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<li class="toctree-l1 current"><a class="current reference internal" href="#"><strong>Doubling Trick for Multi-Armed Bandits</strong></a><ul>
<li class="toctree-l2"><a class="reference internal" href="#for-geometric-sequences">For geometric sequences</a></li>
<li class="toctree-l2"><a class="reference internal" href="#for-exponential-sequences">For exponential sequences</a></li>
<li class="toctree-l2"><a class="reference internal" href="#article">Article</a></li>
<li class="toctree-l2"><a class="reference internal" href="#configuration">Configuration</a></li>
<li class="toctree-l2"><a class="reference internal" href="#how-to-run-the-experiments">How to run the experiments ?</a></li>
<li class="toctree-l2"><a class="reference internal" href="#some-illustrations">Some illustrations</a><ul>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-restart-on-a-simple-bernoulli-problem">Doubling-Trick with restart, on a “simple” Bernoulli problem</a></li>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-restart-on-randomly-taken-bernoulli-problems">Doubling-Trick with restart, on randomly taken Bernoulli problems</a></li>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-restart-on-randomly-taken-gaussian-problems-with-variance-v-1">Doubling-Trick with restart, on randomly taken Gaussian problems with variance <code class="docutils literal notranslate"><span class="pre">$V=1$</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-restart-on-an-easy-gaussian-problems-with-variance-v-1">Doubling-Trick with restart, on an easy Gaussian problems with variance <code class="docutils literal notranslate"><span class="pre">$V=1$</span></code></a></li>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-no-restart-on-randomly-taken-bernoulli-problems">Doubling-Trick with no restart, on randomly taken Bernoulli problems</a></li>
<li class="toctree-l3"><a class="reference internal" href="#doubling-trick-with-no-restart-on-an-simple-bernoulli-problems">Doubling-Trick with no restart, on an “simple” Bernoulli problems</a></li>
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<div class="section" id="doubling-trick-for-multi-armed-bandits">
<h1><strong>Doubling Trick for Multi-Armed Bandits</strong><a class="headerlink" href="#doubling-trick-for-multi-armed-bandits" title="Permalink to this headline">¶</a></h1>
<p>I studied what <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling Trick</a> can and can’t do for multi-armed bandits, to obtain efficient anytime version of non-anytime optimal Multi-Armed Bandits algorithms.</p>
<p>The <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling Trick</a> algorithm, denoted <code class="docutils literal notranslate"><span class="pre">$DT(A,</span> <span class="pre">(T_i))$</span></code> for a diverging increasing sequence <code class="docutils literal notranslate"><span class="pre">$T_i$</span></code>, is the following algorithm:</p>
<p><img alt="Policies/DoublingTrick.py" src="_images/DoublingTrick_algo1.png" /></p>
<p>Long story short, we proved the two following theorems.</p>
<div class="section" id="for-geometric-sequences">
<h2>For geometric sequences<a class="headerlink" href="#for-geometric-sequences" title="Permalink to this headline">¶</a></h2>
<blockquote>
<div><p>It works for minimax regret bounds (in <code class="docutils literal notranslate"><span class="pre">$R_T</span> <span class="pre">=</span> <span class="pre">\mathcal{O}(\sqrt{T}))$</span></code>, with a constant multiplicative loss <code class="docutils literal notranslate"><span class="pre">$\leq</span> <span class="pre">4$</span></code>, but not for logarithmic regret bounds (in <code class="docutils literal notranslate"><span class="pre">$R_T</span> <span class="pre">=</span> <span class="pre">\mathcal{O}(\log</span> <span class="pre">T))$</span></code>.</p>
</div></blockquote>
<p><img alt="https://hal.inria.fr/hal-01736357" src="_images/DoublingTrick_theorem1.png" /></p>
</div>
<div class="section" id="for-exponential-sequences">
<h2>For exponential sequences<a class="headerlink" href="#for-exponential-sequences" title="Permalink to this headline">¶</a></h2>
<blockquote>
<div><p>It works for logarithmic regret bounds (in <code class="docutils literal notranslate"><span class="pre">$R_T</span> <span class="pre">=</span> <span class="pre">\mathcal{O}(\log</span> <span class="pre">T))$</span></code>, but not for minimax regret bounds (in <code class="docutils literal notranslate"><span class="pre">$R_T</span> <span class="pre">=</span> <span class="pre">\mathcal{O}(\sqrt{T}))$</span></code>.</p>
</div></blockquote>
<p><img alt="https://hal.inria.fr/hal-01736357" src="_images/DoublingTrick_theorem2.png" /></p>
</div>
<hr class="docutils" />
<div class="section" id="article">
<h2>Article<a class="headerlink" href="#article" title="Permalink to this headline">¶</a></h2>
<p>I wrote a research article on that topic, it is a better introduction as a self-contained document to explain this idea and the algorithms. Reference: <a class="reference external" href="https://hal.inria.fr/hal-01736357">[What the Doubling Trick Can or Can’t Do for Multi-Armed Bandits, Lilian Besson and Emilie Kaufmann, 2018]</a>.</p>
<blockquote>
<div><p>PDF : <a class="reference external" href="https://hal.inria.fr/hal-01736357/document">BK__ALT_2018.pdf</a> | HAL notice : <a class="reference external" href="https://hal.inria.fr/hal-01736357/">BK__ALT_2018</a> | BibTeX : <a class="reference external" href="https://hal.inria.fr/hal-01736357/bibtex">BK__ALT_2018.bib</a> | <a class="reference external" href="MultiPlayers.html">Source code and documentation</a>
<a class="reference external" href="https://hal.inria.fr/hal-01736357"><img alt="Published" src="https://img.shields.io/badge/Published%3F-waiting-orange.svg" /></a> <a class="reference external" href="https://bitbucket.org/lbesson/what-doubling-tricks-can-and-cant-do-for-multi-armed-bandits/commits/"><img alt="Maintenance" src="https://img.shields.io/badge/Maintained%3F-almost%20finished-orange.svg" /></a> <a class="reference external" href="https://bitbucket.org/lbesson/ama"><img alt="Ask Me Anything !" src="https://img.shields.io/badge/Ask%20me-anything-1abc9c.svg" /></a></p>
</div></blockquote>
</div>
<hr class="docutils" />
<div class="section" id="configuration">
<h2>Configuration<a class="headerlink" href="#configuration" title="Permalink to this headline">¶</a></h2>
<p>A simple python file, <a class="reference external" href="https://smpybandits.github.io/docs/configuration_comparing_doubling_algorithms.html"><code class="docutils literal notranslate"><span class="pre">configuration_comparing_doubling_algorithms.py</span></code></a>, is used to import the <a class="reference external" href="Arms/">arm classes</a>, the <a class="reference external" href="Policies/">policy classes</a> and define the problems and the experiments.</p>
<p>For example, we can compare the standard anytime <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html"><code class="docutils literal notranslate"><span class="pre">klUCB</span></code></a> algorithm against the non-anytime <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html"><code class="docutils literal notranslate"><span class="pre">klUCBPlusPlus</span></code></a> algorithm, as well as 3 versions of <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html"><code class="docutils literal notranslate"><span class="pre">DoublingTrickWrapper</span></code></a> applied to <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html"><code class="docutils literal notranslate"><span class="pre">klUCBPlusPlus</span></code></a>.</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">configuration</span> <span class="o">=</span> <span class="p">{</span>
<span class="s2">"horizon"</span><span class="p">:</span> <span class="mi">10000</span><span class="p">,</span> <span class="c1"># Finite horizon of the simulation</span>
<span class="s2">"repetitions"</span><span class="p">:</span> <span class="mi">100</span><span class="p">,</span> <span class="c1"># number of repetitions</span>
<span class="s2">"n_jobs"</span><span class="p">:</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="c1"># Maximum number of cores for parallelization: use ALL your CPU</span>
<span class="s2">"verbosity"</span><span class="p">:</span> <span class="mi">5</span><span class="p">,</span> <span class="c1"># Verbosity for the joblib calls</span>
<span class="c1"># Environment configuration, you can set up more than one.</span>
<span class="s2">"environment"</span><span class="p">:</span> <span class="p">[</span>
<span class="p">{</span>
<span class="s2">"arm_type"</span><span class="p">:</span> <span class="n">Bernoulli</span><span class="p">,</span>
<span class="s2">"params"</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="mf">0.3</span><span class="p">,</span> <span class="mf">0.4</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.6</span><span class="p">,</span> <span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.8</span><span class="p">,</span> <span class="mf">0.9</span>
<span class="p">}</span>
<span class="p">],</span>
<span class="c1"># Policies that should be simulated, and their parameters.</span>
<span class="s2">"policies"</span><span class="p">:</span> <span class="p">[</span>
<span class="p">{</span><span class="s2">"archtype"</span><span class="p">:</span> <span class="n">UCB</span><span class="p">,</span> <span class="s2">"params"</span><span class="p">:</span> <span class="p">{}</span> <span class="p">},</span>
<span class="p">{</span><span class="s2">"archtype"</span><span class="p">:</span> <span class="n">klUCB</span><span class="p">,</span> <span class="s2">"params"</span><span class="p">:</span> <span class="p">{}</span> <span class="p">},</span>
<span class="p">{</span><span class="s2">"archtype"</span><span class="p">:</span> <span class="n">klUCBPlusPlus</span><span class="p">,</span> <span class="s2">"params"</span><span class="p">:</span> <span class="p">{</span> <span class="s2">"horizon"</span><span class="p">:</span> <span class="mi">10000</span> <span class="p">}</span> <span class="p">},</span>
<span class="p">]</span>
<span class="p">}</span>
</pre></div>
</div>
<p>Then add a <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> bandit algorithm (<a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html"><code class="docutils literal notranslate"><span class="pre">DoublingTrickWrapper</span></code> class</a>), you can use this piece of code:</p>
<div class="highlight-python notranslate"><div class="highlight"><pre><span></span><span class="n">configuration</span><span class="p">[</span><span class="s2">"policies"</span><span class="p">]</span> <span class="o">+=</span> <span class="p">[</span>
<span class="p">{</span>
<span class="s2">"archtype"</span><span class="p">:</span> <span class="n">DoublingTrickWrapper</span><span class="p">,</span>
<span class="s2">"params"</span><span class="p">:</span> <span class="p">{</span>
<span class="s2">"next_horizon"</span><span class="p">:</span> <span class="n">next_horizon</span><span class="p">,</span>
<span class="s2">"full_restart"</span><span class="p">:</span> <span class="n">full_restart</span><span class="p">,</span>
<span class="s2">"policy"</span><span class="p">:</span> <span class="n">BayesUCB</span><span class="p">,</span>
<span class="p">}</span>
<span class="p">}</span>
<span class="k">for</span> <span class="n">full_restart</span> <span class="ow">in</span> <span class="p">[</span> <span class="bp">True</span><span class="p">,</span> <span class="bp">False</span> <span class="p">]</span>
<span class="k">for</span> <span class="n">next_horizon</span> <span class="ow">in</span> <span class="p">[</span>
<span class="n">next_horizon__arithmetic</span><span class="p">,</span>
<span class="n">next_horizon__geometric</span><span class="p">,</span>
<span class="n">next_horizon__exponential_fast</span><span class="p">,</span>
<span class="n">next_horizon__exponential_slow</span><span class="p">,</span>
<span class="n">next_horizon__exponential_generic</span>
<span class="p">]</span>
<span class="p">]</span>
</pre></div>
</div>
</div>
<hr class="docutils" />
<div class="section" id="how-to-run-the-experiments">
<h2><a class="reference internal" href="How_to_run_the_code.html"><span class="doc">How to run the experiments ?</span></a><a class="headerlink" href="#how-to-run-the-experiments" title="Permalink to this headline">¶</a></h2>
<p>You should use the provided <a class="reference external" href="Makefile"><code class="docutils literal notranslate"><span class="pre">Makefile</span></code></a> file to do this simply:</p>
<div class="highlight-bash notranslate"><div class="highlight"><pre><span></span><span class="c1"># if not already installed, otherwise update with 'git pull'</span>
git clone https://github.com/SMPyBandits/SMPyBandits/
<span class="nb">cd</span> SMPyBandits
make install <span class="c1"># install the requirements ONLY ONCE</span>
make comparing_doubling_algorithms <span class="c1"># run and log the main.py script</span>
</pre></div>
</div>
</div>
<hr class="docutils" />
<div class="section" id="some-illustrations">
<h2>Some illustrations<a class="headerlink" href="#some-illustrations" title="Permalink to this headline">¶</a></h2>
<p>Here are some plots illustrating the performances of the different <a class="reference external" href="https://smpybandits.github.io/docs/Policies/">policies</a> implemented in this project, against various problems (with <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Bernoulli.html"><code class="docutils literal notranslate"><span class="pre">Bernoulli</span></code></a> and <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Gaussian.html"><code class="docutils literal notranslate"><span class="pre">UnboundedGaussian</span></code></a> arms only):</p>
<div class="section" id="doubling-trick-with-restart-on-a-simple-bernoulli-problem">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with restart, on a “simple” Bernoulli problem<a class="headerlink" href="#doubling-trick-with-restart-on-a-simple-bernoulli-problem" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with restart, on a "simple" Bernoulli problem" src="_images/main____env1-1_1217677871459230631.png" /></p>
<p>Regret for <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a>, for <code class="docutils literal notranslate"><span class="pre">$K=9$</span></code> <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Bernoulli.html">Bernoulli arms</a>, horizon <code class="docutils literal notranslate"><span class="pre">$T=45678$</span></code>, <code class="docutils literal notranslate"><span class="pre">$n=1000$</span></code> repetitions and <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> taken uniformly in <code class="docutils literal notranslate"><span class="pre">$[0,1]^K$</span></code>.
Geometric doubling (<code class="docutils literal notranslate"><span class="pre">$b=2$</span></code>) and slow exponential doubling (<code class="docutils literal notranslate"><span class="pre">$b=1.1$</span></code>) are too slow, and short first sequences make the regret blow up in the beginning of the experiment.
At <code class="docutils literal notranslate"><span class="pre">$t=40000$</span></code> we see clearly the effect of a new sequence for the best <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">doubling trick</a> (<code class="docutils literal notranslate"><span class="pre">$T_i</span> <span class="pre">=</span> <span class="pre">200</span> <span class="pre">\times</span> <span class="pre">2^i$</span></code>).
As expected, <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> outperforms <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html">kl-UCB</a>, and if the doubling sequence is growing fast enough then <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a>(<a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a>) can perform as well as <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> (see for <code class="docutils literal notranslate"><span class="pre">$t</span> <span class="pre"><</span> <span class="pre">40000$</span></code>).</p>
</div>
<div class="section" id="doubling-trick-with-restart-on-randomly-taken-bernoulli-problems">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with restart, on randomly taken Bernoulli problems<a class="headerlink" href="#doubling-trick-with-restart-on-randomly-taken-bernoulli-problems" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with restart, on randomly taken Bernoulli problems" src="_images/main____env1-1_3633169128724378553.png" /></p>
<p>Similarly but for <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> evenly spaced in <code class="docutils literal notranslate"><span class="pre">$[0,1]^K$</span></code> (<code class="docutils literal notranslate"><span class="pre">${0.1,\ldots,0.9}$</span></code>).
Both <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html">kl-UCB</a> and <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> are very efficient on “easy” problems like this one, and we can check visually that they match the lower bound from Lai & Robbins (1985).
As before we check that slow doubling are too slow to give reasonable performance.</p>
</div>
<div class="section" id="doubling-trick-with-restart-on-randomly-taken-gaussian-problems-with-variance-v-1">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with restart, on randomly taken Gaussian problems with variance <code class="docutils literal notranslate"><span class="pre">$V=1$</span></code><a class="headerlink" href="#doubling-trick-with-restart-on-randomly-taken-gaussian-problems-with-variance-v-1" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with restart, on randomly taken Gaussian problems with variance V=1" src="_images/main____env1-1_2223860464453456415.png" /></p>
<p>Regret for <code class="docutils literal notranslate"><span class="pre">$K=9$</span></code> <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Gaussian.html">Gaussian arms</a> <code class="docutils literal notranslate"><span class="pre">$\mathcal{N}(\mu,</span> <span class="pre">1)$</span></code>, horizon <code class="docutils literal notranslate"><span class="pre">$T=45678$</span></code>, <code class="docutils literal notranslate"><span class="pre">$n=1000$</span></code> repetitions and <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> taken uniformly in <code class="docutils literal notranslate"><span class="pre">$[-5,5]^K$</span></code> and variance <code class="docutils literal notranslate"><span class="pre">$V=1$</span></code>.
On “hard” problems like this one, both <a class="reference external" href="https://smpybandits.github.io/docs/Policies.UCB.html">UCB</a> and <a class="reference external" href="https://smpybandits.github.io/docs/Policies.ApproximatedFHGittins.html">AFHG</a> perform similarly and poorly w.r.t. to the lower bound from Lai & Robbins (1985).
As before we check that geometric doubling (<code class="docutils literal notranslate"><span class="pre">$b=2$</span></code>) and slow exponential doubling (<code class="docutils literal notranslate"><span class="pre">$b=1.1$</span></code>) are too slow, but a fast enough doubling sequence does give reasonable performance for the anytime <a class="reference external" href="https://smpybandits.github.io/docs/Policies.ApproximatedFHGittins.html">AFHG</a> obtained by <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a>.</p>
</div>
<div class="section" id="doubling-trick-with-restart-on-an-easy-gaussian-problems-with-variance-v-1">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with restart, on an easy Gaussian problems with variance <code class="docutils literal notranslate"><span class="pre">$V=1$</span></code><a class="headerlink" href="#doubling-trick-with-restart-on-an-easy-gaussian-problems-with-variance-v-1" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with restart, on an easy Gaussian problems with variance V=1" src="_images/main____env1-1_6979515539977716717.png" /></p>
<p>Regret for <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a>, for <code class="docutils literal notranslate"><span class="pre">$K=9$</span></code> <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Gaussian.html">Gaussian arms</a> <code class="docutils literal notranslate"><span class="pre">$\mathcal{N}(\mu,</span> <span class="pre">1)$</span></code>, horizon <code class="docutils literal notranslate"><span class="pre">$T=45678$</span></code>, <code class="docutils literal notranslate"><span class="pre">$n=1000$</span></code> repetitions and <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> uniformly spaced in <code class="docutils literal notranslate"><span class="pre">$[-5,5]^K$</span></code>.
On “easy” problems like this one, both <a class="reference external" href="https://smpybandits.github.io/docs/Policies.UCB.html">UCB</a> and <a class="reference external" href="https://smpybandits.github.io/docs/Policies.ApproximatedFHGittins.html">AFHG</a> perform similarly and attain near constant regret (identifying the best <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Gaussian.html">Gaussian arm</a> is very easy here as they are sufficiently distinct).
Each <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">doubling trick</a> also appear to attain near constant regret, but geometric doubling (<code class="docutils literal notranslate"><span class="pre">$b=2$</span></code>) and slow exponential doubling (<code class="docutils literal notranslate"><span class="pre">$b=1.1$</span></code>) are slower to converge and thus less efficient.</p>
</div>
<div class="section" id="doubling-trick-with-no-restart-on-randomly-taken-bernoulli-problems">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with no restart, on randomly taken Bernoulli problems<a class="headerlink" href="#doubling-trick-with-no-restart-on-randomly-taken-bernoulli-problems" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with no restart, on randomly taken Bernoulli problems" src="_images/main____env1-1_5964629015089571121.png" /></p>
<p>Regret for <code class="docutils literal notranslate"><span class="pre">$K=9$</span></code> <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Bernoulli.html">Bernoulli arms</a>, horizon <code class="docutils literal notranslate"><span class="pre">$T=45678$</span></code>, <code class="docutils literal notranslate"><span class="pre">$n=1000$</span></code> repetitions and <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> taken uniformly in <code class="docutils literal notranslate"><span class="pre">$[0,1]^K$</span></code>, for <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> no-restart.
Geometric doubling (\eg, <code class="docutils literal notranslate"><span class="pre">$b=2$</span></code>) and slow exponential doubling (\eg, <code class="docutils literal notranslate"><span class="pre">$b=1.1$</span></code>) are too slow, and short first sequences make the regret blow up in the beginning of the experiment.
At <code class="docutils literal notranslate"><span class="pre">$t=40000$</span></code> we see clearly the effect of a new sequence for the best <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">doubling trick</a> (<code class="docutils literal notranslate"><span class="pre">$T_i</span> <span class="pre">=</span> <span class="pre">200</span> <span class="pre">\times</span> <span class="pre">2^i$</span></code>).
As expected, <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> outperforms <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html">kl-UCB</a>, and if the doubling sequence is growing fast enough then <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> no-restart for <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> can perform as well as <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a>.</p>
</div>
<div class="section" id="doubling-trick-with-no-restart-on-an-simple-bernoulli-problems">
<h3><a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> with no restart, on an “simple” Bernoulli problems<a class="headerlink" href="#doubling-trick-with-no-restart-on-an-simple-bernoulli-problems" title="Permalink to this headline">¶</a></h3>
<p><img alt="Doubling-Trick with no restart, on an "simple" Bernoulli problems" src="_images/main____env1-1_5972568793654673752.png" /></p>
<p><code class="docutils literal notranslate"><span class="pre">$K=9$</span></code> <a class="reference external" href="https://smpybandits.github.io/docs/Arms.Bernoulli.html">Bernoulli arms</a> with <code class="docutils literal notranslate"><span class="pre">$\mu_1,\ldots,\mu_K$</span></code> evenly spaced in <code class="docutils literal notranslate"><span class="pre">$[0,1]^K$</span></code>.
On easy problems like this one, both <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html">kl-UCB</a> and <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> are very efficient, and here the geometric allows the <a class="reference external" href="https://smpybandits.github.io/docs/Policies.DoublingTrickWrapper.html">Doubling-Trick</a> no-restart anytime version of <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a> to outperform both <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCB.html">kl-UCB</a> and <a class="reference external" href="https://smpybandits.github.io/docs/Policies.klUCBPlusPlus.html">kl-UCB++</a>.</p>
<blockquote>
<div><p>These illustrations come from my article, <a class="reference external" href="https://hal.inria.fr/hal-01736357">[What the Doubling Trick Can or Can’t Do for Multi-Armed Bandits, Lilian Besson and Emilie Kaufmann, 2018]</a>.</p>
</div></blockquote>
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<h3>📜 License ? <a class="reference external" href="https://github.com/SMPyBandits/SMPyBandits/blob/master/LICENSE"><img alt="GitHub license" src="https://img.shields.io/github/license/SMPyBandits/SMPyBandits.svg" /></a><a class="headerlink" href="#scroll-license-github-license" title="Permalink to this headline">¶</a></h3>
<p><a class="reference external" href="https://lbesson.mit-license.org/">MIT Licensed</a> (file <a class="reference external" href="LICENSE">LICENSE</a>).</p>
<p>© 2016-2018 <a class="reference external" href="https://GitHub.com/Naereen">Lilian Besson</a>.</p>
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