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<title>Improvement on Implementation with Rmosek: Primal and Dual</title>
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<h1 class="title toc-ignore">Improvement on Implementation with <code>Rmosek</code>: Primal and Dual</h1>
<h4 class="author"><em>Lei Sun</em></h4>
<h4 class="date"><em>2017-05-07</em></h4>
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<p><strong>Last updated:</strong> 2018-05-15</p>
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<hr />
<div id="introduction" class="section level2">
<h2>Introduction</h2>
<p>We are experimenting different ways to improve the performance of <code>Rmosek</code>, using typical data sets of correlated <span class="math inline">\(z\)</span> scores.</p>
<pre class="r"><code>z <- read.table("../output/z_null_liver_777.txt")</code></pre>
<pre class="r"><code>sel <- c(32, 327, 355, 483, 778)
ord <- c(4, 9, 9, 4, 4)</code></pre>
<pre class="r"><code>source("../code/gdash.R")</code></pre>
</div>
<div id="algorithm-and-variations" class="section level2">
<h2>Algorithm and variations</h2>
<p>Recall that our <a href="ash_gd.html#optimization_problem">biconvex optimization problem</a> is as follows.</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{\pi,w} & -\sum\limits_{j = 1}^n\log\left(\sum\limits_{k = 1}^K\sum\limits_{l=1}^L\pi_k w_l f_{jkl} + \sum\limits_{k = 1}^K\pi_kf_{jk0}\right) - \sum\limits_{k = 1}^K\left(\lambda_k^\pi - 1\right)\log\pi_k + \sum\limits_{l = 1}^L\lambda_l^w\phi\left(\left|w_l\right|\right)
\\
\text{subject to} & \sum\limits_{k = 1}^K\pi_k = 1\\
& \sum\limits_{l=1}^L w_l \varphi^{(l)}\left(z\right) + \varphi\left(z\right) \geq 0, \forall z\in \mathbb{R}\\
& w_l \text{ decay reasonably fast,}
\end{array}
\]</span> where <span class="math inline">\(- \sum\limits_{k = 1}^K\left(\lambda_k^\pi - 1\right)\log\pi_k\)</span> and <span class="math inline">\(+ \sum\limits_{l = 1}^L\lambda_l^w\phi\left(\left|w_l\right|\right)\)</span> are to regularize <span class="math inline">\(\pi_k\)</span> and <span class="math inline">\(w_l\)</span>, respectively.</p>
<p>This problem can be solved iteratively. Starting with an initial value, each step two convex optimization problems are solved.</p>
<p>With a given <span class="math inline">\(\hat w\)</span>, <span class="math inline">\(\hat\pi\)</span> is solved by</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{\pi} & -\sum\limits_{j = 1}^n\log\left(\sum\limits_{k = 1}^K\pi_k \left(\sum\limits_{l=1}^L \hat w_l f_{jkl} + f_{jk0}\right)\right) - \sum\limits_{k = 1}^K\left(\lambda_k^\pi - 1\right)\log\pi_k\\
\text{subject to} & \sum\limits_{k = 1}^K\pi_k = 1 \ ,\\
\end{array}
\]</span></p>
<p>which is readily available with functions in <code>cvxr</code>.</p>
<p>Meanwhile, with a given <span class="math inline">\(\hat\pi\)</span>, the optimization problem to obtain <span class="math inline">\(\hat w\)</span> becomes</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{w} & -\sum\limits_{j = 1}^n\log\left(\sum\limits_{l=1}^Lw_l\left(\sum\limits_{k = 1}^K\hat\pi_k f_{jkl}\right) + \sum\limits_{k = 1}^K\hat\pi_kf_{jk0}\right) +
\sum\limits_{l = 1}^L\lambda_l^w
\phi
\left(\left|w_l\right|\right)
\\
\text{subject to} & \sum\limits_{l=1}^L w_l \varphi^{(l)}\left(z\right) + \varphi\left(z\right) \geq 0, \forall z\in \mathbb{R}\\
& w_l \text{ decay reasonably fast.}
\end{array}
\]</span> The two constraints are important to prevent <a href="gaussian_derivatives_3.html">numerical instability</a>. Yet they are not readily manifested as convex. Ideally, the regularization <span class="math inline">\(\sum\limits_{l = 1}^L\lambda_l^w\phi\left(\left|w_l\right|\right)\)</span> would be able to capture the essence of these two constaints, and at the same time maintain the convexity. Different ideas will be explored.</p>
<p>In this part, we’ll mainly work on the “basic form” of the <span class="math inline">\(w\)</span> optimization problem; that is, the optimization problem without regularization <span class="math inline">\(\sum\limits_{l = 1}^L\lambda_l^w\phi\left(\left|w_l\right|\right)\)</span>.</p>
<p>In the following, we’ll take a look at the basic form in its primal and dual formulations, optimized by <code>w.mosek()</code> and <code>w.mosek.primal</code> functions. First, they are applied to the correlated null, and their performance can be compared with <a href="gaussian_derivatives_2.html">the previous implementation</a> by <code>cvxr</code>. Then, they are applied to the simulated non-null data sets, and the results are compared with those obtained by <code>ASH</code>, as well as the truth.</p>
</div>
<div id="basic-form-primal" class="section level2">
<h2>Basic form: primal</h2>
<p>In its most basic form, the <span class="math inline">\(w\)</span> estimation problem is as follows.</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{w} &
-\sum\limits_{j = 1}^n
\log\left(\sum\limits_{l=1}^Lw_l a_{jl} + a_{j0}\right)
\\
\text{subject to} & \sum\limits_{l=1}^Lw_l a_{jl} + a_{j0} \geq 0, \forall j \ .\\
\end{array}
\]</span> Let the matrix <span class="math inline">\(A = \left[a_{jl}\right]\)</span>, the vector <span class="math inline">\(a = \left[a_{j0}\right]\)</span>, and we have its equivalent form,</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{w, g} &
-\sum\limits_{j = 1}^n
\log\left(g_j\right)
\\
\text{subject to}
& Aw + a = g \\
& g \geq 0\ .
\end{array}
\]</span></p>
<p>This problem can be coded by <code>Rmosek</code> as a “separable convex optimization” (SCOPT) problem.</p>
<div id="correlated-null" class="section level3">
<h3>Correlated null</h3>
<pre><code>Fitted w: -0.03653526 0.1999284 0.01047094 -0.02012681
Time Cost in Seconds: 7.945 2.884 13.152 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek primal fitting plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.03394291 0.7345604 -0.1532514 0.1883912 -0.05504895 0.02297176 -0.006908406 0.001216017 -0.0003147927
Time Cost in Seconds: 6.902 1.98 6.669 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek primal fitting plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.02262546 0.9213963 0.02180428 0.1760983 -0.01379674 0.003968132 -0.004904973 -0.0006085249 -0.0002990675
Time Cost in Seconds: 5.375 1.932 6.116 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek primal fitting plotting-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.04540484 -0.1275978 0.009150985 0.01004515
Time Cost in Seconds: 19.294 2.066 14.959 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek primal fitting plotting-4.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.005943196 0.3980709 -0.009222458 0.02630697
Time Cost in Seconds: 6.636 1.954 6.815 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek primal fitting plotting-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="signal-correlated-error" class="section level3">
<h3>Signal <span class="math inline">\(+\)</span> correlated error</h3>
<pre><code>Converged: FALSE
Number of Iteration: 101
Time Cost in Seconds: 2181.997 255.051 1644.499 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash primal signal plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: FALSE
Number of Iteration: 101
Time Cost in Seconds: 2634.559 296.365 1970.452 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash primal signal plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 45
Time Cost in Seconds: 553.188 127.588 540.733 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash primal signal plotting-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 49
Time Cost in Seconds: 781.062 126.839 622.938 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash primal signal plotting-4.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: FALSE
Number of Iteration: 101
Time Cost in Seconds: 2161.426 247.628 1565.216 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash primal signal plotting-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="basic-form-dual" class="section level2">
<h2>Basic form: dual</h2>
<p>The primal <span class="math inline">\(w\)</span> optimization problem has <span class="math inline">\(n + L\)</span> variables and <span class="math inline">\(n\)</span> constraints. When <span class="math inline">\(n\)</span> is large, such as <span class="math inline">\(n = 10K\)</span> in our simulations, the time cost is usually substantial. As the authors of <code>REBayes</code> pointed out, it’s better to work on the dual form, which is</p>
<p><span class="math display">\[
\begin{array}{rl}
\min\limits_{v} &
-\sum\limits_{j = 1}^n
\log\left(v_j\right) + a^Tv
\\
\text{subject to}
& A^Tv = 0 \\
& v \geq 0\ .
\end{array}
\]</span> The dual form has <span class="math inline">\(n\)</span> variables and more importantly, only <span class="math inline">\(L\)</span> constraints. As <span class="math inline">\(L \ll n\)</span>, the dual form is far more computationally efficient.</p>
<p><code>Rmosek</code> provides optimal solutions to both primal and dual variables, so <span class="math inline">\(\hat w\)</span> is readily available when working on <span class="math inline">\(v\)</span> optimization. But we need to be careful on which dual variables to use, such as <code>$sol$itr$suc</code>, <code>$sol$itr$slc</code>, or others.</p>
<div id="correlated-null-1" class="section level3">
<h3>Correlated null</h3>
<pre><code>Fitted w: -0.03642797 0.1971637 0.01074601 -0.02181494
Time Cost in Seconds: 0.485 0.049 0.388 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek dual fitting plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.03361434 0.7339189 -0.1544144 0.1875235 -0.0560049 0.02267003 -0.007151714 0.001187682 -0.0003321398
Time Cost in Seconds: 0.524 0.009 0.372 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek dual fitting plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.02273596 0.9206369 0.02224234 0.1750273 -0.01339113 0.003607418 -0.004794795 -0.0006395078 -0.0002908422
Time Cost in Seconds: 0.529 0.008 0.375 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek dual fitting plotting-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.04544396 -0.1272823 0.008811108 0.009915311
Time Cost in Seconds: 0.448 0.006 0.322 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek dual fitting plotting-4.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Fitted w: 0.006084177 0.3976595 -0.009103231 0.02610567
Time Cost in Seconds: 0.489 0.005 0.313 </code></pre>
<p><img src="figure/mosek_reg.rmd/Rmosek dual fitting plotting-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
<div id="signal-correlated-error-1" class="section level3">
<h3>Signal <span class="math inline">\(+\)</span> correlated error</h3>
<pre><code>Converged: TRUE
Number of Iteration: 21
Time Cost in Seconds: 46.978 2.71 29.017 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash dual signal plotting-1.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 21
Time Cost in Seconds: 45.406 5.343 52.01 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash dual signal plotting-2.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 16
Time Cost in Seconds: 34.293 3.519 26.79 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash dual signal plotting-3.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 31
Time Cost in Seconds: 64.198 3.902 39.897 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash dual signal plotting-4.png" width="672" style="display: block; margin: auto;" /></p>
<pre><code>Converged: TRUE
Number of Iteration: 12
Time Cost in Seconds: 27.247 1.446 15.392 </code></pre>
<p><img src="figure/mosek_reg.rmd/gdash dual signal plotting-5.png" width="672" style="display: block; margin: auto;" /></p>
</div>
</div>
<div id="conclusion" class="section level2">
<h2>Conclusion</h2>
<p>Implementation by <code>Rmosek</code> can fully reproduce the results obtained by <code>cvxr</code>. Moreover, the dual form gives the same results as the primal dorm does, with only a fraction of time.</p>
<p>It seems hopeful that we’ll find a key to successfully tackle correlation in simultaneous inference, which has eluded Prof. Brad Efron for more than a decade.</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 High Sierra 10.13.4
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] Rmosek_8.0.69 PolynomF_1.0-1 CVXR_0.95 REBayes_1.2
[5] Matrix_1.2-12 SQUAREM_2017.10-1 EQL_1.0-0 ttutils_1.0-1
loaded via a namespace (and not attached):
[1] Rcpp_0.12.16 knitr_1.20 whisker_0.3-2
[4] magrittr_1.5 workflowr_1.0.1 bit_1.1-12
[7] lattice_0.20-35 R6_2.2.2 stringr_1.3.0
[10] tools_3.4.3 grid_3.4.3 R.oo_1.21.0
[13] git2r_0.21.0 scs_1.1-1 htmltools_0.3.6
[16] bit64_0.9-7 yaml_2.1.18 rprojroot_1.3-2
[19] digest_0.6.15 gmp_0.5-13.1 ECOSolveR_0.4
[22] R.utils_2.6.0 evaluate_0.10.1 rmarkdown_1.9
[25] stringi_1.1.6 Rmpfr_0.6-1 compiler_3.4.3
[28] backports_1.1.2 R.methodsS3_1.7.1</code></pre>
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