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<h1 class="title toc-ignore">Using warmstarts to improve optimization</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>7/26/2018</em></h4>
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<p><strong>Last updated:</strong> 2018-07-27</p>
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Here I turn to the occasional large decreases in the FLASH objective function, an example of which was given in the <a href="init_fn.html">previous investigation</a>. I trace the problem to the call to <code>optim</code> in <code>ebnm_pn</code>.</p>
</div>
<div id="example" class="section level2">
<h2>Example</h2>
<p>I use the same dataset as previously. However, since the example given in the previous investigation is not easily reproducible (due to randomness in <code>udv_si</code>), I re-fit the dataset using <code>udv_svd</code>. A large decrease in the objective function occurs while adding the 23rd factor. The last few lines of verbose output are as follows.</p>
<pre><code>(...)
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</code></pre>
</div>
<div id="analysis-of-problem" class="section level2">
<h2>Analysis of problem</h2>
<p>The problem, as I will show, is that <code>optim</code> fails to find a good solution to the EBNM problem.</p>
<p>I backtrack to just before the bad update using parameter <code>stopAtObj</code> (to reproduce this example, make sure to load branch <code>trackObj</code>).</p>
<p>Since the fits take some time, I pre-run the following code and then load the results from file.</p>
<pre class="r"><code># devtools::install_github("stephenslab/flashr", ref="trackObj")
devtools::load_all("/Users/willwerscheid/GitHub/flashr")</code></pre>
<pre><code>Loading flashr</code></pre>
<pre class="r"><code># devtools::install_github("stephenslab/ebnm")
devtools::load_all("/Users/willwerscheid/GitHub/ebnm")</code></pre>
<pre><code>Loading ebnm</code></pre>
<pre class="r"><code>gtex <- readRDS(gzcon(url("https://github.com/stephenslab/gtexresults/blob/master/data/MatrixEQTLSumStats.Portable.Z.rds?raw=TRUE")))
strong <- t(gtex$strong.z)</code></pre>
<pre class="r"><code># This block was run in advance.
res <- flash_add_greedy(strong, Kmax=50, init_fn="udv_svd", verbose=TRUE)
res <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd", verbose=TRUE)
res <- flash_add_greedy(strong, Kmax=1, f_init=res$f, init_fn="udv_svd",
verbose=TRUE, nullcheck=FALSE, stopAtObj=-1256905)
saveRDS(res, "../data/warmstart/greedy23.rds")</code></pre>
<p>I now step through the factor update code.</p>
<pre class="r"><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)</code></pre>
<pre><code>[1] -1256923</code></pre>
<p>So the objective is indeed worse at this point. I inspect the update to <span class="math inline">\(g_f\)</span>:</p>
<pre class="r"><code>list(before.update = fl.before.update$gf[[k]],
after.update = fl$gf[[k]])</code></pre>
<pre><code>$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</code></pre>
<p>This is a huge change to <span class="math inline">\(g_f\)</span>, much larger than one should reasonably expect from a single update. Indeed, if I initialize <code>ebnm_pn</code> using the current value of <code>gf</code>, I get a much different result.</p>
<pre class="r"><code>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)</code></pre>
<pre><code>[1] -1256901</code></pre>
<p>So the objective improves, as is guaranteed by the theory. The updated value of <span class="math inline">\(g_f\)</span> is:</p>
<pre class="r"><code>fl2$gf[[k]]</code></pre>
<pre><code>$pi0
[1] 0.9213986
$a
[1] 19187.07</code></pre>
</div>
<div id="questions-for-investigation" class="section level2">
<h2>Questions for investigation</h2>
<p>This suggests that decreases in the objective function can be avoided by using warmstarts. I see two possible ways forward: 1. use a warmstart every time (rather than the current default initialization); 2. only use a warmstart if a first attempt at optimization has failed. The first option would be simpler to implement, but could cause FLASH to get stuck in local maxima more easily.</p>
</div>
<div id="results" class="section level2">
<h2>Results</h2>
<p>To determine whether it would be viable to use a warmstart every time, I refit the first 22 factors and compare the time required to optimize and the objective attained.</p>
<pre class="r"><code># This block was run in advance.
res.no.warmstart <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd",
verbose=TRUE)
res.warmstart <- flash_add_greedy(strong, Kmax=22, init_fn="udv_svd",
warmstart=TRUE, verbose=TRUE)
saveRDS(res.no.warmstart, "../data/warmstart/nowarmstart.rds")
saveRDS(res.warmstart, "../data/warmstart/warmstart.rds")</code></pre>
<pre class="r"><code>res.no.warmstart <- readRDS("./data/warmstart/nowarmstart.rds")
res.warmstart <- readRDS("./data/warmstart/warmstart.rds")</code></pre>
<div id="optimization-time" class="section level3">
<h3>Optimization time</h3>
<p>The total time (in seconds) needed to optimize factors is:</p>
<pre class="r"><code>x1 <- unlist(res.no.warmstart$opt_time)
x2 <- unlist(res.warmstart$opt_time)
list(no.warmstart = sum(x1), warmstart = sum(x2))</code></pre>
<pre><code>$no.warmstart
[1] 327.7671
$warmstart
[1] 259.5182</code></pre>
<p>The time required per factor/loading is as follows.</p>
<pre class="r"><code>plot(x1, ylim=c(0, max(x1) + 1), pch=19, col="blue",
xlab="Factor/loading index", ylab="Optimization time (s)")
points(x2, pch=17, col="red")
legend("topleft", c("No warmstart", "Warmstart"),
pch=c(19, 17), col=c("blue", "red"))</code></pre>
<p><img src="figure/warmstart.Rmd/time_per_factor-1.png" width="672" style="display: block; margin: auto;" /></p>
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<p></details></p>
<p>So using a warmstart yields a small (but reliable) speed-up.</p>
</div>
<div id="objective-attained" class="section level3">
<h3>Objective attained</h3>
<p>Using a warmstart yields a slightly worse overall objective.</p>
<pre class="r"><code>list(no.warmstart = flash_get_objective(data, res.no.warmstart$f),
warmstart = flash_get_objective(data, res.warmstart$f))</code></pre>
<pre><code>$no.warmstart
[1] -1256871
$warmstart
[1] -1256873</code></pre>
<p>The 11th factor is the culprit:</p>
<pre class="r"><code>o1 <- sapply(res.no.warmstart$obj,
function(obj) {max(unlist(obj))})
o2 <- sapply(res.warmstart$obj,
function(obj) {max(unlist(obj))})
plot(o2 - o1, type='l',
xlab="Factor/loading index",
ylab="Diff. in obj. using warmstart",
main="Difference in overall objective after adding each factor")</code></pre>
<p><img src="figure/warmstart.Rmd/obj_diff-1.png" width="672" style="display: block; margin: auto;" /></p>
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<p></details></p>
</div>
</div>
<div id="conclusions" class="section level2">
<h2>Conclusions</h2>
<p>Using warmstarts prevents the large decreases in the objective function that I have described here and <a href="init_fn.html">previously</a>. Using a warmstart for all iterations yields a small speed-up, but can give slightly worse results (as measured by the objective function). However, using a warmstart only when the default initialization fails would be complicated to implement. For this reason, I recommend that we use a warmstart for all iterations.</p>
</div>
<div id="coda" class="section level2">
<h2>Coda</h2>
<p>Out of curiosity, I continued running <code>flash_add_greedy</code> using warmstarts to see how many factor/loading pairs it ended up adding. 25 factor/loading pairs were included in the final FLASH fit, which attained an objective of -1255910.7. This is an improvement of 3 factor/loading pairs and 994 log likelihood units over the inital (problematic) fit obtained without using any warmstarts. To verify these results, run the following:</p>
<pre class="r"><code>res.final <- flash_add_greedy(strong, Kmax=50, init_fn="udv_svd",
warmstart=TRUE, verbose=TRUE)
flash_get_nfactors(res.final$f)
flash_get_objective(data, res.final$f) - flash_get_objective(data, res$f)</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
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 </code></pre>
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