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<h1 class="title toc-ignore">Parallelizing MASH v FLASH backfits</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>8/14/2018</em></h4>
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<p><strong>Last updated:</strong> 2018-08-16</p>
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Great! You are using Git for version control. Tracking code development and connecting the code version to the results is critical for reproducibility. The version displayed above was the version of the Git repository at the time these results were generated. <br><br> Note that you need to be careful to ensure that all relevant files for the analysis have been committed to Git prior to generating the results (you can use <code>wflow_publish</code> or <code>wflow_git_commit</code>). workflowr only checks the R Markdown file, but you know if there are other scripts or data files that it depends on. Below is the status of the Git repository when the results were generated:
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<p></details></p>
<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>To further explore the parallel updates implemented in the <a href="parallel.html">previous analysis</a>, I attempt to parallelize the backfits performed in my <a href="https://willwerscheid.github.io/MASHvFLASH/MASHvFLASHgtex3.html">MASH v FLASH GTEx analysis</a>.</p>
<p>While <code>flash_add_greedy</code> and <code>flash_add_factors_from_data</code> both yield factor/loading pairs that are already relatively close to optimal, MASH v FLASH poses more difficult optimization problems in that its “canonical” loadings can be far from optimal, and the data-driven loadings obtained from the “strong” dataset do not necessarily fit the “random” dataset very well. Further, the full set of loadings forms an overcomplete basis for <span class="math inline">\(\mathbb{R}^{44}\)</span>, so that many of the loadings are, in a sense, redundant.</p>
</div>
<div id="experiments" class="section level2">
<h2>Experiments</h2>
<p>I use the three backfitting methods described in the <a href="parallel.html">previous analysis</a> to perform the two backfits described <a href="https://willwerscheid.github.io/MASHvFLASH/MASHvFLASHgtex3.html">here</a>. The code used in this analysis is included in the previous analysis (see <a href="parallel.html#code">here</a>).</p>
</div>
<div id="results" class="section level2">
<h2>Results</h2>
<div id="fitting-priors-to-the-random-dataset" class="section level3">
<h3>Fitting priors to the random dataset</h3>
<p>The first attempt to parallelize the backfit was a disaster. The objective for the first five iterations was:</p>
<pre class="r"><code>res_random_bad <- readRDS("./data/parallel/MASHvFLASHrandom_bad.rds")
knitr::kable(data.frame("Iteration" = 1:5, "Objective" = res_random_bad$parallel_obj))</code></pre>
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<th align="right">Iteration</th>
<th align="right">Objective</th>
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</thead>
<tbody>
<tr class="odd">
<td align="right">1</td>
<td align="right">-1845538</td>
</tr>
<tr class="even">
<td align="right">2</td>
<td align="right">-3964293</td>
</tr>
<tr class="odd">
<td align="right">3</td>
<td align="right">-19971902</td>
</tr>
<tr class="even">
<td align="right">4</td>
<td align="right">-138413167</td>
</tr>
<tr class="odd">
<td align="right">5</td>
<td align="right">-1024826913</td>
</tr>
</tbody>
</table>
<p>The problem, I think, is that since (as mentioned above) the loadings form an overcomplete basis, an update to the equal effects factor could also be achieved (roughly speaking) by updates to each of the unique effects factors. When all are updated simultaneously, the updates overshoot their goal, causing the objective to spiral out of control.</p>
<p>If this reasoning is correct, then better results might be obtained by splitting up the loadings into subsets of loadings that are mutually orthogonal (or nearly so), and then parallelizing the updates to the loadings within each subset. To test this hypothesis, I divided the loadings into an equal effects loading, 44 unique effects loadings (which are of course mutually orthogonal), two data-driven loadings that are distributed across several tissues (loadings 2 and 11 shown <a href="https://willwerscheid.github.io/MASHvFLASH/MASHvFLASHnn.html#multi-tissue_effects">here</a>), and the remaining 11 data-driven loadings (each of which is primarily loaded on two or three tissues).</p>
<p>To update the loadings, then, I updated the equal effects loading, then I updated the unique effects loadings in parallel, then I updated data-driven loadings 2 and 11, and finally I updated the remaining data-driven loadings in parallel. Surprisingly, I was able to achieve a better objective than I achieved using a serial backfit.</p>
<pre class="r"><code>res_random <- readRDS("./data/parallel/MASHvFLASHrandom.rds")
all_obj <- c(res_random$backfit_obj, res_random$parallel_obj)
plot(res_random$backfit_obj, pch=19, col="blue",
xlim=c(1, 20), ylim=c(min(all_obj), max(all_obj)),
xlab="Update", ylab="Objective")
points(res_random$parallel_obj, pch=19, col="red")
legend("bottomright", legend=c("standard", "parallel"),
pch=c(19, 19), col=c("blue", "red"))</code></pre>
<p><img src="figure/parallel2.Rmd/random_obj-1.png" width="672" style="display: block; margin: auto;" /></p>
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<td style="text-align:left;">
Jason Willwerscheid
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2018-08-15
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<p></details></p>
<p>The difference in objective attained (that is, the serial objective minus the parallel objective) is as follows:</p>
<pre class="r"><code>y <- res_random$backfit_obj - res_random$parallel_obj
plot(1:length(y), y, type="l", xlim=c(1, 20), ylim=c(min(y), max(y)),
xlab="Update", ylab="Difference")</code></pre>
<p><img src="figure/parallel2.Rmd/random_diff-1.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/willwerscheid/FLASHvestigations/blob/312981fac71aa857f0259637fd0bfd2ac1309429/docs/figure/parallel2.Rmd/random_diff-1.png" target="_blank">312981f</a>
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Jason Willwerscheid
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<p></details></p>
<p>As expected, the parallel updates are much faster (even after dividing the loadings into four subsets).</p>
<pre class="r"><code>data <- data.frame(standard = res_random$backfit_t,
lapply = res_random$parallel_t,
mclapply = res_random$multicore_t)
boxplot(data, ylim=c(0, max(data)), ylab="Time per iter (s)")</code></pre>
<p><img src="figure/parallel2.Rmd/random_t1-1.png" width="672" style="display: block; margin: auto;" /></p>
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<td style="text-align:left;">
Jason Willwerscheid
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2018-08-15
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<p></details></p>
<p>The total time (in seconds) required for 20 backfitting iterations is:</p>
<pre class="r"><code>colSums(data)</code></pre>
<pre><code>standard lapply mclapply
573.111 286.869 222.333 </code></pre>
</div>
<div id="backfitting-the-strong-dataset" class="section level3">
<h3>Backfitting the strong dataset</h3>
<p>The same trick is also needed to backfit the “strong” dataset using the same loadings as above and using the priors obtained by backfitting the “random” dataset. If I try to update the full set of loadings in parallel, then the objective again diverges to <span class="math inline">\(-\infty\)</span>. But if I do the parallel updates in four chunks (as described above), I obtain an objective that beats the serial method (after 20 iterations, at least):</p>
<pre class="r"><code>res_final <- readRDS("./data/parallel/MASHvFLASHfinal.rds")
all_obj <- c(res_final$backfit_obj, res_final$parallel_obj)
plot(res_final$backfit_obj, pch=19, col="blue",
xlim=c(1, 20), ylim=c(min(all_obj), max(all_obj)),
xlab="Update", ylab="Objective")
points(res_final$parallel_obj, pch=19, col="red")
legend("bottomright", legend=c("standard", "parallel"),
pch=c(19, 19), col=c("blue", "red"))</code></pre>
<p><img src="figure/parallel2.Rmd/final_obj-1.png" width="672" style="display: block; margin: auto;" /></p>
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Jason Willwerscheid
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2018-08-15
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<p></details></p>
<p>The difference in objective attained is as follows:</p>
<pre class="r"><code>y <- res_final$backfit_obj - res_final$parallel_obj
plot(1:length(y), y, type="l", xlim=c(1, 20), ylim=c(min(y), max(y)),
xlab="Update", ylab="Difference")</code></pre>
<p><img src="figure/parallel2.Rmd/final_diff-1.png" width="672" style="display: block; margin: auto;" /></p>
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<a href="https://github.com/willwerscheid/FLASHvestigations/blob/312981fac71aa857f0259637fd0bfd2ac1309429/docs/figure/parallel2.Rmd/final_diff-1.png" target="_blank">312981f</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-15
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The difference in elapsed time is especially dramatic here. The parallel updates take less than a third of the time required by the serial updates:</p>
<pre class="r"><code>data <- data.frame(standard = res_final$backfit_t,
lapply = res_final$parallel_t,
mclapply = res_final$multicore_t)
boxplot(data, ylim=c(0, max(data)), ylab="Time per iter (s)")</code></pre>
<p><img src="figure/parallel2.Rmd/final_t1-1.png" width="672" style="display: block; margin: auto;" /></p>
<details> <summary><em>Expand here to see past versions of final_t1-1.png:</em></summary>
<table style="border-collapse:separate; border-spacing:5px;">
<thead>
<tr>
<th style="text-align:left;">
Version
</th>
<th style="text-align:left;">
Author
</th>
<th style="text-align:left;">
Date
</th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align:left;">
<a href="https://github.com/willwerscheid/FLASHvestigations/blob/312981fac71aa857f0259637fd0bfd2ac1309429/docs/figure/parallel2.Rmd/final_t1-1.png" target="_blank">312981f</a>
</td>
<td style="text-align:left;">
Jason Willwerscheid
</td>
<td style="text-align:left;">
2018-08-15
</td>
</tr>
</tbody>
</table>
<p></details></p>
<p>The total time (in seconds) required is:</p>
<pre class="r"><code>colSums(data)</code></pre>
<pre><code>standard lapply mclapply
322.891 103.770 93.890 </code></pre>
</div>
</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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