Standard Bayesian optimization

Driver:

BayesianOptimization

Download script:

vanilla_bayesian_optimization.py

The target of the study is to run a standard Bayesian optimization of a scalar function. As an example, the 2D Rastrigin function on a circular domain is minimized,

\[ \begin{align}\begin{aligned}&\text{min.}\,& f(x_1,x_2) = 2\cdot10 + \sum_{i=1,2} \left(x_i^2 - 10\cos(2\pi x_i)\right)\\&\text{s.t.}\,& \sqrt{x_1^2 + x_2^2} \leq 1.5.\end{aligned}\end{align} \]

import sys,os
import numpy as np
import time
import pandas as pd

from jcmoptimizer import Server, Client, Study, Obseravtion
server = Server()
client = Client(host=server.host)


# Definition of the search domain
design_space = [
    {'name': 'x1', 'type': 'continuous', 'domain': (-1.5,1.5)}, 
    {'name': 'x2', 'type': 'continuous', 'domain': (-1.5,1.5)},
]

# Definition of fixed environment parameter
environment = [
    {'name': 'radius', 'type': 'fixed', 'domain': 1.5},
]

# Definition of a constraint on the search domain
constraints = [
    {'name': 'circle', 'expression': 'sqrt(x1^2 + x2^2) <= radius'}
]

# Creation of the study object with study_id 'vanilla_bayesian_optimization'
study = client.create_study(
    design_space=design_space,
    environment=environment,
    constraints=constraints,
    driver="BayesianOptimization",
    study_name="Standard Bayesian optimization",
    study_id="vanilla_bayesian_optimization"
)

# Evaluation of the black-box function for specified design parameters
def evaluate(study: Study, x1: float, x2: float, radius: float) -> Observation:

    time.sleep(2) # make objective expensive
    observation = study.new_observation()
    observation.add(10*2
                + (x1**2-10*np.cos(2*np.pi*x1)) 
                + (x2**2-10*np.cos(2*np.pi*x2))
            )
    return observation

# Run the minimization
study.set_evaluator(evaluate)
study.run()

best = study.driver.best_sample
print(f"Best sample at: x1={best['x1']:.3f}, x2={best['x2']:.3f}")
#print information on all found local minima
minima = study.driver.get_minima(num_initial_samples=20)
print(pd.DataFrame(minima))

#get information on best value under uncertain inputs with standard deviation 0.1
study.configure(parameter_distribution=dict(
    distributions =[
        dict(type="normal", parameter="x1", mean=best["x1"], stddev=0.01),
        dict(type="normal", parameter="x2", mean=best["x2"], stddev=0.02)
    ]
))
stats = study.driver.get_statistics(quantiles=[0.5])
print(f"Objective mean {stats['mean'][0]:.2f}, "
      f"standard deviation {np.sqrt(stats['variance'][0]):.2f}, "
      f"median {stats['quantiles'][0][0]:.2f}")