R中梯度下降实现的随机梯度下降

Question

我有一个在 R 中使用梯度下降的多变量线性回归的工作实现。我想看看我是否可以使用我必须运行的随机梯度下降。我不确定这是否真的效率低下。例如，对于每个 α 值，我想执行 500 次 SGD 迭代，并且能够指定每次迭代中随机选取的样本数。这样做会很好，这样我就可以看到样本数量如何影响结果。我在处理小批量时遇到了麻烦，我希望能够轻松绘制结果。

这是我目前所拥有的：

 # Read and process the datasets

# download the files from GitHub
download.file("https://raw.githubusercontent.com/dbouquin/IS_605/master/sgd_ex_data/ex3x.dat", "ex3x.dat", method="curl")
x <- read.table('ex3x.dat')

# we can standardize the x vaules using scale()
x <- scale(x)

download.file("https://raw.githubusercontent.com/dbouquin/IS_605/master/sgd_ex_data/ex3y.dat", "ex3y.dat", method="curl")
y <- read.table('ex3y.dat')

# combine the datasets
data3 <- cbind(x,y)
colnames(data3) <- c("area_sqft", "bedrooms","price")
str(data3)

head(data3)

################ Regular Gradient Descent
# http://www.r-bloggers.com/linear-regression-by-gradient-descent/

# vector populated with 1s for the intercept coefficient
x1 <- rep(1, length(data3$area_sqft))

# appends to dfs
# create x-matrix of independent variables
x <- as.matrix(cbind(x1,x))
# create y-matrix of dependent variables
y <- as.matrix(y)
L <- length(y)

# cost gradient function: independent variables and values of thetas
cost <- function(x,y,theta){
  gradient <- (1/L)* (t(x) %*% ((x%*%t(theta)) - y))
  return(t(gradient)) 
}

# GD simultaneous update algorithm
# https://www.coursera.org/learn/machine-learning/lecture/8SpIM/gradient-descent
GD <- function(x, alpha){
      theta <- matrix(c(0,0,0), nrow=1) 
  for (i in 1:500) {
       theta <- theta - alpha*cost(x,y,theta)  
       theta_r <- rbind(theta_r,theta)    
  }
return(theta_r)
}

# gradient descent α = (0.001, 0.01, 0.1, 1.0) - defined for 500 iterations

alphas <- c(0.001,0.01,0.1,1.0)

# Plot price, area in square feet, and the number of bedrooms

# create empty vector theta_r
theta_r<-c()

for(i in 1:length(alphas)) {

 result <- GD(x, alphas[i])

 # red = price 
 # blue = sq ft 
 # green = bedrooms
 plot(result[,1],ylim=c(min(result),max(result)),col="#CC6666",ylab="Value",lwd=0.35,
      xlab=paste("alpha=", alphas[i]),xaxt="n") #suppress auto x-axis title
      lines(result[,2],type="b",col="#0072B2",lwd=0.35)
      lines(result[,3],type="b",col="#66CC99",lwd=0.35)
}

找到使用sgd()的方法是不是更实用？我似乎无法弄清楚如何使用 sgd 包获得我想要的控制级别

Answer 1

坚持你现在拥有的

## all of this is the same

download.file("https://raw.githubusercontent.com/dbouquin/IS_605/master/sgd_ex_data/ex3x.dat", "ex3x.dat", method="curl")
x <- read.table('ex3x.dat')
x <- scale(x)
download.file("https://raw.githubusercontent.com/dbouquin/IS_605/master/sgd_ex_data/ex3y.dat", "ex3y.dat", method="curl")
y <- read.table('ex3y.dat')
data3 <- cbind(x,y)
colnames(data3) <- c("area_sqft", "bedrooms","price")
x1 <- rep(1, length(data3$area_sqft))
x <- as.matrix(cbind(x1,x))
y <- as.matrix(y)
L <- length(y)
cost <- function(x,y,theta){
  gradient <- (1/L)* (t(x) %*% ((x%*%t(theta)) - y))
  return(t(gradient)) 
}

我将y 添加到您的GD 函数中并创建了一个包装函数myGoD 来调用您的函数，但首先要对数据进行子集化

GD <- function(x, y, alpha){
  theta <- matrix(c(0,0,0), nrow=1)
  theta_r <- NULL
  for (i in 1:500) {
    theta <- theta - alpha*cost(x,y,theta)  
    theta_r <- rbind(theta_r,theta)    
  }
  return(theta_r)
}

myGoD <- function(x, y, alpha, n = nrow(x)) {
  idx <- sample(nrow(x), n)
  y <- y[idx, , drop = FALSE]
  x <- x[idx, , drop = FALSE]
  GD(x, y, alpha)
}

检查以确保它可以正常工作并尝试使用不同的 Ns

all.equal(GD(x, y, 0.001), myGoD(x, y, 0.001))
# [1] TRUE

set.seed(1)
head(myGoD(x, y, 0.001, n = 20), 2)
#          x1        V1       V2
# V1 147.5978  82.54083 29.26000
# V1 295.1282 165.00924 58.48424

set.seed(1)
head(myGoD(x, y, 0.001, n = 40), 2)
#          x1        V1        V2
# V1 290.6041  95.30257  59.66994
# V1 580.9537 190.49142 119.23446

这里是你如何使用它

alphas <- c(0.001,0.01,0.1,1.0)
ns <- c(47, 40, 30, 20, 10)

par(mfrow = n2mfrow(length(alphas)))
for(i in 1:length(alphas)) {

  # result <- myGoD(x, y, alphas[i]) ## original
  result <- myGoD(x, y, alphas[i], ns[i])

  # red = price 
  # blue = sq ft 
  # green = bedrooms
  plot(result[,1],ylim=c(min(result),max(result)),col="#CC6666",ylab="Value",lwd=0.35,
       xlab=paste("alpha=", alphas[i]),xaxt="n") #suppress auto x-axis title
  lines(result[,2],type="b",col="#0072B2",lwd=0.35)
  lines(result[,3],type="b",col="#66CC99",lwd=0.35)
}

你不需要包装函数——你可以稍微改变你的GD。将参数显式传递给您的函数而不是依赖于范围界定始终是一种好习惯。在您假设 y 将从您的全球环境中撤出之前；这里必须给出y，否则你会得到一个错误。这将避免许多令人头疼的问题和错误。

GD <- function(x, y, alpha, n = nrow(x)){
  idx <- sample(nrow(x), n)
  y <- y[idx, , drop = FALSE]
  x <- x[idx, , drop = FALSE]
  theta <- matrix(c(0,0,0), nrow=1)
  theta_r <- NULL

  for (i in 1:500) {
    theta <- theta - alpha*cost(x,y,theta)  
    theta_r <- rbind(theta_r,theta)    
  }
  return(theta_r)
}