Linear regression by gradient descent

linear_regression_by_gradient_descent.R

##
## Linear regression by gradient descent
##
## A learning exercise to help build intuition about gradient descent.
## J. Christopher Bare, 2012
##

# set random seed
set.seed(12345)

# generate random data in which y is a noisy function of x
x <- runif(1000,-5, 5)
y <- x + rnorm(1000) + 3

# fit a linear model
res <- lm(y ~ x)

# plot the data and the model
plot(x, y, col = rgb(0.2, 0.4, 0.6, 0.4), main = 'Linear regression')
abline(res, col = 'blue')

# squared error cost function
cost <- function(X, y, theta) {
  sum((X %*% theta - y) ^ 2) / (2 * length(y))
}

# learning rate and iteration limit
alpha <- 0.01
num_iters <- 1000

# keep history
cost_history <- double(num_iters)
theta_history <- list(num_iters)

# initialize coefficients
theta <- matrix(c(0, 0), nrow = 2)

# add a column of 1's for the intercept coefficient
# to vectorize the calculation.
# X is a matrix of m x n+1 , theta is a vector of m x 1
X <- cbind(1, matrix(x))

h <- function(X, theta) {
  X %*% theta
}
# gradient descent
for (i in 1:num_iters) {
  error <- (h(X, theta) - y)
  delta <- t(X) %*% error / length(y)
  #update all theta
  theta <- theta - alpha * delta
  cost_history[i] <- cost(X, y, theta)
  theta_history[[i]] <- theta
}

# plot data and converging fit
plot(x, y, col = rgb(0.2, 0.4, 0.6, 0.4), main = 'Linear regression by gradient descent')
for (i in c(1, 3, 6, 10, 14, seq(20, num_iters, by = 10))) {
  abline(coef = theta_history[[i]], col = rgb(0.8, 0, 0, 0.3))
}
abline(coef = theta, col = "blue")

# check out the trajectory of the cost function
cost_history[seq(1, num_iters, by = 100)]
plot(
  cost_history,
  type = 'l',
  col = 'blue',
  lwd = 2,
  main = 'Cost function',
  ylab = 'cost',
  xlab = 'Iterations'
)