Using warmstarts to improve optimization

Introduction

Here I turn to the occasional large decreases in the FLASH objective function, an example of which was given in the previous investigation. I trace the problem to the call to optim in ebnm_pn.

Example

I use the same dataset as previously. However, since the example given in the previous investigation is not easily reproducible (due to randomness in udv_si), I re-fit the dataset using udv_svd. A large decrease in the objective function occurs while adding the 23rd factor. The last few lines of verbose output are as follows.

(...)
Objective:-1256928.81965425
Objective:-1256922.94001901
Objective:-1256917.09583091
Objective:-1256911.02417405
Objective:-1256904.83389783
Objective:-1256920.05096899
An iteration decreased the objective. This happens occasionally, perhaps due to numeric reasons. You could ignore this warning, but you might like to check out https://github.com/stephenslab/flashr/issues/26 for more details.performing nullcheck
objective from deleting factor:-1256871.38066608
objective from keeping factor:-1256920.05096899
factor zeroed out

Analysis of problem

The problem, as I will show, is that optim fails to find a good solution to the EBNM problem.

I backtrack to just before the bad update using parameter stopAtObj (to reproduce this example, make sure to load branch trackObj).

Since the fits take some time, I pre-run the following code and then load the results from file.

# devtools::install_github("stephenslab/flashr", ref="trackObj")
devtools::load_all("/Users/willwerscheid/GitHub/flashr")

Loading flashr

# devtools::install_github("stephenslab/ebnm")
devtools::load_all("/Users/willwerscheid/GitHub/ebnm")

Loading ebnm

gtex <- readRDS(gzcon(url("https://github.com/stephenslab/gtexresults/blob/master/data/MatrixEQTLSumStats.Portable.Z.rds?raw=TRUE")))
strong <- t(gtex$strong.z)

# This block was run in advance.

res <- flash_add_greedy(strong, Kmax=50, init_fn="udv_svd", verbose=TRUE)

res <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd", verbose=TRUE)
res <- flash_add_greedy(strong, Kmax=1, f_init=res$f, init_fn="udv_svd",
                       verbose=TRUE, nullcheck=FALSE, stopAtObj=-1256905)
saveRDS(res, "../data/warmstart/greedy23.rds")

I now step through the factor update code.

res <- readRDS("./data/warmstart/greedy23.rds")

fl <- res$f
data <- flash_set_data(strong)
k <- 23
subset <- 1:(flashr:::flash_get_p(fl))

fl <- flash_update_precision(data, fl)

# Get results for factor update:
ebnm_args <- calc_ebnm_f_args(data, fl, k, subset)
a <- do.call("ebnm_pn", list(ebnm_args$x, ebnm_args$s, list()))

# Store results:
fl.before.update <- fl
fl$EF[subset, k] = a$postmean
fl$EF2[subset, k] = a$postmean2
fl$gf[[k]] = a$fitted_g
fl$KL_f[[k]] = a$penloglik - NM_posterior_e_loglik(ebnm_args$x,
                                                   ebnm_args$s,
                                                   a$postmean,
                                                   a$postmean2)

flash_get_objective(data, fl)

[1] -1256923

So the objective is indeed worse at this point. I inspect the update to \(g_f\):

list(before.update = fl.before.update$gf[[k]], 
     after.update = fl$gf[[k]])

$before.update
$before.update$pi0
[1] 0.914959

$before.update$a
[1] 21625.71


$after.update
$after.update$pi0
[1] 0.3275043

$after.update$a
[1] 167827.8

This is a huge change to \(g_f\), much larger than one should reasonably expect from a single update. Indeed, if I initialize ebnm_pn using the current value of gf, I get a much different result.

fl2 <- fl.before.update

a <- do.call("ebnm_pn", list(ebnm_args$x, ebnm_args$s, 
                             list(g=fl2$gf[[k]])))

fl2$EF[subset, k] = a$postmean
fl2$EF2[subset, k] = a$postmean2
fl2$gf[[k]] = a$fitted_g
fl2$KL_f[[k]] = a$penloglik - NM_posterior_e_loglik(ebnm_args$x,
                                                    ebnm_args$s,
                                                    a$postmean,
                                                    a$postmean2)
flash_get_objective(data, fl2)

[1] -1256901

So the objective improves, as is guaranteed by the theory. The updated value of \(g_f\) is:

fl2$gf[[k]]

$pi0
[1] 0.9213986

$a
[1] 19187.07

Questions for investigation

This suggests that decreases in the objective function can be avoided by using warmstarts. I see two possible ways forward: 1. use a warmstart every time (rather than the current default initialization); 2. only use a warmstart if a first attempt at optimization has failed. The first option would be simpler to implement, but could cause FLASH to get stuck in local maxima more easily.

Results

To determine whether it would be viable to use a warmstart every time, I refit the first 22 factors and compare the time required to optimize and the objective attained.

# This block was run in advance.

res.no.warmstart <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd",
                                     verbose=TRUE)
res.warmstart <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd",
                                  warmstart=TRUE, verbose=TRUE)

saveRDS(res.no.warmstart, "../data/warmstart/nowarmstart.rds")
saveRDS(res.warmstart, "../data/warmstart/warmstart.rds")

res.no.warmstart <- readRDS("./data/warmstart/nowarmstart.rds")
res.warmstart <- readRDS("./data/warmstart/warmstart.rds")

x1 <- unlist(res.no.warmstart$opt_time)
x2 <- unlist(res.warmstart$opt_time)
list(no.warmstart = sum(x1), warmstart = sum(x2))

$no.warmstart
[1] 327.7671

$warmstart
[1] 259.5182

plot(x1, ylim=c(0, max(x1) + 1), pch=19, col="blue",
     xlab="Factor/loading index", ylab="Optimization time (s)")
points(x2, pch=17, col="red")
legend("topleft", c("No warmstart", "Warmstart"),
       pch=c(19, 17), col=c("blue", "red"))

list(no.warmstart = flash_get_objective(data, res.no.warmstart$f),
     warmstart = flash_get_objective(data, res.warmstart$f))

$no.warmstart
[1] -1256871

$warmstart
[1] -1256873

o1 <- sapply(res.no.warmstart$obj, 
             function(obj) {max(unlist(obj))})
o2 <- sapply(res.warmstart$obj, 
             function(obj) {max(unlist(obj))})

plot(o2 - o1, type='l',
     xlab="Factor/loading index",
     ylab="Diff. in obj. using warmstart",
     main="Difference in overall objective after adding each factor")

Conclusions

Using warmstarts prevents the large decreases in the objective function that I have described here and previously. Using a warmstart for all iterations yields a small speed-up, but can give slightly worse results (as measured by the objective function). However, using a warmstart only when the default initialization fails would be complicated to implement. For this reason, I recommend that we use a warmstart for all iterations.

Coda

Out of curiosity, I continued running flash_add_greedy using warmstarts to see how many factor/loading pairs it ended up adding. 25 factor/loading pairs were included in the final FLASH fit, which attained an objective of -1255910.7. This is an improvement of 3 factor/loading pairs and 994 log likelihood units over the inital (problematic) fit obtained without using any warmstarts. To verify these results, run the following:

res.final <- flash_add_greedy(strong, Kmax=50, init_fn="udv_svd",
                              warmstart=TRUE, verbose=TRUE)
flash_get_nfactors(res.final$f)
flash_get_objective(data, res.final$f) - flash_get_objective(data, res$f)

Session information

sessionInfo()

R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS High Sierra 10.13.6

Matrix products: default
BLAS: /Library/Frameworks/R.framework/Versions/3.4/Resources/lib/libRblas.0.dylib
LAPACK: /Library/Frameworks/R.framework/Versions/3.4/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] ebnm_0.1-13   flashr_0.5-12

loaded via a namespace (and not attached):
 [1] Rcpp_0.12.17        pillar_1.2.1        plyr_1.8.4         
 [4] compiler_3.4.3      git2r_0.21.0        workflowr_1.0.1    
 [7] R.methodsS3_1.7.1   R.utils_2.6.0       iterators_1.0.9    
[10] tools_3.4.3         testthat_2.0.0      digest_0.6.15      
[13] tibble_1.4.2        evaluate_0.10.1     memoise_1.1.0      
[16] gtable_0.2.0        lattice_0.20-35     rlang_0.2.0        
[19] Matrix_1.2-12       foreach_1.4.4       commonmark_1.4     
[22] yaml_2.1.17         parallel_3.4.3      withr_2.1.1.9000   
[25] stringr_1.3.0       roxygen2_6.0.1.9000 xml2_1.2.0         
[28] knitr_1.20          devtools_1.13.4     rprojroot_1.3-2    
[31] grid_3.4.3          R6_2.2.2            rmarkdown_1.8      
[34] ggplot2_2.2.1       ashr_2.2-10         magrittr_1.5       
[37] whisker_0.3-2       backports_1.1.2     scales_0.5.0       
[40] codetools_0.2-15    htmltools_0.3.6     MASS_7.3-48        
[43] assertthat_0.2.0    softImpute_1.4      colorspace_1.3-2   
[46] stringi_1.1.6       lazyeval_0.2.1      munsell_0.4.3      
[49] doParallel_1.0.11   pscl_1.5.2          truncnorm_1.0-8    
[52] SQUAREM_2017.10-1   R.oo_1.21.0