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<h1 class="title toc-ignore">Accelerating backfits with SQUAREM and DAAREM</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>8/17/2018</em></h4>
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<p><strong>Last updated:</strong> 2018-08-17</p>
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>This analysis is complementary to my investigations into parallelization (see <a href="parallel.html">here</a> and <a href="parallel2.html">here</a>) in that I further explore ways to speed up FLASH backfits.</p>
<p>I use two off-the-shelf EM (and MM) accelerators, SQUAREM and DAAREM. For SQUAREM details, see <a href="https://www.jstor.org/stable/41548597">Varadhan and Roland (2008)</a>. For DAAREM, see <a href="https://arxiv.org/abs/1803.06673">Henderson and Varadhan (2018)</a>.</p>
</div>
<div id="experiments" class="section level2">
<h2>Experiments</h2>
<p>I use the same GTEx dataset that I use <a href="https://willwerscheid.github.io/MASHvFLASH/MASHvFLASHgtex.html">here</a> and in <a href="parallel.html">Investigation 8</a>.</p>
<p>I use <code>flash_add_greedy</code> to create three flash fit objects with, respectively, 5, 10, and 20 factor/loading pairs. I then refine each fit using <code>flash_backfit</code> with: 1. no acceleration; 2. acceleration via SQUAREM; and 3. acceleration via DAAREM.</p>
</div>
<div id="results" class="section level2">
<h2>Results</h2>
<p>Since the experiments take a long time to run, I pre-run the code <a href="#code">below</a> and load the results from file.</p>
<pre class="r"><code>res5 <- readRDS("./data/squarem/res5.rds")
res10 <- readRDS("./data/squarem/res10.rds")
res20 <- readRDS("./data/squarem/res20.rds")</code></pre>
<div id="convergence-behavior" class="section level3">
<h3>Convergence behavior</h3>
<p>In every case, DAAREM takes the fewest iterations to converge and convergence is nearly monotonic. In constrast, SQUAREM evinces highly non-monotonic behavior and tends to take more iterations than backfitting with no acceleration at all.</p>
<pre class="r"><code>plot_obj <- function(res, main) {
data <- c(res$backfit_obj, res$squarem_obj$V1, res$daarem_obj)
plot(1:length(res$backfit_obj), res$backfit_obj,
type='l', col='red', ylim=c(min(data), max(data)),
xlab="Iteration", ylab="Objective", main=main)
lines(1:length(res$squarem_obj$V1), res$squarem_obj$V1,
col='blue')
lines(1:length(res$daarem_obj), res$daarem_obj,
col='green')
legend("bottomright", legend=c("DAAREM", "No acceleration", "SQUAREM"),
lty=1, col=c('green', 'red', 'blue'))
}
plot_obj(res5, "5 factor model")</code></pre>
<p><img src="figure/squarem.Rmd/conv-1.png" width="672" style="display: block; margin: auto;" /></p>
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<pre class="r"><code>plot_obj(res10, "10 factor model")</code></pre>
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<pre class="r"><code>plot_obj(res20, "20 factor model")</code></pre>
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<p></details></p>
</div>
<div id="final-objective" class="section level3">
<h3>Final objective</h3>
<p>Since the final objectives attained are difficult to see in the plots above, I list the differences in the table below. In every case, DAAREM beats the final objective attained with no acceleration. SQUAREM does much worse on the 5- and 10-factor models, but better on the 20-factor model.</p>
<pre class="r"><code>final_obj <- function(res) {
backfit = res$backfit_obj[length(res$backfit_obj)]
return(c(res$daarem_obj[length(res$daarem_obj)] - backfit,
res$squarem_obj$V1[length(res$squarem_obj$V1)] - backfit))
}
table_data <- cbind("5_factors" = final_obj(res5),
"10_factors" = final_obj(res10),
"20_factors" = final_obj(res20))
rownames(table_data) = c("DAAREM (diff from backfit)",
"SQUAREM (diff from backfit)")
knitr::kable(table_data, digits=1)</code></pre>
<table>
<thead>
<tr class="header">
<th></th>
<th align="right">5_factors</th>
<th align="right">10_factors</th>
<th align="right">20_factors</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>DAAREM (diff from backfit)</td>
<td align="right">0.1</td>
<td align="right">3.8</td>
<td align="right">9.1</td>
</tr>
<tr class="even">
<td>SQUAREM (diff from backfit)</td>
<td align="right">-56.5</td>
<td align="right">-48.4</td>
<td align="right">12.4</td>
</tr>
</tbody>
</table>
</div>
<div id="time-per-iteration" class="section level3">
<h3>Time per iteration</h3>
<p>The acceleration methods add a bit of overhead in terms of time required per iteration, but not much. The following are seconds required per iteration:</p>
<pre class="r"><code>times <- function(res) {
return(c(res$backfit_t[3] / length(res$backfit_obj),
res$daarem_t[3] / length(res$daarem_obj),
res$squarem_t[3] / length(res$squarem_obj$V1)))
}
table_data <- cbind("5_factors" = times(res5),
"10_factors" = times(res10),
"20_factors" = times(res20))
rownames(table_data) <- c("backfit", "DAAREM", "SQUAREM")
knitr::kable(table_data, digits = 2)</code></pre>
<table>
<thead>
<tr class="header">
<th></th>
<th align="right">5_factors</th>
<th align="right">10_factors</th>
<th align="right">20_factors</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>backfit</td>
<td align="right">1.35</td>
<td align="right">2.50</td>
<td align="right">5.57</td>
</tr>
<tr class="even">
<td>DAAREM</td>
<td align="right">1.37</td>
<td align="right">2.71</td>
<td align="right">5.53</td>
</tr>
<tr class="odd">
<td>SQUAREM</td>
<td align="right">1.47</td>
<td align="right">2.79</td>
<td align="right">5.60</td>
</tr>
</tbody>
</table>
</div>
<div id="total-iterations-to-convergence" class="section level3">
<h3>Total iterations to convergence</h3>
<p>Perhaps the most important consideration is how quickly we can obtain an estimate that is reasonable close to convergence, where “reasonably close” is defined by some stopping rule. Below I give the number of iterations required given different tolerance parameters (i.e., the number of iterations required before the difference in objective from one iteration to the next is less than <code>tol</code>). (I omit results for SQUAREM since it does so poorly in other respects.)</p>
<pre class="r"><code>niter <- function(res, tols) {
backfit_diff <- (res$backfit_obj[2:length(res$backfit_obj)] -
res$backfit_obj[1:(length(res$backfit_obj) - 1)])
daarem_diff <- (res$daarem_obj[2:length(res$daarem_obj)] -
res$daarem_obj[1:(length(res$daarem_obj) - 1)])
res <- matrix(NA, nrow = length(tols), ncol = 2)
for (i in 1:length(tols)) {
res[i, 1] <- min(which(backfit_diff < tols[i])) + 1
res[i, 2] <- min(which(abs(daarem_diff) < tols[i])) + 1
}
rownames(res) = paste("tol =", as.character(tols))
colnames(res) = c("backfit", "DAAREM")
return(t(res))
}
table_data <- niter(res5, c(0.5, 0.1, 0.05))
knitr::kable(table_data, caption = "5 factors")</code></pre>
<table>
<caption>5 factors</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">tol = 0.5</th>
<th align="right">tol = 0.1</th>
<th align="right">tol = 0.05</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>backfit</td>
<td align="right">240</td>
<td align="right">283</td>
<td align="right">286</td>
</tr>
<tr class="even">
<td>DAAREM</td>
<td align="right">142</td>
<td align="right">142</td>
<td align="right">174</td>
</tr>
</tbody>
</table>
<pre class="r"><code>table_data <- niter(res10, c(0.5, 0.1, 0.05))
knitr::kable(table_data, caption = "10 factors")</code></pre>
<table>
<caption>10 factors</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">tol = 0.5</th>
<th align="right">tol = 0.1</th>
<th align="right">tol = 0.05</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>backfit</td>
<td align="right">192</td>
<td align="right">201</td>
<td align="right">208</td>
</tr>
<tr class="even">
<td>DAAREM</td>
<td align="right">111</td>
<td align="right">116</td>
<td align="right">116</td>
</tr>
</tbody>
</table>
<pre class="r"><code>table_data <- niter(res20, c(0.5, 0.1, 0.05))
knitr::kable(table_data, caption = "20 factors")</code></pre>
<table>
<caption>20 factors</caption>
<thead>
<tr class="header">
<th></th>
<th align="right">tol = 0.5</th>
<th align="right">tol = 0.1</th>
<th align="right">tol = 0.05</th>
</tr>
</thead>
<tbody>
<tr class="odd">
<td>backfit</td>
<td align="right">183</td>
<td align="right">452</td>
<td align="right">483</td>
</tr>
<tr class="even">
<td>DAAREM</td>
<td align="right">127</td>
<td align="right">129</td>
<td align="right">129</td>
</tr>
</tbody>
</table>
</div>
</div>
<div id="code" class="section level2">
<h2>Code</h2>
<p>Click “Code” to view the code used to produce the above results.</p>
<pre class="r"><code># Load packages ---------------------------------------------------------
# devtools::install_github("stephenslab/flashr")
devtools::load_all("/Users/willwerscheid/GitHub/flashr/")
# devtools::install_github("stephenslab/ebnm")
devtools::load_all("/Users/willwerscheid/GitHub/ebnm/")
library(SQUAREM)
library(daarem)
# Load data -------------------------------------------------------------
gtex <- readRDS(gzcon(url("https://github.com/stephenslab/gtexresults/blob/master/data/MatrixEQTLSumStats.Portable.Z.rds?raw=TRUE")))
data <- flash_set_data(t(gtex$strong.z), S = 1)
# Create flash objects for backfitting ----------------------------------
fl_g5 <- flash_add_greedy(data, Kmax = 5,
var_type = "zero", init_fn = "udv_svd")
fl_g10 <- flash_add_greedy(data, Kmax = 5, f_init = fl_g5,
var_type = "zero", init_fn = "udv_svd")
fl_g20 <- flash_add_greedy(data, Kmax = 10, f_init = fl_g10,
var_type = "zero", init_fn = "udv_svd")
# Testing functions -----------------------------------------------------
logit <- function(x) {log(x / (1 - x))}
inv_logit <- function(x) {exp(x) / (1 + exp(x))}
fl.to.param <- function(f) {
c(as.vector(f$EL), as.vector(f$EF),
as.vector(f$EL2), as.vector(f$EF2),
sapply(f$gl, function(k) {logit(k$pi0)}),
sapply(f$gf, function(k) {logit(k$pi0)}),
sapply(f$gl, function(k) {log(k$a)}),
sapply(f$gf, function(k) {log(k$a)}))
}
param.to.fl <- function(param, m, n, k) {
LL = matrix(param[1:(m*k)], ncol=k)
curr_idx = m*k
FF = matrix(param[(curr_idx + 1):(curr_idx + n*k)], ncol=k)
curr_idx = curr_idx + n*k
f = flashr:::flash_init_lf(LL, FF)
f$EL2 = matrix(param[(curr_idx + 1):(curr_idx + m*k)], ncol=k)
curr_idx = curr_idx + m*k
f$EF2 = matrix(param[(curr_idx + 1):(curr_idx + n*k)], ncol=k)
curr_idx = curr_idx + n*k
f$gl = list()
f$gf = list()
for (i in 1:k) {
f$gl[[i]] = list()
f$gl[[i]]$pi0 = inv_logit(param[curr_idx + 1])
curr_idx = curr_idx + 1
}
for (i in 1:k) {
f$gf[[i]] = list()
f$gf[[i]]$pi0 = inv_logit(param[curr_idx + 1])
curr_idx = curr_idx + 1
}
for (i in 1:k) {
f$gl[[i]]$a = exp(param[curr_idx + 1])
curr_idx = curr_idx + 1
}
for (i in 1:k) {
f$gf[[i]]$a = exp(param[curr_idx + 1])
curr_idx = curr_idx + 1
}
return(f)
}
flash.iter <- function(p, data, m, n, k) {
init_fl = param.to.fl(p, m, n, k)
fl = flash_backfit(data, init_fl,
ebnm_fn = "ebnm_pn", var_type = "zero",
nullcheck = FALSE, verbose = FALSE,
maxiter = 1)
obj = flash_get_objective(data, fl)
message(obj)
return(c(fl.to.param(fl), obj))
}
flash.obj <- function(p, data, m, n, k) {
return(p[length(p)])
}
run_test <- function(f_init, data, niter) {
m <- flash_get_n(f_init)
n <- flash_get_p(f_init)
k <- flash_get_k(f_init)
backfit_obj <- rep(NA, niter)
f <- f_init
backfit_t <- system.time({
for (i in 1:niter) {
f <- flash_backfit(data, f,
ebnm_fn = "ebnm_pn", var_type = "zero",
nullcheck = FALSE, verbose = FALSE,
maxiter = 1)
obj <- flash_get_objective(data, f)
message(obj)
backfit_obj[i] <- obj
}
})
message("Sinking SQUAREM results to file...")
zz <- file("tmp.txt", open="wt")
sink(zz, type="message")
squarem_t <- system.time(
squarem_res <- squarem(c(fl.to.param(f_init),
flash_get_objective(data, f_init)),
flash.iter, flash.obj,
data = data, m = m, n = n, k = k,
control = (list(tol=1, maxiter=niter)))
)
sink(type="message")
squarem_obj <- read.csv("tmp.txt", header = FALSE, sep = '\n')
file.remove("tmp.txt")
daarem_t <- system.time(
daarem_res <- daarem(c(fl.to.param(f_init),
flash_get_objective(data, f_init)),
flash.iter, flash.obj,
data = data, m = m, n = n, k = k,
control = (list(tol=1, maxiter=niter)))
)
return(list(backfit_obj = backfit_obj, backfit_t = backfit_t,
squarem_obj = squarem_obj, squarem_t = squarem_t,
daarem_obj = daarem_res$objfn.track, daarem_t = daarem_t))
}
# Run tests -------------------------------------------------------------
fpath <- "./data/squarem/"
# Normal backfit of fl_g5 takes 304 iterations.
res_5 <- run_test(fl_g5, data, niter = 300)
saveRDS(res_5, paste0(fpath, "res5.rds"))
# Normal backfit of fl_g10 takes 228 iterations.
res_10 <- run_test(fl_g10, data, niter = 225)
saveRDS(res_10, paste0(fpath, "res10.rds"))
# Normal backfit of fl_g20 takes 497 iterations.
res_20 <- run_test(fl_g20, data, niter = 495)
saveRDS(res_20, paste0(fpath, "res20.rds"))
# Use this function if gf and gl parameters aren't converted to a
# log/logit scale.
#
# flash.iter <- function(p) {
# init_fl = param.to.fl(p[1:(length(p) - 1)])
# fl = try(flash_backfit(data, init_fl, ebnm_fn="ebnm_pn",
# ebnm_param=list(warmstart=TRUE),
# var_type="zero", nullcheck=F, maxiter=1))
# if (class(fl) == "try-error") {
# fl = flash_backfit(data, init_fl, ebnm_fn="ebnm_pn",
# ebnm_param=list(warmstart=FALSE),
# var_type="zero", nullcheck=F, maxiter=1)
# }
# return(c(f.to.param(fl), flash_get_objective(data, fl)))
# }
plot_obj <- function(res) {
data <- c(res$backfit_obj, res$squarem_obj$V1, res$daarem_obj)
plot(1:length(res$backfit_obj), res$backfit_obj,
type='l', col='red', ylim=c(min(data), max(data)),
xlab="Iteration", ylab="Objective")
lines(1:length(res$squarem_obj$V1), res$squarem_obj$V1,
col='blue')
lines(1:length(res$daarem_obj), res$daarem_obj,
col='green')
}
plot_obj_zoom <- function(res, yrange) {
data <- c(res$backfit_obj, res$squarem_obj$V1, res$daarem_obj)
max_obj <- max(data)
t <- max_obj - yrange
begin_iter <- min(c(which(res$backfit_obj > t),
which(res$squarem_obj > t),
which(res$daarem_obj > t)))
plot(1:length(res$backfit_obj), res$backfit_obj,
type='l', col='red',
xlim=c(begin_iter, length(res$backfit_obj)),
ylim=c(t, max_obj),
xlab="Iteration", ylab="Objective")
lines(1:length(res$squarem_obj$V1), res$squarem_obj$V1,
col='blue')
lines(1:length(res$daarem_obj), res$daarem_obj,
col='green')
}</code></pre>
</div>
<div id="session-information" class="section level2">
<h2>Session information</h2>
<pre class="r"><code>sessionInfo()</code></pre>
<pre><code>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
loaded via a namespace (and not attached):
[1] workflowr_1.0.1 Rcpp_0.12.17 digest_0.6.15
[4] rprojroot_1.3-2 R.methodsS3_1.7.1 backports_1.1.2
[7] git2r_0.21.0 magrittr_1.5 evaluate_0.10.1
[10] highr_0.6 stringi_1.1.6 whisker_0.3-2
[13] R.oo_1.21.0 R.utils_2.6.0 rmarkdown_1.8
[16] tools_3.4.3 stringr_1.3.0 yaml_2.1.17
[19] compiler_3.4.3 htmltools_0.3.6 knitr_1.20 </code></pre>
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