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<li><a class="reference internal" href="#">Bayesian optimization with <code class="docutils literal notranslate"><span class="pre">skopt</span></code></a><ul>
<li><a class="reference internal" href="#problem-statement">Problem statement</a></li>
<li><a class="reference internal" href="#bayesian-optimization-loop">Bayesian optimization loop</a></li>
<li><a class="reference internal" href="#acquisition-functions">Acquisition functions</a></li>
<li><a class="reference internal" href="#toy-example">Toy example</a></li>
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<p>Click <a class="reference internal" href="#sphx-glr-download-auto-examples-bayesian-optimization-py"><span class="std std-ref">here</span></a>
to download the full example code or to run this example in your browser via Binder</p>
</div>
<section class="sphx-glr-example-title" id="bayesian-optimization-with-skopt">
<span id="sphx-glr-auto-examples-bayesian-optimization-py"></span><h1>Bayesian optimization with <code class="docutils literal notranslate"><span class="pre">skopt</span></code><a class="headerlink" href="#bayesian-optimization-with-skopt" title="Permalink to this headline">¶</a></h1>
<p>Gilles Louppe, Manoj Kumar July 2016.
Reformatted by Holger Nahrstaedt 2020</p>
<section id="problem-statement">
<h2>Problem statement<a class="headerlink" href="#problem-statement" title="Permalink to this headline">¶</a></h2>
<p>We are interested in solving</p>
<div class="math notranslate nohighlight">
\[x^* = arg \min_x f(x)\]</div>
<p>under the constraints that</p>
<ul class="simple">
<li><p><span class="math notranslate nohighlight">\(f\)</span> is a black box for which no closed form is known
(nor its gradients);</p></li>
<li><p><span class="math notranslate nohighlight">\(f\)</span> is expensive to evaluate;</p></li>
<li><p>and evaluations of <span class="math notranslate nohighlight">\(y = f(x)\)</span> may be noisy.</p></li>
</ul>
<p><strong>Disclaimer.</strong> If you do not have these constraints, then there
is certainly a better optimization algorithm than Bayesian optimization.</p>
<p>This example uses <a class="reference internal" href="../modules/generated/skopt.plots.plot_gaussian_process.html#skopt.plots.plot_gaussian_process" title="skopt.plots.plot_gaussian_process"><code class="xref py py-class docutils literal notranslate"><span class="pre">plots.plot_gaussian_process</span></code></a> which is available
since version 0.8.</p>
</section>
<section id="bayesian-optimization-loop">
<h2>Bayesian optimization loop<a class="headerlink" href="#bayesian-optimization-loop" title="Permalink to this headline">¶</a></h2>
<p>For <span class="math notranslate nohighlight">\(t=1:T\)</span>:</p>
<ol class="arabic simple">
<li><p>Given observations <span class="math notranslate nohighlight">\((x_i, y_i=f(x_i))\)</span> for <span class="math notranslate nohighlight">\(i=1:t\)</span>, build a
probabilistic model for the objective <span class="math notranslate nohighlight">\(f\)</span>. Integrate out all
possible true functions, using Gaussian process regression.</p></li>
<li><p>optimize a cheap acquisition/utility function <span class="math notranslate nohighlight">\(u\)</span> based on the
posterior distribution for sampling the next point.
<span class="math notranslate nohighlight">\(x_{t+1} = arg \min_x u(x)\)</span>
Exploit uncertainty to balance exploration against exploitation.</p></li>
<li><p>Sample the next observation <span class="math notranslate nohighlight">\(y_{t+1}\)</span> at <span class="math notranslate nohighlight">\(x_{t+1}\)</span>.</p></li>
</ol>
</section>
<section id="acquisition-functions">
<h2>Acquisition functions<a class="headerlink" href="#acquisition-functions" title="Permalink to this headline">¶</a></h2>
<p>Acquisition functions <span class="math notranslate nohighlight">\(u(x)\)</span> specify which sample <span class="math notranslate nohighlight">\(x\)</span>: should be
tried next:</p>
<ul class="simple">
<li><p>Expected improvement (default):
<span class="math notranslate nohighlight">\(-EI(x) = -\mathbb{E} [f(x) - f(x_t^+)]\)</span></p></li>
<li><p>Lower confidence bound: <span class="math notranslate nohighlight">\(LCB(x) = \mu_{GP}(x) + \kappa \sigma_{GP}(x)\)</span></p></li>
<li><p>Probability of improvement: <span class="math notranslate nohighlight">\(-PI(x) = -P(f(x) \geq f(x_t^+) + \kappa)\)</span></p></li>
</ul>
<p>where <span class="math notranslate nohighlight">\(x_t^+\)</span> is the best point observed so far.</p>
<p>In most cases, acquisition functions provide knobs (e.g., <span class="math notranslate nohighlight">\(\kappa\)</span>) for
controlling the exploration-exploitation trade-off.
- Search in regions where <span class="math notranslate nohighlight">\(\mu_{GP}(x)\)</span> is high (exploitation)
- Probe regions where uncertainty <span class="math notranslate nohighlight">\(\sigma_{GP}(x)\)</span> is high (exploration)</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">237</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>
<span class="kn">from</span> <span class="nn">skopt.plots</span> <span class="kn">import</span> <a href="../modules/generated/skopt.plots.plot_gaussian_process.html#skopt.plots.plot_gaussian_process" title="skopt.plots.plot_gaussian_process" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_gaussian_process</span></a>
</pre></div>
</div>
</section>
<section id="toy-example">
<h2>Toy example<a class="headerlink" href="#toy-example" title="Permalink to this headline">¶</a></h2>
<p>Let assume the following noisy function <span class="math notranslate nohighlight">\(f\)</span>:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="n">noise_level</span> <span class="o">=</span> <span class="mf">0.1</span>
<span class="k">def</span> <span class="nf">f</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="n">noise_level</span><span class="p">):</span>
<span class="k">return</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.sin.html#numpy.sin" title="numpy.sin" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">sin</span></a><span class="p">(</span><span class="mi">5</span> <span class="o">*</span> <span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">*</span> <span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <a href="https://numpy.org/doc/stable/reference/generated/numpy.tanh.html#numpy.tanh" title="numpy.tanh" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-data"><span class="n">np</span><span class="o">.</span><span class="n">tanh</span></a><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">**</span> <span class="mi">2</span><span class="p">))</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> <span class="o">*</span> <span class="n">noise_level</span>
</pre></div>
</div>
<p><strong>Note.</strong> In <code class="docutils literal notranslate"><span class="pre">skopt</span></code>, functions <span class="math notranslate nohighlight">\(f\)</span> are assumed to take as input a 1D
vector <span class="math notranslate nohighlight">\(x\)</span>: represented as an array-like and to return a scalar
<span class="math notranslate nohighlight">\(f(x)\)</span>:.</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="c1"># Plot f(x) + contours</span>
<span class="n">x</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">2</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">400</span><span class="p">)</span><span class="o">.</span><span class="n">reshape</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">fx</span> <span class="o">=</span> <span class="p">[</span><span class="n">f</span><span class="p">(</span><span class="n">x_i</span><span class="p">,</span> <span class="n">noise_level</span><span class="o">=</span><span class="mf">0.0</span><span class="p">)</span> <span class="k">for</span> <span class="n">x_i</span> <span class="ow">in</span> <span class="n">x</span><span class="p">]</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.plot.html#matplotlib.pyplot.plot" title="matplotlib.pyplot.plot" 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">plot</span></a><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">fx</span><span class="p">,</span> <span class="s2">"r--"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"True (unknown)"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.fill.html#matplotlib.pyplot.fill" title="matplotlib.pyplot.fill" 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">fill</span></a><span class="p">(</span><a href="https://numpy.org/doc/stable/reference/generated/numpy.concatenate.html#numpy.concatenate" title="numpy.concatenate" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span></a><span class="p">([</span><span class="n">x</span><span class="p">,</span> <span class="n">x</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]]),</span>
<a href="https://numpy.org/doc/stable/reference/generated/numpy.concatenate.html#numpy.concatenate" title="numpy.concatenate" class="sphx-glr-backref-module-numpy sphx-glr-backref-type-py-function"><span class="n">np</span><span class="o">.</span><span class="n">concatenate</span></a><span class="p">(([</span><span class="n">fx_i</span> <span class="o">-</span> <span class="mf">1.9600</span> <span class="o">*</span> <span class="n">noise_level</span> <span class="k">for</span> <span class="n">fx_i</span> <span class="ow">in</span> <span class="n">fx</span><span class="p">],</span>
<span class="p">[</span><span class="n">fx_i</span> <span class="o">+</span> <span class="mf">1.9600</span> <span class="o">*</span> <span class="n">noise_level</span> <span class="k">for</span> <span class="n">fx_i</span> <span class="ow">in</span> <span class="n">fx</span><span class="p">[::</span><span class="o">-</span><span class="mi">1</span><span class="p">]])),</span>
<span class="n">alpha</span><span class="o">=</span><span class="mf">.2</span><span class="p">,</span> <span class="n">fc</span><span class="o">=</span><span class="s2">"r"</span><span class="p">,</span> <span class="n">ec</span><span class="o">=</span><span class="s2">"None"</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.legend.html#matplotlib.pyplot.legend" title="matplotlib.pyplot.legend" 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">legend</span></a><span class="p">()</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.grid.html#matplotlib.pyplot.grid" title="matplotlib.pyplot.grid" 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">grid</span></a><span class="p">()</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" 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">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_bayesian-optimization_001.png" srcset="../_images/sphx_glr_bayesian-optimization_001.png" alt="bayesian optimization" class = "sphx-glr-single-img"/><p>Bayesian optimization based on gaussian process regression is implemented in
<a class="reference internal" href="../modules/generated/skopt.gp_minimize.html#skopt.gp_minimize" title="skopt.gp_minimize"><code class="xref py py-class docutils literal notranslate"><span class="pre">gp_minimize</span></code></a> and can be carried out as follows:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><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="n">res</span> <span class="o">=</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="n">f</span><span class="p">,</span> <span class="c1"># the function to minimize</span>
<span class="p">[(</span><span class="o">-</span><span class="mf">2.0</span><span class="p">,</span> <span class="mf">2.0</span><span class="p">)],</span> <span class="c1"># the bounds on each dimension of x</span>
<span class="n">acq_func</span><span class="o">=</span><span class="s2">"EI"</span><span class="p">,</span> <span class="c1"># the acquisition function</span>
<span class="n">n_calls</span><span class="o">=</span><span class="mi">15</span><span class="p">,</span> <span class="c1"># the number of evaluations of f</span>
<span class="n">n_random_starts</span><span class="o">=</span><span class="mi">5</span><span class="p">,</span> <span class="c1"># the number of random initialization points</span>
<span class="n">noise</span><span class="o">=</span><span class="mf">0.1</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="c1"># the noise level (optional)</span>
<span class="n">random_state</span><span class="o">=</span><span class="mi">1234</span><span class="p">)</span> <span class="c1"># the random seed</span>
</pre></div>
</div>
<p>Accordingly, the approximated minimum is found to be:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="s2">"x^*=</span><span class="si">%.4f</span><span class="s2">, f(x^*)=</span><span class="si">%.4f</span><span class="s2">"</span> <span class="o">%</span> <span class="p">(</span><span class="n">res</span><span class="o">.</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">],</span> <span class="n">res</span><span class="o">.</span><span class="n">fun</span><span class="p">)</span>
</pre></div>
</div>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span>'x^*=-0.3552, f(x^*)=-1.0079'
</pre></div>
</div>
<p>For further inspection of the results, attributes of the <code class="docutils literal notranslate"><span class="pre">res</span></code> named tuple
provide the following information:</p>
<ul class="simple">
<li><p><code class="docutils literal notranslate"><span class="pre">x</span></code> [float]: location of the minimum.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">fun</span></code> [float]: function value at the minimum.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">models</span></code>: surrogate models used for each iteration.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">x_iters</span></code> [array]:
location of function evaluation for each iteration.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">func_vals</span></code> [array]: function value for each iteration.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">space</span></code> [Space]: the optimization space.</p></li>
<li><p><code class="docutils literal notranslate"><span class="pre">specs</span></code> [dict]: parameters passed to the function.</p></li>
</ul>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="n">res</span><span class="p">)</span>
</pre></div>
</div>
<p class="sphx-glr-script-out">Out:</p>
<div class="sphx-glr-script-out highlight-none notranslate"><div class="highlight"><pre><span></span> fun: -1.0079192431413255
func_vals: array([ 0.03716044, 0.00673852, 0.63515442, -0.16042062, 0.10695907,
-0.24436726, -0.5863053 , 0.05238728, -1.00791924, -0.98466748,
-0.86259915, 0.18102445, -0.10782771, 0.00815673, -0.79756402])
models: [GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5) + WhiteKernel(noise_level=0.01),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775)]
random_state: RandomState(MT19937) at 0x7F8FF0D10B40
space: Space([Real(low=-2.0, high=2.0, prior='uniform', transform='normalize')])
specs: {'args': {'func': <function f at 0x7f8fe6520f70>, 'dimensions': Space([Real(low=-2.0, high=2.0, prior='uniform', transform='normalize')]), 'base_estimator': GaussianProcessRegressor(kernel=1**2 * Matern(length_scale=1, nu=2.5),
n_restarts_optimizer=2, noise=0.010000000000000002,
normalize_y=True, random_state=822569775), 'n_calls': 15, 'n_random_starts': 5, 'n_initial_points': 10, 'initial_point_generator': 'random', 'acq_func': 'EI', 'acq_optimizer': 'auto', 'x0': None, 'y0': None, 'random_state': RandomState(MT19937) at 0x7F8FF0D10B40, 'verbose': False, 'callback': None, 'n_points': 10000, 'n_restarts_optimizer': 5, 'xi': 0.01, 'kappa': 1.96, 'n_jobs': 1, 'model_queue_size': None}, 'function': 'base_minimize'}
x: [-0.35518416232959327]
x_iters: [[-0.009345334109402526], [1.2713537644662787], [0.4484475787090836], [1.0854396754496047], [1.4426790855107496], [0.9579248468740373], [-0.45158087416842263], [-0.685948113064442], [-0.35518416232959327], [-0.29315379225502536], [-0.3209941608705478], [-2.0], [2.0], [-1.3373742014126968], [-0.2478422942435552]]
</pre></div>
</div>
<p>Together these attributes can be used to visually inspect the results of the
minimization, such as the convergence trace or the acquisition function at
the last iteration:</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>
<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="n">res</span><span class="p">);</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_bayesian-optimization_002.png" srcset="../_images/sphx_glr_bayesian-optimization_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><AxesSubplot:title={'center':'Convergence plot'}, xlabel='Number of calls $n$', ylabel='$\\min f(x)$ after $n$ calls'>
</pre></div>
</div>
<p>Let us now visually examine</p>
<ol class="arabic simple">
<li><p>The approximation of the fit gp model to the original function.</p></li>
<li><p>The acquisition values that determine the next point to be queried.</p></li>
</ol>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/api/matplotlib_configuration_api.html#matplotlib.rcParams" title="matplotlib.rcParams" class="sphx-glr-backref-module-matplotlib sphx-glr-backref-type-py-data"><span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span></a><span class="p">[</span><span class="s2">"figure.figsize"</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">8</span><span class="p">,</span> <span class="mi">14</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">f_wo_noise</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">f</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="mi">0</span><span class="p">)</span>
</pre></div>
</div>
<p>Plot the 5 iterations following the 5 random points</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><span class="k">for</span> <span class="n">n_iter</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">5</span><span class="p">):</span>
<span class="c1"># Plot true function.</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">n_iter</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
<span class="k">if</span> <span class="n">n_iter</span> <span class="o">==</span> <span class="mi">0</span><span class="p">:</span>
<span class="n">show_legend</span> <span class="o">=</span> <span class="kc">True</span>
<span class="k">else</span><span class="p">:</span>
<span class="n">show_legend</span> <span class="o">=</span> <span class="kc">False</span>
<span class="n">ax</span> <span class="o">=</span> <a href="../modules/generated/skopt.plots.plot_gaussian_process.html#skopt.plots.plot_gaussian_process" title="skopt.plots.plot_gaussian_process" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_gaussian_process</span></a><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">n_calls</span><span class="o">=</span><span class="n">n_iter</span><span class="p">,</span>
<span class="n">objective</span><span class="o">=</span><span class="n">f_wo_noise</span><span class="p">,</span>
<span class="n">noise_level</span><span class="o">=</span><span class="n">noise_level</span><span class="p">,</span>
<span class="n">show_legend</span><span class="o">=</span><span class="n">show_legend</span><span class="p">,</span> <span class="n">show_title</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">show_next_point</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">show_acq_func</span><span class="o">=</span><span class="kc">False</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">""</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">""</span><span class="p">)</span>
<span class="c1"># Plot EI(x)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.subplot.html#matplotlib.pyplot.subplot" title="matplotlib.pyplot.subplot" 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">subplot</span></a><span class="p">(</span><span class="mi">5</span><span class="p">,</span> <span class="mi">2</span><span class="p">,</span> <span class="mi">2</span><span class="o">*</span><span class="n">n_iter</span><span class="o">+</span><span class="mi">2</span><span class="p">)</span>
<span class="n">ax</span> <span class="o">=</span> <a href="../modules/generated/skopt.plots.plot_gaussian_process.html#skopt.plots.plot_gaussian_process" title="skopt.plots.plot_gaussian_process" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_gaussian_process</span></a><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">n_calls</span><span class="o">=</span><span class="n">n_iter</span><span class="p">,</span>
<span class="n">show_legend</span><span class="o">=</span><span class="n">show_legend</span><span class="p">,</span> <span class="n">show_title</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">show_mu</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span> <span class="n">show_acq_func</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span>
<span class="n">show_observations</span><span class="o">=</span><span class="kc">False</span><span class="p">,</span>
<span class="n">show_next_point</span><span class="o">=</span><span class="kc">True</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">""</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">""</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" 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">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_bayesian-optimization_003.png" srcset="../_images/sphx_glr_bayesian-optimization_003.png" alt="bayesian optimization" class = "sphx-glr-single-img"/><p>The first column shows the following:</p>
<ol class="arabic simple">
<li><p>The true function.</p></li>
<li><p>The approximation to the original function by the gaussian process model</p></li>
<li><p>How sure the GP is about the function.</p></li>
</ol>
<p>The second column shows the acquisition function values after every
surrogate model is fit. It is possible that we do not choose the global
minimum but a local minimum depending on the minimizer used to minimize
the acquisition function.</p>
<p>At the points closer to the points previously evaluated at, the variance
dips to zero.</p>
<p>Finally, as we increase the number of points, the GP model approaches
the actual function. The final few points are clustered around the minimum
because the GP does not gain anything more by further exploration:</p>
<div class="highlight-default notranslate"><div class="highlight"><pre><span></span><a href="https://matplotlib.org/api/matplotlib_configuration_api.html#matplotlib.rcParams" title="matplotlib.rcParams" class="sphx-glr-backref-module-matplotlib sphx-glr-backref-type-py-data"><span class="n">plt</span><span class="o">.</span><span class="n">rcParams</span></a><span class="p">[</span><span class="s2">"figure.figsize"</span><span class="p">]</span> <span class="o">=</span> <span class="p">(</span><span class="mi">6</span><span class="p">,</span> <span class="mi">4</span><span class="p">)</span>
<span class="c1"># Plot f(x) + contours</span>
<span class="n">_</span> <span class="o">=</span> <a href="../modules/generated/skopt.plots.plot_gaussian_process.html#skopt.plots.plot_gaussian_process" title="skopt.plots.plot_gaussian_process" class="sphx-glr-backref-module-skopt-plots sphx-glr-backref-type-py-function"><span class="n">plot_gaussian_process</span></a><span class="p">(</span><span class="n">res</span><span class="p">,</span> <span class="n">objective</span><span class="o">=</span><span class="n">f_wo_noise</span><span class="p">,</span>
<span class="n">noise_level</span><span class="o">=</span><span class="n">noise_level</span><span class="p">)</span>
<a href="https://matplotlib.org/api/_as_gen/matplotlib.pyplot.show.html#matplotlib.pyplot.show" title="matplotlib.pyplot.show" 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">show</span></a><span class="p">()</span>
</pre></div>
</div>
<img src="../_images/sphx_glr_bayesian-optimization_004.png" srcset="../_images/sphx_glr_bayesian-optimization_004.png" alt="x* = -0.3552, f(x*) = -1.0079" class = "sphx-glr-single-img"/><p class="sphx-glr-timing"><strong>Total running time of the script:</strong> ( 0 minutes 3.391 seconds)</p>
<p><strong>Estimated memory usage:</strong> 9 MB</p>
<div class="sphx-glr-footer class sphx-glr-footer-example docutils container" id="sphx-glr-download-auto-examples-bayesian-optimization-py">
<div class="binder-badge docutils container">
<a class="reference external image-reference" href="https://mybinder.org/v2/gh/scikit-optimize/scikit-optimize/master?urlpath=lab/tree/notebooks/auto_examples/bayesian-optimization.ipynb"><img alt="Launch binder" src="../_images/binder_badge_logo.svg" width="150px" /></a>
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