.. 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 `_