.. only:: html .. note:: :class: sphx-glr-download-link-note Click :ref:here  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) .. code-block:: default print(__doc__) import numpy as np np.random.seed(237) import matplotlib.pyplot as plt from skopt.plots import plot_gaussian_process Toy example ----------- Let assume the following noisy function :math:f: .. code-block:: default 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 **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):. .. code-block:: default # 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=.2, fc="r", ec="None") plt.legend() plt.grid() plt.show() .. image:: /auto_examples/images/sphx_glr_bayesian-optimization_001.png :alt: bayesian optimization :class: sphx-glr-single-img Bayesian optimization based on gaussian process regression is implemented in :class:gp_minimize and can be carried out as follows: .. code-block:: default 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 Accordingly, the approximated minimum is found to be: .. code-block:: default "x^*=%.4f, f(x^*)=%.4f" % (res.x[0], res.fun) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 'x^*=-0.3552, f(x^*)=-1.0079' 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. .. code-block:: default print(res) .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 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, -0.79756403]) 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 0x7F4690079C40 space: Space([Real(low=-2.0, high=2.0, prior='uniform', transform='normalize')]) 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) at 0x7F4690079C40, '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], [-0.685948113064452], [-0.35518416232959327], [-0.2931537904259709], [-0.32099415962984157], [-2.0], [2.0], [-1.3373741988004628], [-0.2478423111669088]] 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: .. code-block:: default from skopt.plots import plot_convergence plot_convergence(res); .. image:: /auto_examples/images/sphx_glr_bayesian-optimization_002.png :alt: Convergence plot :class: sphx-glr-single-img .. rst-class:: sphx-glr-script-out Out: .. code-block:: none 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. .. code-block:: default plt.rcParams["figure.figsize"] = (8, 14) def f_wo_noise(x): return f(x, noise_level=0) Plot the 5 iterations following the 5 random points .. code-block:: default 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:: /auto_examples/images/sphx_glr_bayesian-optimization_003.png :alt: bayesian optimization :class: sphx-glr-single-img 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: .. code-block:: default 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:: /auto_examples/images/sphx_glr_bayesian-optimization_004.png :alt: x* = -0.3552, f(x*) = -1.0079 :class: sphx-glr-single-img .. rst-class:: sphx-glr-timing **Total running time of the script:** ( 0 minutes 3.555 seconds) **Estimated memory usage:** 8 MB .. _sphx_glr_download_auto_examples_bayesian-optimization.py: .. only :: html .. container:: sphx-glr-footer :class: sphx-glr-footer-example .. container:: binder-badge .. image:: /../../miniconda/envs/testenv/lib/python3.8/site-packages/sphinx_gallery/_static/binder_badge_logo.svg :target: https://mybinder.org/v2/gh/scikit-optimize/scikit-optimize/master?urlpath=lab/tree/notebooks/auto_examples/bayesian-optimization.ipynb :width: 150 px .. container:: sphx-glr-download sphx-glr-download-python :download:Download Python source code: bayesian-optimization.py  .. container:: sphx-glr-download sphx-glr-download-jupyter :download:Download Jupyter notebook: bayesian-optimization.ipynb  .. only:: html .. rst-class:: sphx-glr-signature Gallery generated by Sphinx-Gallery _