Exploitation vs Exploration



In [1]:

    
%matplotlib inline

import numpy as np
import matplotlib.pyplot as plt

from bayes_opt import BayesianOptimization

# use sklearn's default parameters for theta and random_start
gp_params = {"corr": "cubic", "theta0": 0.1, "thetaL": None, "thetaU": None, "random_start": 1}

Target function



In [2]:

    
np.random.seed(42)
xs = np.linspace(-2, 10, 10000)
f = np.exp(-(xs - 2)**2) + np.exp(-(xs - 6)**2/10) + 1/ (xs**2 + 1)

plt.plot(f)
plt.show()

Utility function for plotting



In [3]:

    
def plot_bo(f, bo):
    xs = [x["x"] for x in bo.res["all"]["params"]]
    ys = bo.res["all"]["values"]

    mean, var = bo.gp.predict(np.arange(len(f)).reshape(-1, 1), eval_MSE=True)
    
    plt.figure(figsize=(16, 9))
    plt.plot(f)
    plt.plot(np.arange(len(f)), mean)
    plt.fill_between(np.arange(len(f)), mean+var, mean-var, alpha=0.1)
    plt.scatter(bo.X.flatten(), bo.Y, c="red", s=50, zorder=10)
    plt.xlim(0, len(f))
    plt.ylim(f.min()-0.1*(f.max()-f.min()), f.max()+0.1*(f.max()-f.min()))
    plt.show()

Acquisition Function "Expected Improvement"

Prefer exploitation (xi=0.0)

Note that most points are around the peak(s).



In [4]:

    
bo = BayesianOptimization(f=lambda x: f[int(x)],
                          pbounds={"x": (0, len(f)-1)},
                          verbose=0)

bo.maximize(init_points=2, n_iter=25, acq="ei", xi=0.0, **gp_params)

plot_bo(f, bo)

Prefer exploration (xi=0.1)

Note that the points are more spread out across the whole range.



In [5]:

    
bo = BayesianOptimization(f=lambda x: f[int(x)], pbounds={"x": (0, len(f)-1)}, verbose=0)

bo.maximize(init_points=2, n_iter=25, acq="ei", xi=0.1, **gp_params)

plot_bo(f, bo)

Acquisition Function "Probability of Improvement"

Prefer exploitation (xi=0.0)

Note that most points are around the peak(s).



In [6]:

    
bo = BayesianOptimization(f=lambda x: f[int(x)],
                          pbounds={"x": (0, len(f)-1)},
                          verbose=0)

bo.maximize(init_points=2, n_iter=25, acq="poi", xi=0.0, **gp_params)

plot_bo(f, bo)

Prefer exploration (xi=0.1)

Note that the points are more spread out across the whole range.



In [7]:

    
bo = BayesianOptimization(f=lambda x: f[int(x)], pbounds={"x": (0, len(f)-1)}, verbose=0)

bo.maximize(init_points=2, n_iter=25, acq="poi", xi=0.1, **gp_params)

plot_bo(f, bo)



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