Question

I’m working on a distributed linear regression calculation in R for Hadoop, but before implementing it, I’d like to verify that my calculations agree with the results of the lm function.

I have the following functions which attempt to implement the generic “summation” framework discussed by Andrew Ng et al. in the paper Map-Reduce for Machine Learning on Multicore.

For linear regression, this involves mapping each row y_i and x_i to P_i and Q_i such that:

P_i = x_i * transpose(x_i)
Q_i = x_i * y_i

Then reducing to solve for the coefficients, theta:
theta = (sum(P_i))^-1 * sum(Q_i)

The R functions to do this are:

calculate_p <- function(dat_row) {
  dat_row %*% t(dat_row)
}

calculate_q <- function(dat_row) {
  dat_row[1,1] * dat_row[, -1]
}

calculate_pq <- function(dat_row) {
  c(calculate_p(matrix(dat_row[-1], nrow=1)), calculate_q(matrix(dat_row, nrow=1)))
}

map_pq <- function(dat) {
  t(apply(dat, 1, calculate_pq))
}

reduce_pq <- function(pq) {
  (1 / sum(pq[, 1])) * apply(pq[, -1], 2, sum)
}

You can implement it on some synthetic data by running:

X <- matrix(rnorm(20*5), ncol = 5)
y <- as.matrix(rnorm(20))
reduce_pq(map_pq(cbind(y, X)))
[1]  0.010755882 -0.006339951 -0.034797768  0.067438662 -0.033557351
coef(lm.fit(X, y))
          x1           x2           x3           x4           x5 
-0.038556283 -0.002963991 -0.195897701  0.422552974 -0.029823962

Unfortunately, the outputs don’t match, so obviously I’m doing something wrong. Any ideas how I can fix it?

Answer 1

The inverse you take in reduce_pq needs to be a matrix inverse. Also I changed some of the functions a little bit.

calculate_p <- function(dat_row) { 
    dat_row %*% t(dat_row)
}

calculate_q <- function(dat_row) { 
    dat_row[1] * dat_row[-1] 
}

calculate_pq <- function(dat_row) {
    c(calculate_p(dat_row[-1]), calculate_q(dat_row)) 
}

map_pq <- function(dat) {
    t(apply(dat, 1, calculate_pq))
}

reduce_pq <- function(pq) { 
    solve(matrix(apply(pq[, 1:(ncol(X) * ncol(X))], 2, sum), nrow=ncol(X))) %*% apply(pq[, 1:ncol(X) + ncol(X)*ncol(X)], 2, sum)
}


set.seed(1)
X <- matrix(rnorm(20*5), ncol = 5)
y <- as.matrix(rnorm(20))

t(reduce_pq(map_pq(cbind(y, X))))
          [,1]      [,2]      [,3]       [,4]        [,5]
[1,] 0.1236914 0.2482445 0.5120975 -0.1104451 -0.04080922

coef(lm.fit(X,y))
         x1          x2          x3          x4          x5 
 0.12369137  0.24824449  0.51209753 -0.11044507 -0.04080922 

> all.equal(as.numeric(t(reduce_pq(map_pq(cbind(y, X))))), as.numeric(coef(lm.fit(X,y))))
[1] TRUE