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BayesianMinimization
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BayesianMinimization

[Experimental]

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BayesianMinimization [f,{conf1,conf2,…}]

gives an object representing the result of Bayesian minimization of the function f over the configurations confi.

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minimizes over the region represented by the region specification reg.

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minimizes over configurations obtained by applying the function sampler.

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BayesianMinimization [f,{conf1,conf2,…}nsampler]

applies the function nsampler to successively generate configurations starting from the confi.

Details and Options

  • BayesianMinimization[…] returns a BayesianMinimizationObject[…] whose properties can be obtained using BayesianMinimizationObject[…]["prop"].
  • Possible properties include:
  • "EvaluationHistory"configurations and values explored during minimization
    "Method"method used for Bayesian minimization
    "MinimumConfiguration"configuration found that minimizes the result from f
    "MinimumValue"estimated minimum value obtained from f
    "NextConfiguration"configuration to sample next if minimization were continued
    "PredictorFunction"best prediction model found for the function f
    "Properties"list of all available properties
  • Configurations can be of any form accepted by Predict (single data element, list of data elements, association of data elements, etc.) and of any type accepted by Predict (numerical, textual, sounds, images, etc.).
  • The function f must output a real-number value when applied to a configuration conf.
  • BayesianMinimization[f,…] attempts to find a good minimum using the smallest number of evaluations of f.
  • In BayesianMinimization[f,spec], spec defines the domain of the function f. A domain can be defined by a list of configurations, a geometric region, or a configuration generator function.
  • In BayesianMinimization[f,sampler], sampler[] must output a configuration suitable for f to be applied to it.
  • In BayesianMinimization[f,{conf1,conf2,…}->nsampler], nsampler[conf] must output a configuration.
  • BayesianMinimization takes the following options:
  • AssumeDeterministic Falsewhether to assume that f is deterministic
    InitialEvaluationHistory Noneintial set of configurations and values
    MaxIterations 100maximum number of iterations
    Method Automaticmethod used to determine configurations to evaluate
    ProgressReporting$ProgressReportinghow to report progress
    RandomSeeding1234what seeding of pseudorandom generators should be done internally
  • Possible settings for Method include:
  • Automaticpick the method automatically
    "MaxExpectedImprovement"maximize expected improvement over current best value
    "MaxImprovementProbability"maximize improvement probability over current best value
  • Possible settings for RandomSeeding include:
  • Automaticautomatically reseed every time the function is called
    Inheriteduse externally seeded random numbers
    seeduse an explicit integer or strings as a seed

Examples

open allclose all

Basic Examples  (3)Summary of the most common use cases

Minimize a function over an interval:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-yafonk
Out[1]=1

Use the resulting BayesianMinimizationObject[…] to get the estimated minimum configuration:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-pz88u4
Out[2]=2

Get the estimated minimum function value:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-wwg70u
Out[3]=3

Minimize a function over a set of configurations:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-6zcz9u
Out[1]=1

Get the minimum configuration over the set:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-uz6dpb
Out[2]=2

Minimize a function over a domain defined by a random generator:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-zdwoqt
Out[1]=1

Get the estimated minimum value:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-h4xnun
Out[2]=2

Scope  (3)Survey of the scope of standard use cases

Minimize a function over a region:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-crxktu
In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-dqz944
Out[2]=2

Get the list of available properties to query:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-m05hl3
Out[3]=3

Get the history of evaluations:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-4jx2ii
Out[4]=4

Get information about the method used to determine the configurations to explore:

In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-b6mbml
Out[5]=5

Get the current probabilistic model of the function (this is a PredictorFunction):

In[6]:=6
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https://wolfram.com/xid/0ixi7f7wvbpgx32-dn9cnp
Out[6]=6

Find the best configuration to explore if the minimization were continued:

In[7]:=7
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https://wolfram.com/xid/0ixi7f7wvbpgx32-nbree6
Out[7]=7

Find a list of properties simultaneously:

In[8]:=8
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https://wolfram.com/xid/0ixi7f7wvbpgx32-21tcfr
Out[8]=8

Visualize how well the function is modeled, particularly near the minimum:

In[9]:=9
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https://wolfram.com/xid/0ixi7f7wvbpgx32-xmvbai
Out[9]=9

Minimize a function with initial configurations over a domain defined by a random neighborhood configuration generator:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-beev2s
In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-indum1
Out[2]=2

Get the model of the function:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-19buy9
Out[3]=3

Visualize the model's performance near the minimum:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-9ec9b9
Out[4]=4

Define a function that takes an image and computes the negative probability returned by ImageIdentify of identifying the image as the entity , with the domain defined by a random generator over a corpus of images:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-lz0xfs

Minimize the function above:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-namz5e
Out[2]=2

Get the minimum configuration:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-50yapp
Out[3]=3

Get the evaluation history:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-tm5jl2
Out[4]=4

Options  (4)Common values & functionality for each option

AssumeDeterministic  (1)

Minimize a function over a domain defined by a random generator:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-6yo200
In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-pr37wk
Out[2]=2

The function is assumed to be stochastic; the value from the probabilistic model will differ in general from the function value for evaluated configurations:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-z6eirg
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-ija59a
Out[4]=4

Include information that the function is deterministic, i.e. noise free:

In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-y3xsjp
Out[5]=5

For a deterministic function, the model value and the function value for evaluated configurations agree to a good precision:

In[6]:=6
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https://wolfram.com/xid/0ixi7f7wvbpgx32-dy9h5s
Out[6]=6
In[7]:=7
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https://wolfram.com/xid/0ixi7f7wvbpgx32-0ji5cq
Out[7]=7

InitialEvaluationHistory  (1)

Minimize a function over a disk region with a small number of iterations:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-b2xr57
In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-5hqj2m
Out[2]=2

Use the information from this evaluation in the next:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-ejlq10
Out[3]=3
In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-vl667a
Out[4]=4

Get the estimated minimum configuration now:

In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-p0bco2
Out[5]=5

MaxIterations  (1)

Minimize a function with a domain defined by a random generator:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-bqpg5k
In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-o68e0b
Out[2]=2

Get the number of function evaluations:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-m1n6ee
Out[3]=3

Specify the maximum number of iterations:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-r6xxom
Out[4]=4
In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-2pd5jj
Out[5]=5

Method  (1)

Define a function over an interval region:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-128jpb

Specify the method for exploring configurations:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-2v78q6
Out[2]=2

Specify a different method:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-v0dujq
Out[3]=3

Applications  (2)Sample problems that can be solved with this function

Define a training set to train predictor functions using the Predict function and a test set to measure their performance:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-1kuz80

Create a "loss" function to test the performance of different methods in Predict:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-5ixsxo

Minimize the loss function over a domain defined by a list of different methods for Predict:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-sjwhr1
Out[3]=3

Examine the evaluation history:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-quyr5r
Out[4]=4

Find the best configuration to explore next:

In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-0czb4e
Out[5]=5

Load Fisher's Iris dataset and divide it into a training set and a validation set:

In[1]:=1
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https://wolfram.com/xid/0ixi7f7wvbpgx32-fr84u5

Create "blackbox" functions. Here the functions are the loss (negative log-likelihood) functions for two different methods used in the Classify function. The arguments of the functions are known as hyperparameters.

Train a logistic regression classifier with two hyperparameters, the L1 and L2 regularization coefficients:

In[2]:=2
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https://wolfram.com/xid/0ixi7f7wvbpgx32-t140d

Minimize the loss function for the logistic regression classifier over a domain defined by a rectangular region in the logarithm of the hyperparameters:

In[3]:=3
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https://wolfram.com/xid/0ixi7f7wvbpgx32-7jcyzv
Out[3]=3

Get the model of the function:

In[4]:=4
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https://wolfram.com/xid/0ixi7f7wvbpgx32-cmt4aa
Out[4]=4

Visualize the model of the loss function of the classifier together with the estimated minimum:

In[5]:=5
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https://wolfram.com/xid/0ixi7f7wvbpgx32-f3mjc5
Out[5]=5
In[6]:=6
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https://wolfram.com/xid/0ixi7f7wvbpgx32-uiwp10
Out[6]=6

Now train a support vector machine (SVM) classifier with two hyperparameters, the soft margin parameter and the gamma scaling parameter:

In[7]:=7
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https://wolfram.com/xid/0ixi7f7wvbpgx32-jpmc35

Minimize the loss function for the SVM classifier over a domain defined by a rectangular region in the logarithm of the hyperparameters:

In[8]:=8
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-6enhy6
Out[8]=8

Get the model of the function:

In[9]:=9
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-5l8azx
Out[9]=9

Visualize the model of the loss function together with the estimated minimum:

In[10]:=10
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-78mjwf
Out[10]=10
In[11]:=11
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-da2yjm
Out[11]=11

Possible Issues  (2)Common pitfalls and unexpected behavior

When the domain of the objective function is defined by an initial configuration set and a neighborhood configuration generator, the results depend on the quality of the generator provided.

Minimize a function where the domain is defined as above:

In[1]:=1
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-cwx39c
In[2]:=2
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-6u3b3m
Out[2]=2

Get the model of the function:

In[3]:=3
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-d1lalk
Out[3]=3

Since the starting initial configurations are "far" from the global minimum and the generator takes relatively small steps, the algorithm could converge to a local minimum:

In[4]:=4
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-ij5d68
Out[4]=4

If the neighborhood configuration generator takes steps that are "too large" or "too small", this could lead to problems:

In[1]:=1
✖
https://wolfram.com/xid/0ixi7f7wvbpgx32-jajzx2
Out[1]=1

In this case, at each step the generator gives a configuration value that is the square of the previous one, so the values quickly become extremely large and the probabilistic model does not work properly.

Wolfram Research (2016), BayesianMinimization, Wolfram Language function, https://reference.wolfram.com/language/ref/BayesianMinimization.html (updated 2017).
✖
Wolfram Research (2016), BayesianMinimization, Wolfram Language function, https://reference.wolfram.com/language/ref/BayesianMinimization.html (updated 2017).

Text

Wolfram Research (2016), BayesianMinimization, Wolfram Language function, https://reference.wolfram.com/language/ref/BayesianMinimization.html (updated 2017).

✖
Wolfram Research (2016), BayesianMinimization, Wolfram Language function, https://reference.wolfram.com/language/ref/BayesianMinimization.html (updated 2017).

CMS

Wolfram Language. 2016. "BayesianMinimization." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/BayesianMinimization.html.

✖
Wolfram Language. 2016. "BayesianMinimization." Wolfram Language & System Documentation Center. Wolfram Research. Last Modified 2017. https://reference.wolfram.com/language/ref/BayesianMinimization.html.

APA

Wolfram Language. (2016). BayesianMinimization. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BayesianMinimization.html

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Wolfram Language. (2016). BayesianMinimization. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/BayesianMinimization.html

BibTeX

@misc{reference.wolfram_2025_bayesianminimization, author="Wolfram Research", title="{BayesianMinimization}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/BayesianMinimization.html}", note=[Accessed: 23-May-2025 ]}

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@misc{reference.wolfram_2025_bayesianminimization, author="Wolfram Research", title="{BayesianMinimization}", year="2017", howpublished="\url{https://reference.wolfram.com/language/ref/BayesianMinimization.html}", note=[Accessed: 23-May-2025 ]}

BibLaTeX

@online{reference.wolfram_2025_bayesianminimization, organization={Wolfram Research}, title={BayesianMinimization}, year={2017}, url={https://reference.wolfram.com/language/ref/BayesianMinimization.html}, note=[Accessed: 23-May-2025 ]}

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@online{reference.wolfram_2025_bayesianminimization, organization={Wolfram Research}, title={BayesianMinimization}, year={2017}, url={https://reference.wolfram.com/language/ref/BayesianMinimization.html}, note=[Accessed: 23-May-2025 ]}