In [1]:
import numpy as np
np.random.seed(777)
%matplotlib inline
import matplotlib.pyplot as plt
plt.rcParams["figure.figsize"] = (10, 6)
We are interested in solving $$x^* = \arg \min_x f(x)$$ under the constraints that
Disclaimer. If you do not have these constraints, then there is certainly a better optimization algorithm than Bayesian optimization.
For $t=1:T$:
Given observations $(x_i, y_i=f(x_i))$ for $i=1:t$, build a probabilistic model for the objective $f$. Integrate out all possible true functions, using Gaussian process regression.
optimize a cheap acquisition/utility function $u$ based on the posterior distribution for sampling the next point. $$x_{t+1} = \arg \min_x u(x)$$ Exploit uncertainty to balance exploration against exploitation.
Sample the next observation $y_{t+1}$ at $x_{t+1}$.
Acquisition functions $\text{u}(x)$ specify which sample $x$ should be tried next:
where $x_t^+$ is the best point observed so far.
In most cases, acquisition functions provide knobs (e.g., $\kappa$) for controlling the exploration-exploitation trade-off.
In [2]:
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 $f$ are assumed to take as input a 1D vector $x$ represented as an array-like and to return a scalar $f(x)$.
In [3]:
# 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()
Bayesian optimization based on gaussian process regression is implemented in skopt.gp_minimize and can be carried out as follows:
In [4]:
from skopt import gp_minimize
from skopt.acquisition import gaussian_lcb
from sklearn.gaussian_process import GaussianProcessRegressor
from sklearn.gaussian_process.kernels import Matern
# Note that we have fixed the hyperparameters of the kernel, because it is
# sufficient for this easy problem.
gp = GaussianProcessRegressor(kernel=Matern(length_scale_bounds="fixed"),
alpha=noise_level**2, random_state=0)
res = gp_minimize(f, # the function to minimize
[(-2.0, 2.0)], # the bounds on each dimension of x
x0=[0.], # the starting point
acq="LCB", # the acquisition function (optional)
base_estimator=gp, # a GP estimator (optional)
n_calls=15, # the number of evaluations of f including at x0
n_random_starts=0, # the number of random initialization points
random_state=777)
Accordingly, the approximated minimum is found to be:
In [5]:
"x^*=%.4f, f(x^*)=%.4f" % (res.x[0], res.fun)
Out[5]:
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.
In [6]:
for key, value in sorted(res.items()):
print(key, "=", value)
print()
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:
In [7]:
from skopt.plots import plot_convergence
plot_convergence(res)
Out[7]:
Let us visually examine
At the points closer to the points previously evaluated at, the variance dips to zero.
The first column shows the following:
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.
In [8]:
plt.rcParams["figure.figsize"] = (20, 20)
x = np.linspace(-2, 2, 400).reshape(-1, 1)
fx = np.array([f(x_i, noise_level=0.0) for x_i in x])
# Plot first five iterations.
for n_iter in range(5):
gp = res.models[n_iter]
curr_x_iters = res.x_iters[: n_iter+1]
curr_func_vals = res.func_vals[: n_iter+1]
# Plot true function.
plt.subplot(5, 2, 2*n_iter+1)
plt.plot(x, fx, "r--", label="True (unknown)")
plt.fill(np.concatenate([x, x[::-1]]),
np.concatenate([fx - 1.9600 * noise_level, fx[::-1] + 1.9600 * noise_level]),
alpha=.2, fc="r", ec="None")
# Plot GP(x) + contours
y_pred, sigma = gp.predict(x, return_std=True)
plt.plot(x, y_pred, "g--", label=r"$\mu_{GP}(x)$")
plt.fill(np.concatenate([x, x[::-1]]),
np.concatenate([y_pred - 1.9600 * sigma,
(y_pred + 1.9600 * sigma)[::-1]]),
alpha=.2, fc="g", ec="None")
# Plot sampled points
plt.plot(curr_x_iters, curr_func_vals,
"r.", markersize=15, label="Observations")
plt.title(r"$x_{%d} = %.4f, f(x_{%d}) = %.4f$" % (
n_iter, res.x_iters[n_iter][0], n_iter, res.func_vals[n_iter]))
plt.grid()
if n_iter == 0:
plt.legend(loc="best", prop={'size': 8}, numpoints=1)
plt.subplot(5, 2, 2*n_iter+2)
acq = gaussian_lcb(x, gp)
plt.plot(x, acq, "b", label="LCB(x)")
plt.fill_between(x.ravel(), -2.0, acq.ravel(), alpha=0.3, color='blue')
next_x = np.asarray(res.x_iters[n_iter + 1])
next_acq = gaussian_lcb(next_x.reshape(-1, 1), gp)
plt.plot(next_x[0], next_acq, "bo", markersize=10, label="Next query point")
plt.grid()
if n_iter == 0:
plt.legend(loc="best", prop={'size': 12}, numpoints=1)
plt.suptitle("Sequential model-based minimization using gp_minimize.", fontsize=20)
plt.show()
Finally, as we increase the number of points, the GP model approaches the actual function. The final few points are cluttered around the minimum because the GP does not gain anything more by further exploration.
In [9]:
# Plot f(x) + contours
plt.rcParams["figure.figsize"] = (10, 6)
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")
# Plot GP(x) + concours
gp = res.models[-1]
y_pred, sigma = gp.predict(x, return_std=True)
plt.plot(x, y_pred, "g--", label=r"$\mu_{GP}(x)$")
plt.fill(np.concatenate([x, x[::-1]]),
np.concatenate([y_pred - 1.9600 * sigma,
(y_pred + 1.9600 * sigma)[::-1]]),
alpha=.2, fc="g", ec="None")
# Plot sampled points
plt.plot(res.x_iters,
res.func_vals,
"r.", markersize=15, label="Observations")
# Plot LCB(x) + next query point
acq = gaussian_lcb(x, gp)
plt.plot(x, gaussian_lcb(x, gp), "b", label="LCB(x)")
next_x = np.argmin(acq)
plt.plot([x[next_x]], [acq[next_x]], "b.", markersize=15, label="Next query point")
plt.title(r"$x^* = %.4f, f(x^*) = %.4f$" % (res.x[0], res.fun))
plt.legend(loc="best")
plt.grid()
plt.show()