ExpectedImprovement

Expected improvement acquisition function.

Overview

The ExpectedImprovement acquisition function for parallel active learning (Bayesian optimization) is given by:

$(1)var element = document.getElementById("moose-equation-56935d26-3862-4a2f-9c12-e773efd52421");katex.render(" a(\\pmb{x}) = (\\mu-f(\\pmb{x}^*)-\\psi)~\\Phi\\Big(\\frac{\\mu-f(\\pmb{x}^*)-\\psi}{\\sigma}\\Big)+\\sigma~\\phi\\Big(\\frac{\\mu-f(\\pmb{x}^*)-\\psi}{\\sigma}\\Big)", element, {displayMode:true,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$

where, $var element = document.getElementById("moose-equation-78043e51-7515-4bd8-8306-bcbc27d4d3d7");katex.render("\\psi", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a tuning parameter to boost exploration or exploitation, $var element = document.getElementById("moose-equation-f09805ee-a681-4a84-a942-6643be28751a");katex.render("f(\\pmb{x}^*)", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the computational model output at the best point thus far $var element = document.getElementById("moose-equation-67a32df7-d152-4d09-af0a-28c8623e5625");katex.render("\\pmb{x}^*", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ , $var element = document.getElementById("moose-equation-bb883d47-8d98-4d0f-8ebb-5acfdc572195");katex.render("\\Phi", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a standard Normal CDF, $var element = document.getElementById("moose-equation-0151e8ba-f12b-421a-a334-8a92fe1d198d");katex.render("\\phi", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is a standard Normal PDF, $var element = document.getElementById("moose-equation-8f5a293d-5439-4de1-b3f2-aeada0469ef6");katex.render("\\mu", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Gaussian process mean prediction, and $var element = document.getElementById("moose-equation-93842fe1-fd59-4ebc-9e0c-f54755879c04");katex.render("\\sigma", element, {displayMode:false,throwOnError:false,macros:{"\\eqc":"\\,,","\\eqp":"\\,.","\\pd":"\\frac{\\partial #1}{\\partial #2}","\\pr":"\\left(#1\\right)","\\ddt":"\\frac{d #1}{d t}"}});$ is the Gaussian process standard deviation.

Input Parameters

tuning0.001Tuning parameter to control exploration vs exploitation.
Default:0.001
C++ Type:double
Unit:(no unit assumed)
Range:tuning > 0
Controllable:No
Description:Tuning parameter to control exploration vs exploitation.

Optional Parameters

control_tagsAdds user-defined labels for accessing object parameters via control logic.
C++ Type:std::vector<std::string>
Controllable:No
Description:Adds user-defined labels for accessing object parameters via control logic.
enableTrueSet the enabled status of the MooseObject.
Default:True
C++ Type:bool
Controllable:No
Description:Set the enabled status of the MooseObject.

Advanced Parameters

Input Files

(modules/stochastic_tools/test/tests/samplers/ParallelActiveLearning/parallelAL.i)

(modules/stochastic_tools/test/tests/samplers/ParallelActiveLearning/parallelAL.i)

[StochasticTools]
[]

[Distributions]
  [left]
    type = Normal
    mean = 0.0
    standard_deviation = 1.0
  []
  [right]
    type = Normal
    mean = 0.0
    standard_deviation = 1.0
  []
[]

[ParallelAcquisition]
  [expectedimprovement]
    type = ExpectedImprovement
  []
[]

[Samplers]
  [sample]
    type = GenericActiveLearningSampler
    distributions = 'left right'
    sorted_indices = 'conditional/sorted_indices'
    num_parallel_proposals = 5
    num_tries = 1000
    seed = 100
    num_random_seeds = 100
    initial_values = '0.1 0.1'
    execute_on = 'PRE_MULTIAPP_SETUP timestep_end'
  []
[]

[MultiApps]
  [sub]
    type = SamplerFullSolveMultiApp
    input_files = sub.i
    sampler = sample
  []
[]

[Transfers]
  [reporter_transfer]
    type = SamplerReporterTransfer
    from_reporter = 'average/value'
    stochastic_reporter = 'constant'
    from_multi_app = sub
    sampler = sample
  []
[]

[Controls]
  [cmdline]
    type = MultiAppSamplerControl
    multi_app = sub
    sampler = sample
    param_names = 'left_bc right_bc'
  []
[]

[Reporters]
  [constant]
    type = StochasticReporter
  []
  [conditional]
    type = GenericActiveLearner
    output_value = constant/reporter_transfer:average:value
    sampler = sample
    al_gp = GP_al_trainer
    gp_evaluator = GP_eval
    acquisition = 'expectedimprovement'
    execute_on = 'timestep_begin'
  []
[]

[Trainers]
  [GP_al_trainer]
    type = ActiveLearningGaussianProcess
    covariance_function = 'covar'
    standardize_params = 'true'
    standardize_data = 'true'
    tune_parameters = 'covar:signal_variance covar:length_factor'
    num_iters = 500
    learning_rate = 0.01
    # show_every_nth_iteration = 100
    batch_size = 350
  []
[]

[Surrogates]
  [GP_eval]
    type = GaussianProcessSurrogate
    trainer = GP_al_trainer
  []
[]

[Covariance]
  [covar]
    type = SquaredExponentialCovariance
    signal_variance = 4.0
    noise_variance = 1e-6
    length_factor = '4.0 4.0'
  []
[]

[Executioner]
  type = Transient
  num_steps = 5
[]

[Outputs]
  file_base = 'al1'
  [out1_parallelAL]
    type = JSON
    execute_system_information_on = NONE
  []
[]

Overview
Input Parameters
Input Files