Last updated: 2018-07-15

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Introduction

Here I begin to look into why the FLASH objective function can decrease after an iteration.

Illustration of problem

I’m using the “strong” tests from the MASH paper GTEx dataset. The first problem appears when fitting the fourth factor. Notice that in the final iteration, the objective decreases by a very small amount and a warning is displayed.

# 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 <- gtex$strong.z
res <- flash_add_greedy(strong, Kmax=3, verbose=FALSE)
fitting factor/loading 1
fitting factor/loading 2
fitting factor/loading 3
res <- flash_add_greedy(strong, f_init=res$f, Kmax=1, verbose=TRUE)
fitting factor/loading 1
Objective:-1298710.77860735
Objective:-1297543.73149909
Objective:-1297376.91372722
Objective:-1297290.91328428
Objective:-1297238.97898488
Objective:-1297206.99734743
Objective:-1297186.95839066
Objective:-1297174.12546441
Objective:-1297165.70006397
Objective:-1297160.02771237
Objective:-1297156.13460765
Objective:-1297153.44144442
Objective:-1297151.57875474
Objective:-1297150.29283403
Objective:-1297149.40714787
Objective:-1297148.79985992
Objective:-1297148.38694209
Objective:-1297148.11014097
Objective:-1297147.92880915
Objective:-1297147.81438557
Objective:-1297147.74670432
Objective:-1297147.71143507
Objective:-1297147.69841607
Objective:-1297147.70039797
Warning in r1_opt(flash_get_Rk(data, f, k), flash_get_R2k(data, f, k), f
$EL[, : 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:-1301896.25041515
objective from keeping factor:-1297147.70039797
nullcheck complete, objective:-1297147.70039797

Analysis

A more granular tracking of the objective function reveals a larger problem. Recall that there are three steps in each iteration: updating the precision matrix, updating the factors (via the prior \(g_f\)), and updating the loadings (via \(g_l\)). Plotting the objective after each step rather than each iteration reveals a sawtooth pattern. (See branch trackObj, file r1_opt.R for the code used to obtain these results.)

obj_data <- as.vector(rbind(res$obj[[1]]$after_tau,
                            res$obj[[1]]$after_f,
                            res$obj[[1]]$after_l))
max_obj <- max(obj_data)
obj_data <- obj_data - max_obj
iter <- 1:length(obj_data) / 3

plt_xlab = "Iteration"
plt_ylab = "Diff. from maximum obj."
plot(iter, obj_data, type='l', xlab=plt_xlab, ylab=plt_ylab)

Expand here to see past versions of plot-1.png:
Version Author Date
0b30bef Jason Willwerscheid 2018-07-15

Discarding the first 8 iterations in order to zoom in on the problem area:

obj_data <- obj_data[-(1:24)]
iter <- iter[-(1:24)]
plt_colors <- c("indianred1", "indianred3", "indianred4")
plt_pch <- c(16, 17, 15)

plot(iter, obj_data, col=plt_colors, pch=plt_pch,
     xlab=plt_xlab, ylab=plt_ylab)
legend("bottomright", c("after tau", "after f", "after l"),
       col=plt_colors, pch=plt_pch)

Expand here to see past versions of plot2-1.png:
Version Author Date
0b30bef Jason Willwerscheid 2018-07-15

I backtrack to just before the “bad” update.

res2 <- flash_add_greedy(strong, Kmax=4, stopAtObj=-1297147.7)
fitting factor/loading 1
fitting factor/loading 2
fitting factor/loading 3
fitting factor/loading 4
flash_get_objective(strong, res2$f) - flash_get_objective(strong, res$f)
[1] 0.002208033

So at this point, the objective is indeed better than for the flash object attained above. The component parts of the objective are:

fl <- res2$f
data <- flash_set_data(strong)
k <- 4

KL_l <- fl$KL_l[[k]]
KL_f <- fl$KL_f[[k]]
loglik <- flashr:::e_loglik(data, fl)
list(KL_l = KL_l, KL_f = KL_f, loglik = loglik)
$KL_l
[1] -8324.579

$KL_f
[1] -128.9953

$loglik
[1] -1227372

First I update the precision (I follow the code in r1_opt). Only the “loglik” component is affected by this update:

init_fl = fl
init_KL_l = KL_l
init_KL_f = KL_f
init_loglik = loglik

R2 = flashr:::flash_get_R2(data, fl)
fl$tau = flashr:::compute_precision(R2, data$missing, 
                                    "by_column", data$S)
flashr:::e_loglik(data, fl) - init_loglik
[1] 0.04309978

So the overall objective indeed increases. Now I update the loadings (FLASH updates factors first, but the order of updates is not supposed to affect the monotonicity of the objective function).

s2 = 1/(fl$EF2[, k] %*% t(fl$tau))
s = sqrt(s2)
Rk = flashr:::flash_get_Rk(data, fl, k)
x = fl$EF[, k] %*% t(Rk * fl$tau) * s2
ebnm_l = flashr:::ebnm_pn(x, s, list())
KL_l = (ebnm_l$penloglik 
        - flashr:::NM_posterior_e_loglik(x, s, ebnm_l$postmean,
                                         ebnm_l$postmean2))

fl$EL[, k] = ebnm_l$postmean
fl$EL2[, k] = ebnm_l$postmean2
fl$gl[[k]] = ebnm_l$fitted_g
fl$KL_l[[k]] = KL_l
flash_get_objective(data, fl) - flash_get_objective(data, init_fl)
[1] -0.1154585

So the objective has in fact gotten worse. And tightening the control parameters or changing the initialization for the ebnm function does not help matters. For example:

s2 = 1/(fl$EF2[, k] %*% t(fl$tau))
s = sqrt(s2)
Rk = flashr:::flash_get_Rk(data, fl, k)
x = fl$EF[, k] %*% t(Rk * fl$tau) * s2
ebnm_l = flashr:::ebnm_pn(x, s, list(startpar=c(5,5),
                                     control=list(factr=100)))
KL_l = (ebnm_l$penloglik 
        - flashr:::NM_posterior_e_loglik(x, s, ebnm_l$postmean,
                                         ebnm_l$postmean2))

fl$EL[, k] = ebnm_l$postmean
fl$EL2[, k] = ebnm_l$postmean2
fl$gl[[k]] = ebnm_l$fitted_g
fl$KL_l[[k]] = KL_l
flash_get_objective(data, fl) - flash_get_objective(data, init_fl)
[1] -0.1154585

Conclusions and questions

The decrease appears too large to be explained by numerical error. Indeed, it would be very surprising to me if EL and EL2 could only be trusted to five digits or so (as would have to be the case to produce errors of the above magnitude).

More seriously, the sawtooth pattern depicted above points to a more regular feature of the optimization. Indeed, it appears that all of the triangles (objectives after updating factors) are biased upwards and all of the squares (objectives after updating loadings) are biased slightly downwards.

The theory appears to be sound, so what is going on here?

Session information

sessionInfo()
R version 3.4.3 (2017-11-30)
Platform: x86_64-apple-darwin15.6.0 (64-bit)
Running under: macOS Sierra 10.12.6

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_0.1-12   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

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