Decreases in the FLASH objective function

Last updated: 2018-07-20

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File	Version	Author	Date	Message
Rmd	d1a6b4f	Jason Willwerscheid	2018-07-20	wflow_publish(c(“analysis/objective.Rmd”,
html	b7b744f	Jason Willwerscheid	2018-07-20	Build site.
Rmd	7a2c0ba	Jason Willwerscheid	2018-07-20	wflow_publish(“analysis/objective.Rmd”)
html	456972e	Jason Willwerscheid	2018-07-16	Build site.
Rmd	055d572	Jason Willwerscheid	2018-07-16	wflow_publish(“analysis/objective.Rmd”)
html	7db12a1	Jason Willwerscheid	2018-07-16	Build site.
Rmd	87a47fd	Jason Willwerscheid	2018-07-16	wflow_publish(“analysis/objective.Rmd”, republish = T)
html	c7b5ab1	Jason Willwerscheid	2018-07-15	Build site.
Rmd	8258811	Jason Willwerscheid	2018-07-15	wflow_publish(“analysis/objective.Rmd”)
html	0b30bef	Jason Willwerscheid	2018-07-15	Build site.
Rmd	db11cbb	Jason Willwerscheid	2018-07-15	wflow_publish(“analysis/objective.Rmd”)

Introduction

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

Example

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:-1298710.64322328

Objective:-1297543.71231454

Objective:-1297377.02562144

Objective:-1297291.11358246

Objective:-1297239.23330907

Objective:-1297207.28393276

Objective:-1297187.26387941

Objective:-1297174.44135814

Objective:-1297166.02132697

Objective:-1297160.35163403

Objective:-1297156.46023221

Objective:-1297153.76860079

Objective:-1297151.90730432

Objective:-1297150.62250311

Objective:-1297149.73767208

Objective:-1297149.13102524

Objective:-1297148.71858473

Objective:-1297148.44214222

Objective:-1297148.26107547

Objective:-1297148.14684887

Objective:-1297148.07930637

Objective:-1297148.04415067

Objective:-1297148.0312139

Objective:-1297148.033256

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:-1301896.87271162

objective from keeping factor:-1297148.033256

nullcheck complete, objective:-1297148.033256

Illustration of problem

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. I discard the first 8 iterations in order to zoom in on the problem area. (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)

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 plot-1.png:

Version	Author	Date
b7b744f	Jason Willwerscheid	2018-07-20
7db12a1	Jason Willwerscheid	2018-07-16
0b30bef	Jason Willwerscheid	2018-07-15

Analysis

I backtrack to just before the “bad” update.

res2 <- flash_add_greedy(strong, Kmax=4, stopAtObj = -1297148.032)

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] 0.002265139

So at this point, the objective is indeed better than for the flash fit obtained above.

Now, I update the precision.

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

R2 = flashr:::flash_get_R2(data, fl)
fl$tau = flashr:::compute_precision(R2, data$missing,
                                    "by_column", data$S)
flash_get_objective(strong, fl) - flash_get_objective(strong, init_fl)

[1] 0.04306088

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

last_fl = fl

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, last_fl)

[1] -0.1585035

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, tightening the tolerance parameter by adding factr = 100 does not change anything:

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(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, last_fl)

[1] -0.1585035

Conclusions

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. 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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