import numpy as np
from math import inf
from spotpython.fun.objectivefunctions import analytical
from spotpython.spot import spot
import matplotlib.pyplot as plt
from spotpython.utils.init import fun_control_init, get_spot_tensorboard_path
from spotpython.utils.init import fun_control_init, design_control_init, surrogate_control_init
= "09" PREFIX
14 Optimal Computational Budget Allocation in Spot
This chapter demonstrates how noisy functions can be handled with Optimal Computational Budget Allocation (OCBA) by Spot
.
14.1 Example: Spot
, OCBA, and the Noisy Sphere Function
14.1.1 The Objective Function: Noisy Sphere
The spotpython
package provides several classes of objective functions. We will use an analytical objective function with noise, i.e., a function that can be described by a (closed) formula: \[f(x) = x^2 + \epsilon\]
Since sigma
is set to 0.1
, noise is added to the function:
= analytical().fun_sphere
fun = fun_control_init(
fun_control =PREFIX,
PREFIX=0.1) sigma
A plot illustrates the noise:
= np.linspace(-1,1,100).reshape(-1,1)
x = fun(x, fun_control=fun_control)
y
plt.figure()"k")
plt.plot(x,y, plt.show()
Spot
is adopted as follows to cope with noisy functions:
fun_repeats
is set to a value larger than 1 (here: 2)noise
is set totrue
. Therefore, a nugget (Lambda
) term is added to the correlation matrixinit size
(of thedesign_control
dictionary) is set to a value larger than 1 (here: 2)
= spot.Spot(fun=fun,
spot_1_noisy =fun_control_init(
fun_control= np.array([-1]),
lower = np.array([1]),
upper = 20,
fun_evals = 2,
fun_repeats ="ei",
infill_criterion= True,
noise =0.0,
tolerance_x= 1,
ocba_delta =True),
show_models=design_control_init(init_size=3, repeats=2),
design_control=surrogate_control_init(noise=True)) surrogate_control
spot_1_noisy.run()
spotpython tuning: 0.00891934134603014 [####------] 40.00%
spotpython tuning: 0.00891934134603014 [#####-----] 50.00%
spotpython tuning: 6.820812967131544e-05 [######----] 60.00%
spotpython tuning: 3.692698488681066e-06 [#######---] 70.00%
spotpython tuning: 1.4235007051487162e-06 [########--] 80.00%
spotpython tuning: 1.4235007051487162e-06 [#########-] 90.00%
spotpython tuning: 1.4235007051487162e-06 [##########] 100.00% Done...
14.2 Print the Results
spot_1_noisy.print_results()
min y: 1.4235007051487162e-06
min mean y: 1.4235007051487162e-06
x0: -0.0011931054878545804
[['x0', -0.0011931054878545804]]
=False) spot_1_noisy.plot_progress(log_y
14.3 Noise and Surrogates: The Nugget Effect
14.3.1 The Noisy Sphere
14.3.1.1 The Data
We prepare some data first:
import numpy as np
import spotpython
from spotpython.fun.objectivefunctions import analytical
from spotpython.spot import spot
from spotpython.design.spacefilling import SpaceFilling
from spotpython.build.kriging import Kriging
import matplotlib.pyplot as plt
= SpaceFilling(1)
gen = np.random.RandomState(1)
rng = np.array([-10])
lower = np.array([10])
upper = analytical().fun_sphere
fun = fun_control_init(
fun_control =2,
sigma=125)
seed= gen.scipy_lhd(10, lower=lower, upper = upper)
X = fun(X, fun_control=fun_control)
y = X.reshape(-1,1)
X_train = y y_train
A surrogate without nugget is fitted to these data:
= Kriging(name='kriging',
S =123,
seed=50,
log_level=1,
n_theta=False)
noise
S.fit(X_train, y_train)
= np.linspace(start=-13, stop=13, num=1000).reshape(-1, 1)
X_axis = S.predict(X_axis, return_val="all")
mean_prediction, std_prediction, ei
="Observations")
plt.scatter(X_train, y_train, label="mue")
plt.plot(X_axis, mean_prediction, label
plt.legend()"$x$")
plt.xlabel("$f(x)$")
plt.ylabel(= plt.title("Sphere: Gaussian process regression on noisy dataset") _
In comparison to the surrogate without nugget, we fit a surrogate with nugget to the data:
= Kriging(name='kriging',
S_nug =123,
seed=50,
log_level=1,
n_theta=True)
noise
S_nug.fit(X_train, y_train)= np.linspace(start=-13, stop=13, num=1000).reshape(-1, 1)
X_axis = S_nug.predict(X_axis, return_val="all")
mean_prediction, std_prediction, ei ="Observations")
plt.scatter(X_train, y_train, label="mue")
plt.plot(X_axis, mean_prediction, label
plt.legend()"$x$")
plt.xlabel("$f(x)$")
plt.ylabel(= plt.title("Sphere: Gaussian process regression with nugget on noisy dataset") _
The value of the nugget term can be extracted from the model as follows:
S.Lambda
S_nug.Lambda
9.867760027597887e-05
We see:
- the first model
S
has no nugget, - whereas the second model has a nugget value (
Lambda
) larger than zero.
14.4 Exercises
14.4.1 Noisy fun_cubed
Analyse the effect of noise on the fun_cubed
function with the following settings:
= analytical().fun_cubed
fun = fun_control_init(
fun_control =10,
sigma=123)
seed= np.array([-10])
lower = np.array([10]) upper
14.4.2 fun_runge
Analyse the effect of noise on the fun_runge
function with the following settings:
= np.array([-10])
lower = np.array([10])
upper = analytical().fun_runge
fun = fun_control_init(
fun_control =0.25,
sigma=123) seed
14.4.3 fun_forrester
Analyse the effect of noise on the fun_forrester
function with the following settings:
= np.array([0])
lower = np.array([1])
upper = analytical().fun_forrester
fun = {"sigma": 5,
fun_control "seed": 123}
14.4.4 fun_xsin
Analyse the effect of noise on the fun_xsin
function with the following settings:
= np.array([-1.])
lower = np.array([1.])
upper = analytical().fun_xsin
fun = fun_control_init(
fun_control =0.5,
sigma=123) seed