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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml" lang="en" xml:lang="en"><head>
<meta charset="utf-8">
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<body class="fullcontent">
<div id="quarto-content" class="page-columns page-rows-contents page-layout-article">
<main class="content" id="quarto-document-content">
<section id="the-4x3-grid-world-example" class="level1">
<h1>The 4x3 Grid World Example</h1>
<p>The used example is the simple 4x3 grid world described in AIMA Figure 12.1 and used again in 22.1 as:</p>
<div class="quarto-figure quarto-figure-center">
<figure class="figure">
<p><img src="figures/AIMA_Figure_17_1.png" class="img-fluid figure-img"></p>
<figcaption class="figure-caption">AIMA Figure 17.1: (a) A simple, stochastic <span class="math inline">\(4 \times 3\)</span> environment that presents the agent with a sequential decision problem. (b) Illustration of the transition model of the environment: the “intended” outcome occurs with probability 0.8, but with probability 0.2 the agent moves at right angles to the intended direction. A collision with a wall results in no movement. Transitions into the two terminal states have reward +1 and -1, respectively, and all other transitions have a reward of -0.04.</figcaption>
</figure>
</div>
<p>In the following, we implement states, actions, the reward function,and the transition model. We also show how to represent a policy and how to estimate the expected utility using simulation.</p>
<section id="states" class="level2">
<h2 class="anchored" data-anchor-id="states">States</h2>
<p>We define the atomic state space <span class="math inline">\(S\)</span> by labeling the states <span class="math inline">\(1, 2, ...\)</span>. We convert coordinates <code>(rows, columns)</code> to the state label.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb1"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb1-1"><a href="#cb1-1" aria-hidden="true" tabindex="-1"></a><span class="co"># I use capitalized variables as global constants</span></span>
<span id="cb1-2"><a href="#cb1-2" aria-hidden="true" tabindex="-1"></a>COLS <span class="ot"><-</span> <span class="dv">4</span></span>
<span id="cb1-3"><a href="#cb1-3" aria-hidden="true" tabindex="-1"></a>ROWS <span class="ot"><-</span> <span class="dv">3</span></span>
<span id="cb1-4"><a href="#cb1-4" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-5"><a href="#cb1-5" aria-hidden="true" tabindex="-1"></a>S <span class="ot">=</span> <span class="fu">seq_len</span>(ROWS <span class="sc">*</span> COLS)</span>
<span id="cb1-6"><a href="#cb1-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb1-7"><a href="#cb1-7" aria-hidden="true" tabindex="-1"></a>LAYOUT <span class="ot"><-</span> <span class="fu">matrix</span>(S, <span class="at">nrow =</span> ROWS , <span class="at">ncol =</span> COLS)</span>
<span id="cb1-8"><a href="#cb1-8" aria-hidden="true" tabindex="-1"></a>LAYOUT</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [,1] [,2] [,3] [,4]
[1,] 1 4 7 10
[2,] 2 5 8 11
[3,] 3 6 9 12</code></pre>
</div>
</div>
<p>Note that the rows are displayed upside-down compared to the text book, so we use a function to display them in reverse order.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb3"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb3-1"><a href="#cb3-1" aria-hidden="true" tabindex="-1"></a>show_layout <span class="ot"><-</span> <span class="cf">function</span>(x) {</span>
<span id="cb3-2"><a href="#cb3-2" aria-hidden="true" tabindex="-1"></a> x <span class="ot"><-</span> <span class="fu">matrix</span>(x, <span class="at">ncol =</span> COLS, <span class="at">nrow =</span> ROWS, </span>
<span id="cb3-3"><a href="#cb3-3" aria-hidden="true" tabindex="-1"></a> <span class="at">dimnames =</span> <span class="fu">list</span>(<span class="at">row =</span> <span class="fu">seq_len</span>(ROWS), <span class="at">col =</span> <span class="fu">seq_len</span>(COLS)))</span>
<span id="cb3-4"><a href="#cb3-4" aria-hidden="true" tabindex="-1"></a> x[<span class="fu">rev</span>(<span class="fu">seq_len</span>(ROWS)), ]</span>
<span id="cb3-5"><a href="#cb3-5" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb3-6"><a href="#cb3-6" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb3-7"><a href="#cb3-7" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(LAYOUT)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 3 6 9 12
2 2 5 8 11
1 1 4 7 10</code></pre>
</div>
</div>
<p>Convert between coordinates and state labels.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb5"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb5-1"><a href="#cb5-1" aria-hidden="true" tabindex="-1"></a>rc_to_s <span class="ot"><-</span> <span class="cf">function</span>(rc)</span>
<span id="cb5-2"><a href="#cb5-2" aria-hidden="true" tabindex="-1"></a> LAYOUT[<span class="fu">rbind</span>(rc)]</span>
<span id="cb5-3"><a href="#cb5-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb5-4"><a href="#cb5-4" aria-hidden="true" tabindex="-1"></a>s_to_rc <span class="ot"><-</span></span>
<span id="cb5-5"><a href="#cb5-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>(s)</span>
<span id="cb5-6"><a href="#cb5-6" aria-hidden="true" tabindex="-1"></a> <span class="fu">drop</span>(<span class="fu">which</span>(LAYOUT <span class="sc">==</span> s, <span class="at">arr.ind =</span> <span class="cn">TRUE</span>, <span class="at">useNames =</span> <span class="cn">FALSE</span>))</span>
<span id="cb5-7"><a href="#cb5-7" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb5-8"><a href="#cb5-8" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb5-9"><a href="#cb5-9" aria-hidden="true" tabindex="-1"></a><span class="fu">rc_to_s</span>(<span class="fu">c</span>(<span class="dv">3</span>, <span class="dv">4</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 12</code></pre>
</div>
<div class="sourceCode cell-code" id="cb7"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb7-1"><a href="#cb7-1" aria-hidden="true" tabindex="-1"></a><span class="fu">s_to_rc</span>(<span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 3 4</code></pre>
</div>
</div>
<p>Start state</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb9"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb9-1"><a href="#cb9-1" aria-hidden="true" tabindex="-1"></a>start <span class="ot"><-</span> 1L</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Define terminal states.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb10"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb10-1"><a href="#cb10-1" aria-hidden="true" tabindex="-1"></a>is_terminal <span class="ot"><-</span> <span class="cf">function</span>(s) s <span class="sc">%in%</span> <span class="fu">c</span>(<span class="dv">5</span>, <span class="dv">11</span>, <span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
</section>
<section id="actions" class="level2">
<h2 class="anchored" data-anchor-id="actions">Actions</h2>
<p>The complete set of actions is <span class="math inline">\(A = \{\mathrm{'Up', 'Right', 'Down', 'Left', 'None'}\}\)</span>. Not all actions are available in every state. Also, action <code>None</code> is added as the only possible action in an absorbing state.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb11"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb11-1"><a href="#cb11-1" aria-hidden="true" tabindex="-1"></a>A <span class="ot">=</span> <span class="fu">c</span>(<span class="st">'Up'</span>, <span class="st">'Right'</span>, <span class="st">'Down'</span>, <span class="st">'Left'</span>, <span class="st">'None'</span>)</span>
<span id="cb11-2"><a href="#cb11-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb11-3"><a href="#cb11-3" aria-hidden="true" tabindex="-1"></a>actions <span class="ot"><-</span> <span class="cf">function</span>(s) { </span>
<span id="cb11-4"><a href="#cb11-4" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb11-5"><a href="#cb11-5" aria-hidden="true" tabindex="-1"></a> <span class="co"># absorbing states</span></span>
<span id="cb11-6"><a href="#cb11-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(s <span class="sc">==</span> <span class="dv">11</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">12</span>) <span class="fu">return</span>(<span class="st">'None'</span>)</span>
<span id="cb11-7"><a href="#cb11-7" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb11-8"><a href="#cb11-8" aria-hidden="true" tabindex="-1"></a> <span class="co"># illegal state</span></span>
<span id="cb11-9"><a href="#cb11-9" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(s <span class="sc">==</span> <span class="dv">5</span>) <span class="fu">return</span>(<span class="st">'None'</span>)</span>
<span id="cb11-10"><a href="#cb11-10" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb11-11"><a href="#cb11-11" aria-hidden="true" tabindex="-1"></a> <span class="fu">c</span>(<span class="st">'Up'</span>, <span class="st">'Right'</span>, <span class="st">'Down'</span>, <span class="st">'Left'</span>)</span>
<span id="cb11-12"><a href="#cb11-12" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb11-13"><a href="#cb11-13" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb11-14"><a href="#cb11-14" aria-hidden="true" tabindex="-1"></a><span class="fu">lapply</span>(S, actions)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[[1]]
[1] "Up" "Right" "Down" "Left"
[[2]]
[1] "Up" "Right" "Down" "Left"
[[3]]
[1] "Up" "Right" "Down" "Left"
[[4]]
[1] "Up" "Right" "Down" "Left"
[[5]]
[1] "None"
[[6]]
[1] "Up" "Right" "Down" "Left"
[[7]]
[1] "Up" "Right" "Down" "Left"
[[8]]
[1] "Up" "Right" "Down" "Left"
[[9]]
[1] "Up" "Right" "Down" "Left"
[[10]]
[1] "Up" "Right" "Down" "Left"
[[11]]
[1] "None"
[[12]]
[1] "None"</code></pre>
</div>
</div>
</section>
<section id="rewards" class="level2">
<h2 class="anchored" data-anchor-id="rewards">Rewards</h2>
<p><span class="math inline">\(R(s, a, s')\)</span> define the reward for the transition from <span class="math inline">\(s\)</span> to <span class="math inline">\(s'\)</span> with action <span class="math inline">\(a\)</span>.</p>
<p>For the textbook example we have:</p>
<ul>
<li>Any move costs utility (a reward of -0.04).</li>
<li>Going to state 12 has a reward of +1</li>
<li>Going to state 11 has a reward of -1.</li>
</ul>
<p>Note that once you are in an absorbing state (11 or 12), then the problem is over and there is no more reward!</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb13"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb13-1"><a href="#cb13-1" aria-hidden="true" tabindex="-1"></a>R <span class="ot"><-</span> <span class="cf">function</span>(s, a, s_prime) {</span>
<span id="cb13-2"><a href="#cb13-2" aria-hidden="true" tabindex="-1"></a> <span class="do">## no more reward when we in 11 or 12.</span></span>
<span id="cb13-3"><a href="#cb13-3" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(a <span class="sc">==</span> <span class="st">'None'</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">11</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">12</span>) <span class="fu">return</span>(<span class="dv">0</span>)</span>
<span id="cb13-4"><a href="#cb13-4" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb13-5"><a href="#cb13-5" aria-hidden="true" tabindex="-1"></a> <span class="do">## transition to the absorbing states.</span></span>
<span id="cb13-6"><a href="#cb13-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(s_prime <span class="sc">==</span> <span class="dv">12</span>) <span class="fu">return</span>(<span class="sc">+</span><span class="dv">1</span>)</span>
<span id="cb13-7"><a href="#cb13-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(s_prime <span class="sc">==</span> <span class="dv">11</span>) <span class="fu">return</span>(<span class="sc">-</span><span class="dv">1</span>)</span>
<span id="cb13-8"><a href="#cb13-8" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb13-9"><a href="#cb13-9" aria-hidden="true" tabindex="-1"></a> <span class="do">## cost for each move</span></span>
<span id="cb13-10"><a href="#cb13-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(<span class="sc">-</span><span class="fl">0.04</span>)</span>
<span id="cb13-11"><a href="#cb13-11" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb13-12"><a href="#cb13-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb13-13"><a href="#cb13-13" aria-hidden="true" tabindex="-1"></a><span class="fu">R</span>(<span class="dv">1</span>, <span class="st">'Up'</span>, <span class="dv">2</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] -0.04</code></pre>
</div>
<div class="sourceCode cell-code" id="cb15"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb15-1"><a href="#cb15-1" aria-hidden="true" tabindex="-1"></a><span class="fu">R</span>(<span class="dv">9</span>, <span class="st">'Right'</span>, <span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 1</code></pre>
</div>
<div class="sourceCode cell-code" id="cb17"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb17-1"><a href="#cb17-1" aria-hidden="true" tabindex="-1"></a><span class="fu">R</span>(<span class="dv">12</span>, <span class="st">'None'</span>, <span class="dv">12</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0</code></pre>
</div>
</div>
</section>
<section id="transition-model" class="level2">
<h2 class="anchored" data-anchor-id="transition-model">Transition Model</h2>
<p><span class="math inline">\(P(s' | s, a)\)</span> is the probability of going from state <span class="math inline">\(s\)</span> to <span class="math inline">\(s'\)</span> by when taking action <span class="math inline">\(a\)</span>. We will create a matrix <span class="math inline">\(P_a(s' | s)\)</span> for each action.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb19"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb19-1"><a href="#cb19-1" aria-hidden="true" tabindex="-1"></a>calc_transition <span class="ot"><-</span> <span class="cf">function</span>(s, action) {</span>
<span id="cb19-2"><a href="#cb19-2" aria-hidden="true" tabindex="-1"></a> action <span class="ot"><-</span> <span class="fu">match.arg</span>(action, <span class="at">choices =</span> A)</span>
<span id="cb19-3"><a href="#cb19-3" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-4"><a href="#cb19-4" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(<span class="fu">length</span>(s) <span class="sc">></span> <span class="dv">1</span>) <span class="fu">return</span>(<span class="fu">t</span>(<span class="fu">sapply</span>(s, calc_transition, <span class="at">action =</span> action)))</span>
<span id="cb19-5"><a href="#cb19-5" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-6"><a href="#cb19-6" aria-hidden="true" tabindex="-1"></a> <span class="co"># deal with absorbing and illegal state</span></span>
<span id="cb19-7"><a href="#cb19-7" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(s <span class="sc">==</span> <span class="dv">11</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">12</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">5</span> <span class="sc">||</span> action <span class="sc">==</span> <span class="st">'None'</span>) {</span>
<span id="cb19-8"><a href="#cb19-8" aria-hidden="true" tabindex="-1"></a> P <span class="ot"><-</span> <span class="fu">rep</span>(<span class="dv">0</span>, <span class="fu">length</span>(S))</span>
<span id="cb19-9"><a href="#cb19-9" aria-hidden="true" tabindex="-1"></a> P[s] <span class="ot"><-</span> <span class="dv">1</span></span>
<span id="cb19-10"><a href="#cb19-10" aria-hidden="true" tabindex="-1"></a> <span class="fu">return</span>(P) </span>
<span id="cb19-11"><a href="#cb19-11" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb19-12"><a href="#cb19-12" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-13"><a href="#cb19-13" aria-hidden="true" tabindex="-1"></a> action_to_delta <span class="ot"><-</span> <span class="fu">list</span>(</span>
<span id="cb19-14"><a href="#cb19-14" aria-hidden="true" tabindex="-1"></a> <span class="st">'Up'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="sc">+</span><span class="dv">1</span>, <span class="dv">0</span>),</span>
<span id="cb19-15"><a href="#cb19-15" aria-hidden="true" tabindex="-1"></a> <span class="st">'Down'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">0</span>),</span>
<span id="cb19-16"><a href="#cb19-16" aria-hidden="true" tabindex="-1"></a> <span class="st">'Right'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="sc">+</span><span class="dv">1</span>),</span>
<span id="cb19-17"><a href="#cb19-17" aria-hidden="true" tabindex="-1"></a> <span class="st">'Left'</span> <span class="ot">=</span> <span class="fu">c</span>(<span class="dv">0</span>, <span class="sc">-</span><span class="dv">1</span>)</span>
<span id="cb19-18"><a href="#cb19-18" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb19-19"><a href="#cb19-19" aria-hidden="true" tabindex="-1"></a> delta <span class="ot"><-</span> action_to_delta[[action]]</span>
<span id="cb19-20"><a href="#cb19-20" aria-hidden="true" tabindex="-1"></a> dr <span class="ot"><-</span> delta[<span class="dv">1</span>]</span>
<span id="cb19-21"><a href="#cb19-21" aria-hidden="true" tabindex="-1"></a> dc <span class="ot"><-</span> delta[<span class="dv">2</span>]</span>
<span id="cb19-22"><a href="#cb19-22" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-23"><a href="#cb19-23" aria-hidden="true" tabindex="-1"></a> rc <span class="ot"><-</span> <span class="fu">s_to_rc</span>(s)</span>
<span id="cb19-24"><a href="#cb19-24" aria-hidden="true" tabindex="-1"></a> r <span class="ot"><-</span> rc[<span class="dv">1</span>]</span>
<span id="cb19-25"><a href="#cb19-25" aria-hidden="true" tabindex="-1"></a> c <span class="ot"><-</span> rc[<span class="dv">2</span>]</span>
<span id="cb19-26"><a href="#cb19-26" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-27"><a href="#cb19-27" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(dr <span class="sc">!=</span> <span class="dv">0</span> <span class="sc">&&</span> dc <span class="sc">!=</span> <span class="dv">0</span>) </span>
<span id="cb19-28"><a href="#cb19-28" aria-hidden="true" tabindex="-1"></a> <span class="fu">stop</span>(<span class="st">"You can only go up/down or right/left!"</span>)</span>
<span id="cb19-29"><a href="#cb19-29" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-30"><a href="#cb19-30" aria-hidden="true" tabindex="-1"></a> P <span class="ot"><-</span> <span class="fu">matrix</span>(<span class="dv">0</span>, <span class="at">nrow =</span> ROWS, <span class="at">ncol =</span> COLS)</span>
<span id="cb19-31"><a href="#cb19-31" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-32"><a href="#cb19-32" aria-hidden="true" tabindex="-1"></a> <span class="co"># UP/DOWN</span></span>
<span id="cb19-33"><a href="#cb19-33" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(dr <span class="sc">!=</span> <span class="dv">0</span>) {</span>
<span id="cb19-34"><a href="#cb19-34" aria-hidden="true" tabindex="-1"></a> new_r <span class="ot"><-</span> r <span class="sc">+</span> dr</span>
<span id="cb19-35"><a href="#cb19-35" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(new_r <span class="sc">></span> ROWS <span class="sc">||</span> new_r <span class="sc"><</span> <span class="dv">1</span>) new_r <span class="ot"><-</span> r</span>
<span id="cb19-36"><a href="#cb19-36" aria-hidden="true" tabindex="-1"></a> <span class="do">## can't got to (2, 2)</span></span>
<span id="cb19-37"><a href="#cb19-37" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(new_r <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&&</span> c <span class="sc">==</span> <span class="dv">2</span>) new_r <span class="ot"><-</span> r</span>
<span id="cb19-38"><a href="#cb19-38" aria-hidden="true" tabindex="-1"></a> P[new_r, c] <span class="ot"><-</span> .<span class="dv">8</span></span>
<span id="cb19-39"><a href="#cb19-39" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-40"><a href="#cb19-40" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(c <span class="sc"><</span> COLS <span class="sc">&</span> <span class="sc">!</span>(r <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&</span> (c <span class="sc">+</span> <span class="dv">1</span>) <span class="sc">==</span> <span class="dv">2</span>)) </span>
<span id="cb19-41"><a href="#cb19-41" aria-hidden="true" tabindex="-1"></a> P[r, c <span class="sc">+</span> <span class="dv">1</span>] <span class="ot"><-</span> .<span class="dv">1</span> <span class="cf">else</span> P[r, c] <span class="ot"><-</span> P[r, c] <span class="sc">+</span> .<span class="dv">1</span> </span>
<span id="cb19-42"><a href="#cb19-42" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(c <span class="sc">></span> <span class="dv">1</span> <span class="sc">&</span> <span class="sc">!</span>(r <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&</span> (c <span class="sc">-</span> <span class="dv">1</span>) <span class="sc">==</span> <span class="dv">2</span>)) </span>
<span id="cb19-43"><a href="#cb19-43" aria-hidden="true" tabindex="-1"></a> P[r, c <span class="sc">-</span> <span class="dv">1</span>] <span class="ot"><-</span> .<span class="dv">1</span> <span class="cf">else</span> P[r, c] <span class="ot"><-</span> P[r, c] <span class="sc">+</span> .<span class="dv">1</span> </span>
<span id="cb19-44"><a href="#cb19-44" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb19-45"><a href="#cb19-45" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-46"><a href="#cb19-46" aria-hidden="true" tabindex="-1"></a> <span class="co"># RIGHT/LEFT</span></span>
<span id="cb19-47"><a href="#cb19-47" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(dc <span class="sc">!=</span> <span class="dv">0</span>) {</span>
<span id="cb19-48"><a href="#cb19-48" aria-hidden="true" tabindex="-1"></a> new_c <span class="ot"><-</span> c <span class="sc">+</span> dc</span>
<span id="cb19-49"><a href="#cb19-49" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(new_c <span class="sc">></span> COLS <span class="sc">||</span> new_c <span class="sc"><</span> <span class="dv">1</span>) new_c <span class="ot"><-</span> c</span>
<span id="cb19-50"><a href="#cb19-50" aria-hidden="true" tabindex="-1"></a> <span class="do">## can't got to (2, 2)</span></span>
<span id="cb19-51"><a href="#cb19-51" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(r <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&&</span> new_c <span class="sc">==</span> <span class="dv">2</span>) new_c <span class="ot"><-</span> c</span>
<span id="cb19-52"><a href="#cb19-52" aria-hidden="true" tabindex="-1"></a> P[r, new_c] <span class="ot"><-</span> .<span class="dv">8</span></span>
<span id="cb19-53"><a href="#cb19-53" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-54"><a href="#cb19-54" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(r <span class="sc"><</span> ROWS <span class="sc">&</span> <span class="sc">!</span>((r <span class="sc">+</span> <span class="dv">1</span>) <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&</span> c <span class="sc">==</span> <span class="dv">2</span>)) </span>
<span id="cb19-55"><a href="#cb19-55" aria-hidden="true" tabindex="-1"></a> P[r <span class="sc">+</span> <span class="dv">1</span>, c] <span class="ot"><-</span> .<span class="dv">1</span> <span class="cf">else</span> P[r, c] <span class="ot"><-</span> P[r, c] <span class="sc">+</span> .<span class="dv">1</span> </span>
<span id="cb19-56"><a href="#cb19-56" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span>(r <span class="sc">></span> <span class="dv">1</span> <span class="sc">&</span> <span class="sc">!</span>((r <span class="sc">-</span> <span class="dv">1</span>) <span class="sc">==</span> <span class="dv">2</span> <span class="sc">&</span> c <span class="sc">==</span> <span class="dv">2</span>)) </span>
<span id="cb19-57"><a href="#cb19-57" aria-hidden="true" tabindex="-1"></a> P[r <span class="sc">-</span> <span class="dv">1</span>, c] <span class="ot"><-</span> .<span class="dv">1</span> <span class="cf">else</span> P[r, c] <span class="ot"><-</span> P[r, c] <span class="sc">+</span> .<span class="dv">1</span> </span>
<span id="cb19-58"><a href="#cb19-58" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb19-59"><a href="#cb19-59" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb19-60"><a href="#cb19-60" aria-hidden="true" tabindex="-1"></a> <span class="fu">as.vector</span>(P)</span>
<span id="cb19-61"><a href="#cb19-61" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Try to go up from state 1 (this is (1,1), the bottom left corner). Note: we cannot go left so there is a .1 chance to stay in place.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb20"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb20-1"><a href="#cb20-1" aria-hidden="true" tabindex="-1"></a><span class="fu">calc_transition</span>(<span class="dv">1</span>, <span class="st">'Up'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [1] 0.1 0.8 0.0 0.1 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0</code></pre>
</div>
<div class="sourceCode cell-code" id="cb22"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb22-1"><a href="#cb22-1" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(<span class="fu">calc_transition</span>(<span class="dv">1</span>, <span class="st">'Up'</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 0.0 0.0 0 0
2 0.8 0.0 0 0
1 0.1 0.1 0 0</code></pre>
</div>
</div>
<p>Try to go right from (2,1). Since right is blocked, there is a .8 probability of staying in place.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb24"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb24-1"><a href="#cb24-1" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(<span class="fu">calc_transition</span>(<span class="dv">2</span>, <span class="st">'Right'</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 0.1 0 0 0
2 0.8 0 0 0
1 0.1 0 0 0</code></pre>
</div>
</div>
<p>Calculate transitions for each state to each other state. Each row represents a state <span class="math inline">\(s\)</span> and each column a state <span class="math inline">\(s'\)</span> so we get a complete definition for <span class="math inline">\(P_a(s' | s)\)</span>. Note that the matrix is stochastic (all rows add up to 1).</p>
<p>Create a matrix for each action.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb26"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb26-1"><a href="#cb26-1" aria-hidden="true" tabindex="-1"></a>P_matrices <span class="ot"><-</span> <span class="fu">lapply</span>(A, <span class="at">FUN =</span> <span class="cf">function</span>(a) <span class="fu">calc_transition</span>(S, a))</span>
<span id="cb26-2"><a href="#cb26-2" aria-hidden="true" tabindex="-1"></a><span class="fu">names</span>(P_matrices) <span class="ot"><-</span> A</span>
<span id="cb26-3"><a href="#cb26-3" aria-hidden="true" tabindex="-1"></a><span class="fu">str</span>(P_matrices)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>List of 5
$ Up : num [1:12, 1:12] 0.1 0 0 0.1 0 0 0 0 0 0 ...
$ Right: num [1:12, 1:12] 0.1 0.1 0 0 0 0 0 0 0 0 ...
$ Down : num [1:12, 1:12] 0.9 0.8 0 0.1 0 0 0 0 0 0 ...
$ Left : num [1:12, 1:12] 0.9 0.1 0 0.8 0 0 0 0 0 0 ...
$ None : num [1:12, 1:12] 1 0 0 0 0 0 0 0 0 0 ...</code></pre>
</div>
</div>
<p>Create a function interface for <span class="math inline">\(P(s' | s, a)\)</span>.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb28"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb28-1"><a href="#cb28-1" aria-hidden="true" tabindex="-1"></a>P <span class="ot"><-</span> <span class="cf">function</span>(sp, s, a) P_matrices[[a]][s, sp]</span>
<span id="cb28-2"><a href="#cb28-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb28-3"><a href="#cb28-3" aria-hidden="true" tabindex="-1"></a><span class="fu">P</span>(<span class="dv">2</span>, <span class="dv">1</span>, <span class="st">'Up'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.8</code></pre>
</div>
<div class="sourceCode cell-code" id="cb30"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb30-1"><a href="#cb30-1" aria-hidden="true" tabindex="-1"></a><span class="fu">P</span>(<span class="dv">5</span>, <span class="dv">4</span>, <span class="st">'Up'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0</code></pre>
</div>
</div>
</section>
<section id="policy" class="level2">
<h2 class="anchored" data-anchor-id="policy">Policy</h2>
<p>The solution to an MDP is a policy <span class="math inline">\(\pi\)</span> which defines which action to take in each state. We represent deterministic policies as a vector of actions. I make up a policy that always goes up and then to the right once the agent hits the top.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb32"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb32-1"><a href="#cb32-1" aria-hidden="true" tabindex="-1"></a>pi_manual <span class="ot"><-</span> <span class="fu">rep</span>(<span class="st">'Up'</span>, <span class="at">times =</span> <span class="fu">length</span>(S))</span>
<span id="cb32-2"><a href="#cb32-2" aria-hidden="true" tabindex="-1"></a>pi_manual[<span class="fu">c</span>(<span class="dv">3</span>, <span class="dv">6</span>, <span class="dv">9</span>)] <span class="ot"><-</span> <span class="st">'Right'</span></span>
<span id="cb32-3"><a href="#cb32-3" aria-hidden="true" tabindex="-1"></a>pi_manual</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> [1] "Up" "Up" "Right" "Up" "Up" "Right" "Up" "Up" "Right"
[10] "Up" "Up" "Up" </code></pre>
</div>
<div class="sourceCode cell-code" id="cb34"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb34-1"><a href="#cb34-1" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(pi_manual)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 "Right" "Right" "Right" "Up"
2 "Up" "Up" "Up" "Up"
1 "Up" "Up" "Up" "Up"</code></pre>
</div>
</div>
<p>We can also create a random policy by randomly choosing from the available actions for each state.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb36"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb36-1"><a href="#cb36-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1234</span>)</span>
<span id="cb36-2"><a href="#cb36-2" aria-hidden="true" tabindex="-1"></a>create_random_policy <span class="ot"><-</span></span>
<span id="cb36-3"><a href="#cb36-3" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>()</span>
<span id="cb36-4"><a href="#cb36-4" aria-hidden="true" tabindex="-1"></a> <span class="fu">sapply</span>(</span>
<span id="cb36-5"><a href="#cb36-5" aria-hidden="true" tabindex="-1"></a> S,</span>
<span id="cb36-6"><a href="#cb36-6" aria-hidden="true" tabindex="-1"></a> <span class="at">FUN =</span> <span class="cf">function</span>(s)</span>
<span id="cb36-7"><a href="#cb36-7" aria-hidden="true" tabindex="-1"></a> <span class="fu">sample</span>(<span class="fu">actions</span>(s), <span class="dv">1</span>)</span>
<span id="cb36-8"><a href="#cb36-8" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb36-9"><a href="#cb36-9" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb36-10"><a href="#cb36-10" aria-hidden="true" tabindex="-1"></a>pi_random <span class="ot"><-</span> <span class="fu">create_random_policy</span>()</span>
<span id="cb36-11"><a href="#cb36-11" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(pi_random)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 "Right" "Left" "Up" "None"
2 "Left" "None" "Up" "None"
1 "Left" "Right" "Down" "Right"</code></pre>
</div>
</div>
<p>Stochastic policies use probabilities of actions in each state. We use as simple table with probabilities where each row is a state and the columns are the actions.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb38"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb38-1"><a href="#cb38-1" aria-hidden="true" tabindex="-1"></a><span class="fu">set.seed</span>(<span class="dv">1234</span>)</span>
<span id="cb38-2"><a href="#cb38-2" aria-hidden="true" tabindex="-1"></a>create_random_soft_policy <span class="ot"><-</span></span>
<span id="cb38-3"><a href="#cb38-3" aria-hidden="true" tabindex="-1"></a> <span class="cf">function</span>() {</span>
<span id="cb38-4"><a href="#cb38-4" aria-hidden="true" tabindex="-1"></a> p <span class="ot"><-</span> <span class="fu">matrix</span>(<span class="dv">0</span>, <span class="at">nrow =</span> <span class="fu">length</span>(S), <span class="at">ncol =</span> <span class="fu">length</span>(A), <span class="at">dimnames =</span> <span class="fu">list</span>(S, A)) </span>
<span id="cb38-5"><a href="#cb38-5" aria-hidden="true" tabindex="-1"></a> <span class="cf">for</span> (s <span class="cf">in</span> S)</span>
<span id="cb38-6"><a href="#cb38-6" aria-hidden="true" tabindex="-1"></a> p[s, <span class="fu">actions</span>(s)] <span class="ot"><-</span> <span class="fu">runif</span>(<span class="at">n =</span> <span class="fu">length</span>(<span class="fu">actions</span>(s)))</span>
<span id="cb38-7"><a href="#cb38-7" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb38-8"><a href="#cb38-8" aria-hidden="true" tabindex="-1"></a> <span class="co"># make stochastic (rows sum up to 1)</span></span>
<span id="cb38-9"><a href="#cb38-9" aria-hidden="true" tabindex="-1"></a>p <span class="ot"><-</span> <span class="fu">sweep</span>(p, <span class="at">MARGIN =</span> <span class="dv">1</span>, <span class="at">STATS =</span> <span class="fu">rowSums</span>(p), <span class="at">FUN =</span> <span class="st">'/'</span>)</span>
<span id="cb38-10"><a href="#cb38-10" aria-hidden="true" tabindex="-1"></a>p</span>
<span id="cb38-11"><a href="#cb38-11" aria-hidden="true" tabindex="-1"></a>}</span>
<span id="cb38-12"><a href="#cb38-12" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb38-13"><a href="#cb38-13" aria-hidden="true" tabindex="-1"></a>pi_random_soft <span class="ot"><-</span> <span class="fu">create_random_soft_policy</span>()</span>
<span id="cb38-14"><a href="#cb38-14" aria-hidden="true" tabindex="-1"></a>pi_random_soft</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> Up Right Down Left None
1 0.05775684 0.3161035 0.309487501 0.3166521 0
2 0.49385022 0.3673038 0.005447087 0.1333988 0
3 0.27536627 0.2125970 0.286738186 0.2252985 0
4 0.12104469 0.3953429 0.125147070 0.3584653 0
5 0.00000000 0.0000000 0.000000000 0.0000000 1
6 0.26618674 0.1862791 0.231674078 0.3158601 0
7 0.42009540 0.2207333 0.055508653 0.3036627 0
8 0.26298502 0.1705537 0.296745408 0.2697159 0
9 0.04270727 0.4255694 0.247440113 0.2842832 0
10 0.30758515 0.1098004 0.460595778 0.1220187 0
11 0.00000000 0.0000000 0.000000000 0.0000000 1
12 0.00000000 0.0000000 0.000000000 0.0000000 1</code></pre>
</div>
</div>
</section>
<section id="expected-utility" class="level2">
<h2 class="anchored" data-anchor-id="expected-utility">Expected Utility</h2>
<p>The expected utility can be calculated by</p>
<p><span class="math inline">\(U^\pi = E\left[\sum_{t=0}^\infty \gamma^t R(s_t, \pi(s_t), s_{t+1})\right]\)</span></p>
<p>We need to define the discount factor.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb40"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb40-1"><a href="#cb40-1" aria-hidden="true" tabindex="-1"></a>GAMMA <span class="ot"><-</span> <span class="dv">1</span></span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>We can evaluate the utility of a policy using simulation. For the stochastic transition model, we need to be able to sample the state <span class="math inline">\(s'\)</span> the system transitions to when using action <span class="math inline">\(a\)</span> in state <span class="math inline">\(s\)</span>.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb41"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb41-1"><a href="#cb41-1" aria-hidden="true" tabindex="-1"></a>sample_transition <span class="ot"><-</span> <span class="cf">function</span>(s, a)</span>
<span id="cb41-2"><a href="#cb41-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">sample</span>(S, <span class="at">size =</span> <span class="dv">1</span>, <span class="at">prob =</span> P_matrices[[a]][s,])</span>
<span id="cb41-3"><a href="#cb41-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb41-4"><a href="#cb41-4" aria-hidden="true" tabindex="-1"></a><span class="fu">sample_transition</span>(<span class="dv">1</span>, <span class="st">'Up'</span>)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 4</code></pre>
</div>
<div class="sourceCode cell-code" id="cb43"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb43-1"><a href="#cb43-1" aria-hidden="true" tabindex="-1"></a><span class="fu">table</span>(<span class="fu">replicate</span>(<span class="at">n =</span> <span class="dv">100</span>, <span class="fu">sample_transition</span>(<span class="dv">1</span>, <span class="st">'Up'</span>)))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>
1 2 4
11 83 6 </code></pre>
</div>
</div>
<p>We can now simulate the utility for one episode.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb45"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb45-1"><a href="#cb45-1" aria-hidden="true" tabindex="-1"></a>simulate_utility <span class="ot"><-</span> <span class="cf">function</span>(pi, <span class="at">s0 =</span> <span class="dv">1</span>, <span class="at">max_t =</span> <span class="dv">100</span>) {</span>
<span id="cb45-2"><a href="#cb45-2" aria-hidden="true" tabindex="-1"></a> s <span class="ot"><-</span> s0</span>
<span id="cb45-3"><a href="#cb45-3" aria-hidden="true" tabindex="-1"></a> U <span class="ot"><-</span> <span class="dv">0</span></span>
<span id="cb45-4"><a href="#cb45-4" aria-hidden="true" tabindex="-1"></a> t <span class="ot"><-</span> <span class="dv">0</span></span>
<span id="cb45-5"><a href="#cb45-5" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-6"><a href="#cb45-6" aria-hidden="true" tabindex="-1"></a> <span class="cf">while</span> (<span class="cn">TRUE</span>) {</span>
<span id="cb45-7"><a href="#cb45-7" aria-hidden="true" tabindex="-1"></a> <span class="do">## get action from policy (matrix means it is a stochastic policy)</span></span>
<span id="cb45-8"><a href="#cb45-8" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (<span class="sc">!</span><span class="fu">is.matrix</span>(pi))</span>
<span id="cb45-9"><a href="#cb45-9" aria-hidden="true" tabindex="-1"></a> a <span class="ot"><-</span> pi[s]</span>
<span id="cb45-10"><a href="#cb45-10" aria-hidden="true" tabindex="-1"></a> <span class="cf">else</span></span>
<span id="cb45-11"><a href="#cb45-11" aria-hidden="true" tabindex="-1"></a> a <span class="ot"><-</span> <span class="fu">sample</span>(A, <span class="at">size =</span> <span class="dv">1</span>, <span class="at">prob =</span> pi[s, ])</span>
<span id="cb45-12"><a href="#cb45-12" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-13"><a href="#cb45-13" aria-hidden="true" tabindex="-1"></a> <span class="do">## sample a transition given the action from the policy</span></span>
<span id="cb45-14"><a href="#cb45-14" aria-hidden="true" tabindex="-1"></a> s_prime <span class="ot"><-</span> <span class="fu">sample_transition</span>(s, a)</span>
<span id="cb45-15"><a href="#cb45-15" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-16"><a href="#cb45-16" aria-hidden="true" tabindex="-1"></a> <span class="do">##</span></span>
<span id="cb45-17"><a href="#cb45-17" aria-hidden="true" tabindex="-1"></a> U <span class="ot"><-</span> U <span class="sc">+</span> GAMMA <span class="sc">^</span> t <span class="sc">*</span> <span class="fu">R</span>(s, a, s_prime)</span>
<span id="cb45-18"><a href="#cb45-18" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-19"><a href="#cb45-19" aria-hidden="true" tabindex="-1"></a> s <span class="ot"><-</span> s_prime</span>
<span id="cb45-20"><a href="#cb45-20" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-21"><a href="#cb45-21" aria-hidden="true" tabindex="-1"></a> <span class="do">## reached an absorbing state?</span></span>
<span id="cb45-22"><a href="#cb45-22" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (s <span class="sc">==</span> <span class="dv">11</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">12</span> <span class="sc">||</span> s <span class="sc">==</span> <span class="dv">5</span>)</span>
<span id="cb45-23"><a href="#cb45-23" aria-hidden="true" tabindex="-1"></a> <span class="cf">break</span></span>
<span id="cb45-24"><a href="#cb45-24" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-25"><a href="#cb45-25" aria-hidden="true" tabindex="-1"></a> t <span class="ot"><-</span> t <span class="sc">+</span> <span class="dv">1</span></span>
<span id="cb45-26"><a href="#cb45-26" aria-hidden="true" tabindex="-1"></a> <span class="cf">if</span> (t <span class="sc">>=</span> max_t)</span>
<span id="cb45-27"><a href="#cb45-27" aria-hidden="true" tabindex="-1"></a> <span class="cf">break</span></span>
<span id="cb45-28"><a href="#cb45-28" aria-hidden="true" tabindex="-1"></a> }</span>
<span id="cb45-29"><a href="#cb45-29" aria-hidden="true" tabindex="-1"></a> </span>
<span id="cb45-30"><a href="#cb45-30" aria-hidden="true" tabindex="-1"></a> U</span>
<span id="cb45-31"><a href="#cb45-31" aria-hidden="true" tabindex="-1"></a>}</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>Simulate the <code>N</code> episodes.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb46"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb46-1"><a href="#cb46-1" aria-hidden="true" tabindex="-1"></a>simulate_utilities <span class="ot"><-</span> <span class="cf">function</span>(pi, <span class="at">s0 =</span> <span class="dv">1</span>, <span class="at">N =</span> <span class="dv">1000</span>, <span class="at">max_t =</span> <span class="dv">100</span>)</span>
<span id="cb46-2"><a href="#cb46-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">replicate</span>(N, <span class="fu">simulate_utility</span>(pi, s0, max_t))</span>
<span id="cb46-3"><a href="#cb46-3" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb46-4"><a href="#cb46-4" aria-hidden="true" tabindex="-1"></a>utility_manual <span class="ot"><-</span> <span class="fu">simulate_utilities</span>(pi_manual)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
</div>
<p>The expected utility for starting from state <span class="math inline">\(s_0 = 1\)</span> is.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb47"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb47-1"><a href="#cb47-1" aria-hidden="true" tabindex="-1"></a><span class="fu">mean</span>(utility_manual)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>[1] 0.69924</code></pre>
</div>
<div class="sourceCode cell-code" id="cb49"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb49-1"><a href="#cb49-1" aria-hidden="true" tabindex="-1"></a><span class="fu">hist</span>(utility_manual, <span class="at">xlim =</span> <span class="fu">c</span>(<span class="sc">-</span><span class="dv">1</span>, <span class="dv">1</span>))</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output-display">
<p><img src="_AIMA-4x3-gridworld_files/figure-html/unnamed-chunk-20-1.png" class="img-fluid" width="672"></p>
</div>
</div>
<p>Compare with the utility of the random policy.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb50"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb50-1"><a href="#cb50-1" aria-hidden="true" tabindex="-1"></a>utility_random <span class="ot"><-</span> <span class="fu">simulate_utilities</span>(pi_random, <span class="at">max_t =</span> <span class="dv">100</span>)</span>
<span id="cb50-2"><a href="#cb50-2" aria-hidden="true" tabindex="-1"></a></span>
<span id="cb50-3"><a href="#cb50-3" aria-hidden="true" tabindex="-1"></a><span class="fu">table</span>(utility_random)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code>utility_random
-4
1000 </code></pre>
</div>
</div>
<p>The random policy performs really bad and most likely always stumbles around for <code>max_t</code> moves at a cost of .04 each. The manually created policy is obviously better.</p>
<p>We can use simulation to estimate the expected utility for starting from each state following the policy.</p>
<div class="cell">
<div class="sourceCode cell-code" id="cb52"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb52-1"><a href="#cb52-1" aria-hidden="true" tabindex="-1"></a>U_manual <span class="ot"><-</span></span>
<span id="cb52-2"><a href="#cb52-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">sapply</span>(</span>
<span id="cb52-3"><a href="#cb52-3" aria-hidden="true" tabindex="-1"></a> S,</span>
<span id="cb52-4"><a href="#cb52-4" aria-hidden="true" tabindex="-1"></a> <span class="at">FUN =</span> <span class="cf">function</span>(s)</span>
<span id="cb52-5"><a href="#cb52-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(<span class="fu">simulate_utilities</span>(pi_manual, <span class="at">s0 =</span> s))</span>
<span id="cb52-6"><a href="#cb52-6" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb52-7"><a href="#cb52-7" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(U_manual)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 0.86044 0.89644 0.95420 0.00000
2 0.79396 -0.04000 0.71252 0.00000
1 0.71736 0.35884 0.51804 -0.86708</code></pre>
</div>
</div>
<div class="cell">
<div class="sourceCode cell-code" id="cb54"><pre class="sourceCode r code-with-copy"><code class="sourceCode r"><span id="cb54-1"><a href="#cb54-1" aria-hidden="true" tabindex="-1"></a>U_random <span class="ot"><-</span></span>
<span id="cb54-2"><a href="#cb54-2" aria-hidden="true" tabindex="-1"></a> <span class="fu">sapply</span>(</span>
<span id="cb54-3"><a href="#cb54-3" aria-hidden="true" tabindex="-1"></a> S,</span>
<span id="cb54-4"><a href="#cb54-4" aria-hidden="true" tabindex="-1"></a> <span class="at">FUN =</span> <span class="cf">function</span>(s)</span>
<span id="cb54-5"><a href="#cb54-5" aria-hidden="true" tabindex="-1"></a> <span class="fu">mean</span>(<span class="fu">simulate_utilities</span>(pi_random, <span class="at">s0 =</span> s))</span>
<span id="cb54-6"><a href="#cb54-6" aria-hidden="true" tabindex="-1"></a> )</span>
<span id="cb54-7"><a href="#cb54-7" aria-hidden="true" tabindex="-1"></a><span class="fu">show_layout</span>(U_random)</span></code><button title="Copy to Clipboard" class="code-copy-button"><i class="bi"></i></button></pre></div>
<div class="cell-output cell-output-stdout">
<pre><code> col
row 1 2 3 4
3 -4 -4.00000 -1.60504 0.00000
2 -4 0.00000 -1.44744 0.00000
1 -4 -1.84944 -1.79828 -1.37408</code></pre>
</div>
</div>
</section>
</section>
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if (window.bootstrap) {
button.setAttribute("data-bs-toggle", "tooltip");
button.setAttribute("data-bs-placement", "left");
button.setAttribute("data-bs-title", "Copied!");
tooltip = new bootstrap.Tooltip(button,
{ trigger: "manual",
customClass: "code-copy-button-tooltip",
offset: [0, -8]});
tooltip.show();
}
setTimeout(function() {
if (tooltip) {
tooltip.hide();
button.removeAttribute("data-bs-title");
button.removeAttribute("data-bs-toggle");
button.removeAttribute("data-bs-placement");
}
button.setAttribute("title", currentTitle);
button.classList.remove('code-copy-button-checked');
}, 1000);
// clear code selection
e.clearSelection();
});
function tippyHover(el, contentFn) {
const config = {
allowHTML: true,
content: contentFn,
maxWidth: 500,
delay: 100,
arrow: false,
appendTo: function(el) {
return el.parentElement;
},
interactive: true,
interactiveBorder: 10,
theme: 'quarto',
placement: 'bottom-start'
};
window.tippy(el, config);
}
const noterefs = window.document.querySelectorAll('a[role="doc-noteref"]');
for (var i=0; i<noterefs.length; i++) {
const ref = noterefs[i];
tippyHover(ref, function() {
// use id or data attribute instead here
let href = ref.getAttribute('data-footnote-href') || ref.getAttribute('href');
try { href = new URL(href).hash; } catch {}
const id = href.replace(/^#\/?/, "");
const note = window.document.getElementById(id);
return note.innerHTML;
});
}
let selectedAnnoteEl;
const selectorForAnnotation = ( cell, annotation) => {
let cellAttr = 'data-code-cell="' + cell + '"';
let lineAttr = 'data-code-annotation="' + annotation + '"';
const selector = 'span[' + cellAttr + '][' + lineAttr + ']';
return selector;
}
const selectCodeLines = (annoteEl) => {
const doc = window.document;
const targetCell = annoteEl.getAttribute("data-target-cell");
const targetAnnotation = annoteEl.getAttribute("data-target-annotation");
const annoteSpan = window.document.querySelector(selectorForAnnotation(targetCell, targetAnnotation));
const lines = annoteSpan.getAttribute("data-code-lines").split(",");
const lineIds = lines.map((line) => {
return targetCell + "-" + line;
})
let top = null;
let height = null;
let parent = null;
if (lineIds.length > 0) {
//compute the position of the single el (top and bottom and make a div)
const el = window.document.getElementById(lineIds[0]);
top = el.offsetTop;
height = el.offsetHeight;
parent = el.parentElement.parentElement;
if (lineIds.length > 1) {
const lastEl = window.document.getElementById(lineIds[lineIds.length - 1]);
const bottom = lastEl.offsetTop + lastEl.offsetHeight;
height = bottom - top;
}
if (top !== null && height !== null && parent !== null) {
// cook up a div (if necessary) and position it
let div = window.document.getElementById("code-annotation-line-highlight");
if (div === null) {
div = window.document.createElement("div");
div.setAttribute("id", "code-annotation-line-highlight");
div.style.position = 'absolute';
parent.appendChild(div);
}
div.style.top = top - 2 + "px";
div.style.height = height + 4 + "px";
let gutterDiv = window.document.getElementById("code-annotation-line-highlight-gutter");
if (gutterDiv === null) {
gutterDiv = window.document.createElement("div");
gutterDiv.setAttribute("id", "code-annotation-line-highlight-gutter");
gutterDiv.style.position = 'absolute';
const codeCell = window.document.getElementById(targetCell);
const gutter = codeCell.querySelector('.code-annotation-gutter');
gutter.appendChild(gutterDiv);
}
gutterDiv.style.top = top - 2 + "px";
gutterDiv.style.height = height + 4 + "px";
}
selectedAnnoteEl = annoteEl;
}
};
const unselectCodeLines = () => {
const elementsIds = ["code-annotation-line-highlight", "code-annotation-line-highlight-gutter"];
elementsIds.forEach((elId) => {
const div = window.document.getElementById(elId);
if (div) {
div.remove();
}
});
selectedAnnoteEl = undefined;
};
// Attach click handler to the DT
const annoteDls = window.document.querySelectorAll('dt[data-target-cell]');
for (const annoteDlNode of annoteDls) {
annoteDlNode.addEventListener('click', (event) => {
const clickedEl = event.target;
if (clickedEl !== selectedAnnoteEl) {
unselectCodeLines();
const activeEl = window.document.querySelector('dt[data-target-cell].code-annotation-active');
if (activeEl) {
activeEl.classList.remove('code-annotation-active');
}
selectCodeLines(clickedEl);
clickedEl.classList.add('code-annotation-active');
} else {
// Unselect the line
unselectCodeLines();
clickedEl.classList.remove('code-annotation-active');
}
});
}
const findCites = (el) => {
const parentEl = el.parentElement;
if (parentEl) {
const cites = parentEl.dataset.cites;
if (cites) {
return {
el,
cites: cites.split(' ')
};
} else {
return findCites(el.parentElement)
}
} else {
return undefined;
}
};
var bibliorefs = window.document.querySelectorAll('a[role="doc-biblioref"]');
for (var i=0; i<bibliorefs.length; i++) {
const ref = bibliorefs[i];
const citeInfo = findCites(ref);
if (citeInfo) {
tippyHover(citeInfo.el, function() {
var popup = window.document.createElement('div');
citeInfo.cites.forEach(function(cite) {
var citeDiv = window.document.createElement('div');
citeDiv.classList.add('hanging-indent');
citeDiv.classList.add('csl-entry');
var biblioDiv = window.document.getElementById('ref-' + cite);
if (biblioDiv) {
citeDiv.innerHTML = biblioDiv.innerHTML;
}
popup.appendChild(citeDiv);
});
return popup.innerHTML;
});
}
}
});
</script>
</div> <!-- /content -->
</body></html>