Bayesian optimization tutorial

Basic Bayesian Optimization¶

In this tutorial we demonstrate the use of Xopt to preform Bayesian Optimization on a simple test problem.

Define the test problem¶

Here we define a simple optimization problem, where we attempt to minimize the sin function in the domian [0,2*pi]. Note that the function used to evaluate the objective function takes a dictionary as input and returns a dictionary as the output.

In [1]:

from xopt.vocs import VOCS
import math

# define variables and function objectives
vocs = VOCS(
    variables={"x": [0, 2 * math.pi]},
    objectives={"f": "MINIMIZE"},
)

from xopt.vocs import VOCS import math # define variables and function objectives vocs = VOCS( variables={"x": [0, 2 * math.pi]}, objectives={"f": "MINIMIZE"}, )

In [2]:

# define a test function to optimize
import numpy as np

def sin_function(input_dict):
    return {"f": np.sin(input_dict["x"])}

# define a test function to optimize import numpy as np def sin_function(input_dict): return {"f": np.sin(input_dict["x"])}

Create Xopt objects¶

Create the evaluator to evaluate our test function and create a generator that uses the Upper Confidence Bound acquisition function to perform Bayesian Optimization.

In [3]:

from xopt.evaluator import Evaluator
from xopt.generators.bayesian import UpperConfidenceBoundGenerator
from xopt import Xopt

evaluator = Evaluator(function=sin_function)
generator = UpperConfidenceBoundGenerator(vocs=vocs)
X = Xopt(evaluator=evaluator, generator=generator, vocs=vocs)

from xopt.evaluator import Evaluator from xopt.generators.bayesian import UpperConfidenceBoundGenerator from xopt import Xopt evaluator = Evaluator(function=sin_function) generator = UpperConfidenceBoundGenerator(vocs=vocs) X = Xopt(evaluator=evaluator, generator=generator, vocs=vocs)

Generate and evaluate initial points¶

To begin optimization, we must generate some random initial data points. The first call to X.step() will generate and evaluate a number of randomly points specified by the generator. Note that if we add data to xopt before calling X.step() by assigning the data to X.data, calls to X.step() will ignore the random generation and proceed to generating points via Bayesian optimization.

In [4]:

# call X.random_evaluate() to generate + evaluate 3 initial points
X.random_evaluate(2)

# inspect the gathered data
X.data

# call X.random_evaluate() to generate + evaluate 3 initial points X.random_evaluate(2) # inspect the gathered data X.data

Out[4]:

	x	f	xopt_runtime	xopt_error
0	3.871049	-0.666464	0.000018	False
1	5.255789	-0.855955	0.000002	False

Do bayesian optimization steps¶

To perform optimization we simply call X.step() in a loop. This allows us to do intermediate tasks in between optimization steps, such as examining the model and acquisition function at each step (as we demonstrate here).

In [5]:

import torch
import matplotlib.pyplot as plt

n_steps = 5

# test points for plotting
test_x = torch.linspace(*X.vocs.bounds.flatten(), 50).double()

for i in range(n_steps):
    # get the Gaussian process model from the generator
    model = X.generator.train_model()

    # get acquisition function from generator
    acq = X.generator.get_acquisition(model)

    # calculate model posterior and acquisition function at each test point
    # NOTE: need to add a dimension to the input tensor for evaluating the
    # posterior and another for the acquisition function, see
    # https://botorch.org/docs/batching for details
    # NOTE: we use the `torch.no_grad()` environment to speed up computation by
    # skipping calculations for backpropagation
    with torch.no_grad():
        posterior = model.posterior(test_x.unsqueeze(1))
        acq_val = acq(test_x.reshape(-1, 1, 1))

    # get mean function and confidence regions
    mean = posterior.mean
    l,u = posterior.mvn.confidence_region()

    # plot model and acquisition function
    fig,ax = plt.subplots(2, 1, sharex="all")

    # plot model posterior
    ax[0].plot(test_x, mean, label="Posterior mean")
    ax[0].fill_between(test_x, l, u, alpha=0.25, label="Posterior confidence region")

    # add data to model plot
    ax[0].plot(X.data["x"],X.data["f"],"C1o", label="Training data")

    # plot true function
    true_f = sin_function({"x": test_x})["f"]
    ax[0].plot(test_x, true_f,'--', label="Ground truth")

    # add legend
    ax[0].legend()

    # plot acquisition function
    ax[1].plot(test_x, acq_val.flatten())

    ax[0].set_ylabel("f")
    ax[1].set_ylabel(r"$\alpha(x)$")
    ax[1].set_xlabel("x")

    # do the optimization step
    X.step()

import torch import matplotlib.pyplot as plt n_steps = 5 # test points for plotting test_x = torch.linspace(*X.vocs.bounds.flatten(), 50).double() for i in range(n_steps): # get the Gaussian process model from the generator model = X.generator.train_model() # get acquisition function from generator acq = X.generator.get_acquisition(model) # calculate model posterior and acquisition function at each test point # NOTE: need to add a dimension to the input tensor for evaluating the # posterior and another for the acquisition function, see # https://botorch.org/docs/batching for details # NOTE: we use the `torch.no_grad()` environment to speed up computation by # skipping calculations for backpropagation with torch.no_grad(): posterior = model.posterior(test_x.unsqueeze(1)) acq_val = acq(test_x.reshape(-1, 1, 1)) # get mean function and confidence regions mean = posterior.mean l,u = posterior.mvn.confidence_region() # plot model and acquisition function fig,ax = plt.subplots(2, 1, sharex="all") # plot model posterior ax[0].plot(test_x, mean, label="Posterior mean") ax[0].fill_between(test_x, l, u, alpha=0.25, label="Posterior confidence region") # add data to model plot ax[0].plot(X.data["x"],X.data["f"],"C1o", label="Training data") # plot true function true_f = sin_function({"x": test_x})["f"] ax[0].plot(test_x, true_f,'--', label="Ground truth") # add legend ax[0].legend() # plot acquisition function ax[1].plot(test_x, acq_val.flatten()) ax[0].set_ylabel("f") ax[1].set_ylabel(r"$\alpha(x)$") ax[1].set_xlabel("x") # do the optimization step X.step()

In [6]:

# access the collected data
X.data

# access the collected data X.data

Out[6]:

	x	f	xopt_runtime	xopt_error
0	3.871049	-6.664642e-01	0.000018	False
1	5.255789	-8.559555e-01	0.000002	False
2	6.283185	-2.449294e-16	0.000008	False
3	0.000000	0.000000e+00	0.000008	False
4	4.665832	-9.989164e-01	0.000009	False
5	4.850349	-9.904987e-01	0.000009	False
6	4.712748	-9.999999e-01	0.000008	False

Getting the optimization result¶

To get the best point (without evaluating it) we ask the generator to predict the optimum based on the posterior mean.

In [7]:

X.generator.get_optimum()

X.generator.get_optimum()

Out[7]:

	x
0	4.720811

Customizing optimization¶

Each generator has a set of options that can be modified to effect optimization behavior

In [8]:

X.generator.dict()

X.generator.dict()

Out[8]:

{'model': ModelListGP(
   (models): ModuleList(
     (0): SingleTaskGP(
       (likelihood): GaussianLikelihood(
         (noise_covar): HomoskedasticNoise(
           (noise_prior): GammaPrior()
           (raw_noise_constraint): GreaterThan(1.000E-04)
         )
       )
       (mean_module): ConstantMean()
       (covar_module): ScaleKernel(
         (base_kernel): MaternKernel(
           (lengthscale_prior): GammaPrior()
           (raw_lengthscale_constraint): Positive()
         )
         (outputscale_prior): GammaPrior()
         (raw_outputscale_constraint): Positive()
       )
       (outcome_transform): Standardize()
       (input_transform): Normalize()
     )
   )
   (likelihood): LikelihoodList(
     (likelihoods): ModuleList(
       (0): GaussianLikelihood(
         (noise_covar): HomoskedasticNoise(
           (noise_prior): GammaPrior()
           (raw_noise_constraint): GreaterThan(1.000E-04)
         )
       )
     )
   )
 ),
 'n_monte_carlo_samples': 128,
 'turbo_controller': None,
 'use_cuda': False,
 'gp_constructor': {'name': 'standard',
  'use_low_noise_prior': True,
  'covar_modules': {},
  'mean_modules': {},
  'trainable_mean_keys': [],
  'transform_inputs': True,
  'custom_noise_prior': None},
 'numerical_optimizer': {'name': 'LBFGS',
  'n_restarts': 20,
  'max_iter': 2000,
  'max_time': None},
 'max_travel_distances': None,
 'fixed_features': None,
 'computation_time':    training  acquisition_optimization
 0  0.073200                  0.018838
 1  0.093547                  0.061379
 2  0.067075                  0.035901
 3  0.087797                  0.060589
 4  0.079382                  0.042273,
 'log_transform_acquisition_function': False,
 'n_interpolate_points': None,
 'n_candidates': 1,
 'beta': 2.0}

In [8]: