From cff2ae54c4d81255391bbf1ca0034ad6982e9cf6 Mon Sep 17 00:00:00 2001 From: jiangzhonglian Date: Wed, 19 Apr 2017 17:06:39 +0800 Subject: [PATCH] =?UTF-8?q?=E6=9B=B4=E6=96=B0=206.SVM=E5=85=AC=E5=BC=8F?= MIME-Version: 1.0 Content-Type: text/plain; charset=UTF-8 Content-Transfer-Encoding: 8bit --- docs/6.支持向量机.md | 45 +++++++++++----------- images/6.SVM/SVM_5_Lagrangemultiplier.png | Bin 81895 -> 104623 bytes 2 files changed, 23 insertions(+), 22 deletions(-) diff --git a/docs/6.支持向量机.md b/docs/6.支持向量机.md index 8b90446d..5e397435 100644 --- a/docs/6.支持向量机.md +++ b/docs/6.支持向量机.md @@ -53,53 +53,54 @@ This is the simplest kind of SVM (Called an LSVM) Support Vectors are those data ``` * 选择D会比B、C分隔的效果要好很多,原因是上述的5个结论。 +* 所有的点看作地雷吧,那么我们(超平面)得找到最近所有的地雷,并保证我们离它最远。 ![线性可分](/images/6.SVM/SVM_3_linearly-separable.jpg) ### 怎么寻找最大间隔 > 点到超平面的距离 -* 分隔超平面`函数间距`: $$y(x)=w^Tx+b$$ -* 分类的结果: $$f(x)=sign(w^Tx+b)$$ (sign表示>0为1,<0为-1,=0为0) -* 点到超平面的`几何间距`: $$d(x)=(w^Tx+b)/||w||$$ [ ||w||表示w矩阵的二范式=> sqrt(w*w^T), 点到超平面的距离也是类似的,需要复习一下向量的知识 ] +* 分隔超平面`函数间距`: \\(y(x)=w^Tx+b\\) +* 分类的结果: \\(f(x)=sign(w^Tx+b)\\) (sign表示>0为1,<0为-1,=0为0) +* 点到超平面的`几何间距`: \\(d(x)=(w^Tx+b)/||w||\\) (||w||表示w矩阵的二范式=> \\(\sqrt{w*w^T}\\), 点到超平面的距离也是类似的,需要复习一下向量的知识) ![点到直线的几何距离](/images/6.SVM/SVM_4_point2line-distance.png) > 拉格朗日乘子法的使用 -* 类别标签用-1、1,是为了后期方便`lable*(w^Tx+b)`的标识和距离计算;如果`lable*(w^Tx+b)>0`表示预测正确,否则预测错误。 +* 类别标签用-1、1,是为了后期方便 \\(lable*(w^Tx+b)\\) 的标识和距离计算;如果 \\(lable*(w^Tx+b)>0\\) 表示预测正确,否则预测错误。 * 现在目标很明确,就是要找到`w`和`b`,因此我们必须要找到最小间隔的数据点,也就是前面所说的`支持向量`。 * 也就说,让最小的距离取最大.(最小的距离:就是最小间隔的数据点;最大:就是最大间距,为了找出最优超平面--最终就是支持向量) * 怎么理解呢? 例如: 所有的点看作地雷吧,那么我们(超平面)得找到最近所有的地雷,并保证我们离它最远。 - * 目标函数:`arg max{min(lable*(w^Tx+b)/||w||)}` - * 1.如果`lable*(w^Tx+b)>0`表示预测正确,`||w||`可以理解为归一化,我们始终可以找到一个阈值让`lable*(w^Tx+b)>=1` - * 2.所以令`lable*(w^Tx+b)=1`,我们本质上是求 `arg max[关于w, b] (1/||w||)`;也就说,我们约束(前提)条件是`lable*(w^Tx+b)>=1` -* 新的目标函数求解: `arg max[关于w, b] (1/||w||)` - * => 就是求: `arg min[关于w, b] (||w||)` (求矩阵会比较麻烦,如果x只是1/2*X^2的偏导数,那么。。同样是求最小值) - * => 就是求: `arg min[关于w, b] (1/2*||w||^2)` (二次函数求导,求极值,平方也方便计算) + * 目标函数:\\(arg: max\ \{min\ [lable*(w^Tx+b)/||w||]\}\\) + * 1.如果 \\(lable*(w^Tx+b)>0\\) 表示预测正确,也称`函数间隔`,\\(||w||\\) 可以理解为归一化,也称`几何间隔`,我们始终可以找到一个阈值让 \\(lable*(w^Tx+b)>=1\\) + * 2.所以令 \\(lable*(w^Tx+b)=1\\),我们本质上是求 \\(arg: max\{关于w, b\}(1/||w||)\\);也就说,我们约束(前提)条件是: \\(lable*(w^Tx+b)>=1\\) +* 新的目标函数求解: \\(arg: max\{关于w, b\}\ (1/||w||)\\) + * => 就是求: \\(arg: min\{关于w, b\}\ (||w||)\\) (求矩阵会比较麻烦,如果x只是1/2*X^2的偏导数,那么。。同样是求最小值) + * => 就是求: \\(arg: min\{关于w, b\}\ (\frac{1}{2}*||w||^2)\\) (二次函数求导,求极值,平方也方便计算) * 本质上就是求线性不等式的二次优化问题(求分隔超平面,等价于求解相应的凸二次规划问题。) * 通过拉格朗日乘子法,求二次优化问题 - * 假设需要求极值的目标函数 (objective function) 为 f(x,y),限制条件为 φ(x,y)=M - * 设g(x,y)=M-φ(x,y) # 临时φ(x,y)表示下文中`label*(w^Tx+b)` + * 假设需要求极值的目标函数 (objective function) 为 f(x,y),限制条件为 φ(x,y)=M # M>=1 + * 设g(x,y)=M-φ(x,y) # 临时φ(x,y)表示下文中 \\(label*(w^Tx+b)\\) * 定义一个新函数: F(x,y,λ)=f(x,y)+λg(x,y) * a为λ,代表要引入的拉格朗日乘子(Lagrange multiplier) - * 那么: `L(w,b,a)=1/2||w||^2 + Σ a*[1-label*(w^Tx+b)]` => `L(w,b,a)=1/2||w||^2 - Σ a*[label*(w^Tx+b)-1]` - * 因为:`label*(w^Tx+b)>=1, a>0` , 所以`a*[label*(w^Tx+b)-1]>=0`, Σ 0~n的加和也都大于0 - * `max[关于a] (L(w,b,a)) = 1/2*||w||^2` - * 如果求: `min(1/2*||w||^2)`, 也就是要求: `arg min[关于w, b]{ max[关于a] (L(w,b,a)) }` + * 那么: \\(L(w,b,\alpha)=\frac{1}{2} * ||w||^2 + \sum_{i=1}^{n} \alpha * [1 - label * (w^Tx+b)]\\) => \\(L(w,b,\alpha)=\frac{1}{2} * ||w||^2 - \sum_{i=1}^{n} \alpha * [label * (w^Tx+b) - 1]\\) + * 因为:\\(label*(w^Tx+b)>=1, \alpha>0\\) , 所以 \\(\alpha*[label*(w^Tx+b)-1]>=0\\), \\(\sum_{i=1}^{n} \alpha * [label * (w^Tx+b) - 1]>=0\\) + * \\(max\{关于\alpha\}\ L(w,b,\alpha) = \frac{1}{2} *||w||^2\\) + * 如果求: \\(min\{关于w, b\}\ \frac{1}{2} *||w||^2\\) , 也就是要求: \\(min\{关于w, b\}\ [max\{关于\alpha\}\ L(w,b,\alpha)]\\) * 现在转化到对偶问题的求解 - * `min[关于w, b]{ max[关于w, b] (L(w,b,a)) } >= max[关于w, b] { min[关于w, b] (L(w,b,a)) }` + * \\(min\{关于w, b\}\ [max\{关于\alpha\}\ L(w,b,\alpha)]\\) >= \\(max\{关于\alpha\}\ [min\{关于w, b\}\ L(w,b,\alpha)]\\) * 现在分2步 - * 先求:`min[关于w, b] (L(w,b,a))=1/2||w||^2 + Σ a*[1-label*(w^Tx+b)]` + * 先求: \\(min\{关于w, b\}\ L(w,b,\alpha)=\frac{1}{2} * ||w||^2 + \sum_{i=1}^{n} \alpha * [1 - label * (w^Tx+b)]\\) * 就是求`L(w,b,a)`关于[w, b]的偏导数, 得到`w和b的值`,并化简为:`L和a的方程`。 - * 参考: 1.【计算拉格朗日函数的对偶函数-下图是参考小象学院】 2.如果公式推导还是不懂,也可以参考《统计学习方法》李航-P103<学习的对偶算法> + * 参考: 如果公式推导还是不懂,也可以参考《统计学习方法》李航-P103<学习的对偶算法> * ![计算拉格朗日函数的对偶函数](/images/6.SVM/SVM_5_Lagrangemultiplier.png) -* 终于得到课本上的公式:`max(Σ(i=>n) a[i] - 1/2*Σ(i,j=>n) a[i]*a[j]]*label(i)*label(j)*)` -* 约束条件: `a>=0 and Σ a[i]*label(i)=0` +* 终于得到课本上的公式: \\(max\ \{\alpha\}\ [\sum_{i=1}^{m} \alpha - \frac{1}{2} \sum_{i, j=1}^{m} label_i·label_j·\alpha_i·\alpha_j·]\\) +* 约束条件: \\(a>=0,\ and\ \sum_{i=1}^{m} a_i·label_i=0\\) > 松弛变量(slack variable) * 我们知道几乎所有的数据都不那么干净, 通过引入松弛变量来允许数据点可以处于分隔面错误的一侧。 -* 约束条件: `C>=a>=0 and Σ a[i]*label(i)=0` +* 约束条件: \\(C>=a>=0,\ and\ \sum_{i=1}^{m} a_i·label_i=0\\) * 这里常量C用于控制“最大化间隔”和“保证大部分点的函数间隔小于1.0” 这两个目标的权重。 * 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