clip_image012_1]，而是y=0的特征[![clip\_image014][clip_image014]][clip_image014_clip_image014]。Logistic回归就是要学习得到[![clip\_image016][clip_image016]][clip_image016_clip_image016]，使得正例的特征远大于0，负例的特征远小于0，强调在全部训练实例上达到这个目标。

图形化表示如下：

[![clip\_image017][clip_image017]][clip_image017_clip_image017]

中间那条线是[![clip\_image019][clip_image019]][clip_image019_clip_image019]，logistic回顾强调所有点尽可能地远离中间那条线。学习出的结果也就中间那条线。考虑上面3个点A、B和C。从图中我们可以确定A是×类别的，然而C我们是不太确定的，B还算能够确定。这样我们可以得出结论，我们更应该关心靠近中间分割线的点，让他们尽可能地远离中间线，而不是在所有点上达到最优。因为那样的话，要使得一部分点靠近中间线来换取另外一部分点更加远离中间线。我想这就是支持向量机的思路和logistic回归的不同点，一个考虑局部（不关心已经确定远离的点），一个考虑全局（已经远离的点可能通过调整中间线使其能够更加远离）。这是我的个人直观理解。

##### 3 形式化表示 #####

我们这次使用的结果标签是y=-1,y=1，替换在logistic回归中使用的y=0和y=1。同时将[![clip\_image016\[1\]][clip_image016_1]][clip_image016_1_clip_image016_1]替换成w和b。以前的[![clip\_image021][clip_image021]][clip_image021_clip_image021]，其中认为[![clip\_image023][clip_image023]][clip_image023_clip_image023]。现在我们替换[![clip\_image025][clip_image025]][clip_image025_clip_image025]为b，后面替换[![clip\_image027][clip_image027]][clip_image027_clip_image027]为[![clip\_image029][clip_image029]][clip_image029_clip_image029]（即[![clip\_image031][clip_image031]][clip_image031_clip_image031]）。这样，我们让[![clip\_image033][clip_image033]][clip_image033_clip_image033]，进一步[![clip\_image035][clip_image035]][clip_image035_clip_image035]。也就是说除了y由y=0变为y=-1，只是标记不同外，与logistic回归的形式化表示没区别。再明确下假设函数

[![clip\_image037][clip_image037]][clip_image037_clip_image037]

上一节提到过我们只需考虑[![clip\_image008\[4\]][clip_image008_4]][clip_image008_4_clip_image008_4]的正负问题，而不用关心g(z)，因此我们这里将g(z)做一个简化，将其简单映射到y=-1和y=1上。映射关系如下：

[![clip\_image039][clip_image039]][clip_image039_clip_image039]

##### 4 函数间隔（functional margin）和几何间隔（geometric margin） #####

给定一个训练样本[![clip\_image041][clip_image041]][clip_image041_clip_image041]，x是特征，y是结果标签。i表示第i个样本。我们定义函数间隔如下：

[![clip\_image043][clip_image043]][clip_image043_clip_image043]

可想而知，当[![clip\_image045][clip_image045]][clip_image045_clip_image045]时，在我们的g(z)定义中，[![clip\_image047][clip_image047]][clip_image047_clip_image047]，[![clip\_image049][clip_image049]][clip_image049_clip_image049]的值实际上就是[![clip\_image051][clip_image051]][clip_image051_clip_image051]。反之亦然。为了使函数间隔最大（更大的信心确定该例是正例还是反例），当[![clip\_image045\[1\]][clip_image045_1]][clip_image045_1_clip_image045_1]时，[![clip\_image053][clip_image053]][clip_image053_clip_image053]应该是个大正数，反之是个大负数。因此函数间隔代表了我们认为特征是正例还是反例的确信度。

继续考虑w和b，如果同时加大w和b，比如在[![clip\_image055][clip_image055]][clip_image055_clip_image055]前面乘个系数比如2，那么所有点的函数间隔都会增大二倍，这个对求解问题来说不应该有影响，因为我们要求解的是[![clip\_image057][clip_image057]][clip_image057_clip_image057]，同时扩大w和b对结果是无影响的。这样，我们为了限制w和b，可能需要加入归一化条件，毕竟求解的目标是确定唯一一个w和b，而不是多组线性相关的向量。这个归一化一会再考虑。

刚刚我们定义的函数间隔是针对某一个样本的，现在我们定义全局样本上的函数间隔

[![clip\_image058][clip_image058]][clip_image058_clip_image058]

说白了就是在训练样本上分类正例和负例确信度最小那个函数间隔。

接下来定义几何间隔，先看图

[![clip\_image059][clip_image059]][clip_image059_clip_image059]

假设我们有了B点所在的[![clip\_image057\[1\]][clip_image057_1]][clip_image057_1_clip_image057_1]分割面。任何其他一点，比如A到该面的距离以[![clip\_image061][clip_image061]][clip_image061_clip_image061]表示，假设B就是A在分割面上的投影。我们知道向量BA的方向是[![clip\_image063][clip_image063]][clip_image063_clip_image063]（分割面的梯度），单位向量是[![clip\_image065][clip_image065]][clip_image065_clip_image065]。A点是[![clip\_image041\[1\]][clip_image041_1]][clip_image041_1_clip_image041_1]，所以B点是x=[![clip\_image067][clip_image067]][clip_image067_clip_image067]（利用初中的几何知识），带入[![clip\_image057\[2\]][clip_image057_2]][clip_image057_2_clip_image057_2]得，

[![clip\_image069][clip_image069]][clip_image069_clip_image069]

进一步得到

[![clip\_image070][clip_image070]][clip_image070_clip_image070]

[![clip\_image061\[1\]][clip_image061_1]][clip_image061_1_clip_image061_1]实际上就是点到平面距离。

再换种更加优雅的写法：

[![clip\_image071][clip_image071]][clip_image071_clip_image071]

当[![clip\_image073][clip_image073]][clip_image073_clip_image073]时，不就是函数间隔吗？是的，前面提到的函数间隔归一化结果就是几何间隔。他们为什么会一样呢？因为函数间隔是我们定义的，在定义的时候就有几何间隔的色彩。同样，同时扩大w和b，w扩大几倍，[![clip\_image075][clip_image075]][clip_image075_clip_image075]就扩大几倍，结果无影响。同样定义全局的几何间隔[![clip\_image076][clip_image076]][clip_image076_clip_image076]

##### 5 最优间隔分类器（optimal margin classifier） #####

回想前面我们提到我们的目标是寻找一个超平面，使得离超平面比较近的点能有更大的间距。也就是我们不考虑所有的点都必须远离超平面，我们关心求得的超平面能够让所有点中离它最近的点具有最大间距。形象的说，我们将上面的图看作是一张纸，我们要找一条折线，按照这条折线折叠后，离折线最近的点的间距比其他折线都要大。形式化表示为：

[![clip\_image077][clip_image077]][clip_image077_clip_image077]

这里用[![clip\_image075\[1\]][clip_image075_1]][clip_image075_1_clip_image075_1]=1规约w，使得[![clip\_image079][clip_image079]][clip_image079_clip_image079]是几何间隔。

到此，我们已经将模型定义出来了。如果求得了w和b，那么来一个特征x，我们就能够分类了，称为最优间隔分类器。接下的问题就是如何求解w和b的问题了。

由于[![clip\_image081][clip_image081]][clip_image081_clip_image081]不是凸函数，我们想先处理转化一下，考虑几何间隔和函数间隔的关系，[![clip\_image083][clip_image083]][clip_image083_clip_image083]，我们改写一下上面的式子：

[![clip\_image084][clip_image084]][clip_image084_clip_image084]

这时候其实我们求的最大值仍然是几何间隔，只不过此时的w不受[![clip\_image081\[1\]][clip_image081_1]][clip_image081_1_clip_image081_1]的约束了。然而这个时候目标函数仍然不是凸函数，没法直接代入优化软件里计算。我们还要改写。前面说到同时扩大w和b对结果没有影响，但我们最后要求的仍然是w和b的确定值，不是他们的一组倍数值，因此，我们需要对[![clip\_image086][clip_image086]][clip_image086_clip_image086]做一些限制，以保证我们解是唯一的。这里为了简便我们取[![clip\_image088][clip_image088]][clip_image088_clip_image088]。这样的意义是将全局的函数间隔定义为1，也即是将离超平面最近的点的距离定义为[![clip\_image090][clip_image090]][clip_image090_clip_image090]。由于求[![clip\_image090\[1\]][clip_image090_1]][clip_image090_1_clip_image090_1]的最大值相当于求[![clip\_image092][clip_image092]][clip_image092_clip_image092]的最小值，因此改写后结果为：

[![clip\_image093][clip_image093]][clip_image093_clip_image093]

这下好了，只有线性约束了，而且是个典型的二次规划问题（目标函数是自变量的二次函数）。代入优化软件可解。

到这里发现，这个讲义虽然没有像其他讲义一样先画好图，画好分类超平面，在图上标示出间隔那么直观，但每一步推导有理有据，依靠思路的流畅性来推导出目标函数和约束。

接下来介绍的是手工求解的方法了，一种更优的求解方法。

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