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<h1 class="title toc-ignore">Decreases in the Objective Function</h1>
<h4 class="author"><em>Jason Willwerscheid</em></h4>
<h4 class="date"><em>7/14/2018</em></h4>
</div>
<p><strong>Last updated:</strong> 2018-07-15</p>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>Here I begin to look into why the FLASH objective function can decrease after an iteration.</p>
</div>
<div id="illustration-of-problem" class="section level2">
<h2>Illustration of problem</h2>
<p>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.</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 <- gtex$strong.z
res <- flash_add_greedy(strong, Kmax=3, verbose=FALSE)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre class="r"><code>res <- flash_add_greedy(strong, f_init=res$f, Kmax=1, verbose=TRUE)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>Objective:-1298710.77860735</code></pre>
<pre><code>Objective:-1297543.73149909</code></pre>
<pre><code>Objective:-1297376.91372722</code></pre>
<pre><code>Objective:-1297290.91328428</code></pre>
<pre><code>Objective:-1297238.97898488</code></pre>
<pre><code>Objective:-1297206.99734743</code></pre>
<pre><code>Objective:-1297186.95839066</code></pre>
<pre><code>Objective:-1297174.12546441</code></pre>
<pre><code>Objective:-1297165.70006397</code></pre>
<pre><code>Objective:-1297160.02771237</code></pre>
<pre><code>Objective:-1297156.13460765</code></pre>
<pre><code>Objective:-1297153.44144442</code></pre>
<pre><code>Objective:-1297151.57875474</code></pre>
<pre><code>Objective:-1297150.29283403</code></pre>
<pre><code>Objective:-1297149.40714787</code></pre>
<pre><code>Objective:-1297148.79985992</code></pre>
<pre><code>Objective:-1297148.38694209</code></pre>
<pre><code>Objective:-1297148.11014097</code></pre>
<pre><code>Objective:-1297147.92880915</code></pre>
<pre><code>Objective:-1297147.81438557</code></pre>
<pre><code>Objective:-1297147.74670432</code></pre>
<pre><code>Objective:-1297147.71143507</code></pre>
<pre><code>Objective:-1297147.69841607</code></pre>
<pre><code>Objective:-1297147.70039797</code></pre>
<pre><code>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.</code></pre>
<pre><code>performing nullcheck</code></pre>
<pre><code>objective from deleting factor:-1301896.25041515</code></pre>
<pre><code>objective from keeping factor:-1297147.70039797</code></pre>
<pre><code>nullcheck complete, objective:-1297147.70039797</code></pre>
</div>
<div id="analysis" class="section level2">
<h2>Analysis</h2>
<p>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 <span class="math inline">\(g_f\)</span>), and updating the loadings (via <span class="math inline">\(g_l\)</span>). Plotting the objective after each step rather than each iteration reveals a sawtooth pattern. (See branch <code>trackObj</code>, file <code>r1_opt.R</code> for the code used to obtain these results.)</p>
<pre class="r"><code>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)</code></pre>
<p><img src="figure/objective.Rmd/plot-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>Discarding the first 8 iterations in order to zoom in on the problem area:</p>
<pre class="r"><code>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)</code></pre>
<p><img src="figure/objective.Rmd/plot2-1.png" width="672" style="display: block; margin: auto;" /></p>
<p>I backtrack to just before the “bad” update.</p>
<pre class="r"><code>res2 <- flash_add_greedy(strong, Kmax=4, stopAtObj=-1297147.7)</code></pre>
<pre><code>fitting factor/loading 1</code></pre>
<pre><code>fitting factor/loading 2</code></pre>
<pre><code>fitting factor/loading 3</code></pre>
<pre><code>fitting factor/loading 4</code></pre>
<pre class="r"><code>flash_get_objective(strong, res2$f) - flash_get_objective(strong, res$f)</code></pre>
<pre><code>[1] 0.002208033</code></pre>
<p>So at this point, the objective is indeed better than for the flash object attained above. The component parts of the objective are:</p>
<pre class="r"><code>fl <- res2$f
data <- flash_set_data(strong)
k <- 4
KL_l <- fl$KL_l[[k]]
KL_f <- fl$KL_f[[k]]
loglik <- flashr:::e_loglik(data, fl)
list(KL_l = KL_l, KL_f = KL_f, loglik = loglik)</code></pre>
<pre><code>$KL_l
[1] -8324.579
$KL_f
[1] -128.9953
$loglik
[1] -1227372</code></pre>
<p>First I update the precision (I follow the code in <code>r1_opt</code>). Only the “loglik” component is affected by this update:</p>
<pre class="r"><code>init_fl = fl
init_KL_l = KL_l
init_KL_f = KL_f
init_loglik = loglik
R2 = flashr:::flash_get_R2(data, fl)
fl$tau = flashr:::compute_precision(R2, data$missing,
"by_column", data$S)
flashr:::e_loglik(data, fl) - init_loglik</code></pre>
<pre><code>[1] 0.04309978</code></pre>
<p>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).</p>
<pre class="r"><code>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)</code></pre>
<pre><code>[1] -0.1154585</code></pre>
<p>So the objective has in fact gotten worse. And tightening the control parameters or changing the initialization for the <code>ebnm</code> function does not help matters. For example:</p>
<pre class="r"><code>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)</code></pre>
<pre><code>[1] -0.1154585</code></pre>
</div>
<div id="perturbation-analysis" class="section level2">
<h2>Perturbation analysis</h2>
<p>It’s possible that numerical error is responsible for the decrease, but it seems unlikely to me that this is the whole story.</p>
<p>Indeed, assume that numerical error is sufficient to explain the decrease. Recall that the objective consists of a part that is calculated from <code>R2</code> and <code>tau</code>, a part that comes from <code>KL_l</code>, and a part that comes from <code>KL_f</code>. The first part is coded as <code>-0.5 * sum(log((2 * pi)/tau) + tau * R2)</code>, and R2 is updated as <code>R2k - 2 * outer(l, f) * Rk + outer(l2, f2)</code> (where <code>Rk</code> is residuals for all factors but the <code>k</code>th and similarly for <code>R2k</code>). The <em>updated</em> parts of the objective have magnitude:</p>
<pre class="r"><code>sum(fl$tau * outer(fl$EL[, k], fl$EF[, k]) * Rk)</code></pre>
<pre><code>[1] 43462.72</code></pre>
<pre class="r"><code>-0.5 * sum(fl$tau * outer(fl$EL2[, k], fl$EF2[, k]))</code></pre>
<pre><code>[1] -21594.06</code></pre>
<p>So, errors in the sixth digit of either of these components could explain the decrease in the objective function. Let there be errors in the updates to <code>EL2</code> and consider the latter part of the objective: <span class="math display">\[ -\frac{1}{2} \sum_{i, j} \tau_{i, j} \left( \bar{l^2}_i + \epsilon_i \right) \bar{f^2}_j
= -\frac{1}{2} \sum_i \bar{l^2}_i \sum_j \tau_{i, j} \bar{f^2}_j -\frac{1}{2} \sum_i \epsilon_i \sum_j \tau_{i, j} \bar{f^2}_j \]</span> so we’d need to see errors in (roughly) the sixth digit of <code>EL2</code>. A similar calculation shows that errors in the sixth digit of <code>EL</code> could suffice to explain the decrease.</p>
<p>To test this hypothesis, I check to see what happens if only five digits are retained when performing the above calculations.</p>
<pre class="r"><code>last_obj = flash_get_objective(data, fl)
digits = 5
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] = signif(ebnm_l$postmean, digits=digits)
fl$EL2[, k] = signif(ebnm_l$postmean2, digits=digits)
fl$gl[[k]] = ebnm_l$fitted_g
fl$KL_l[[k]] = KL_l
flash_get_objective(data, fl) - last_obj</code></pre>
<pre><code>[1] -0.01432302</code></pre>
<p>So an overall error that is <em>roughly</em> on the scale of the decrease in objective function is produced.</p>
</div>
<div id="conclusions-and-questions" class="section level2">
<h2>Conclusions and questions</h2>
<p>Still, the error is not quite as large, and it would be very surprising to me if <code>EL</code> and <code>EL2</code> could only be trusted to five digits. More seriously, the sawtooth pattern discussed 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. Still, this would not explain the decrease in the objective that occurs after a complete iteration.</p>
</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 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 </code></pre>
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