.. DO NOT EDIT. .. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY. .. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE: .. "auto_examples\bayesian-optimization.py" .. LINE NUMBERS ARE GIVEN BELOW. .. only:: html .. note:: :class: sphx-glr-download-link-note :ref:`Go to the end ` to download the full example code or to run this example in your browser via Binder .. rst-class:: sphx-glr-example-title .. _sphx_glr_auto_examples_bayesian-optimization.py: ================================== Bayesian optimization with `skopt` ================================== Gilles Louppe, Manoj Kumar July 2016. Reformatted by Holger Nahrstaedt 2020 .. currentmodule:: skopt Problem statement ----------------- We are interested in solving .. math:: x^* = arg \\min_x f(x) under the constraints that - :math:`f` is a black box for which no closed form is known (nor its gradients); - :math:`f` is expensive to evaluate; - and evaluations of :math:`y = f(x)` may be noisy. **Disclaimer.** If you do not have these constraints, then there is certainly a better optimization algorithm than Bayesian optimization. This example uses :class:`plots.plot_gaussian_process` which is available since version 0.8. Bayesian optimization loop -------------------------- For :math:`t=1:T`: 1. Given observations :math:`(x_i, y_i=f(x_i))` for :math:`i=1:t`, build a probabilistic model for the objective :math:`f`. Integrate out all possible true functions, using Gaussian process regression. 2. optimize a cheap acquisition/utility function :math:`u` based on the posterior distribution for sampling the next point. :math:`x_{t+1} = arg \\min_x u(x)` Exploit uncertainty to balance exploration against exploitation. 3. Sample the next observation :math:`y_{t+1}` at :math:`x_{t+1}`. Acquisition functions --------------------- Acquisition functions :math:`u(x)` specify which sample :math:`x`: should be tried next: - Expected improvement (default): :math:`-EI(x) = -\\mathbb{E} [f(x) - f(x_t^+)]` - Lower confidence bound: :math:`LCB(x) = \\mu_{GP}(x) + \\kappa \\sigma_{GP}(x)` - Probability of improvement: :math:`-PI(x) = -P(f(x) \\geq f(x_t^+) + \\kappa)` where :math:`x_t^+` is the best point observed so far. In most cases, acquisition functions provide knobs (e.g., :math:`\\kappa`) for controlling the exploration-exploitation trade-off. - Search in regions where :math:`\\mu_{GP}(x)` is high (exploitation) - Probe regions where uncertainty :math:`\\sigma_{GP}(x)` is high (exploration) .. GENERATED FROM PYTHON SOURCE LINES 67-77 .. code-block:: Python print(__doc__) import numpy as np np.random.seed(237) import matplotlib.pyplot as plt from skopt.plots import plot_gaussian_process .. GENERATED FROM PYTHON SOURCE LINES 78-82 Toy example ----------- Let assume the following noisy function :math:`f`: .. GENERATED FROM PYTHON SOURCE LINES 82-90 .. code-block:: Python noise_level = 0.1 def f(x, noise_level=noise_level): return np.sin(5 * x[0]) * (1 - np.tanh(x[0] ** 2)) + np.random.randn() * noise_level .. GENERATED FROM PYTHON SOURCE LINES 91-94 **Note.** In `skopt`, functions :math:`f` are assumed to take as input a 1D vector :math:`x`: represented as an array-like and to return a scalar :math:`f(x)`:. .. GENERATED FROM PYTHON SOURCE LINES 94-115 .. code-block:: Python # Plot f(x) + contours x = np.linspace(-2, 2, 400).reshape(-1, 1) fx = [f(x_i, noise_level=0.0) for x_i in x] plt.plot(x, fx, "r--", label="True (unknown)") plt.fill( np.concatenate([x, x[::-1]]), np.concatenate( ( [fx_i - 1.9600 * noise_level for fx_i in fx], [fx_i + 1.9600 * noise_level for fx_i in fx[::-1]], ) ), alpha=0.2, fc="r", ec="None", ) plt.legend() plt.grid() plt.show() .. image-sg:: /auto_examples/images/sphx_glr_bayesian-optimization_001.png :alt: bayesian optimization :srcset: /auto_examples/images/sphx_glr_bayesian-optimization_001.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 116-118 Bayesian optimization based on gaussian process regression is implemented in :class:`gp_minimize` and can be carried out as follows: .. GENERATED FROM PYTHON SOURCE LINES 118-131 .. code-block:: Python from skopt import gp_minimize res = gp_minimize( f, # the function to minimize [(-2.0, 2.0)], # the bounds on each dimension of x acq_func="EI", # the acquisition function n_calls=15, # the number of evaluations of f n_random_starts=5, # the number of random initialization points noise=0.1**2, # the noise level (optional) random_state=1234, ) # the random seed .. GENERATED FROM PYTHON SOURCE LINES 132-133 Accordingly, the approximated minimum is found to be: .. GENERATED FROM PYTHON SOURCE LINES 133-136 .. code-block:: Python f"x^*={res.x[0]:.4f}, f(x^*)={res.fun:.4f}" .. rst-class:: sphx-glr-script-out .. code-block:: none 'x^*=-0.3552, f(x^*)=-1.0079' .. GENERATED FROM PYTHON SOURCE LINES 137-148 For further inspection of the results, attributes of the `res` named tuple provide the following information: - `x` [float]: location of the minimum. - `fun` [float]: function value at the minimum. - `models`: surrogate models used for each iteration. - `x_iters` [array]: location of function evaluation for each iteration. - `func_vals` [array]: function value for each iteration. - `space` [Space]: the optimization space. - `specs` [dict]: parameters passed to the function. .. GENERATED FROM PYTHON SOURCE LINES 148-151 .. code-block:: Python print(res) .. rst-class:: sphx-glr-script-out .. code-block:: none fun: -1.007919274002016 x: [-0.35518414273753307] func_vals: [ 3.716e-02 6.739e-03 ... 8.157e-03 -7.976e-01] x_iters: [[-0.009345334109402526], [1.2713537644662787], [0.4484475787090836], [1.0854396754496047], [1.4426790855107496], [0.9579248468740365], [-0.4515808656811222], [-0.6859481043850504], [-0.35518414273753307], [-0.29315377717222235], [-0.32099415298782463], [-2.0], [2.0], [-1.3373742019079444], [-0.24784228664930108]] 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)] space: Space([Real(low=-2.0, high=2.0, prior='uniform', transform='normalize')]) random_state: RandomState(MT19937) specs: args: func: 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) verbose: False callback: None n_points: 10000 n_restarts_optimizer: 5 xi: 0.01 kappa: 1.96 n_jobs: 1 model_queue_size: None space_constraint: None function: base_minimize .. GENERATED FROM PYTHON SOURCE LINES 152-155 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: .. GENERATED FROM PYTHON SOURCE LINES 155-160 .. code-block:: Python from skopt.plots import plot_convergence plot_convergence(res) .. image-sg:: /auto_examples/images/sphx_glr_bayesian-optimization_002.png :alt: Convergence plot :srcset: /auto_examples/images/sphx_glr_bayesian-optimization_002.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out .. code-block:: none .. GENERATED FROM PYTHON SOURCE LINES 161-165 Let us now visually examine 1. The approximation of the fit gp model to the original function. 2. The acquisition values that determine the next point to be queried. .. GENERATED FROM PYTHON SOURCE LINES 165-173 .. code-block:: Python plt.rcParams["figure.figsize"] = (8, 14) def f_wo_noise(x): return f(x, noise_level=0) .. GENERATED FROM PYTHON SOURCE LINES 174-175 Plot the 5 iterations following the 5 random points .. GENERATED FROM PYTHON SOURCE LINES 175-214 .. code-block:: Python for n_iter in range(5): # Plot true function. plt.subplot(5, 2, 2 * n_iter + 1) if n_iter == 0: show_legend = True else: show_legend = False ax = plot_gaussian_process( res, n_calls=n_iter, objective=f_wo_noise, noise_level=noise_level, show_legend=show_legend, show_title=False, show_next_point=False, show_acq_func=False, ) ax.set_ylabel("") ax.set_xlabel("") # Plot EI(x) plt.subplot(5, 2, 2 * n_iter + 2) ax = plot_gaussian_process( res, n_calls=n_iter, show_legend=show_legend, show_title=False, show_mu=False, show_acq_func=True, show_observations=False, show_next_point=True, ) ax.set_ylabel("") ax.set_xlabel("") plt.show() .. image-sg:: /auto_examples/images/sphx_glr_bayesian-optimization_003.png :alt: bayesian optimization :srcset: /auto_examples/images/sphx_glr_bayesian-optimization_003.png :class: sphx-glr-single-img .. GENERATED FROM PYTHON SOURCE LINES 215-232 The first column shows the following: 1. The true function. 2. The approximation to the original function by the gaussian process model 3. How sure the GP is about the function. 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. At the points closer to the points previously evaluated at, the variance dips to zero. 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: .. GENERATED FROM PYTHON SOURCE LINES 232-239 .. code-block:: Python plt.rcParams["figure.figsize"] = (6, 4) # Plot f(x) + contours _ = plot_gaussian_process(res, objective=f_wo_noise, noise_level=noise_level) plt.show() .. image-sg:: /auto_examples/images/sphx_glr_bayesian-optimization_004.png :alt: x* = -0.3552, f(x*) = -1.0079 :srcset: /auto_examples/images/sphx_glr_bayesian-optimization_004.png :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** (0 minutes 3.907 seconds) .. _sphx_glr_download_auto_examples_bayesian-optimization.py: .. only:: html .. container:: sphx-glr-footer sphx-glr-footer-example .. container:: binder-badge .. image:: images/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/holgern/scikit-optimize/master?urlpath=lab/tree/notebooks/auto_examples/bayesian-optimization.ipynb :alt: Launch binder :width: 150 px .. container:: sphx-glr-download sphx-glr-download-jupyter :download:`Download Jupyter notebook: bayesian-optimization.ipynb ` .. container:: sphx-glr-download sphx-glr-download-python :download:`Download Python source code: bayesian-optimization.py ` .. only:: html .. rst-class:: sphx-glr-signature `Gallery generated by Sphinx-Gallery `_