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html a2aeadb zihao12 2019-10-02 Build site.
Rmd 15e4e99 zihao12 2019-10-02 experiment with softmax in r

Goal

I need to compute softmax along an axis in 3d array (see equation below), but it often takes too long for a moderately sized array.

\[ \begin{align} \frac{exp(X_{ijk})}{\sum_k exp(X_{ijk})} \end{align} \]

## x is 3d array, and we want to do softmax along the 3rd axis, so sum_k prob_ijk = 1
softmax <- function(x){
  score.exp <- exp(x)
  probs <-sweep(score.exp, MARGIN = c(1,2), apply(score.exp, MARGIN = c(1,2), sum), '/')
  return(probs)
}

# n = 1000
# p = 5000
# k = 10
# x = array(runif(n*p*k), dim = c(n, p,k)) ## this step also takes a long time!!
# start = proc.time()
# x.softmax = softmax(x)
# runtime = proc.time() - start
# print(runtime[[3]])
## [1] 12.928
# x.softmax.sum =  apply(x.softmax, MARGIN = c(1,2), sum)
# print(all.equal(x.softmax.sum, matrix(replicate(n*p, 1), nrow = n)))
## TRUE

Here is how scipy implements it: np.exp(x - logsumexp(x, axis=axis, keepdims=True))

So do that in the log space first:

\[ \begin{align} exp(X_{ijk} - \sum_k logsumexp(X_{ijk})) \end{align} \]

library(matrixStats)
softmax2 <- function(x){
  k = dim(x)[3]
  x.logsumexp = apply(x, c(1,2), logSumExp)
  x.softmax = exp(x - replicate(k, x.logsumexp))
  return(x.softmax)
}
# n = 1000
# p = 5000
# k = 10
# x = array(runif(n*p*k), dim = c(n, p,k)) ## this step also takes a long time!!
# start = proc.time()
# x.softmax = softmax2(x)
# runtime = proc.time() - start
# print(runtime[[3]])
## [1] 16.445
# x.softmax.sum =  apply(x.softmax, MARGIN = c(1,2), sum)
# print(all.equal(x.softmax.sum, matrix(replicate(n*p, 1), nrow = n)))
## [1] TRUE

It is even slower. Python code is around 1.88 seconds.

From Matthew:

softmax <- function(x){
  score.exp <- exp(x)
  probs <-sweep(score.exp, MARGIN = c(1,2), apply(score.exp, MARGIN = c(1,2), sum), '/')
  return(probs)
}
softmax2 <- function(x){
  score.exp <- exp(x)
  probs <-sweep(score.exp, MARGIN = c(1,2), rowSums(score.exp,dims=2), '/')
  return(probs)
}
softmax3 <- function(x){
  score.exp <- exp(x)
  probs <-as.vector(score.exp)/as.vector(rowSums(score.exp,dims=2))
  dim(probs) <- dim(x)
  return(probs)
}
# n = 1000
# p = 5000
# k = 10
# x = array(runif(n*p*k), dim = c(n, p,k)) ## this step also takes a long time!!
# start = proc.time()
# x.softmax = softmax(x)
# runtime = proc.time() - start
# print(runtime[[3]])
# start = proc.time()
# x.softmax2 = softmax2(x)
# runtime = proc.time() - start
# print(runtime[[3]])
# start = proc.time()
# x.softmax3 = softmax3(x)
# runtime = proc.time() - start
# print(runtime[[3]])
# identical(x.softmax3,x.softmax)
# identical(x.softmax2,x.softmax)
# 
# # [1] 11.188
# # [1] 2.662
# # [1] 1.685
# # [1] TRUE
# # [1] TRUE


n = 100
p = 200
k = 2
x = array(runif(n*p*k), dim = c(n, p,k)) ## this step also takes a long time!!
start = proc.time()
x.softmax = softmax(x)
runtime = proc.time() - start
print(runtime[[3]])
[1] 0.027
start = proc.time()
x.softmax2 = softmax2(x)
runtime = proc.time() - start
print(runtime[[3]])
[1] 0.01
start = proc.time()
x.softmax3 = softmax3(x)
runtime = proc.time() - start
print(runtime[[3]])
[1] 0.004
identical(x.softmax3,x.softmax)
[1] TRUE
identical(x.softmax2,x.softmax)
[1] TRUE

Lesson

  • try to avoid “apply”: rowSums(score.exp,dims=2) is much faster than apply(score.exp, MARGIN = c(1,2), sum)
  • R is column major, so be aware of what as.vector does
  • operations between vectors of different lengths: shorter vectors are recycled until it matches the lengths of the longer vector

sessionInfo()
R version 3.5.1 (2018-07-02)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS  10.14

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.5/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.5/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] matrixStats_0.54.0

loaded via a namespace (and not attached):
 [1] workflowr_1.4.0 Rcpp_1.0.2      digest_0.6.21   rprojroot_1.3-2
 [5] backports_1.1.5 git2r_0.25.2    magrittr_1.5    evaluate_0.14  
 [9] stringi_1.4.3   fs_1.3.1        whisker_0.3-2   rmarkdown_1.13 
[13] tools_3.5.1     stringr_1.4.0   glue_1.3.1      xfun_0.8       
[17] yaml_2.2.0      compiler_3.5.1  htmltools_0.3.6 knitr_1.25