Benchmarks

This is a short effort to give users an idea of how long the functions take to process. The benchmarks were performed using the default R install on Travis CI.

Note: the benchmarks – particularly ADMMsigma – run significantly faster on my personal machine (MacBook Pro Late 2016). In most cases, the processes take \(\approx\) 25% of the time.

We will be estimating a tri-diagonal precision matrix with dimension \(p = 100\):

library(ADMMsigma)
library(microbenchmark)

#  generate data from tri-diagonal (sparse) matrix
# compute covariance matrix (can confirm inverse is tri-diagonal)
S = matrix(0, nrow = 100, ncol = 100)

for (i in 1:100){
  for (j in 1:100){
    S[i, j] = 0.7^(abs(i - j))
  }
}

# generate 1000 x 100 matrix with rows drawn from iid N_p(0, S)
set.seed(123)
Z = matrix(rnorm(1000*100), nrow = 1000, ncol = 100)
out = eigen(S, symmetric = TRUE)
S.sqrt = out$vectors %*% diag(out$values^0.5) %*% t(out$vectors)
X = Z %*% S.sqrt

# calculate sample covariance matrix
Sample = (nrow(X) - 1)/nrow(X)*cov(X)

Default convergence tolerance with specified tuning parameters (no cross validation):

# benchmark ADMMsigma - default tolerance
microbenchmark(ADMMsigma(S = Sample, lam = 0.1, alpha = 1, tol.abs = 1e-4, tol.rel = 1e-4, trace = "none"))

## Unit: milliseconds
##                                                                                                expr
##  ADMMsigma(S = Sample, lam = 0.1, alpha = 1, tol.abs = 1e-04,      tol.rel = 1e-04, trace = "none")
##       min       lq     mean  median       uq      max neval
##  353.3034 355.8327 359.1908 357.401 361.0537 378.2791   100

Stricter convergence tolerance with specified tuning parameters (no cross validation):

# benchmark ADMMsigma - tolerance 1e-8
microbenchmark(ADMMsigma(S = Sample, lam = 0.1, alpha = 1, tol.abs = 1e-8, tol.rel = 1e-8, trace = "none"))

## Unit: seconds
##                                                                                                expr
##  ADMMsigma(S = Sample, lam = 0.1, alpha = 1, tol.abs = 1e-08,      tol.rel = 1e-08, trace = "none")
##       min       lq     mean   median       uq      max neval
##  1.261917 1.280508 1.291236 1.287888 1.301755 1.333052   100

Default convergence tolerance with cross validation for both lam and alpha:

# benchmark ADMMsigma CV - default parameter grid
microbenchmark(ADMMsigma(X, trace = "none"), times = 5)

## Unit: seconds
##                          expr      min       lq     mean  median       uq
##  ADMMsigma(X, trace = "none") 30.38821 30.53333 30.85469 30.5974 31.23629
##       max neval
##  31.51823     5

Parallel (cores = 2) cross validation:

# benchmark ADMMsigma parallel CV
microbenchmark(ADMMsigma(X, cores = 2, trace = "none"), times = 5)

## Unit: seconds
##                                     expr      min       lq     mean
##  ADMMsigma(X, cores = 2, trace = "none") 13.34387 13.40334 13.42437
##    median       uq      max neval
##  13.42051 13.44604 13.50807     5

Cross validation with log-likelihood convergence criteria:

# benchmark ADMMsigma CV - likelihood convergence criteria
microbenchmark(ADMMsigma(X, crit = "loglik", trace = "none"), times = 5)

## Unit: seconds
##                                           expr      min      lq     mean
##  ADMMsigma(X, crit = "loglik", trace = "none") 20.38331 21.0669 24.60484
##    median       uq      max neval
##  21.53833 29.96147 30.07418     5

RIDGEsigma with fine parameter grid:

# benchmark RIDGEsigma CV
microbenchmark(RIDGEsigma(X, lam = 10^seq(-8, 8, 0.01), trace = "none"), times = 5)

## Unit: seconds
##                                                      expr      min
##  RIDGEsigma(X, lam = 10^seq(-8, 8, 0.01), trace = "none") 36.30071
##        lq     mean   median       uq      max neval
##  37.11547 40.06527 40.18395 40.69736 46.02888     5

Matt Galloway