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<li><a class="reference internal" href="#">Comparing surrogate models</a><ul>
<li><a class="reference internal" href="#toy-model">Toy model</a></li>
<li><a class="reference internal" href="#objective">Objective</a></li>
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<section class="sphx-glr-example-title" id="comparing-surrogate-models">
<span id="sphx-glr-auto-examples-strategy-comparison-py"></span><h1>Comparing surrogate models<a class="headerlink" href="#comparing-surrogate-models" title="Permalink to this headline">¶</a></h1>
<p>Tim Head, July 2016.
Reformatted by Holger Nahrstaedt 2020</p>
<p>Bayesian optimization or sequential model-based optimization uses a surrogate
model to model the expensive to evaluate function <code class="docutils literal notranslate"><span class="pre">func</span></code>. There are several
choices for what kind of surrogate model to use. This notebook compares the
performance of:</p>
<ul class="simple">
<li><p>gaussian processes,</p></li>
<li><p>extra trees, and</p></li>
<li><p>random forests</p></li>
</ul>
<p>as surrogate models. A purely random optimization strategy is also used as
a baseline.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="vm">__doc__</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.seed.html#numpy.random.seed" title="numpy.random.seed" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">seed</span></a><span class="p">(</span><span class="mi">123</span><span class="p">)</span>
<span class="kn">import</span> <span class="nn">matplotlib.pyplot</span> <span class="k">as</span> <span class="nn">plt</span>
</pre></div>
</div>
<section id="toy-model">
<h2>Toy model<a class="headerlink" href="#toy-model" title="Permalink to this headline">¶</a></h2>
<p>We will use the <a class="reference internal" href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin"><code class="xref py py-class docutils literal notranslate"><span class="pre">benchmarks.branin</span></code></a> function as toy model for the expensive function.
In a real world application this function would be unknown and expensive
to evaluate.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">skopt.benchmarks</span> <span class="kn">import</span> <span class="n">branin</span> <span class="k">as</span> <a href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin" class="sphx-glr-backref-module-skopt-benchmarks sphx-glr-backref-type-py-function"><span class="n">_branin</span></a>
<span class="k">def</span> <span class="nf">branin</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">noise_level</span><span class="o">=</span><span class="mf">0.</span><span class="p">):</span>
<span class="k">return</span> <a href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin" class="sphx-glr-backref-module-skopt-benchmarks sphx-glr-backref-type-py-function"><span class="n">_branin</span></a><span class="p">(</span><span class="n">x</span><span class="p">)</span> <span class="o">+</span> <span class="n">noise_level</span> <span class="o">*</span> <a href="https://numpy.org/doc/stable/reference/random/generated/numpy.random.randn.html#numpy.random.randn" title="numpy.random.randn" class="sphx-glr-backref-module-numpy-random sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">randn</span></a><span class="p">()</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">matplotlib.colors</span> <span class="kn">import</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.colors.LogNorm.html#matplotlib.colors.LogNorm" title="matplotlib.colors.LogNorm" class="sphx-glr-backref-module-matplotlib-colors sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogNorm</span></a>
<span class="k">def</span> <span class="nf">plot_branin</span><span class="p">():</span>
<span class="n">fig</span><span class="p">,</span> <span class="n">ax</span> <span class="o">=</span> <a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplots.html#matplotlib.pyplot.subplots" title="matplotlib.pyplot.subplots" class="sphx-glr-backref-module-matplotlib-pyplot sphx-glr-backref-type-py-function"><span class="n">plt</span><span class="o">.</span><span class="n">subplots</span></a><span class="p">()</span>
<span class="n">x1_values</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">x2_values</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.linspace.html#numpy.linspace" title="numpy.linspace" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">linspace</span></a><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
<span class="n">x_ax</span><span class="p">,</span> <span class="n">y_ax</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.meshgrid.html#numpy.meshgrid" title="numpy.meshgrid" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">meshgrid</span></a><span class="p">(</span><span class="n">x1_values</span><span class="p">,</span> <span class="n">x2_values</span><span class="p">)</span>
<span class="n">vals</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.c_.html#numpy.c_" title="numpy.c_" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">c_</span></a><span class="p">[</span><span class="n">x_ax</span><span class="o">.</span><span class="n">ravel</span><span class="p">(),</span> <span class="n">y_ax</span><span class="o">.</span><span class="n">ravel</span><span class="p">()]</span>
<span class="n">fx</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.reshape.html#numpy.reshape" title="numpy.reshape" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">reshape</span></a><span class="p">([</span><span class="n">branin</span><span class="p">(</span><span class="n">val</span><span class="p">)</span> <span class="k">for</span> <span class="n">val</span> <span class="ow">in</span> <span class="n">vals</span><span class="p">],</span> <span class="p">(</span><span class="mi">100</span><span class="p">,</span> <span class="mi">100</span><span class="p">))</span>
<span class="n">cm</span> <span class="o">=</span> <span class="n">ax</span><span class="o">.</span><span class="n">pcolormesh</span><span class="p">(</span><span class="n">x_ax</span><span class="p">,</span> <span class="n">y_ax</span><span class="p">,</span> <span class="n">fx</span><span class="p">,</span>
<span class="n">norm</span><span class="o">=</span><a href="https://matplotlib.org/api/_as_gen/matplotlib.colors.LogNorm.html#matplotlib.colors.LogNorm" title="matplotlib.colors.LogNorm" class="sphx-glr-backref-module-matplotlib-colors sphx-glr-backref-type-py-class sphx-glr-backref-instance"><span class="n">LogNorm</span></a><span class="p">(</span><span class="n">vmin</span><span class="o">=</span><span class="n">fx</span><span class="o">.</span><span class="n">min</span><span class="p">(),</span>
<span class="n">vmax</span><span class="o">=</span><span class="n">fx</span><span class="o">.</span><span class="n">max</span><span class="p">()),</span>
<span class="n">cmap</span><span class="o">=</span><span class="s1">'viridis_r'</span><span class="p">)</span>
<span class="n">minima</span> <span class="o">=</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.array.html#numpy.array" title="numpy.array" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">array</span></a><span class="p">([[</span><span class="o">-</span><a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a><span class="p">,</span> <span class="mf">12.275</span><span class="p">],</span> <span class="p">[</span><span class="o">+</span><a href="https://numpy.org/doc/stable/reference/constants.html#numpy.pi" title="numpy.pi" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">pi</span></a><span class="p">,</span> <span class="mf">2.275</span><span class="p">],</span> <span class="p">[</span><span class="mf">9.42478</span><span class="p">,</span> <span class="mf">2.475</span><span class="p">]])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">minima</span><span class="p">[:,</span> <span class="mi">0</span><span class="p">],</span> <span class="n">minima</span><span class="p">[:,</span> <span class="mi">1</span><span class="p">],</span> <span class="s2">"r."</span><span class="p">,</span> <span class="n">markersize</span><span class="o">=</span><span class="mi">14</span><span class="p">,</span>
<span class="n">lw</span><span class="o">=</span><span class="mi">0</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"Minima"</span><span class="p">)</span>
<span class="n">cb</span> <span class="o">=</span> <span class="n">fig</span><span class="o">.</span><span class="n">colorbar</span><span class="p">(</span><span class="n">cm</span><span class="p">)</span>
<span class="n">cb</span><span class="o">.</span><span class="n">set_label</span><span class="p">(</span><span class="s2">"f(x)"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">,</span> <span class="n">numpoints</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlabel</span><span class="p">(</span><span class="s2">"X1"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_xlim</span><span class="p">([</span><span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">10</span><span class="p">])</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylabel</span><span class="p">(</span><span class="s2">"X2"</span><span class="p">)</span>
<span class="n">ax</span><span class="o">.</span><span class="n">set_ylim</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span> <span class="mi">15</span><span class="p">])</span>
<span class="n">plot_branin</span><span class="p">()</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_strategy-comparison_001.png" srcset="../_images/sphx_glr_strategy-comparison_001.png" alt="strategy comparison" class = "sphx-glr-single-img"/><p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>/home/circleci/project/examples/strategy-comparison.py:56: MatplotlibDeprecationWarning: shading='flat' when X and Y have the same dimensions as C is deprecated since 3.3. Either specify the corners of the quadrilaterals with X and Y, or pass shading='auto', 'nearest' or 'gouraud', or set rcParams['pcolor.shading']. This will become an error two minor releases later.
cm = ax.pcolormesh(x_ax, y_ax, fx,
</pre></div>
</div>
<p>This shows the value of the two-dimensional branin function and
the three minima.</p>
</section>
<section id="objective">
<h2>Objective<a class="headerlink" href="#objective" title="Permalink to this headline">¶</a></h2>
<p>The objective of this example is to find one of these minima in as
few iterations as possible. One iteration is defined as one call
to the <a class="reference internal" href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin"><code class="xref py py-class docutils literal notranslate"><span class="pre">benchmarks.branin</span></code></a> function.</p>
<p>We will evaluate each model several times using a different seed for the
random number generator. Then compare the average performance of these
models. This makes the comparison more robust against models that get
“lucky”.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">functools</span> <span class="kn">import</span> <a href="https://docs.python.org/3/library/functools.html#functools.partial" title="functools.partial" class="sphx-glr-backref-module-functools sphx-glr-backref-type-py-function"><span class="n">partial</span></a>
<span class="kn">from</span> <span class="nn">skopt</span> <span class="kn">import</span> <a href="../modules/generated/skopt.gp_minimize.html#skopt.gp_minimize" title="skopt.gp_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">gp_minimize</span></a><span class="p">,</span> <a href="../modules/generated/skopt.forest_minimize.html#skopt.forest_minimize" title="skopt.forest_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">forest_minimize</span></a><span class="p">,</span> <a href="../modules/generated/skopt.dummy_minimize.html#skopt.dummy_minimize" title="skopt.dummy_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">dummy_minimize</span></a>
<span class="n">func</span> <span class="o">=</span> <a href="https://docs.python.org/3/library/functools.html#functools.partial" title="functools.partial" class="sphx-glr-backref-module-functools sphx-glr-backref-type-py-function"><span class="n">partial</span></a><span class="p">(</span><span class="n">branin</span><span class="p">,</span> <span class="n">noise_level</span><span class="o">=</span><span class="mf">2.0</span><span class="p">)</span>
<span class="n">bounds</span> <span class="o">=</span> <span class="p">[(</span><span class="o">-</span><span class="mf">5.0</span><span class="p">,</span> <span class="mf">10.0</span><span class="p">),</span> <span class="p">(</span><span class="mf">0.0</span><span class="p">,</span> <span class="mf">15.0</span><span class="p">)]</span>
<span class="n">n_calls</span> <span class="o">=</span> <span class="mi">60</span>
</pre></div>
</div>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">def</span> <span class="nf">run</span><span class="p">(</span><span class="n">minimizer</span><span class="p">,</span> <span class="n">n_iter</span><span class="o">=</span><span class="mi">5</span><span class="p">):</span>
<span class="k">return</span> <span class="p">[</span><span class="n">minimizer</span><span class="p">(</span><span class="n">func</span><span class="p">,</span> <span class="n">bounds</span><span class="p">,</span> <span class="n">n_calls</span><span class="o">=</span><span class="n">n_calls</span><span class="p">,</span> <span class="n">random_state</span><span class="o">=</span><span class="n">n</span><span class="p">)</span>
<span class="k">for</span> <span class="n">n</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n_iter</span><span class="p">)]</span>
<span class="c1"># Random search</span>
<span class="n">dummy_res</span> <span class="o">=</span> <span class="n">run</span><span class="p">(</span><a href="../modules/generated/skopt.dummy_minimize.html#skopt.dummy_minimize" title="skopt.dummy_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">dummy_minimize</span></a><span class="p">)</span>
<span class="c1"># Gaussian processes</span>
<span class="n">gp_res</span> <span class="o">=</span> <span class="n">run</span><span class="p">(</span><a href="../modules/generated/skopt.gp_minimize.html#skopt.gp_minimize" title="skopt.gp_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">gp_minimize</span></a><span class="p">)</span>
<span class="c1"># Random forest</span>
<span class="n">rf_res</span> <span class="o">=</span> <span class="n">run</span><span class="p">(</span><a href="https://docs.python.org/3/library/functools.html#functools.partial" title="functools.partial" class="sphx-glr-backref-module-functools sphx-glr-backref-type-py-function"><span class="n">partial</span></a><span class="p">(</span><a href="../modules/generated/skopt.forest_minimize.html#skopt.forest_minimize" title="skopt.forest_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">forest_minimize</span></a><span class="p">,</span> <span class="n">base_estimator</span><span class="o">=</span><span class="s2">"RF"</span><span class="p">))</span>
<span class="c1"># Extra trees</span>
<span class="n">et_res</span> <span class="o">=</span> <span class="n">run</span><span class="p">(</span><a href="https://docs.python.org/3/library/functools.html#functools.partial" title="functools.partial" class="sphx-glr-backref-module-functools sphx-glr-backref-type-py-function"><span class="n">partial</span></a><span class="p">(</span><a href="../modules/generated/skopt.forest_minimize.html#skopt.forest_minimize" title="skopt.forest_minimize" class="sphx-glr-backref-module-skopt sphx-glr-backref-type-py-function"><span class="n">forest_minimize</span></a><span class="p">,</span> <span class="n">base_estimator</span><span class="o">=</span><span class="s2">"ET"</span><span class="p">))</span>
</pre></div>
</div>
<p>Note that this can take a few minutes.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="kn">from</span> <span class="nn">skopt.plots</span> <span class="kn">import</span> <a href="../modules/generated/skopt.plots.plot_convergence.html#skopt.plots.plot_convergence" title="skopt.plots.plot_convergence" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_convergence</span></a>
<span class="n">plot</span> <span class="o">=</span> <a href="../modules/generated/skopt.plots.plot_convergence.html#skopt.plots.plot_convergence" title="skopt.plots.plot_convergence" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_convergence</span></a><span class="p">((</span><span class="s2">"dummy_minimize"</span><span class="p">,</span> <span class="n">dummy_res</span><span class="p">),</span>
<span class="p">(</span><span class="s2">"gp_minimize"</span><span class="p">,</span> <span class="n">gp_res</span><span class="p">),</span>
<span class="p">(</span><span class="s2">"forest_minimize('rf')"</span><span class="p">,</span> <span class="n">rf_res</span><span class="p">),</span>
<span class="p">(</span><span class="s2">"forest_minimize('et)"</span><span class="p">,</span> <span class="n">et_res</span><span class="p">),</span>
<span class="n">true_minimum</span><span class="o">=</span><span class="mf">0.397887</span><span class="p">,</span> <span class="n">yscale</span><span class="o">=</span><span class="s2">"log"</span><span class="p">)</span>
<span class="n">plot</span><span class="o">.</span><span class="n">legend</span><span class="p">(</span><span class="n">loc</span><span class="o">=</span><span class="s2">"best"</span><span class="p">,</span> <span class="n">prop</span><span class="o">=</span><span class="p">{</span><span class="s1">'size'</span><span class="p">:</span> <span class="mi">6</span><span class="p">},</span> <span class="n">numpoints</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_strategy-comparison_002.png" srcset="../_images/sphx_glr_strategy-comparison_002.png" alt="Convergence plot" class = "sphx-glr-single-img"/><p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span><matplotlib.legend.Legend object at 0x7f8fe67bb340>
</pre></div>
</div>
<p>This plot shows the value of the minimum found (y axis) as a function
of the number of iterations performed so far (x axis). The dashed red line
indicates the true value of the minimum of the <a class="reference internal" href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin"><code class="xref py py-class docutils literal notranslate"><span class="pre">benchmarks.branin</span></code></a> function.</p>
<p>For the first ten iterations all methods perform equally well as they all
start by creating ten random samples before fitting their respective model
for the first time. After iteration ten the next point at which
to evaluate <a class="reference internal" href="../modules/generated/skopt.benchmarks.branin.html#skopt.benchmarks.branin" title="skopt.benchmarks.branin"><code class="xref py py-class docutils literal notranslate"><span class="pre">benchmarks.branin</span></code></a> is guided by the model, which is where differences
start to appear.</p>
<p>Each minimizer only has access to noisy observations of the objective
function, so as time passes (more iterations) it will start observing
values that are below the true value simply because they are fluctuations.</p>
<p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 2 minutes 48.806 seconds)</p>
<p><strong>Estimated memory usage:</strong> 69 MB</p>
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