Stochastic gradient descent from gradient descent implementation in R

Question

I have a working implementation of multivariable linear regression using gradient descent in R. I'd like to see if I can use what I have to run a stochastic gradient descent. I'm not sure if this is really inefficient or not. For example, for each value of α I want to perform 500 SGD iterations and be able to specify the number of randomly picked samples in each iteration. It would be nice to do this so I could see how the number of samples influences the results. I'm having trouble through with the mini-batching and I want to be able to easily plot the results.

This is what I have so far:

 # 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)
}

Is it more practical to find a way to use sgd()? I can't seem to figure out how to have the level of control I'm looking for with the sgd package

Answer 1

Sticking with what you have now

## 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)) 
}

I added y to your GD function and created a wrapper function, myGoD, to call yours but first subsetting the data

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)
}

Check to make sure it works and try with different 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

Here is how you can use it

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)
}

You don't need the wrapper function--you can just change your GD slightly. It is always good practice to explicitly pass arguments to your functions rather than relying on scoping. Before you were assuming that y would be pulled from your global environment; here y must be given or you will get an error. This will avoid many headaches and mistakes down the road.

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)
}