Speeding up repeated generation of Exponential random variables in loop - r

Question

I am implementing an algorithm, and as part of that, I need to generate exponential random variables. Unfortunately though, I can't really avoid looping, as each generated random variable depends on the previous one so I think vectorisation is out of the question. There are some calculations that I do around the generation, but the bottleneck (at present) is the generation. At this point I am assuming N will be large (N >= 1,000,000).
Here is some example code:
N <- 1e7
#Preallocate
x <- rep(0, times=N)
#Set a starting seed
x[1] <- runif(1)
for(i in 2:N) {
#Do some calculations
x[i] <- x[i-1] + rexp(1, x[i-1]) #Bottleneck
#Do some more calculations
}
How can I speed this up? I've tried implementing in Rcpp, but it doesn't seem to do much in this case. Is there another clever way I can get around the rexp() call in each iteration?

Answer 1

We can use the fact that if X ~ Exp(λ) then kX ~ Exp(λ/k) (source: Wikipedia) to speed up the code. This way we can do all the random draws with rate = 1 up front and then just divide within the loop to scale them appropriately.
draws = rexp(N, rate = 1)
x <- rep(0, times = N)
x[1] <- runif(1)
for(i in 2:N) {
#Do some calculations
x[i] <- x[i-1] + draws[i] / x[i-1]
#Do some more calculations
}
A microbenchmark with N = 1e6 values show this to be about 14x faster:
N <- 1e6
draws = rexp(N, rate = 1)
x <- rep(0, times = N)
x[1] <- runif(1)
microbenchmark::microbenchmark(
draw_up_front = {
draws = rexp(N, rate = 1)
for (i in 2:N)
x[i] <- x[i - 1] + draws[i] / x[i - 1]
},
draw_one_at_time = {
for (i in 2:N)
x[i] <- x[i - 1] + rexp(1, x[i - 1])
},
times = 10
)
# Unit: milliseconds
# expr min lq mean median uq max neval cld
# draw_up_front 153.9547 156.6552 159.9622 160.1901 161.9803 167.2831 10 a
# draw_one_at_time 2207.1997 2212.0460 2280.1265 2236.5197 2332.9913 2478.5104 10 b

Answer 2

A brute-force Rcpp solution:
#include <Rcpp.h>
using namespace Rcpp;
// [[Rcpp::export]]
NumericVector genExp(int N) {
NumericVector res(N);
double prev;
res[0] = unif_rand();
for (int i = 1; i < N; i++) {
prev = res[i-1];
res[i] = prev + exp_rand() / prev;
}
return res;
}
Microbenchmark with N = 1e6:
Unit: milliseconds
expr min lq mean median uq max neval
draw_up_front 167.17031 168.57345 170.62292 170.18072 171.73782 175.46868 20
draw_one_at_time 1415.01898 1465.57139 1510.81220 1502.15753 1550.07829 1623.70831 20
rcpp 28.25466 29.33682 33.52528 29.89636 30.74908 94.38009 20
With N = 1e7:
Unit: milliseconds
expr min lq mean median uq max neval
draw_up_front 1698.730 1708.739 1737.8633 1716.1345 1752.3276 1923.3940 20
rcpp 297.142 319.794 338.6755 327.6626 364.6308 398.1554 20