机器学习基石---第二周PLA

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  台大《机器学习基石》第二周课的笔记，只整理部分重要内容。希望能把课上学的，做一个精简的记录。

变量说明

  存在两类数据，标记为$y$，取值为${-1,1}$。特征向量记为$x$，$x=(x_0,x_1,x_2,...,x_d)$。其中$x_0$为常量1，其余为具体特征值。存在超平面$w^Tx=0$，其中$w=(w_0,w_1,...,w_d)$，可以正确分开两类数据。共有$N$个样本数据。

迭代过程

  PLA采取知错就改的策略。遍历所有样本，如果发现分类错误，采用如下方式如下方式更新$w$
  

$$\begin{array}{l} For\;t = 0,1,...N\\ 1.\;find\;a\;mistake\;of\;{w_t}\;called\;\left( {{x_{n(t)}},{y_{n(t)}}} \right)\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;sign(w_t^T{x_{n(t)}}) \ne {y_{n(t)}}\\ 2.\;\left( {try\;to} \right)\;correct\;the\;mistake\;by\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{w_{t + 1}} \leftarrow \;{w_t} + \;{y_{n(t)}}{x_{n(t)}}\\ ...\;until\;no\;more\;mistakes\\ return\;last\;w\left( {called\;{w_{PLA}}} \right)\;as\;g \end{array}$$

更新理由

  判断类别的公式：

$$sign(w_t^T{x_{n(t)}}) = sign(\left\| {w_t^T} \right\|\left\| {{x_{n(t)}}} \right\|\cos \left( \theta \right))$$
  如果正类被误判，则$cos(\theta)<0$，即$\theta \in (\frac{\pi }{2},\pi )$，所以要缩小法向量和特征向量之间的夹角。故采用上图方法迭代$w$的值。

证明

  证明线性可分数据集，PLA算法一定能够经过有限次的迭代，得到一个完美的分割超平面。

每一次迭代$w_t$更接近$w_f$

  1. $w_f$为完美分类器
  2. $(x_n,y_n)$为错分的样本
  3. $(x_{n(t)},y_{n(t)})$为第t次迭代时，$w_t$错分的样本
  因为$w_f$是完美分类器，则一定有：

$$y_{n(t)}w_f^Tx_{n(t)}\geq \mathop{min}_ny_nw_f^Tx_n> 0$$

  利用任意一个错判样本$(x_{n(t)},y_{n(t)})$进行第$t+1$次迭代之后,计算:

$$\begin{array}{l} \frac{{w_f^T{w_{t + 1}}}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}} = \frac{{w_f^T({w_t} + {y_{n(t)}}{x_{n(t)}})}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; = \frac{{w_f^T{w_t} + {y_{n(t)}}w_f^T{x_{n(t)}}}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \frac{{w_f^T{w_t} + \mathop {\min }\limits_n {y_{n(t)}}w_f^T{x_{n(t)}}}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; > \frac{{w_f^T{w_t} + 0}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}} = \frac{{w_f^T{w_t}}}{{\left\| {w_f^T} \right\|\left\| {{w_{t + 1}}} \right\|}} \end{array}$$
  从余弦相似度的角度看，通过错判样本对$w_t$的修正，使得迭代后的$w$更接近于完美的分割超平面。

每一次迭代$w_t$的模增长较小

$$\begin{array}{l} {\left\| {{w_{t + 1}}} \right\|^2}\; = \;\;{\left\| {{w_t} + {y_{n(t)}}{x_{n(t)}}} \right\|^2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; = \;\;{\left\| {{w_t}} \right\|^2} + 2{y_{n(t)}}w_t^T{x_{n(t)}} + {\left\| {{y_{n(t)}}{x_{n(t)}}} \right\|^2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \;\;{\left\| {{w_t}} \right\|^2} + 0 + {\left\| {{y_{n(t)}}{x_{n(t)}}} \right\|^2}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\; \le \;\;{\left\| {{w_t}} \right\|^2} + \mathop {\max }\limits_n {\left\| {{y_n}{x_n}} \right\|^2}\; \end{array}$$

迭代次数有限

  假设$w_0=0$,经过$T$次迭代之后:

$$\begin{array}{l} \frac{{w_f^T{w_T}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\;\;\; = \;\;\;\frac{{w_f^T({w_{T - 1}} + {y_{n(T - 1)}}{x_{n(T - 1)}})}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ = }}\;\;\;\frac{{w_f^T({w_{T - 1}} + {y_{n(T - 1)}}{x_{n(T - 1)}})}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;{\rm{ = }}\;\;\;\frac{{w_f^T{w_{T - 1}} + {y_{n(T - 1)}}w_f^T{x_{n(T - 1)}}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \;\;\;\frac{{w_f^T{w_{T - 1}} + \mathop {\min }\limits_n {y_n}w_f^T{x_n}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \;\;\;\frac{{w_f^T{w_{T - 2}} + {y_{n(T - 2)}}w_f^T{x_{n(T - 2)}} + \mathop {\min }\limits_n {y_n}w_f^T{x_n}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \;\;\;\frac{{w_f^T{w_{T - 2}} + 2\mathop {\min }\limits_n {y_n}w_f^T{x_n}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \cdots \\ \;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \ge \;\;\;\frac{{T\mathop {\min }\limits_n {y_n}w_f^T{x_n}}}{{\left\| {{w_f}} \right\|\left\| {{w_T}} \right\|}}\\ {\rm{Further}}:\\ w_f^T{w_T}\; \ge \;\;T\mathop {\min }\limits_n {y_n}w_f^T{x_n}\\ T\;\;\;\; \le \;\;\frac{{w_f^T{w_T}\;}}{{\mathop {\min }\limits_n {y_n}w_f^T{x_n}}}\\ {T^{\rm{2}}}\;\; \le \;\;\frac{{{{(w_f^T{w_T})}^2}\;}}{{{{(\mathop {\min }\limits_n {y_n}w_f^T{x_n})}^{\rm{2}}}}}\; = \;\frac{{{{\left\| {{w_f}} \right\|}^2}{{\left\| {{w_T}} \right\|}^2}{{\sin }^2}(\theta )}}{{{{(\mathop {\min }\limits_n {y_n}w_f^T{x_n})}^{\rm{2}}}}}\\ \;\;\;\;\;\; \le \;\;\frac{{{{\left\| {{w_f}} \right\|}^2}{{\left\| {{w_T}} \right\|}^2}}}{{{{(\mathop {\min }\limits_n {y_n}w_f^T{x_n})}^{\rm{2}}}}}\\ \;\;\;\;\;\; \le \;\;\frac{{{{\left\| {{w_f}} \right\|}^2}*{{\mathop {\max \left\| {{y_n}{x_n}} \right\|}\limits_n }^2}}}{{{{(\mathop {\min }\limits_n {y_n}w_f^T{x_n})}^{\rm{2}}}}}\; = \;\frac{{{{\left\| {{w_f}} \right\|}^2}*{{\mathop {\max \left\| {{x_n}} \right\|}\limits_n }^2}}}{{{{(\mathop {\min }\limits_n {y_n}w_f^T{x_n})}^{\rm{2}}}}} \end{array}$$
  所以迭代次数有上界。

案例

构造数据集

  构造数据集，验证算法。

x11 <- 1:10
x21 <- x11 + runif(10, 0, 1) + 3
x22 <- x11 - runif(10, 0, 1)
example_data <- data.frame(x1 = rep(x11, 2),
x2 = c(x21, x22),
label = rep(c(1, -1), each = 10))
example_data$label <- as.factor(example_data$label)
library(ggplot2)
ggplot(data = example_data, aes(
x = x1,
y = x2,
color = label,
shape = label
)) +
geom_point()

PLA算法

## 参数:数据集、标签名称

PLA_f <- function(dataset, label) {
  ## 样本数
  row_num <-  nrow(dataset)
  w <- rep(1, ncol(dataset))
  w0 <- matrix(w, 1, 3, byrow = T)
  real_label <- as.numeric(as.vector(dataset[, label]))
  feature_matrix <-
    as.matrix(data.frame(x0 = rep(1, row_num), cbind(dataset[, setdiff(colnames(dataset), label)])))
  i <- 1
  j <- 0
  while (i < row_num & j == 0) {
    i <- 1
    j <- 0
    for (i in 1:row_num) {
      ## 判断是否有误判
      if (as.vector(feature_matrix[i,] %*% t(w0)) * real_label[i] <= 0) {
        ## 存在误判，修正w0
        w0 <- w0 + real_label[i] * feature_matrix[i,]
        w <- c(w, w0)
        j <- 1
      }
      if(j == 1){
        j <- 0
        i <- row_num-1
        break()}
    }
  }
  w_data <- data.frame(matrix(w,ncol=ncol(dataset),byrow = TRUE))
  colnames(w_data) <- paste0("x",0:(ncol(feature_matrix)-1))
  w_data <- dplyr::mutate(w_data,
                          slope = -x1 / x2,
                          intercept = -x0 / x2)
  return(w_data)
}

求解

w_data <- PLA_f(dataset = example_data, label = "label")
w_data

   x0 x1           x2        slope    intercept
1   1  1  1.000000000   -1.0000000   -1.0000000
2   0  0  0.495471116    0.0000000    0.0000000
3  -1 -1 -0.009057768 -110.4024725 -110.4024725
4   0  0  4.912654036    0.0000000    0.0000000
5  -1 -1  4.408125152    0.2268538    0.2268538
6  -2 -2  3.903596268    0.5123481    0.5123481
7  -3 -4  1.915120282    2.0886417    1.5664812
8  -2 -1  8.363856425    0.1195621    0.2391241
9  -3 -2  7.859327541    0.2544747    0.3817120
10 -4 -4  5.870851555    0.6813322    0.6813322
11 -5 -9  1.747566727    5.1500179    2.8611211
12 -4 -8  6.669278532    1.1995300    0.5997650

动图

library(animation)
## 指定ImageMagic目录位置，注意是magick.exe，之前版本貌似一致是convert.exe
ani.options(convert = "D:/ImageMagic/ImageMagick-7.0.7-Q16/magick.exe")
saveGIF(
  expr = {
    library(ggplot2)
    for (i in 1:nrow(w_data)) {plot(
      x = example_data$x1[1:10],
      y = example_data$x2[1:10],
      pch = 15,
      col = "red",
      xlim = c(0, 20),
      ylim = c(0, 15),
      xlab = "x1",
      ylab = "x2",main = paste0("Picture",i)
    )
      lines(x = example_data$x1[11:20],
            y = example_data$x2[11:20],
            type = "p",
            pch = 17,
            col = "blue")
      abline(coef=c(w_data$intercept[i],w_data$slope[i]),lwd=2)
      }
  },
  ## GIF文件名，注意文件后缀名要加上
  movie.name = "PLA.gif",
  ## 时间间隔
  interval = 1,
  ## 图形设置
  ani.width = 600,
  ani.height = 600,
  ## 文件输出在当前目录
  outdir = getwd()
)

Ref

[1]课程PPT

2017-12-19于杭州