Last updated: 2021-05-26

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Introduction

I was suprised to hear that lasso complexity for full solution path is O(np min(n,p)). I always thought it was O(np) per iteration and did not think about how the number of iterations required might increase with n and p. I wanted to do a quick simulation check.

Empirically the log-log plot for time vs n is approximately linear with slope near 1.25, suggesting a practical scaling of O(n^1.25 p). Of course this is very much just a quick initial assessment.

library("glmnet")
Warning: package 'glmnet' was built under R version 3.6.2
Loading required package: Matrix
Loaded glmnet 4.1
n_seq = c(100, 200, 500, 1000, 2000, 5000, 10000)
p = 10000
nmax = 10000
X = matrix(rnorm(nmax*p),nrow=nmax)
b = rnorm(p)
time = c()

for(n in n_seq){
  
  y = X[1:n,] %*% b + rnorm(n)
  time = c(time,system.time(fit <- glmnet(X[1:n,],y))[1]) # user time
  #print(time)
  
}

plot(log(n_seq),log(time), main = "log(user time) vs log(n)")

Version Author Date
222582b Matthew Stephens 2021-05-26 
slope = (log(time)[7]-log(time)[1])/(log(n_seq)[7]-log(n_seq)[1])
print(slope)
user.self 
 1.260601 

sessionInfo()
R version 3.6.0 (2019-04-26)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.16

Matrix products: default
BLAS:   /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.6/Resources/lib/libRlapack.dylib

locale:
[1] en_US.UTF-8/en_US.UTF-8/en_US.UTF-8/C/en_US.UTF-8/en_US.UTF-8

attached base packages:
[1] stats     graphics  grDevices utils     datasets  methods   base     

other attached packages:
[1] glmnet_4.1    Matrix_1.2-18

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