Easy-to-use Bayesian optimization library made available for either closed-loop or user-driven (manual) optimization of either known or unknown objective functions. Drawing on PyTorch (GPyTorch), BOTorch and with proprietary extensions.

A short primer on Bayesian optimization is provided in this section.

Features

• Handles continuous, integer and categorical covariates.
• Optimization of either known or unknown functions. The allows for optimization of e.g. real-world experiments without specifically requiring a model of the system be defined a priori.
• Simple interface with focus on ease of use: only few lines of code required for full Bayesian optimization.
• Erroneous observations of either covariates or response can be overwritten during optimization.
• Well-documented code with detailed end-to-end examples of use, see examples.
• Optimization can start from scratch or repurpose existing data.

Design decisions

• Multivariate covariates, univariate system response: It is assumed that input covariates (the independent variables) can be either multivariate or univariate, while the system response (the dependent variable) is only univariate.
• Optimizing across continuous, integer and categorical covariates: Problems can depend on any of these types of variables, in any combination. Special attention is given to implementation of integer and categorical variables which are handled via the method of Garrido-Merchán and Hernandéz-Lobato (E.C. Garrido-Merchán and D. Hernandéz-Lobato, Neurocomputing, see References).
• System-generated or manual input: Observations of covariates and responses during optimization can be provided both programmatically or manually via prompt input.
• Optimizes known and unknown response functions: Both cases where the response function can be formulated mathematically and cases where the response can only be measured (e.g. a real-life experiment) can be optimized.
• Observed covariates can vary from the proposed covariates: The optimization routine at each iteration proposes new covariate data points to investigate, but there is no requirement that this is also the observed data point. At each iteration step, proposed covariates, observed covariates and observed response are 3 separate entities. That that noisy or unexpected measurement points will be fully useful (no introduce any errors), even if they vary a lot from the proposed covariate data points.
• Data stored in class instance: Data for proposed covariate data points, observed covariates and observed responses is stored in the instantiated class object.
• Data format and type validation: Input data is validated at each iteration.
• Observations of covariates and response can be overridden during execution: If an observation of either covariates or response seems incorrect, the framework allows overriding the previous observation.
• Consistency in number of covariates and observations: It is assumed that there is consistency in the number of observations of covariates and responses: at each step a new covariate data point is proposed, before observations of covariates and response for this iteration are reported (specifically the number of proposed data points cannot exceed the number of observed covariates by more than 1, and the number of observed covariates also cannot exceed the number of observed responses by more than 1). If additional data is provided for either observed covariates or observed response, this will override the last provided data.

Installation

Via pip

The library is available on https://pypi.org/, so to install simply run

pip install greattunes

From source

You can also download the library source code and install it from there.

First install torch dependencies

Installing torch dependencies is not always a requirement. Unfortunately, in some cases torch-libraries have to be installed outside normal bulk pip install -r requirements.txt. First try to install directly via steps 1-3 in Install library below, and only install torch libraries manually if direct installation fails.

To find the right installation command for torch, use this link to determine the details and add as a separate command in the github actions yaml. As an example, the following is the install command on my local system (an Ubuntu-based system with pip and without CUDA access)

pip install torch==1.6.0+cpu torchvision==0.7.0+cpu -f https://download.pytorch.org/whl/torch_stable.html

Install library

Currently the code is not available on any repo servers except the private GitHub account. The best way to install the code (after adding torch and torchvision) is follow this series of steps.

1. Upgrade local versions of packaging libraries

pip install --upgrade setuptools wheel

2. Clone this repo
3. Do local installation

python -m pip install https://github.com/svedel/greattunes/

Step 3 will install by running greattunes/setup.py locally and installing. This step can also be broken into two, which might improve debugging

python3 https://github.com/svedel/greattunes/ setup.py bdist_wheel
python -m pip install https://github.com/svedel/greattunes/dist/greattunes-<version>-py3-none-any.whl

where <version> is the latest version in normal python format of MAJOR.MINOR[.MICRO] (check /dist-folder to see which one to pick).

Using the framework

All capabilities of the framework are described below.

For readers wanting to skip directly to working with the framework, a number of examples of how to use the framework end-to-end are included as Jupyter notebooks in examples.

Solving a problem

Solving an optimization problem consists of two steps in this framework:

1. Define the input variables (covariates), the surrogate model type and the acquisition function. Also define the response function if this is known
2. Optimize based on closed-loop or iterative interface

Here's a simple illustration of how to do this for a known function f.

Step 1: Define the problem

The critical things to define in this step are

• The number of covariates. Upper and lower limits must be provided for each covariate to constraint the search space, and initial guess for each to be provided as well. Works for both univariate and multivariate covariate structures.
• The type of surrogate model. The model will be fitted at each step of the optimization.
• The type of acquisitition function. This will also be fitted at each step of the optimization.

# import library
from greattunes import TuneSession

# === Step 1: define the input ===

# specify covariate. For each covariate of the model provide best guess of starting point together with upper and lower
# limit 
x_start = 0.5  # initial guess
x_min = 0  # lower limit
x_max = 1  # upper limit
covars = [(x_start, x_min, x_max)]

# initialize the class
cls = TuneSession(covars=covars, model="SingleTaskGP", acq_func="ExpectedImprovement")

Step 2: Solve the problem

In order to optimize, we must first describe which function we want to do this for. The framework works both when this function can be formulated mathematically and when it can only be sampled (e.g. through examples) but cannot be formulated. For an illustrate of the latter see Example 2 under examples.

Here we will work with a known objective function to optimize

# === Step 2: solve the problem ===

# univariate function to optimize
import numpy as np

def f(x):
    return -(6 * x - 2) ** 2 * np.sin(12 * x - 4)

Beware that the number of covariates (including their range) specified by covars under Step 1 must comply with the functional dependence of the objective function (x in the case above).

We are now ready to solve the problem. We will run for max_iter=20 iterations.

# run the auto-method
cls.auto(response_samp_func=f, max_iter=max_iter)

Had we worked with an objective function f which could not be formulated explicitly, the right entrypoint would have been to use the .ask-.tell methods instead of .auto.

Key attributes and methods

User-facing attributes/methods: easily-accessible covariates and response

The following key attributes are stored for each optimization as part of the instantiated class. These primary data structures for users are stored in pandas dataframes in pretty format. current_best and best_predicted are methods which print their output to the prompt.

Attribute/method	Comments
x_data	All observed covariates with dimensions, one row per observation. If no names have been added to the covariates they will take the naems "covar0", "covar1", ... . Dimensions num_observations X num_covariates.
y_data	All observed responses corresponding to the covariate points (rows) in x_data. Dimensions num_observations X 1.
best_response	Best observed response value during optimization run, including current iteration. Dimensions num_observations X 1.
covars_best_response	Observed covariates for best response value during optimization run, i.e. each row in covars_best_response generated the same row in best_response. Dimensions num_observations X num_covariates.
current_best()	Returns the best observed response of the objective up to the current iteration.
best_predicted()	Best predicted response from the surrogate model. Calculates for the mean model as well as for the lower confidence region (e.g. mean minus one standard deviation) of the full model. For both cases also returns the covariates resulting in the maximum.

Backend attributes

In the backend the framework makes use of different data structures based on the tensor structure from torch which also handles one-hot encoding of categorical variables. The key backend attributes are listed in the table below.

Attribute	Comments
train_X	All observed covariates with dimensions num_observations X num_covariates. Backend equivalent to x_data.
proposed_X	All proposed covariate datapoints to investigate, with dimensions num_observations X num_covariates.
train_Y	All observed responses corresponding to the covariate points in train_X. Dimensions num_observations X 1. Backend equivalent to y_data.
best_response_value	Best observed response value during optimization run, including current iteration. Dimensions num_observations X 1. Backend equivalent to best_response.
covars_best_response_value	Observed covariates for best response value during optimization run, i.e. each row in covars_best_response_value generated the same row in best_response_value. Dimensions num_observations X num_covariates. Backend equivalent to covars_best_response.

Covariates: the free parameters which are adjusted by the framework during optimization

The user must detail which covariates the framework can adjust in order to optimize (maximize/minimize) the response. This is a mandatory part of class initialization and set via covars input variable; without any knowledge of the covariates, the framework cannot proceed to optimization. Here's an example for a problem with two covariates

covars = [(0.5, 0, 1), (2,1,4)]  # each tuple defines one covariate; the tuple entries are (initial guess, min, max)

# initialize the class
cls = TuneSession(covars=covars, ...)

This is also illustrated for a single-variable situation in Step 1: Define the problem above.

Supported types: Handling continuous, integer and categorical covariates

The following three types of covariates are supported.

• Continuous: Variables which can take any numerical value, i.e. can take values which include decimals. The data type of a continuous variable will be among float types. Typical examples of continuous covariates will be weights in a model and time thresholds (imagine a case where total runtime was a parameter).
• Integer: Variables which can only take integer values; the data types of these variables will be among int types. Special consideration must be taken during optimization because these variables only can update in discrete steps, resulting in step changes of the response. Examples of integer covariates include number of layers in a neural network and number of eggs in a recipe.
• Categorical: Variables that can take different discrete values, which, contrary to integers do not even have any internal relation in terms of size. An example is a variable which can take the values {green,blue,red} where there clearly is no direct numerical relationship between the potential values; in contrast, a numerical relationship does exist for integer variables (e.g. 5 is bigger than 2). In addition to the color example above, another example of a categorical variable can be one which determines the make of a car (e.g. take values volvo, lincoln, fiat etc)

The framework follows the method of Garrido-Merchán and Hernandéz-Lobato (see References) to integrate the different types of covariates and bring them to a form that is consistent with using continuous Gaussian processes to drive the optimization. Briefly, the method relies on adding a transformation of variables in the correlation (kernel) function of the Gaussian processes with the following properties: integer covariates are rounded to nearest integer and categorical variables are one-hot encoded and only the one with highest numerical value is carried forward in each round by adjusting the value of its associated one-hot encoded variable to 1 and setting all other one-hot encoded variables to 0.

Two approaches to defining covariates in framework: working with named covariates and setting data types

Two ways are offered to provide covariate details to the framework: the simple way which assigns names to covariates and infers their data types from the provided data in covars (used so far), and an elaborate way which allows for naming covariates and gives more control to specify data types. In either case, the information is given to the framework via the covars input variable.

Simple approach: faster, but no control over covariate names and data types

Each covariate is defined by a tuple, and the order of the tuples defines the order of the covariates. The same order must be used later if covariates are manually reported via the .tell-method.

Covariate data types

Covariate data type is critical because it impacts how to handle the covariate during the optimization. In this simple approach, data types are inferred from the provided data in covars as indicated by the table below.

Data type	How report	Example	Comments
Integer	(<initial_guess>,<parameter_minimum>, <parameter_maximum>)	(2, 0, 5)	All tuple entries must be of data type int for covariate to be taken as integer
Continuous	(<initial_guess>,<parameter_minimum>, <parameter_maximum>)	(2.0, -1.2, 2.5)	Only one tuple entry has to be a float for the covariate to be set to continuous
Categorical	(<initial_guess>,<option_1>, <option_2>, ...)	(volvo, fiat, aston martin, ford, toyota)	Covariate is taken as categorical if any entry has data type str. There must be at least one other option than <initial_guess>, but otherwise no limit to the number of entries.

Here's an example of how to use the simple approach to define the covars-variable to communicate covariates of different data types. This covars could be used to initialize a class instantiation

covars = [
            (1, 0, 2),  # will be taken as INTEGER (type: int)
            (1.0, 0.0, 2.0),  # will be taken as CONTINUOUS (type: float)
            (1, 0, 2.0),  # will be taken as CONTINUOUS (type: float)
            ("red", "green", "blue", "yellow"),  # will be taken as CATEGORICAL (type: str)
            ("volvo", "chevrolet", "ford"),  # will be taken as CATEGORICAL (type: str)
            ("sunny", "cloudy"),  # will be taken as CATEGORICAL (type: str)
        ]

Covariate names

Covariates are assigned names behind the scenes of the type covar1, covar2 etc. with numbers added in the order in which the variable is processed from the covars list of tuples during class initialization (beware that this order may not be preserved). Covariate names are visible as the column names in the x_data attribute.

Elaborate approach: allows for specifying names and data types of covariates

This approach requires a bit more details to be provided, but also offers much more flexibility.

In this approach, all covariates are defined in a dictionary which is fed via the covars parameter, and each covariate is defined by their own dictionary nested within the outer dictionary specifying all covariates. An example, which will be elaborated further in the following, is given below for 3 covariates to make this concrete

covars = {
            'variable1':  # type: integer
                {
                    'guess': 1,
                    'min': 0,
                    'max': 2,
                    'type': int,
                },
            'variable2':  # type: continuous (float)
                {
                    'guess': 12.2,
                    'min': -3.4,
                    'max': 30.8,
                    'type': float,
                },
            'variable3':  # type: categorical (str)
                {
                    'guess': 'red',
                    'options': {'red', 'blue', 'green'},
                    'type': str,
                }
        }

Each nested dictionary gives the details of an individual covariate, and the name of these nested dictionaries are used to name the covariate.

Covariate names: Anything that's permissable as a python string is a valid covariate name. These names are used throughout the framework (will be inherited into x_data).

Specifying data type: The variable type indicates the type of the covariate. The framework uses the following types

• int: integer covariate
• float: continuous covariate
• str: categorical covariate Beware that the data type (and not a string) is used to define the type (i.e. use e.g. str not 'str' to indicate a categorical variable).

Required information for each covariate: Requirements vary with the covariate data type. The following is required for each type of covariate

• Integer ('type': int): Required fields are guess, min and max (all single entries of type: int), as well as type (must be int to specify categorical).
• Continuous ('type': float): Required fields are guess, min and max (all single entries of types int or float), as well as type (must be float to specify categorical).
• Categorical ('type': str): Required fields are guess (a single entry, type: str), options (dictionary of str, one for each option the covariate can take. Must also include the element in guess) and type (must be str to specify categorical).

The example above shows 3 covariates but the framework can handle any number of covariates. Simply adjust the number of nested dictionaries to meet the need (and use appropriate naming and covariate specification for your application).

Multivariate covariates

Multivariate covariates are set via the (mandatory) covars parameter during class initialization. Each covariate is given as a 3-tuple of parameters (<initial_guess>,<parameter_minimum>, <parameter_maximum>) (the order matters!), with covars being a list of these tuples. As an example, for a cases with 3 covariates, the covars parameter would be

covars = [(1, 0, 4.4), (5.2, 1.5, 7.0), (4, 2.2, 5.1)]

The order of the covariates matters since framework does not work with named covariates. Hence, the parameter defined by the first tuple in covars will always have to be reported as the first covariate when iterating during optimization, the second covariate will be initialized by the second tuple in covars etc.

Observations of multivariate covariates are specified as columns in the train_X attribute (format: torch.tensor), with observations added as rows. As an example, the initial guess for the three covariates defined by covars above would be

train_X = torch.tensor([[1, 5.2, 4]], dtype=torch.double)

Initialization options

Starting with historical data

If historical data for pairs of covariates and response is available for your system, this can be added during initialization. In this case the optimization framework will have a better starting position and will likely converge more quickly.

Historical data is added during class initialization. The number of observations (rows) of covariates and response must match. Historical training data is added during class instantiation via arguments train_X=<> and train_Y=<> as illustrated below for the following cases

1. Multiple observations of multivariate system
2. Single observation of univariate system
3. Single observation of multivariate system

# import
import torch
from greattunes import TuneSession

### ------ Case 1 - multiple observations (multivariate) ------ ###

# set range of data
covars = [(1, 0, 4.4), (5.2, 1.5, 7.0), (4, 2.2, 5.1)]

# define initial data
X = torch.tensor([[1, 2, 3],[3, 4.4, 5]], dtype=torch.double)
Y = torch.tensor([[33],[37.8]], dtype=torch.double)

# initialize class
cls = TuneSession(covars=covars,train_X=X, train_Y=Y)

### ------ Case 2 - single observation (univariate) ------ ###

# set range of data
covars = [(1, 0, 4.4)]

# define initial data
X = torch.tensor([[1]], dtype=torch.double)
Y = torch.tensor([[33]], dtype=torch.double)

# initialize class
cls = TuneSession(covars=covars,train_X=X, train_Y=Y)

### ------ Case 3 - single observation (multivariate) ------ ###

# set range of data
covars = [(1, 0, 4.4), (5.2, 1.5, 7.0), (4, 2.2, 5.1)]

# define initial data
X = torch.tensor([[1, 2, 3]], dtype=torch.double)
Y = torch.tensor([[33]], dtype=torch.double)

# initialize class
cls = TuneSession(covars=covars,train_X=X, train_Y=Y)

Random initialization

Starting from a few randomly sampled datapoints typically increases the convergence of the optimization because it makes it less likely that the algorithm locks onto a local maximum without consideration for an unknown global one. Furthermore, in the absence of historical data, random sampling is the best option is to start.

Random initialization is enabled via the parameter random_start during initialization and can be applied both in case historical data has been added or not (default is random_start = True).

# import
import torch
from greattunes import TuneSession

### ------ Case 1 - No historical data ------ ###

# set range of data
covars = [(1, 0, 4.4), (5.2, 1.5, 7.0), (4, 2.2, 5.1)]

# define initial data
X = torch.tensor([[1, 2, 3],[3, 4.4, 5]], dtype=torch.double)
Y = torch.tensor([[33],[37.8]], dtype=torch.double)

# initialize class
cls = TuneSession(covars=covars, random_start=True)

### ------ Case 2 - With historical data ------ ###

# set range of data
covars = [(1, 0, 4.4), (5.2, 1.5, 7.0), (4, 2.2, 5.1)]

# define initial data
X = torch.tensor([[1, 2, 3],[3, 4.4, 5]], dtype=torch.double)
Y = torch.tensor([[33],[37.8]], dtype=torch.double)

# initialize class
cls = TuneSession(covars=covars,train_X=X, train_Y=Y, random_start=True)

Parameters for random start

Number of random datapoints: The number of random datapoints to be sampled is set via the kwarg num_initial_random during initialization. This defaults to the closest integer to $\sqrt{d}$ for a problem with $d$ covariates unless a value is provided.

Sampling method: Two sampling methods are available:

• random: Fully random sampling within the whole hypercube specified by covars.
• latin_hcs: Latin hypercube sampling within the hypercube specified by covars. The sampling method is determined by the kwarg random_sampling_method during class initialization.

Improved convergence: adding randomly sampled points during optimization

Just like random initialization helps with convergence, best practice also prescribes adding randomly sampled points during the optimization run.

This is easily done within this framework. The parameter random_step_cadence determines the cadence between randomly sampled datapoints (in between points sampled via Bayesian optimization).

Kernels for Gaussian process surrogate model

The following kernels for Gaussian process surrogate model are implemented. Model type and listed parameters are provided as input to class initialization, i.e. during initialization of TuneSession

Model name	Parameters	Comments
"SingleTaskGP"	N/A	A single-task exact kernel for Gaussian process regression. Follow this link for more details.
"FixedNoiseGP"	train_Yvar	A single-task exact kernel for Gaussian process regression assuming a fixed noise level. Follow this link for more details.
"HeteroskedasticSingleTaskGP"	train_Yvar	A single-task exact kernel for Gaussian process regression using a heteroskedastic noise model. Follow this link for more details.
"SimpleCustomMaternGP"	nu	A custom Matérn kernel with parameter nu (a float). For more details on Matérn kernels see wiki page, and see the source code for the model in greattunes\custom_models.

Acquisition functions

These acquisition functions are currently available. Parameters (if any) are provided during initialization of the TuneSession class instance.

Acquisition function name	Parameter	Comments
"ExpectedImprovement"	N/A	Expected improvement acquisition function. This is the default for greattunes. For more details see here or Section 2 in this paper.
"NoisyExpectedImprovement"	num_fantasies (default: 20)	Expected improvement acquisition averaged over num_fantasies realizations of a single but noisy model. Requires that the Gaussian process model is of the type FixedNoiseGP. For more details see here.
"qExpectedImprovement"	sampler (default: botorch.sampling.SobolQMCNormalSampler)	Monte Carlo-based expected improvement function. For more details see here.
"qNoisyExpectedImprovement"	sampler (default: botorch.sampling.SobolQMCNormalSampler)	Monte Carlo-based noisy expected improvement function. For more details see here.
"PosteriorMean"	N/A	Posterior mean. Requires the surrogate (Gaussian process) model to have a mean property (all implemented models do). For more details see here.
"ProbabilityOfImprovement"	N/A	Probability of improvement over the current best observed value, computed using the analytic formula under a Normal posterior distribution. Requires the outcome to be Gaussian. For more details see here.
"qProbabilityOfImprovement"	sampler (default: botorch.sampling.SobolQMCNormalSampler)	Monte Carlo based probability of improvement method. For more details see here.
"qSimpleRegret"	sampler (default: botorch.sampling.SobolQMCNormalSampler)	Monte Carlo method for simple regret. For more details see here.
"UpperConfidenceBound"	beta (default: 0.2)	Analytic upper confidence bound that comprises of the posterior mean plus an additional term: the posterior standard deviation weighted by a trade-off parameter, beta. For more details see here.
"qUpperConfidenceBound"	beta (default: 0.2), sampler (default: botorch.sampling.SobolQMCNormalSampler)	Monte carlo based Upper Confidence Bound method. For more details see here or here.
"qKnowledgeGradient"	num_fantasies (default: 20)	Computes the Knowledge Gradient using realizations ("fantasies") for the outer expectation and either the model posterior mean or MC-sampling for the inner expectation. For a fixed number of realizations ("fantasies"), optimizes in a “one-shot” fashion. For more details see here or here.
"qMaxValueEntropy"	N/A	Uses max-value entropy search. This acquisition function computes the mutual information of max values and a candidate point. For more details see here or here.
"qMultiFidelityMaxValueEntropy"	N/A	Multi-fidelity max-value entropy search. For more details see here or here.

Closed-loop: the .auto method

Closed-loop optimization refers to situations where the function is known and therefore can iterate itself to optimality. These are addressed via the .auto method, which takes a function handle response_samp_func as well as a maximum number of iterations max_iter as input parameters. See the example above as illustration of how to use the method.

Stopping based on relative improvement in best observed response: rel_tol and rel_tol_steps

The optimization can be stopped before max_iter steps have been taken by specifying the limit on the relative improvement in best observed response value (best_response_value). This is invoked by providing the parameter rel_tol to the .auto method.

# some function to optimize
def f(x):
  ...

# parameters
max_iter = 100
rel_tol = 1e-10

# run the auto-method
cls.auto(response_samp_func=f, max_iter=max_iter, rel_tol=rel_tol)

In most cases the best results are found by requiring the rel_tol limit to be satisfied for multiple consecutive iterations. This can be achieved by also providing the number of consecutive steps required rel_tol_steps. If rel_tol_steps is not provided, the limit on relative improvement only needs to be reached once for convergence.

# some function to optimize
def f(x):
  ...

# parameters
max_iter = 100
rel_tol = 1e-10
rel_tol_steps = 5

# run the auto-method
cls.auto(response_samp_func=f, max_iter=max_iter, rel_tol=rel_tol, rel_tol_steps=rel_tol_steps)

Best practises on using rel_tol and rel_tol_steps are provided in Example 5 in examples.

Iterative: the .ask and .tell methods

The true value of Bayesian optimization is its ability to optimize problems which cannot be formulated mathematically. The mathematical method can work as long as a response can be generated, and in fact makes no assumptions on the nature of the problem (except that a maximum is present). Thus, whether the response is generated as a measurement from an experiment, the feedback from users or the output of a defined mathematical function does not matter; all can be optimized via the framework.

Optimization of unknown functions is handled by the methods .ask and .tell.

• .ask provides a best guess of the next covariate data point to sample, given the history of previously sampled points for the problem (that is, .ask provides the output of the acquisition function)
• .tell is the method to report the observed covariate data point and the associated response One call to .ask followed by a call to .tell performs one iteration of .auto from the point of view of the Bayesian optimization; the difference is only in how to interface with it. Examples 2 and 3 in examples shows how to use .ask-.tell to solve problems end-to-end.

To solve a problem, apply these problems iteratively: in each iteration start by calling .ask, then use the proposed new data point to sample the system response and provide both this value and the actually sampled covariate values (can be different from proposed values) back via .tell.

# in below, "cc" is an instantiated version of TuneSession class (identical initialization as when using .auto method) 
max_iter = 20

for i in range(max_iter):
  
    # generate candidate
    cls.ask()  # new candidate is last row in cc.proposed_X

    # sample response (beware results must be formulated as torch tensors)
    observed_covars = <from measurement or from cc.proposed_X>
    observed_response = <from measurement or from specified objective function>

    # report response
    cls.tell(covars=observed_covars, response=observed_response)

Providing input via prompt

Observations of covariates and response can be provided manually to .tell. To do so, simply call .tell without any arguments at each iteration (all book keeping will be handled on backend)

# in below, "cc" is an instantiated version of TuneSession class (identical initialization as when using .auto method) 
max_iter = 20

for i in range(max_iter):
  
    # generate candidate
    cls.ask()  # new candidate is last row in cc.proposed_X

    # report response
    cls.tell()

In this case, the user will be prompted to provide input manually. There will be 3 attempts to provide covariates (another 3 for response), and the method will stop if not successful within these attempts. Provided input data will be validated for number of variables and data type as part of these cycles.

Any of covars and response not provided as (named) parameter to .tell the user will be requested to provide via manual input in prompt. It is thus possible to get e.g. covariates automatically but manually read off response values from an instrument.

Overriding reported values of covariates or response

Observed covariates and observed responses are sometimes off. To override the latest datapoint for either, simply provide it again in the same iteration. This will automatically override the latest reported value

# in below, "cc" is an instantiated version of TuneSession class (identical initialization as when using .auto method) 
# further assumes that at least on full iteration has been taken

# define a response
def f(x):
  ...

# generate candidate
cls.ask()  # new candidate is last row in cc.proposed_X

# first result
observed_results = torch.tensor([[it.item() for it in cc.proposed_X[-1]]], dtype=torch.double)
observed_response = torch.tensor([[f(cc.proposed_X[-1]).item()]], dtype=torch.double)

# report first response
cls.tell(covars=observed_results, response=observed_response)

# second result
observed_response_second = observed_response + 1

# update response
cls.tell(covars=observed_results, response=observed_response_second)

Plotting and results presentation

Some standard plots and standard methods for presenting the results have been included.

Pre-defined plots

• plot_1d_latest(): plots the latest retrained surrogate model (mean and variance), including all sampled data points.
• plot_convergence(): plots the relative error between consecutive iterations.
• plot_best_objective(): plots the best recorded value of the objective function as a function of the number of iterations.

Result summaries

These methods print their results to the prompt.

• current_best(): returns the largest observed response value (observed in either previous or current iteration). Also returns the corresponding values of the covariates.
• best_predicted(): returns the largest response predicted from the surrogate model trained on all available data. Two values are returned: the largest mean and the largest of the lower confidence region, i.e. the largest value of the mean minus the first standard deviation (note: heteroskedacticity is allowed, so the standard deviation will vary across different covariates). Also returns the corresponding covariate values. Uses the Nelder-Mead method, a multivariate equivalent to bisection, to find maximum value of surrogate model.

Contributing

We are happy if you would like to invest time in this project! Details are given in CONTRIBUTING.md on how to get started.

Examples

A number of examples showing how to use the framework in jupyter notebooks is available in the examples folder. This includes both closed-loop and iterative usages, as well as a few real-world examples (latter to come!)

References

• E.C. Garrido-Merchán and D. Hernandéz-Lobato: Dealing with categorical and integer-valued variables in Bayesian Optimization with Gaussian processes, Neurocomputing vol. 380, 7 March 2020, pp. 20-35, ArXiv preprint

A primer on Bayesian optimization

A number of good resources are available for Bayesian optimization, so below follows only a short primer. Interested readers are referred to the references listed below for more information.

Basics of Bayesian optimization

Briefly and heuristically, Bayesian optimization works as follows.

1. Define a objective function. The goal of the optimization is to maximize this function.
2. Define a surrogate model. This is an approximation of the actual functional dependencies underlying the objective function. Because Bayesian optimization builds its own model there is no requirement that the objective function can be written as a mathematical expression.
3. Define an acquisition function. This function is applied to the surrogate model to identify the next datapoint to sample (as such, the acquisition function is actually a functional)
4. Iterate:
   • Use the acquisition function to identify the next data point to sample.
   • Observe the response of the objective function at the proposed point
   • Based on all observed covariates and responses of the objective function, update the surrogate model via Bayes theorem and repeat.

Surrogate models

A typical choice of surrogate model class is the Gaussian process, but this is not a strict requirement. Examples exist in which both random forest and various types of neural networks have been used.

Formally, Bayesian optimization considers the function to be optimized as unknown and instead places a Bayesian prior distribution over it. This is the initial surrogate model. Upon observing the response, the prior model is updated to obtain the posterior distribution of functions.

The benefit of Gaussian process models is their explicit modeling of the uncertainty and ease of obtaining the posterior.

Acquisition functions

Acquisition functions (functionals) propose the best point to sample for a particular problem, given the prior distribution of the surrogate model.

A number of different functions exist, with some typical ones provided in Peter Frazier's Tutorial on Bayesian Optimization. They typically balance exploration and exploitation in different ways.

References

A list of Bayesian optimization references for later use

• Wikipedia entry on Bayesian optimization
• BoTorch introduction to Bayesian optimization
• borealis.ai
• bayesopt, SigOpt page
• Towards Data Science
• Gaussian processes for dummies
• Peter Frazier, Cornell, Bayesian Optimization expert
• Tutorial on Bayesian Optimization
• Bayesian Optimization, Martin Krasser's blog
• Bayesian Optimization with inequality constraints
• Bayesian deep learning