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<section id="tutorials">
<h1>Tutorials<a class="headerlink" href="#tutorials" title="Link to this heading">#</a></h1>
<p>In the following, we propose some short tutorials which we hope can help the users to better understand the deployment of some of the fundamental functionalities of <code class="docutils literal notranslate"><span class="pre">madupite</span></code>.</p>
<section id="writing-and-running-your-code-using-madupite">
<h2>Writing and running your code using <code class="docutils literal notranslate"><span class="pre">madupite</span></code><a class="headerlink" href="#writing-and-running-your-code-using-madupite" title="Link to this heading">#</a></h2>
<p><code class="docutils literal notranslate"><span class="pre">madupite</span></code> can be imported and used as any other Python package. Though, if you want to run your code it in a parallel/distributed set up, the code should be contained inside a <code class="docutils literal notranslate"><span class="pre">main</span></code> function as this guarantees a correct finalization of the MPI ranks.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">madupite</span> <span class="k">as</span> <span class="nn">md</span>
<span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
<span class="c1"># main body of your code</span>
</pre></div>
</div>
<p>In this way, you can save your code into a <code class="docutils literal notranslate"><span class="pre">.py</span></code> file, <em>e.g.</em> <code class="docutils literal notranslate"><span class="pre">mycode.py</span></code>, and then run it smoothly with the command <code class="docutils literal notranslate"><span class="pre">mpirun</span> <span class="pre">-n</span> <span class="pre"><number_of_ranks></span> <span class="pre">python</span> <span class="pre">mycode.py</span></code>. There is no need to use the <code class="docutils literal notranslate"><span class="pre">main</span></code> function if you instead just want to run your code in a single-core setting with the command <code class="docutils literal notranslate"><span class="pre">python</span> <span class="pre">mycode.py</span></code>.</p>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Note that as of <code class="docutils literal notranslate"><span class="pre">madupite</span></code> V1.0, if you do not use the <code class="docutils literal notranslate"><span class="pre">main</span></code> function and then run your code with <code class="docutils literal notranslate"><span class="pre">mpirun</span> <span class="pre">-n</span> <span class="pre"><number_of_ranks></span> <span class="pre">python</span> <span class="pre">mycode.py</span></code>, then your code will still run and output the desired results, but an error will be raised at the end of the execution because the finalization of the ranks is not handled appropriately.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span>nanobind: leaked 1 instances!
nanobind: leaked 1 keep_alive records!
nanobind: leaked 1 types!
- leaked type "madupite._madupite_impl.Madupite"
nanobind: this is likely caused by a reference counting issue in the binding code.
</pre></div>
</div>
</div>
</section>
<section id="loading-and-reading-data">
<h2>Loading and reading data<a class="headerlink" href="#loading-and-reading-data" title="Link to this heading">#</a></h2>
<p>In this tutorial, we want to show how to load and read MDP data that is stored in files. The data itself must be stored as a PETSc binary file (explained <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatLoad/">here</a>). <code class="docutils literal notranslate"><span class="pre">madupite</span></code> provides a method to save numpy or scipy matrices to PETSc binary files (<a class="reference internal" href="_autosummary/madupite.writePETScBinary.html#madupite.writePETScBinary" title="madupite.writePETScBinary"><code class="xref py py-func docutils literal notranslate"><span class="pre">madupite.writePETScBinary()</span></code></a>).</p>
<p>Assuming the stage cost matrix and transition probability tensor are stored as <code class="docutils literal notranslate"><span class="pre">g.bin</span></code> and <code class="docutils literal notranslate"><span class="pre">P.bin</span></code> we can load them with <code class="xref py py-func docutils literal notranslate"><span class="pre">madupite.Matrix.fromFile()</span></code>. We need to specify whether it is a stage cost matrix (<code class="docutils literal notranslate"><span class="pre">md.MatrixCategory.Cost</span></code>) or a transition probability tensor (<code class="docutils literal notranslate"><span class="pre">md.MatrixCategory.Dynamics</span></code>) to ensure that the number of states and actions is correctly inferred.</p>
<p>Furthermore, you can specify whether the matrix is sparse or dense using the <code class="docutils literal notranslate"><span class="pre">md.MatrixType</span></code> enum. Sparse matrices are stored in a compressed format, which can save memory and speed up computations.</p>
<p>Notice that, unlike with function simulations, defining an object for matrix preallocation is not necessary when loading from files since the information about non-zero elements is already contained in the binary file.</p>
<p>In the following code snippet, we show how to save numpy matrices into <em>.bin</em> files and then correctly create the madupite stage-cost and transition probability objects from those files.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">madupite</span> <span class="k">as</span> <span class="nn">md</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
<span class="c1"># Write matrices to file</span>
<span class="n">numStates</span> <span class="o">=</span> <span class="mi">10</span>
<span class="n">numActions</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">cost_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">numStates</span><span class="p">,</span> <span class="n">numActions</span><span class="p">)</span>
<span class="n">md</span><span class="o">.</span><span class="n">writePETScBinary</span><span class="p">(</span><span class="n">cost_matrix</span><span class="p">,</span> <span class="s2">"g.bin"</span><span class="p">)</span>
<span class="n">prob_matrix</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">rand</span><span class="p">(</span><span class="n">numStates</span> <span class="o">*</span> <span class="n">numActions</span><span class="p">,</span> <span class="n">numStates</span><span class="p">)</span>
<span class="c1"># normalize rows to 1, in order to have a valid transition probability matrix</span>
<span class="n">prob_matrix</span> <span class="o">/=</span> <span class="n">prob_matrix</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">)[:,</span> <span class="kc">None</span><span class="p">]</span>
<span class="n">md</span><span class="o">.</span><span class="n">writePETScBinary</span><span class="p">(</span><span class="n">prob_matrix</span><span class="p">,</span> <span class="s2">"P.bin"</span><span class="p">)</span>
<span class="c1"># Load matrices from file</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">Matrix</span><span class="o">.</span><span class="n">fromFile</span><span class="p">(</span>
<span class="n">filename</span> <span class="o">=</span> <span class="s2">"g.bin"</span><span class="p">,</span>
<span class="n">category</span><span class="o">=</span><span class="n">md</span><span class="o">.</span><span class="n">MatrixCategory</span><span class="o">.</span><span class="n">Cost</span><span class="p">,</span>
<span class="nb">type</span><span class="o">=</span><span class="n">md</span><span class="o">.</span><span class="n">MatrixType</span><span class="o">.</span><span class="n">Dense</span>
<span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">Matrix</span><span class="o">.</span><span class="n">fromFile</span><span class="p">(</span>
<span class="n">filename</span> <span class="o">=</span> <span class="s2">"P.bin"</span><span class="p">,</span>
<span class="n">category</span><span class="o">=</span><span class="n">md</span><span class="o">.</span><span class="n">MatrixCategory</span><span class="o">.</span><span class="n">Dynamics</span><span class="p">,</span>
<span class="nb">type</span><span class="o">=</span><span class="n">md</span><span class="o">.</span><span class="n">MatrixType</span><span class="o">.</span><span class="n">Sparse</span>
<span class="p">)</span>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
<span class="n">main</span><span class="p">()</span>
</pre></div>
</div>
<div class="admonition warning">
<p class="admonition-title">Warning</p>
<p>Note that as of <code class="docutils literal notranslate"><span class="pre">madupite</span></code> V1.0, the files themselves must contain the data in a sparse format because PETSc does not support reading dense matrices from binary files. By specifying the matrix type as dense, the data will be read as a sparse matrix and then converted to a dense matrix. This is recommended for stage cost matrices to benefit from data locality and speed up computations.</p>
</div>
</section>
<section id="generating-data">
<h2>Generating data<a class="headerlink" href="#generating-data" title="Link to this heading">#</a></h2>
<p>Depending on the problem, creating the MDP data with numpy and reading them with <code class="docutils literal notranslate"><span class="pre">madupite</span></code> is often slower than generating them directly with <code class="docutils literal notranslate"><span class="pre">madupite</span></code>. This is because <code class="docutils literal notranslate"><span class="pre">madupite</span></code> can generate the transition probabilities in parallel and in the correct format, which avoids the need to convert the data.</p>
<p>In the following example, we show how to generate the stage cost matrix and transition probability tensor with <code class="docutils literal notranslate"><span class="pre">madupite</span></code>. We define a cost function and a probability function that are used to generate the data. The cost function takes a state and an action as input and returns the associated cost. The probability function takes a state and an action as input and returns two lists: the first list containts the non-zero transition probabilities values, and the second list contains their associated next state indices.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">madupite</span> <span class="k">as</span> <span class="nn">md</span>
<span class="k">def</span> <span class="nf">costfunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="n">s</span> <span class="o">+</span> <span class="n">a</span>
<span class="k">def</span> <span class="nf">probfunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
<span class="n">transition_probabilities</span> <span class="o">=</span> <span class="p">[</span><span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">]</span>
<span class="n">state_indices</span> <span class="o">=</span> <span class="p">[</span><span class="n">s</span><span class="p">,</span> <span class="p">(</span><span class="n">s</span> <span class="o">+</span> <span class="n">a</span><span class="p">)</span> <span class="o">%</span> <span class="mi">50</span><span class="p">]</span>
<span class="k">return</span> <span class="n">transition_probabilities</span><span class="p">,</span> <span class="n">state_indices</span>
<span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
<span class="n">num_states</span> <span class="o">=</span> <span class="mi">50</span>
<span class="n">num_actions</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createStageCostMatrix</span><span class="p">(</span>
<span class="n">numStates</span><span class="o">=</span><span class="n">num_states</span><span class="p">,</span> <span class="n">numActions</span><span class="o">=</span><span class="n">num_actions</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="n">costfunc</span>
<span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createTransitionProbabilityTensor</span><span class="p">(</span>
<span class="n">numStates</span><span class="o">=</span><span class="n">num_states</span><span class="p">,</span>
<span class="n">numActions</span><span class="o">=</span><span class="n">num_actions</span><span class="p">,</span>
<span class="n">func</span><span class="o">=</span><span class="n">probfunc</span>
<span class="p">)</span>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
<span class="n">main</span><span class="p">()</span>
</pre></div>
</div>
</section>
<section id="matrix-preallocation">
<h2>Matrix preallocation<a class="headerlink" href="#matrix-preallocation" title="Link to this heading">#</a></h2>
<p>For large MDPs with sparse transition probability tensors, it is often beneficial to preallocate the matrices to avoid reallocations during the computation. This can be done by specifying the <code class="docutils literal notranslate"><span class="pre">preallocation</span></code> argument. The method takes an instance of the <a class="reference internal" href="_autosummary/madupite.MatrixPreallocation.html#madupite.MatrixPreallocation" title="madupite.MatrixPreallocation"><code class="xref py py-class docutils literal notranslate"><span class="pre">madupite.MatrixPreallocation</span></code></a> class, which specifies the number of non-zero elements per row in the diagonal and off-diagonal block. See the example below for more details (adapted from <a class="reference external" href="https://petsc.org/release/manualpages/Mat/MatMPIAIJSetPreallocation/">PETSc</a>).</p>
<p>Consider the following 8x8 matrix with 34 non-zero values, that is
assembled across 3 ranks. Let’s assume that rank0 owns 3 rows,
rank1 owns 3 rows, rank2 owns 2 rows. This division can be shown
as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span> <span class="mi">1</span> <span class="mi">2</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">3</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">4</span>
<span class="n">rank0</span> <span class="mi">0</span> <span class="mi">5</span> <span class="mi">6</span> <span class="o">|</span> <span class="mi">7</span> <span class="mi">0</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">8</span> <span class="mi">0</span>
<span class="mi">9</span> <span class="mi">0</span> <span class="mi">10</span> <span class="o">|</span> <span class="mi">11</span> <span class="mi">0</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">12</span> <span class="mi">0</span>
<span class="o">-------------------------------------</span>
<span class="mi">13</span> <span class="mi">0</span> <span class="mi">14</span> <span class="o">|</span> <span class="mi">15</span> <span class="mi">16</span> <span class="mi">17</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">0</span>
<span class="n">rank1</span> <span class="mi">0</span> <span class="mi">18</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">19</span> <span class="mi">20</span> <span class="mi">21</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">0</span>
<span class="mi">0</span> <span class="mi">0</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">22</span> <span class="mi">23</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">24</span> <span class="mi">0</span>
<span class="o">-------------------------------------</span>
<span class="n">rank2</span> <span class="mi">25</span> <span class="mi">26</span> <span class="mi">27</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">0</span> <span class="mi">28</span> <span class="o">|</span> <span class="mi">29</span> <span class="mi">0</span>
<span class="mi">30</span> <span class="mi">0</span> <span class="mi">0</span> <span class="o">|</span> <span class="mi">31</span> <span class="mi">32</span> <span class="mi">33</span> <span class="o">|</span> <span class="mi">0</span> <span class="mi">34</span>
</pre></div>
</div>
<p>This can be represented as a collection of submatrices as:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">A</span> <span class="n">B</span> <span class="n">C</span>
<span class="n">D</span> <span class="n">E</span> <span class="n">F</span>
<span class="n">G</span> <span class="n">H</span> <span class="n">I</span>
</pre></div>
</div>
<p>Where the submatrices A, B, C are owned by rank0, D, E, F are
owned by rank1, G, H, I are owned by rank2.</p>
<p>The DIAGONAL submatrices corresponding to rank0, rank1, rank2 are
submatrices [A], [E], [I], respectively. The OFF-DIAGONAL submatrices
corresponding to rank0, rank1, rank2 are [BC], [DF], [GH], respectively.</p>
<p>When <code class="docutils literal notranslate"><span class="pre">d_nz</span></code>, <code class="docutils literal notranslate"><span class="pre">o_nz</span></code> parameters are specified, <code class="docutils literal notranslate"><span class="pre">d_nz</span></code> storage elements are
allocated for every row of the local diagonal submatrix, and <code class="docutils literal notranslate"><span class="pre">o_nz</span></code>
storage locations are allocated for every row of the OFF-DIAGONAL submatrix.
Typically one chooses <code class="docutils literal notranslate"><span class="pre">d_nz</span></code> and <code class="docutils literal notranslate"><span class="pre">o_nz</span></code> as the max non-zeros per local
rows for each of the local DIAGONAL, and the OFF-DIAGONAL submatrices.
In this case, the values of <code class="docutils literal notranslate"><span class="pre">d_nz</span></code>, <code class="docutils literal notranslate"><span class="pre">o_nz</span></code> are:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">rank0</span> <span class="n">d_nz</span> <span class="o">=</span> <span class="mi">2</span><span class="p">,</span> <span class="n">o_nz</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">rank1</span> <span class="n">d_nz</span> <span class="o">=</span> <span class="mi">3</span><span class="p">,</span> <span class="n">o_nz</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">rank2</span> <span class="n">d_nz</span> <span class="o">=</span> <span class="mi">1</span><span class="p">,</span> <span class="n">o_nz</span> <span class="o">=</span> <span class="mi">4</span>
</pre></div>
</div>
<p>When <code class="docutils literal notranslate"><span class="pre">d_nnz</span></code>, <code class="docutils literal notranslate"><span class="pre">o_nnz</span></code> parameters are specified, the storage is specified
for every row, corresponding to both DIAGONAL and OFF-DIAGONAL submatrices.
In the above case the values for <code class="docutils literal notranslate"><span class="pre">d_nnz</span></code>, <code class="docutils literal notranslate"><span class="pre">o_nnz</span></code> are:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">rank0</span> <span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="ow">and</span> <span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span>
<span class="n">rank1</span> <span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span><span class="mi">3</span><span class="p">,</span><span class="mi">2</span><span class="p">]</span> <span class="ow">and</span> <span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span>
<span class="n">rank2</span> <span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">]</span> <span class="ow">and</span> <span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span><span class="mi">4</span><span class="p">]</span>
</pre></div>
</div>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">madupite</span> <span class="k">as</span> <span class="nn">md</span>
<span class="c1"># ...</span>
<span class="n">rank</span><span class="p">,</span> <span class="n">size</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">mpi_rank_size</span><span class="p">()</span>
<span class="c1"># Option 1</span>
<span class="n">pc</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">MatrixPreallocation</span><span class="p">()</span>
<span class="k">if</span> <span class="n">rank</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">pc</span><span class="o">.</span><span class="n">d_nz</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">pc</span><span class="o">.</span><span class="n">o_nz</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">elif</span> <span class="n">rank</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">pc</span><span class="o">.</span><span class="n">d_nz</span> <span class="o">=</span> <span class="mi">3</span>
<span class="n">pc</span><span class="o">.</span><span class="n">o_nz</span> <span class="o">=</span> <span class="mi">2</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">pc</span><span class="o">.</span><span class="n">d_nz</span> <span class="o">=</span> <span class="mi">1</span>
<span class="n">pc</span><span class="o">.</span><span class="n">o_nz</span> <span class="o">=</span> <span class="mi">4</span>
<span class="c1"># Option 2</span>
<span class="n">pc2</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">MatrixPreallocation</span><span class="p">()</span>
<span class="k">if</span> <span class="n">rank</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="k">elif</span> <span class="n">rank</span> <span class="o">==</span> <span class="mi">1</span><span class="p">:</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">3</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">2</span><span class="p">]</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="p">,</span> <span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">d_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">]</span>
<span class="n">pc2</span><span class="o">.</span><span class="n">o_nnz</span> <span class="o">=</span> <span class="p">[</span><span class="mi">4</span><span class="p">,</span> <span class="mi">4</span><span class="p">]</span>
<span class="k">def</span> <span class="nf">probfunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="mi">0</span><span class="p">]</span>
<span class="n">P1</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createTransitionProbabilityTensor</span><span class="p">(</span>
<span class="n">numStates</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
<span class="n">numActions</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">func</span><span class="o">=</span><span class="n">probfunc</span><span class="p">,</span>
<span class="n">preallocation</span><span class="o">=</span><span class="n">pc</span>
<span class="p">)</span>
<span class="n">P2</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createTransitionProbabilityTensor</span><span class="p">(</span>
<span class="n">numStates</span><span class="o">=</span><span class="mi">8</span><span class="p">,</span>
<span class="n">numActions</span><span class="o">=</span><span class="mi">1</span><span class="p">,</span>
<span class="n">func</span><span class="o">=</span><span class="n">probfunc</span><span class="p">,</span>
<span class="n">preallocation</span><span class="o">=</span><span class="n">pc2</span>
<span class="p">)</span>
</pre></div>
</div>
<div class="admonition note">
<p class="admonition-title">Note</p>
<p>Using the preallocation greatly improves performance. At the same time, it can be quite cumbersome to compute these parameters for larger settings. We suggest to use the parameters <code class="docutils literal notranslate"><span class="pre">d_nz</span></code> and <code class="docutils literal notranslate"><span class="pre">o_nz</span></code> and set their values to the maximum number of non-zero entries per-row of the full-matrix.</p>
</div>
</section>
<section id="data-format">
<h2>Data format<a class="headerlink" href="#data-format" title="Link to this heading">#</a></h2>
<p>The data format for the MDP is defined by the stage cost matrix and the transition probability tensor. The stage cost matrix is a matrix of size <code class="docutils literal notranslate"><span class="pre">numStates</span> <span class="pre">x</span> <span class="pre">numActions</span></code>, where each element (s, a) represents the cost of taking action a in state s. The transition probabilities are usually expressed as a tensor of size <code class="docutils literal notranslate"><span class="pre">numStates</span> <span class="pre">x</span> <span class="pre">numActions</span> <span class="pre">x</span> <span class="pre">numStates</span></code>, where each element (s, a, s’) represents the probability of transitioning from state s to state s’ after applying action a. For <code class="docutils literal notranslate"><span class="pre">madupite</span></code> the tensor is flattened to a matrix of size <code class="docutils literal notranslate"><span class="pre">numStates*numActions</span> <span class="pre">x</span> <span class="pre">numStates</span></code>, where each row i represents the transition probabilities from state i // numStates to state s’ after applying action i % numStates.</p>
<p>The tensor can be reshaped as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="gp">>>> </span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="gp">>>> </span><span class="n">numStates</span> <span class="o">=</span> <span class="mi">3</span>
<span class="gp">>>> </span><span class="n">numActions</span> <span class="o">=</span> <span class="mi">2</span>
<span class="gp">>>> </span><span class="n">P</span><span class="o">=</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span>
<span class="gp">... </span> <span class="p">[[[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.0</span> <span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.25</span><span class="p">,</span> <span class="mf">0.33</span><span class="p">,</span> <span class="mf">0.42</span><span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span><span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span> <span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">,</span> <span class="mf">0.4</span> <span class="p">]],</span>
<span class="gp">...</span>
<span class="gp">... </span> <span class="p">[[</span><span class="mf">0.6</span> <span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.3</span> <span class="p">],</span>
<span class="gp">... </span> <span class="p">[</span><span class="mf">0.7</span> <span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span> <span class="p">]]])</span>
<span class="gp">>>></span>
<span class="gp">>>> </span><span class="n">P</span><span class="o">.</span><span class="n">reshape</span><span class="p">((</span><span class="n">numStates</span><span class="o">*</span><span class="n">numActions</span><span class="p">,</span> <span class="n">numStates</span><span class="p">))</span>
<span class="go">array([[0.5 , 0.5 , 0. ],</span>
<span class="go"> [0.25, 0.33, 0.42],</span>
<span class="go"> [0.3 , 0.3 , 0.4 ],</span>
<span class="go"> [0.4 , 0.2 , 0.4 ],</span>
<span class="go"> [0.6 , 0.1 , 0.3 ],</span>
<span class="go"> [0.7 , 0.1 , 0.2 ]])</span>
</pre></div>
</div>
</section>
<section id="the-mdp-class">
<h2>The MDP-class<a class="headerlink" href="#the-mdp-class" title="Link to this heading">#</a></h2>
<p>Now that all the main ingredients are explained, we are ready to introduce the MDP-class, which is basically where all the magic of <code class="docutils literal notranslate"><span class="pre">madupite</span></code> happens! This class allows you to create and solve your own MDP, and it comes with a lot of options that you can use to customize your MDP. The code snippet down below exemplifies how you use instances of this class to create and solve an MDP. In particular, it simulates via functions, creates and solves a dense random MDP with <code class="docutils literal notranslate"><span class="pre">madupite</span></code>.
The optimal policy and stats are saved into <cite>policy.out</cite> and <cite>stats.json</cite> files at the end.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">madupite</span> <span class="k">as</span> <span class="nn">md</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="n">num_states</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">num_actions</span> <span class="o">=</span> <span class="mi">5</span>
<span class="k">def</span> <span class="nf">probfunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span><span class="n">a</span><span class="p">):</span>
<span class="n">prob_sprime</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="n">size</span><span class="o">=</span><span class="n">num_states</span><span class="p">)</span>
<span class="n">values</span> <span class="o">=</span> <span class="p">(</span><span class="n">prob_sprime</span><span class="o">/</span><span class="n">prob_sprime</span><span class="o">.</span><span class="n">sum</span><span class="p">())</span><span class="o">.</span><span class="n">tolist</span><span class="p">()</span>
<span class="k">return</span> <span class="n">values</span><span class="p">,</span> <span class="nb">list</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="n">num_states</span><span class="p">))</span>
<span class="k">def</span> <span class="nf">costfunc</span><span class="p">(</span><span class="n">s</span><span class="p">,</span> <span class="n">a</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">main</span><span class="p">():</span>
<span class="n">mdp</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">MDP</span><span class="p">()</span>
<span class="n">prealloc</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">MatrixPreallocation</span><span class="p">()</span>
<span class="n">prealloc</span><span class="o">.</span><span class="n">d_nz</span> <span class="o">=</span> <span class="n">num_states</span>
<span class="n">prealloc</span><span class="o">.</span><span class="n">o_nz</span> <span class="o">=</span> <span class="n">num_states</span>
<span class="n">g</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createStageCostMatrix</span><span class="p">(</span>
<span class="n">name</span><span class="o">=</span><span class="s2">"g"</span><span class="p">,</span> <span class="n">numStates</span><span class="o">=</span><span class="n">num_states</span><span class="p">,</span> <span class="n">numActions</span><span class="o">=</span><span class="n">num_actions</span><span class="p">,</span> <span class="n">func</span><span class="o">=</span><span class="n">costfunc</span>
<span class="p">)</span>
<span class="n">P</span> <span class="o">=</span> <span class="n">md</span><span class="o">.</span><span class="n">createTransitionProbabilityTensor</span><span class="p">(</span>
<span class="n">name</span><span class="o">=</span><span class="s2">"P"</span><span class="p">,</span>
<span class="n">numStates</span><span class="o">=</span><span class="n">num_states</span><span class="p">,</span>
<span class="n">numActions</span><span class="o">=</span><span class="n">num_actions</span><span class="p">,</span>
<span class="n">func</span><span class="o">=</span><span class="n">probfunc</span><span class="p">,</span>
<span class="n">preallocation</span><span class="o">=</span><span class="n">prealloc</span>
<span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setStageCostMatrix</span><span class="p">(</span><span class="n">g</span><span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setTransitionProbabilityTensor</span><span class="p">(</span><span class="n">P</span><span class="p">)</span>
<span class="c1">#mandatory options to select</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setOption</span><span class="p">(</span><span class="s2">"-mode"</span><span class="p">,</span> <span class="s2">"MINCOST"</span><span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setOption</span><span class="p">(</span><span class="s2">"-discount_factor"</span><span class="p">,</span> <span class="s2">"0.9"</span><span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setOption</span><span class="p">(</span><span class="s2">"-file_policy"</span><span class="p">,</span> <span class="s2">"policy.out"</span><span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">setOption</span><span class="p">(</span><span class="s2">"-file_stats"</span><span class="p">,</span> <span class="s2">"stats.json"</span><span class="p">)</span>
<span class="n">mdp</span><span class="o">.</span><span class="n">solve</span><span class="p">()</span>
<span class="k">if</span> <span class="vm">__name__</span> <span class="o">==</span> <span class="s2">"__main__"</span><span class="p">:</span>
<span class="n">main</span><span class="p">()</span>
</pre></div>
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