Log-transformed optimization

Log-Transformed Acqusition Functions¶

A common problem when using acquisition functions is vanishing gradients in regions of parameter space that are nearly zero. This happens often when using Expected Improvement or Constraint probability weighting. To address this, [1] suggests taking the log of the acquisition function to make numerical optimization more robust. We demonstrate this feature in Xopt. Compare acquisition function plots shown here to those in the constrained BO tutorial.

[1] Ament, Sebastian, et al. "Unexpected improvements to expected improvement for bayesian optimization." arXiv preprint arXiv:2310.20708 (2023).

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 ExpectedImprovementGenerator
from xopt import Xopt

evaluator = Evaluator(function=sin_function)
generator = ExpectedImprovementGenerator(vocs=vocs, log_transform_acquisition_function=True)
X = Xopt(evaluator=evaluator, generator=generator, vocs=vocs)

from xopt.evaluator import Evaluator from xopt.generators.bayesian import ExpectedImprovementGenerator from xopt import Xopt evaluator = Evaluator(function=sin_function) generator = ExpectedImprovementGenerator(vocs=vocs, log_transform_acquisition_function=True) 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	4.579408	-0.991171	0.000004	False
1	1.149406	0.912521	0.000001	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

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()

    # visualize model
    fig, ax = generator.visualize_model(n_grid=len(test_x))
    
    # plot true function
    true_f = sin_function({"x": test_x})["f"]
    gt = ax[0].plot(test_x, true_f, "C1--")

    # do the optimization step
    X.step()

import torch 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() # visualize model fig, ax = generator.visualize_model(n_grid=len(test_x)) # plot true function true_f = sin_function({"x": test_x})["f"] gt = ax[0].plot(test_x, true_f, "C1--") # 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	4.579408	-9.911710e-01	0.000004	False
1	1.149406	9.125210e-01	0.000001	False
2	6.283185	-2.449294e-16	0.000004	False
3	3.728141	-5.534894e-01	0.000003	False
4	4.944302	-9.732284e-01	0.000003	False
5	4.722263	-9.999513e-01	0.000003	False
6	4.721998	-9.999538e-01	0.000004	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	4.717988

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  0.041450                  0.012696
 1  0.039781                  0.016518
 2  0.036743                  0.022948
 3  0.038187                  0.032903
 4  0.044792                  0.077229,
 'log_transform_acquisition_function': True,
 'n_interpolate_points': None,
 'n_candidates': 1}

In [ ]: