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<section id="algorithm">
<h1>Algorithm<a class="headerlink" href="#algorithm" title="Link to this heading">#</a></h1>
<p><strong>Dynamic programming</strong> (DP) comprises algorithms to compute the optimal cost and an optimal policy of an MDP. The main DP methods are <em>value iteration</em> (VI) and <em>policy iteration</em> (PI) <a class="reference internal" href="#bertsekas" id="id1"><span>[Bertsekas]</span></a>. As dynamic programming suffers from the so called <em>curse of dimensionality</em>, there is need for solution methods which scale and are distributable.</p>
<p>Various approximation methods have been developed to address this issue, <em>e.g.</em>, <em>optimistic/modified policy iteration</em> (OPI) approximates the expensive policy evaluation step in PI using a finite number VI iterations.</p>
<p><code class="docutils literal notranslate"><span class="pre">madupite</span></code> relies on a novel class of algorithms called <em>inexact policy iteration</em> (iPI) methods which are based on an inexact policy evaluation step. We refer to Sec. 4 in <a class="reference internal" href="examples.html#gargiani2024" id="id2"><span>[Gargiani2024]</span></a> for a detailed description of iPI. A key advantage of iPI’s flexibility is that depending on the problem structure, one can choose the most suitable approximation method for the policy evaluation step.</p>
<p><code class="docutils literal notranslate"><span class="pre">madupite</span></code> exploits this flexibiliy by allowing the user to select any iterative method available in PETSc via the <code class="docutils literal notranslate"><span class="pre">-ksp_type</span></code> option (see <a class="reference internal" href="options.html"><span class="doc">Options</span></a>). Available methods include GMRES, BiCGStab, TFQMR etc. A full list of available solvers can be found <a class="reference external" href="https://petsc.org/release/manualpages/KSP/KSPType/">here</a>. Some of these methods show superior convergence properties for almost undiscounted MDPs <span class="math notranslate nohighlight">\((\gamma \approx 1)\)</span> than VI.</p>
<section id="does-madupite-still-support-standard-dynamic-programming-algorithms-such-as-vi-and-pi">
<h2>Does <code class="docutils literal notranslate"><span class="pre">madupite</span></code> still support standard dynamic programming algorithms such as VI and PI?<a class="headerlink" href="#does-madupite-still-support-standard-dynamic-programming-algorithms-such-as-vi-and-pi" title="Link to this heading">#</a></h2>
<p>Yes, but be aware of their performance limitations. Since iPI is a more general class of algorithms, you can retrieve PI, VI and OPI by setting the inner solver and tolerances accordingly.</p>
<p><strong>Policy Iteration</strong> (PI): To evaluate the policy exactly, use singular value decomposition as a preconditioner. The choice of inner solver (KSP type) in this case becomes irrelevant. You can therefore simply leave it unspecified which defaults to GMRES. <em>PETSc only supports SVD for sequential execution!</em></p>
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>$<span class="w"> </span>python<span class="w"> </span>ex1.py<span class="w"> </span>-pc_type<span class="w"> </span>svd
</pre></div>
</div>
<p><strong>Value Iteration</strong> (VI): Since a VI step for policy evaluation is equivalent to one iteration of Richardson method with a scaling factor of 1, you can retrieve VI by limiting the number of inner iterations to 1 and selecting Richardson method as inner solver (Proposition 19 in <a class="reference internal" href="examples.html#gargiani2024" id="id3"><span>[Gargiani2024]</span></a>).</p>
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>$<span class="w"> </span>python<span class="w"> </span>ex1.py<span class="w"> </span>-ksp_type<span class="w"> </span>richardson<span class="w"> </span>-ksp_richardson_scale<span class="w"> </span><span class="m">1</span>.0<span class="w"> </span>-alpha<span class="w"> </span>1e-40<span class="w"> </span>-max_iter_ksp<span class="w"> </span><span class="m">1</span>
</pre></div>
</div>
<p><strong>Optimistic Policy Iteration</strong> (OPI): You can retrieve OPI by selecting the same parameters as for VI, but with a number of inner iterations that can potentially be greater than 1. In particular, set the value of <code class="docutils literal notranslate"><span class="pre">-max_iter_ksp</span></code> to the desired number of inner VI iterations you would like to be performed, <em>e.g.</em>, 50. Finally, be sure to set the inner tolerance parameter to a very small value such that it does not impact on the execution of the inner loop.</p>
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span>$<span class="w"> </span>python<span class="w"> </span>ex1.py<span class="w"> </span>-ksp_type<span class="w"> </span>richardson<span class="w"> </span>-ksp_richardson_scale<span class="w"> </span><span class="m">1</span>.0<span class="w"> </span>-alpha<span class="w"> </span>1e-40<span class="w"> </span>-max_iter_ksp<span class="w"> </span><span class="m">50</span>
</pre></div>
</div>
<p class="rubric">References</p>
<div role="list" class="citation-list">
<div class="citation" id="bertsekas" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span><a role="doc-backlink" href="#id1">Bertsekas</a><span class="fn-bracket">]</span></span>
<p>Bertsekas, D. P. <em>Dynamic Programming and Optimal Control</em>, Vol. II, Athena Scientific, Belmont, Massachusets, 4th edition, 2012.</p>
</div>
<div class="citation" id="gargiani2024" role="doc-biblioentry">
<span class="label"><span class="fn-bracket">[</span>Gargiani2024<span class="fn-bracket">]</span></span>
<span class="backrefs">(<a role="doc-backlink" href="#id2">1</a>,<a role="doc-backlink" href="#id3">2</a>)</span>
<p>Gargiani, M.; Sieber. R.; Balta, E.; Liao-McPherson, D.; Lygeros, J. <em>Inexact Policy Iteration Methods for Large-Scale Markov Decision Processes</em>. <a class="reference external" href="https://arxiv.org/abs/2404.06136">https://arxiv.org/abs/2404.06136</a>.</p>
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<li class="toc-h2 nav-item toc-entry"><a class="reference internal nav-link" href="#does-madupite-still-support-standard-dynamic-programming-algorithms-such-as-vi-and-pi">Does <code class="docutils literal notranslate"><span class="pre">madupite</span></code> still support standard dynamic programming algorithms such as VI and PI?</a></li>
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