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# skopt module

Scikit-Optimize, or skopt, is a simple and efficient library to minimize (very) expensive and noisy black-box functions. It implements several methods for sequential model-based optimization. skopt is reusable in many contexts and accessible.

## Install

pip install scikit-optimize


## Getting started

Find the minimum of the noisy function f(x) over the range -2 < x < 2 with skopt:

import numpy as np
from skopt import gp_minimize

def f(x):
return (np.sin(5 * x[0]) * (1 - np.tanh(x[0] ** 2)) *
np.random.randn() * 0.1)

res = gp_minimize(f, [(-2.0, 2.0)])


For more read our introduction to bayesian optimization and the other examples.

## Development

The library is still experimental and under heavy development.

The development version can be installed through:

git clone https://github.com/scikit-optimize/scikit-optimize.git
cd scikit-optimize
pip install -r requirements.txt
python setup.py develop


Run the tests by executing pytest in the top level directory.

"""
Scikit-Optimize, or skopt, is a simple and efficient library to
minimize (very) expensive and noisy black-box functions. It implements
several methods for sequential model-based optimization. skopt is reusable
in many contexts and accessible.

[![Build Status](https://travis-ci.org/scikit-optimize/scikit-optimize.svg?branch=master)](https://travis-ci.org/scikit-optimize/scikit-optimize)

## Install


pip install scikit-optimize


## Getting started

Find the minimum of the noisy function f(x) over the range -2 < x < 2
with skopt:

python
import numpy as np
from skopt import gp_minimize

def f(x):
return (np.sin(5 * x[0]) * (1 - np.tanh(x[0] ** 2)) *
np.random.randn() * 0.1)

res = gp_minimize(f, [(-2.0, 2.0)])


For more read our [introduction to bayesian optimization](https://scikit-optimize.github.io/notebooks/bayesian-optimization.html)
and the other [examples](https://github.com/scikit-optimize/scikit-optimize/tree/master/examples).

## Development

The library is still experimental and under heavy development.

The development version can be installed through:

git clone https://github.com/scikit-optimize/scikit-optimize.git
cd scikit-optimize
pip install -r requirements.txt
python setup.py develop

Run the tests by executing pytest in the top level directory.
"""

from . import acquisition
from . import benchmarks
from . import callbacks
from . import learning
from . import optimizer
from . import plots
from . import space
from .optimizer import dummy_minimize
from .optimizer import forest_minimize
from .optimizer import gbrt_minimize
from .optimizer import gp_minimize
from .optimizer import Optimizer
from .utils import dump
from .utils import expected_minimum

__version__ = "0.3"

__all__ = (
"acquisition",
"benchmarks",
"callbacks",
"learning",
"optimizer",
"plots",
"space",
"gp_minimize",
"dummy_minimize",
"forest_minimize",
"gbrt_minimize",
"Optimizer",
"dump",
"expected_minimum"
)


## Functions

def dummy_minimize(

func, dimensions, n_calls=100, x0=None, y0=None, random_state=None, verbose=False, callback=None)

Random search by uniform sampling within the given bounds.

## Parameters

• func [callable]: Function to minimize. Should take a array of parameters and return the function values.

• dimensions [list, shape=(n_dims,)]: List of search space dimensions. Each search dimension can be defined either as

• a (upper_bound, lower_bound) tuple (for Real or Integer dimensions),
• a (upper_bound, lower_bound, prior) tuple (for Real dimensions),
• as a list of categories (for Categorical dimensions), or
• an instance of a Dimension object (Real, Integer or Categorical).
• n_calls [int, default=100]: Number of calls to func to find the minimum.

• x0 [list, list of lists or None]: Initial input points.

• If it is a list of lists, use it as a list of input points.
• If it is a list, use it as a single initial input point.
• If it is None, no initial input points are used.
• y0 [list, scalar or None]: Evaluation of initial input points.

• If it is a list, then it corresponds to evaluations of the function at each element of x0 : the i-th element of y0 corresponds to the function evaluated at the i-th element of x0.
• If it is a scalar, then it corresponds to the evaluation of the function at x0.
• If it is None and x0 is provided, then the function is evaluated at each element of x0.
• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

• verbose [boolean, default=False]: Control the verbosity. It is advised to set the verbosity to True for long optimization runs.

• callback [callable, list of callables, optional] If callable then callback(res) is called after each call to func. If list of callables, then each callable in the list is called.

## Returns

• res [OptimizeResult, scipy object]: The optimization result returned as a OptimizeResult object. Important attributes are:

• x [list]: location of the minimum.
• fun [float]: function value at the minimum.
• x_iters [list of lists]: location of function evaluation for each iteration.
• func_vals [array]: function value for each iteration.
• space [Space]: the optimisation space.
• specs [dict]: the call specifications.
• rng [RandomState instance]: State of the random state at the end of minimization.

For more details related to the OptimizeResult object, refer http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html

def dummy_minimize(func, dimensions, n_calls=100, x0=None, y0=None,
random_state=None, verbose=False, callback=None):
"""Random search by uniform sampling within the given bounds.

Parameters
----------
* func [callable]:
Function to minimize. Should take a array of parameters and
return the function values.

* dimensions [list, shape=(n_dims,)]:
List of search space dimensions.
Each search dimension can be defined either as

- a (upper_bound, lower_bound) tuple (for Real or Integer
dimensions),
- a (upper_bound, lower_bound, prior) tuple (for Real
dimensions),
- as a list of categories (for Categorical dimensions), or
- an instance of a Dimension object (Real, Integer or
Categorical).

* n_calls [int, default=100]:
Number of calls to func to find the minimum.

* x0 [list, list of lists or None]:
Initial input points.

- If it is a list of lists, use it as a list of input points.
- If it is a list, use it as a single initial input point.
- If it is None, no initial input points are used.

* y0 [list, scalar or None]:
Evaluation of initial input points.

- If it is a list, then it corresponds to evaluations of the function
at each element of x0 : the i-th element of y0 corresponds
to the function evaluated at the i-th element of x0.
- If it is a scalar, then it corresponds to the evaluation of the
function at x0.
- If it is None and x0 is provided, then the function is evaluated
at each element of x0.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

* verbose [boolean, default=False]:
Control the verbosity. It is advised to set the verbosity to True
for long optimization runs.

* callback [callable, list of callables, optional]
If callable then callback(res) is called after each call to func.
If list of callables, then each callable in the list is called.

Returns
-------
* res [OptimizeResult, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:

- x [list]: location of the minimum.
- fun [float]: function value at the minimum.
- x_iters [list of lists]: location of function evaluation for each
iteration.
- func_vals [array]: function value for each iteration.
- space [Space]: the optimisation space.
- specs [dict]: the call specifications.
- rng [RandomState instance]: State of the random state
at the end of minimization.

For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
# Save call args
specs = {"args": copy.copy(inspect.currentframe().f_locals),
"function": inspect.currentframe().f_code.co_name}

# Check params
rng = check_random_state(random_state)
space = Space(dimensions)

if x0 is None:
x0 = []
elif not isinstance(x0[0], list):
x0 = [x0]

if not isinstance(x0, list):
raise ValueError("x0 should be a list, got %s" % type(x0))

n_init_func_calls = 0
if len(x0) > 0 and y0 is not None:
if isinstance(y0, Iterable):
y0 = list(y0)
elif isinstance(y0, numbers.Number):
y0 = [y0]
else:
raise ValueError("y0 should be an iterable or a scalar, got %s"
% type(y0))
if len(x0) != len(y0):
raise ValueError("x0 and y0 should have the same length")

if not all(map(np.isscalar, y0)):
raise ValueError("y0 elements should be scalars")

elif len(x0) > 0 and y0 is None:
y0 = []
n_calls -= len(x0)
n_init_func_calls = len(x0)

elif len(x0) == 0 and y0 is not None:
raise ValueError("x0cannot be None when y0 is provided")

else:  # len(x0) == 0 and y0 is None
y0 = []

callbacks = check_callback(callback)
if verbose:
callbacks.append(VerboseCallback(
n_init=n_init_func_calls, n_total=n_calls))

X = x0
y = y0

# Random search
X = X + space.rvs(n_samples=n_calls, random_state=rng)
first = True
result = None

for i in range(len(y0), len(X)):
y_i = func(X[i])

if first:
first = False
if not np.isscalar(y_i):
raise ValueError("func should return a scalar")

y.append(y_i)
result = create_result(X[:i + 1], y, space, rng, specs)
if eval_callbacks(callbacks, result):
break

y = np.array(y)
return create_result(X, y, space, rng, specs)


def dump(

res, filename, store_objective=True, **kwargs)

Store an skopt optimization result into a file.

## Parameters

• res [OptimizeResult, scipy object]: Optimization result object to be stored.

• filename [string or pathlib.Path]: The path of the file in which it is to be stored. The compression method corresponding to one of the supported filename extensions ('.z', '.gz', '.bz2', '.xz' or '.lzma') will be used automatically.

• store_objective [boolean, default=True]: Whether the objective function should be stored. Set store_objective to False if your objective function (.specs['args']['func']) is unserializable (i.e. if an exception is raised when trying to serialize the optimization result).

Notice that if store_objective is set to False, a deep copy of the optimization result is created, potentially leading to performance problems if res is very large. If the objective function is not critical, one can delete it before calling skopt.dump() and thus avoid deep copying of res.

• **kwargs [other keyword arguments]: All other keyword arguments will be passed to joblib.dump.

def dump(res, filename, store_objective=True, **kwargs):
"""
Store an skopt optimization result into a file.

Parameters
----------
* res [OptimizeResult, scipy object]:
Optimization result object to be stored.

* filename [string or pathlib.Path]:
The path of the file in which it is to be stored. The compression
method corresponding to one of the supported filename extensions ('.z',
'.gz', '.bz2', '.xz' or '.lzma') will be used automatically.

* store_objective [boolean, default=True]:
Whether the objective function should be stored. Set store_objective
to False if your objective function (.specs['args']['func']) is
unserializable (i.e. if an exception is raised when trying to serialize
the optimization result).

Notice that if store_objective is set to False, a deep copy of the
optimization result is created, potentially leading to performance
problems if res is very large. If the objective function is not
critical, one can delete it before calling skopt.dump() and thus
avoid deep copying of res.

* **kwargs [other keyword arguments]:
All other keyword arguments will be passed to joblib.dump.
"""
if store_objective:
dump_(res, filename, **kwargs)

elif 'func' in res.specs['args']:
# If the user does not want to store the objective and it is indeed
# present in the provided object, then create a deep copy of it and
# remove the objective function before dumping it with joblib.dump.
res_without_func = deepcopy(res)
del res_without_func.specs['args']['func']
dump_(res_without_func, filename, **kwargs)

else:
# If the user does not want to store the objective and it is already
# missing in the provided object, dump it without copying.
dump_(res, filename, **kwargs)


def expected_minimum(

res, n_random_starts=20, random_state=None)

Compute the minimum over the predictions of the last surrogate model.

Note that the returned minimum may not necessarily be an accurate prediction of the minimum of the true objective function.

## Parameters

• res [OptimizeResult, scipy object]: The optimization result returned by a skopt minimizer.

• n_random_starts [int, default=20]: The number of random starts for the minimization of the surrogate model.

• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

## Returns

• x [list]: location of the minimum.

• fun [float]: the surrogate function value at the minimum.

def expected_minimum(res, n_random_starts=20, random_state=None):
"""
Compute the minimum over the predictions of the last surrogate model.

Note that the returned minimum may not necessarily be an accurate
prediction of the minimum of the true objective function.

Parameters
----------
* res  [OptimizeResult, scipy object]:
The optimization result returned by a skopt minimizer.

* n_random_starts [int, default=20]:
The number of random starts for the minimization of the surrogate
model.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

Returns
-------
* x [list]: location of the minimum.

* fun [float]: the surrogate function value at the minimum.
"""
def func(x):
reg = res.models[-1]
return reg.predict(x.reshape(1, -1))[0]

xs = [res.x]
if n_random_starts > 0:
xs.extend(res.space.rvs(n_random_starts, random_state=random_state))

best_x = None
best_fun = np.inf

for x0 in xs:
r = sp_minimize(func, x0=x0, bounds=res.space.bounds)

if r.fun < best_fun:
best_x = r.x
best_fun = r.fun

return [v for v in best_x], best_fun


def forest_minimize(

func, dimensions, base_estimator='ET', n_calls=100, n_random_starts=10, acq_func='EI', x0=None, y0=None, random_state=None, verbose=False, callback=None, n_points=10000, xi=0.01, kappa=1.96, n_jobs=1)

Sequential optimisation using decision trees.

A tree based regression model is used to model the expensive to evaluate function func. The model is improved by sequentially evaluating the expensive function at the next best point. Thereby finding the minimum of func with as few evaluations as possible.

The total number of evaluations, n_calls, are performed like the following. If x0 is provided but not y0, then the elements of x0 are first evaluated, followed by n_random_starts evaluations. Finally, n_calls - len(x0) - n_random_starts evaluations are made guided by the surrogate model. If x0 and y0 are both provided then n_random_starts evaluations are first made then n_calls - n_random_starts subsequent evaluations are made guided by the surrogate model.

## Parameters

• func [callable]: Function to minimize. Should take a array of parameters and return the function values.

• dimensions [list, shape=(n_dims,)]: List of search space dimensions. Each search dimension can be defined either as

• a (upper_bound, lower_bound) tuple (for Real or Integer dimensions),
• a (upper_bound, lower_bound, prior) tuple (for Real dimensions),
• as a list of categories (for Categorical dimensions), or
• an instance of a Dimension object (Real, Integer or Categorical).

NOTE: The upper and lower bounds are inclusive for Integer dimensions.

• base_estimator [string or Regressor, default="ET"]: The regressor to use as surrogate model. Can be either

• "RF" for random forest regressor
• "ET" for extra trees regressor
• instance of regressor with support for return_std in its predict method

The predefined models are initilized with good defaults. If you want to adjust the model parameters pass your own instance of a regressor which returns the mean and standard deviation when making predictions.

• n_calls [int, default=100]: Number of calls to func.

• n_random_starts [int, default=10]: Number of evaluations of func with random points before approximating it with base_estimator.

• acq_func [string, default="LCB"]: Function to minimize over the forest posterior. Can be either

• "LCB" for lower confidence bound.
• "EI" for negative expected improvement.
• "PI" for negative probability of improvement.
• x0 [list, list of lists or None]: Initial input points.

• If it is a list of lists, use it as a list of input points.
• If it is a list, use it as a single initial input point.
• If it is None, no initial input points are used.
• y0 [list, scalar or None]: Evaluation of initial input points.

• If it is a list, then it corresponds to evaluations of the function at each element of x0 : the i-th element of y0 corresponds to the function evaluated at the i-th element of x0.
• If it is a scalar, then it corresponds to the evaluation of the function at x0.
• If it is None and x0 is provided, then the function is evaluated at each element of x0.
• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

• verbose [boolean, default=False]: Control the verbosity. It is advised to set the verbosity to True for long optimization runs.

• callback [callable, optional] If provided, then callback(res) is called after call to func.

• n_points [int, default=10000]: Number of points to sample when minimizing the acquisition function.

• xi [float, default=0.01]: Controls how much improvement one wants over the previous best values. Used when the acquisition is either "EI" or "PI".

• kappa [float, default=1.96]: Controls how much of the variance in the predicted values should be taken into account. If set to be very high, then we are favouring exploration over exploitation and vice versa. Used when the acquisition is "LCB".

• n_jobs [int, default=1]: The number of jobs to run in parallel for fit and predict. If -1, then the number of jobs is set to the number of cores.

## Returns

• res [OptimizeResult, scipy object]: The optimization result returned as a OptimizeResult object. Important attributes are:

• x [list]: location of the minimum.
• fun [float]: function value at the minimum.
• models: surrogate models used for each iteration.
• x_iters [list of lists]: location of function evaluation for each iteration.
• func_vals [array]: function value for each iteration.
• space [Space]: the optimization space.
• specs [dict]: the call specifications.

For more details related to the OptimizeResult object, refer http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html

def forest_minimize(func, dimensions, base_estimator="ET", n_calls=100,
n_random_starts=10, acq_func="EI",
x0=None, y0=None, random_state=None, verbose=False,
callback=None, n_points=10000, xi=0.01, kappa=1.96,
n_jobs=1):
"""Sequential optimisation using decision trees.

A tree based regression model is used to model the expensive to evaluate
function func. The model is improved by sequentially evaluating
the expensive function at the next best point. Thereby finding the
minimum of func with as few evaluations as possible.

The total number of evaluations, n_calls, are performed like the
following. If x0 is provided but not y0, then the elements of x0
are first evaluated, followed by n_random_starts evaluations.
Finally, n_calls - len(x0) - n_random_starts evaluations are
made guided by the surrogate model. If x0 and y0 are both
provided then n_random_starts evaluations are first made then
n_calls - n_random_starts subsequent evaluations are made
guided by the surrogate model.

Parameters
----------
* func [callable]:
Function to minimize. Should take a array of parameters and
return the function values.

* dimensions [list, shape=(n_dims,)]:
List of search space dimensions.
Each search dimension can be defined either as

- a (upper_bound, lower_bound) tuple (for Real or Integer
dimensions),
- a (upper_bound, lower_bound, prior) tuple (for Real
dimensions),
- as a list of categories (for Categorical dimensions), or
- an instance of a Dimension object (Real, Integer or
Categorical).

NOTE: The upper and lower bounds are inclusive for Integer
dimensions.

* base_estimator [string or Regressor, default="ET"]:
The regressor to use as surrogate model. Can be either

- "RF" for random forest regressor
- "ET" for extra trees regressor
- instance of regressor with support for return_std in its predict
method

The predefined models are initilized with good defaults. If you
a regressor which returns the mean and standard deviation when
making predictions.

* n_calls [int, default=100]:
Number of calls to func.

* n_random_starts [int, default=10]:
Number of evaluations of func with random points before
approximating it with base_estimator.

* acq_func [string, default="LCB"]:
Function to minimize over the forest posterior. Can be either

- "LCB" for lower confidence bound.
- "EI" for negative expected improvement.
- "PI" for negative probability of improvement.

* x0 [list, list of lists or None]:
Initial input points.

- If it is a list of lists, use it as a list of input points.
- If it is a list, use it as a single initial input point.
- If it is None, no initial input points are used.

* y0 [list, scalar or None]:
Evaluation of initial input points.

- If it is a list, then it corresponds to evaluations of the function
at each element of x0 : the i-th element of y0 corresponds
to the function evaluated at the i-th element of x0.
- If it is a scalar, then it corresponds to the evaluation of the
function at x0.
- If it is None and x0 is provided, then the function is evaluated
at each element of x0.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

* verbose [boolean, default=False]:
Control the verbosity. It is advised to set the verbosity to True
for long optimization runs.

* callback [callable, optional]
If provided, then callback(res) is called after call to func.

* n_points [int, default=10000]:
Number of points to sample when minimizing the acquisition function.

* xi [float, default=0.01]:
Controls how much improvement one wants over the previous best
values. Used when the acquisition is either "EI" or "PI".

* kappa [float, default=1.96]:
Controls how much of the variance in the predicted values should be
taken into account. If set to be very high, then we are favouring
exploration over exploitation and vice versa.
Used when the acquisition is "LCB".

* n_jobs [int, default=1]:
The number of jobs to run in parallel for fit and predict.
If -1, then the number of jobs is set to the number of cores.

Returns
-------
* res [OptimizeResult, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:

- x [list]: location of the minimum.
- fun [float]: function value at the minimum.
- models: surrogate models used for each iteration.
- x_iters [list of lists]: location of function evaluation for each
iteration.
- func_vals [array]: function value for each iteration.
- space [Space]: the optimization space.
- specs [dict]: the call specifications.

For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
rng = check_random_state(random_state)

# Default estimator
if isinstance(base_estimator, str):
if base_estimator not in ("RF", "ET"):
raise ValueError(
"Valid strings for the base_estimator parameter"
" are: 'RF' or 'ET', not '%s'" % base_estimator)

if base_estimator == "RF":
base_estimator = RandomForestRegressor(n_estimators=100,
min_samples_leaf=3,
n_jobs=n_jobs,
random_state=rng)

elif base_estimator == "ET":
base_estimator = ExtraTreesRegressor(n_estimators=100,
min_samples_leaf=3,
n_jobs=n_jobs,
random_state=rng)

return base_minimize(func, dimensions, base_estimator,
n_calls=n_calls, n_points=n_points,
n_random_starts=n_random_starts,
x0=x0, y0=y0, random_state=random_state,
acq_func=acq_func,
xi=xi, kappa=kappa, verbose=verbose,
callback=callback, acq_optimizer="sampling")


def gbrt_minimize(

func, dimensions, base_estimator=None, n_calls=100, n_random_starts=10, acq_func='EI', acq_optimizer='auto', x0=None, y0=None, random_state=None, verbose=False, callback=None, n_points=10000, xi=0.01, kappa=1.96, n_jobs=1)

Sequential optimization using gradient boosted trees.

Gradient boosted regression trees are used to model the (very) expensive to evaluate function func. The model is improved by sequentially evaluating the expensive function at the next best point. Thereby finding the minimum of func with as few evaluations as possible.

The total number of evaluations, n_calls, are performed like the following. If x0 is provided but not y0, then the elements of x0 are first evaluated, followed by n_random_starts evaluations. Finally, n_calls - len(x0) - n_random_starts evaluations are made guided by the surrogate model. If x0 and y0 are both provided then n_random_starts evaluations are first made then n_calls - n_random_starts subsequent evaluations are made guided by the surrogate model.

## Parameters

• func [callable]: Function to minimize. Should take a array of parameters and return the function values.

• dimensions [list, shape=(n_dims,)]: List of search space dimensions. Each search dimension can be defined either as

• a (upper_bound, lower_bound) tuple (for Real or Integer dimensions),
• a (upper_bound, lower_bound, "prior") tuple (for Real dimensions),
• as a list of categories (for Categorical dimensions), or
• an instance of a Dimension object (Real, Integer or Categorical).
• base_estimator [GradientBoostingQuantileRegressor]: The regressor to use as surrogate model

• n_calls [int, default=100]: Number of calls to func.

• n_random_starts [int, default=10]: Number of evaluations of func with random points before approximating it with base_estimator.

• acq_func [string, default="LCB"]: Function to minimize over the forest posterior. Can be either

• "LCB" for lower confidence bound.
• "EI" for negative expected improvement.
• "PI" for negative probability of improvement.
• x0 [list, list of lists or None]: Initial input points.

• If it is a list of lists, use it as a list of input points.
• If it is a list, use it as a single initial input point.
• If it is None, no initial input points are used.
• y0 [list, scalar or None]: Evaluation of initial input points.

• If it is a list, then it corresponds to evaluations of the function at each element of x0 : the i-th element of y0 corresponds to the function evaluated at the i-th element of x0.
• If it is a scalar, then it corresponds to the evaluation of the function at x0.
• If it is None and x0 is provided, then the function is evaluated at each element of x0.
• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

• verbose [boolean, default=False]: Control the verbosity. It is advised to set the verbosity to True for long optimization runs.

• callback [callable, optional] If provided, then callback(res) is called after call to func.

• n_points [int, default=10000]: Number of points to sample when minimizing the acquisition function.

• xi [float, default=0.01]: Controls how much improvement one wants over the previous best values. Used when the acquisition is either "EI" or "PI".

• kappa [float, default=1.96]: Controls how much of the variance in the predicted values should be taken into account. If set to be very high, then we are favouring exploration over exploitation and vice versa. Used when the acquisition is "LCB".

• n_jobs [int, default=1]: The number of jobs to run in parallel for fit and predict. If -1, then the number of jobs is set to the number of cores.

## Returns

• res [OptimizeResult, scipy object]: The optimization result returned as a OptimizeResult object. Important attributes are:

• x [list]: location of the minimum.
• fun [float]: function value at the minimum.
• models: surrogate models used for each iteration.
• x_iters [list of lists]: location of function evaluation for each iteration.
• func_vals [array]: function value for each iteration.
• space [Space]: the optimization space.
• specs [dict]: the call specifications.
• rng [RandomState instance]: State of the random state at the end of minimization.

For more details related to the OptimizeResult object, refer http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html

def gbrt_minimize(func, dimensions, base_estimator=None,
n_calls=100, n_random_starts=10,
acq_func="EI", acq_optimizer="auto",
x0=None, y0=None, random_state=None, verbose=False,
callback=None, n_points=10000, xi=0.01, kappa=1.96,
n_jobs=1):
"""Sequential optimization using gradient boosted trees.

Gradient boosted regression trees are used to model the (very)
expensive to evaluate function func. The model is improved
by sequentially evaluating the expensive function at the next
best point. Thereby finding the minimum of func with as
few evaluations as possible.

The total number of evaluations, n_calls, are performed like the
following. If x0 is provided but not y0, then the elements of x0
are first evaluated, followed by n_random_starts evaluations.
Finally, n_calls - len(x0) - n_random_starts evaluations are
made guided by the surrogate model. If x0 and y0 are both
provided then n_random_starts evaluations are first made then
n_calls - n_random_starts subsequent evaluations are made
guided by the surrogate model.

Parameters
----------
* func [callable]:
Function to minimize. Should take a array of parameters and
return the function values.

* dimensions [list, shape=(n_dims,)]:
List of search space dimensions.
Each search dimension can be defined either as

- a (upper_bound, lower_bound) tuple (for Real or Integer
dimensions),
- a (upper_bound, lower_bound, "prior") tuple (for Real
dimensions),
- as a list of categories (for Categorical dimensions), or
- an instance of a Dimension object (Real, Integer or
Categorical).

* base_estimator [GradientBoostingQuantileRegressor]:
The regressor to use as surrogate model

* n_calls [int, default=100]:
Number of calls to func.

* n_random_starts [int, default=10]:
Number of evaluations of func with random points before
approximating it with base_estimator.

* acq_func [string, default="LCB"]:
Function to minimize over the forest posterior. Can be either

- "LCB" for lower confidence bound.
- "EI" for negative expected improvement.
- "PI" for negative probability of improvement.

* x0 [list, list of lists or None]:
Initial input points.

- If it is a list of lists, use it as a list of input points.
- If it is a list, use it as a single initial input point.
- If it is None, no initial input points are used.

* y0 [list, scalar or None]:
Evaluation of initial input points.

- If it is a list, then it corresponds to evaluations of the function
at each element of x0 : the i-th element of y0 corresponds
to the function evaluated at the i-th element of x0.
- If it is a scalar, then it corresponds to the evaluation of the
function at x0.
- If it is None and x0 is provided, then the function is evaluated
at each element of x0.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

* verbose [boolean, default=False]:
Control the verbosity. It is advised to set the verbosity to True
for long optimization runs.

* callback [callable, optional]
If provided, then callback(res) is called after call to func.

* n_points [int, default=10000]:
Number of points to sample when minimizing the acquisition function.

* xi [float, default=0.01]:
Controls how much improvement one wants over the previous best
values. Used when the acquisition is either "EI" or "PI".

* kappa [float, default=1.96]:
Controls how much of the variance in the predicted values should be
taken into account. If set to be very high, then we are favouring
exploration over exploitation and vice versa.
Used when the acquisition is "LCB".

* n_jobs [int, default=1]:
The number of jobs to run in parallel for fit and predict.
If -1, then the number of jobs is set to the number of cores.

Returns
-------
* res [OptimizeResult, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:

- x [list]: location of the minimum.
- fun [float]: function value at the minimum.
- models: surrogate models used for each iteration.
- x_iters [list of lists]: location of function evaluation for each
iteration.
- func_vals [array]: function value for each iteration.
- space [Space]: the optimization space.
- specs [dict]: the call specifications.
- rng [RandomState instance]: State of the random state
at the end of minimization.

For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
# Check params
rng = check_random_state(random_state)

# Default estimator
if base_estimator is None:
n_jobs=n_jobs,
random_state=rng)

return base_minimize(func, dimensions, base_estimator,
n_calls=n_calls, n_points=n_points,
n_random_starts=n_random_starts,
x0=x0, y0=y0, random_state=random_state, xi=xi,
kappa=kappa, acq_func=acq_func, verbose=verbose,
callback=callback, acq_optimizer="sampling")


def gp_minimize(

func, dimensions, base_estimator=None, n_calls=100, n_random_starts=10, acq_func='gp_hedge', acq_optimizer='lbfgs', x0=None, y0=None, random_state=None, verbose=False, callback=None, n_points=10000, n_restarts_optimizer=5, xi=0.01, kappa=1.96, noise='gaussian', n_jobs=1)

Bayesian optimization using Gaussian Processes.

If every function evaluation is expensive, for instance when the parameters are the hyperparameters of a neural network and the function evaluation is the mean cross-validation score across ten folds, optimizing the hyperparameters by standard optimization routines would take for ever!

The idea is to approximate the function using a Gaussian process. In other words the function values are assumed to follow a multivariate gaussian. The covariance of the function values are given by a GP kernel between the parameters. Then a smart choice to choose the next parameter to evaluate can be made by the acquisition function over the Gaussian prior which is much quicker to evaluate.

The total number of evaluations, n_calls, are performed like the following. If x0 is provided but not y0, then the elements of x0 are first evaluated, followed by n_random_starts evaluations. Finally, n_calls - len(x0) - n_random_starts evaluations are made guided by the surrogate model. If x0 and y0 are both provided then n_random_starts evaluations are first made then n_calls - n_random_starts subsequent evaluations are made guided by the surrogate model.

## Parameters

• func [callable]: Function to minimize. Should take a array of parameters and return the function values.

• dimensions [list, shape=(n_dims,)]: List of search space dimensions. Each search dimension can be defined either as

• a (upper_bound, lower_bound) tuple (for Real or Integer dimensions),
• a (upper_bound, lower_bound, "prior") tuple (for Real dimensions),
• as a list of categories (for Categorical dimensions), or
• an instance of a Dimension object (Real, Integer or Categorical).

NOTE: The upper and lower bounds are inclusive for Integer dimensions.

• base_estimator [a Gaussian process estimator]: The Gaussian process estimator to use for optimization. By default, a Matern kernel is used with the following hyperparameters tuned.

• All the length scales of the Matern kernel.
• The covariance amplitude that each element is multiplied with.
• Noise that is added to the matern kernel. The noise is assumed to be iid gaussian.
• n_calls [int, default=100]: Number of calls to func.

• n_random_starts [int, default=10]: Number of evaluations of func with random points before approximating it with base_estimator.

• acq_func [string, default="EI"]: Function to minimize over the gaussian prior. Can be either

• "LCB" for lower confidence bound.
• "EI" for negative expected improvement.
• "PI" for negative probability of improvement.
• "gp_hedge" Probabilistically choose one of the above three acquisition functions at every iteration. The weightage given to these gains can be set by \eta through acq_func_kwargs.
• The gains g_i are initialized to zero.
• At every iteration,
• Each acquisition function is optimised independently to propose an candidate point X_i.
• Out of all these candidate points, the next point X_best is chosen by softmax(\eta g_i)
• After fitting the surrogate model with (X_best, y_best), the gains are updated such that g_i -= \mu(X_i)

Reference: https://dslpitt.org/uai/papers/11/p327-hoffman.pdf

• acq_optimizer [string, "sampling" or "lbfgs", default="lbfgs"]: Method to minimize the acquistion function. The fit model is updated with the optimal value obtained by optimizing acq_func with acq_optimizer.

The acq_func is computed at n_points sampled randomly.

• If set to "sampling", then the point among these n_points where the acq_func is minimum is the next candidate minimum.
• If set to "lbfgs", then
• The n_restarts_optimizer no. of points which the acquisition function is least are taken as start points.
• "lbfgs" is run for 20 iterations with these points as initial points to find local minima.
• The optimal of these local minima is used to update the prior.
• x0 [list, list of lists or None]: Initial input points.

• If it is a list of lists, use it as a list of input points.
• If it is a list, use it as a single initial input point.
• If it is None, no initial input points are used.
• y0 [list, scalar or None] Evaluation of initial input points.

• If it is a list, then it corresponds to evaluations of the function at each element of x0 : the i-th element of y0 corresponds to the function evaluated at the i-th element of x0.
• If it is a scalar, then it corresponds to the evaluation of the function at x0.
• If it is None and x0 is provided, then the function is evaluated at each element of x0.
• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

• verbose [boolean, default=False]: Control the verbosity. It is advised to set the verbosity to True for long optimization runs.

• callback [callable, list of callables, optional] If callable then callback(res) is called after each call to func. If list of callables, then each callable in the list is called.

• n_points [int, default=10000]: Number of points to sample to determine the next "best" point. Useless if acq_optimizer is set to "lbfgs".

• n_restarts_optimizer [int, default=5]: The number of restarts of the optimizer when acq_optimizer is "lbfgs".

• kappa [float, default=1.96]: Controls how much of the variance in the predicted values should be taken into account. If set to be very high, then we are favouring exploration over exploitation and vice versa. Used when the acquisition is "LCB".

• xi [float, default=0.01]: Controls how much improvement one wants over the previous best values. Used when the acquisition is either "EI" or "PI".

• noise [float, default="gaussian"]:

• Use noise="gaussian" if the objective returns noisy observations. The noise of each observation is assumed to be iid with mean zero and a fixed variance.
• If the variance is known before-hand, this can be set directly to the variance of the noise.
• Set this to a value close to zero (1e-10) if the function is noise-free. Setting to zero might cause stability issues.
• n_jobs [int, default=1] Number of cores to run in parallel while running the lbfgs optimizations over the acquisition function. Valid only when acq_optimizer is set to "lbfgs." Defaults to 1 core. If n_jobs=-1, then number of jobs is set to number of cores.

## Returns

• res [OptimizeResult, scipy object]: The optimization result returned as a OptimizeResult object. Important attributes are:

• x [list]: location of the minimum.
• fun [float]: function value at the minimum.
• models: surrogate models used for each iteration.
• x_iters [list of lists]: location of function evaluation for each iteration.
• func_vals [array]: function value for each iteration.
• space [Space]: the optimization space.
• specs [dict]: the call specifications.
• rng [RandomState instance]: State of the random state at the end of minimization.

For more details related to the OptimizeResult object, refer http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html

def gp_minimize(func, dimensions, base_estimator=None,
n_calls=100, n_random_starts=10,
acq_func="gp_hedge", acq_optimizer="lbfgs", x0=None, y0=None,
random_state=None, verbose=False, callback=None,
n_points=10000, n_restarts_optimizer=5, xi=0.01, kappa=1.96,
noise="gaussian", n_jobs=1):
"""Bayesian optimization using Gaussian Processes.

If every function evaluation is expensive, for instance
when the parameters are the hyperparameters of a neural network
and the function evaluation is the mean cross-validation score across
ten folds, optimizing the hyperparameters by standard optimization
routines would take for ever!

The idea is to approximate the function using a Gaussian process.
In other words the function values are assumed to follow a multivariate
gaussian. The covariance of the function values are given by a
GP kernel between the parameters. Then a smart choice to choose the
next parameter to evaluate can be made by the acquisition function
over the Gaussian prior which is much quicker to evaluate.

The total number of evaluations, n_calls, are performed like the
following. If x0 is provided but not y0, then the elements of x0
are first evaluated, followed by n_random_starts evaluations.
Finally, n_calls - len(x0) - n_random_starts evaluations are
made guided by the surrogate model. If x0 and y0 are both
provided then n_random_starts evaluations are first made then
n_calls - n_random_starts subsequent evaluations are made
guided by the surrogate model.

Parameters
----------
* func [callable]:
Function to minimize. Should take a array of parameters and
return the function values.

* dimensions [list, shape=(n_dims,)]:
List of search space dimensions.
Each search dimension can be defined either as

- a (upper_bound, lower_bound) tuple (for Real or Integer
dimensions),
- a (upper_bound, lower_bound, "prior") tuple (for Real
dimensions),
- as a list of categories (for Categorical dimensions), or
- an instance of a Dimension object (Real, Integer or
Categorical).

NOTE: The upper and lower bounds are inclusive for Integer
dimensions.

* base_estimator [a Gaussian process estimator]:
The Gaussian process estimator to use for optimization.
By default, a Matern kernel is used with the following
hyperparameters tuned.
- All the length scales of the Matern kernel.
- The covariance amplitude that each element is multiplied with.
- Noise that is added to the matern kernel. The noise is assumed
to be iid gaussian.

* n_calls [int, default=100]:
Number of calls to func.

* n_random_starts [int, default=10]:
Number of evaluations of func with random points before
approximating it with base_estimator.

* acq_func [string, default="EI"]:
Function to minimize over the gaussian prior. Can be either

- "LCB" for lower confidence bound.
- "EI" for negative expected improvement.
- "PI" for negative probability of improvement.
- "gp_hedge" Probabilistically choose one of the above three
acquisition functions at every iteration. The weightage
given to these gains can be set by \eta through acq_func_kwargs.
- The gains g_i are initialized to zero.
- At every iteration,
- Each acquisition function is optimised independently to propose an
candidate point X_i.
- Out of all these candidate points, the next point X_best is
chosen by softmax(\eta g_i)
- After fitting the surrogate model with (X_best, y_best),
the gains are updated such that g_i -= \mu(X_i)

Reference: https://dslpitt.org/uai/papers/11/p327-hoffman.pdf

* acq_optimizer [string, "sampling" or "lbfgs", default="lbfgs"]:
Method to minimize the acquistion function. The fit model
is updated with the optimal value obtained by optimizing acq_func
with acq_optimizer.

The acq_func is computed at n_points sampled randomly.

- If set to "sampling", then the point among these n_points
where the acq_func is minimum is the next candidate minimum.
- If set to "lbfgs", then
- The n_restarts_optimizer no. of points which the acquisition
function is least are taken as start points.
- "lbfgs" is run for 20 iterations with these points as initial
points to find local minima.
- The optimal of these local minima is used to update the prior.

* x0 [list, list of lists or None]:
Initial input points.

- If it is a list of lists, use it as a list of input points.
- If it is a list, use it as a single initial input point.
- If it is None, no initial input points are used.

* y0 [list, scalar or None]
Evaluation of initial input points.

- If it is a list, then it corresponds to evaluations of the function
at each element of x0 : the i-th element of y0 corresponds
to the function evaluated at the i-th element of x0.
- If it is a scalar, then it corresponds to the evaluation of the
function at x0.
- If it is None and x0 is provided, then the function is evaluated
at each element of x0.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

* verbose [boolean, default=False]:
Control the verbosity. It is advised to set the verbosity to True
for long optimization runs.

* callback [callable, list of callables, optional]
If callable then callback(res) is called after each call to func.
If list of callables, then each callable in the list is called.

* n_points [int, default=10000]:
Number of points to sample to determine the next "best" point.
Useless if acq_optimizer is set to "lbfgs".

* n_restarts_optimizer [int, default=5]:
The number of restarts of the optimizer when acq_optimizer
is "lbfgs".

* kappa [float, default=1.96]:
Controls how much of the variance in the predicted values should be
taken into account. If set to be very high, then we are favouring
exploration over exploitation and vice versa.
Used when the acquisition is "LCB".

* xi [float, default=0.01]:
Controls how much improvement one wants over the previous best
values. Used when the acquisition is either "EI" or "PI".

* noise [float, default="gaussian"]:
- Use noise="gaussian" if the objective returns noisy observations.
The noise of each observation is assumed to be iid with
mean zero and a fixed variance.
- If the variance is known before-hand, this can be set directly
to the variance of the noise.
- Set this to a value close to zero (1e-10) if the function is
noise-free. Setting to zero might cause stability issues.

* n_jobs [int, default=1]
Number of cores to run in parallel while running the lbfgs
optimizations over the acquisition function. Valid only
when acq_optimizer is set to "lbfgs."
Defaults to 1 core. If n_jobs=-1, then number of jobs is set
to number of cores.

Returns
-------
* res [OptimizeResult, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:

- x [list]: location of the minimum.
- fun [float]: function value at the minimum.
- models: surrogate models used for each iteration.
- x_iters [list of lists]: location of function evaluation for each
iteration.
- func_vals [array]: function value for each iteration.
- space [Space]: the optimization space.
- specs [dict]: the call specifications.
- rng [RandomState instance]: State of the random state
at the end of minimization.

For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
# Check params
rng = check_random_state(random_state)

dim_types = [check_dimension(d) for d in dimensions]
is_cat = all([isinstance(check_dimension(d), Categorical)
for d in dim_types])
if is_cat:
transformed_dims = [check_dimension(d, transform="identity")
for d in dimensions]
else:
transformed_dims = []
for dim_type, dim in zip(dim_types, dimensions):
if isinstance(dim_type, Categorical):
transformed_dims.append(
check_dimension(dim, transform="onehot")
)
# To make sure that GP operates in the [0, 1] space
else:
transformed_dims.append(
check_dimension(dim, transform="normalize")
)

space = Space(transformed_dims)
# Default GP
if base_estimator is None:
cov_amplitude = ConstantKernel(1.0, (0.01, 1000.0))

if is_cat:
other_kernel = HammingKernel(
length_scale=np.ones(space.transformed_n_dims))
acq_optimizer = "sampling"
else:
other_kernel = Matern(
length_scale=np.ones(space.transformed_n_dims),
length_scale_bounds=[(0.01, 100)] * space.transformed_n_dims,
nu=2.5)

base_estimator = GaussianProcessRegressor(
kernel=cov_amplitude * other_kernel,
normalize_y=True, random_state=rng, alpha=0.0, noise=noise,
n_restarts_optimizer=2)

return base_minimize(
func, dimensions, base_estimator=base_estimator,
acq_func=acq_func,
xi=xi, kappa=kappa, acq_optimizer=acq_optimizer, n_calls=n_calls,
n_points=n_points, n_random_starts=n_random_starts,
n_restarts_optimizer=n_restarts_optimizer,
x0=x0, y0=y0, random_state=random_state, verbose=verbose,
callback=callback, n_jobs=n_jobs)


filename, **kwargs)

Reconstruct a skopt optimization result from a file persisted with skopt.dump.

Notice that the loaded optimization result can be missing the objective function (.specs['args']['func']) if dump was called with store_objective=False.

## Parameters

• filename [string or pathlib.Path]: The path of the file from which to load the optimization result.

• **kwargs [other keyword arguments]: All other keyword arguments will be passed to joblib.load.

## Returns

• res [OptimizeResult, scipy object]: Reconstructed OptimizeResult instance.
def load(filename, **kwargs):
"""
Reconstruct a skopt optimization result from a file
persisted with skopt.dump.

Notice that the loaded optimization result can be missing
the objective function (.specs['args']['func']) if skopt.dump
was called with store_objective=False.

Parameters
----------
* filename [string or pathlib.Path]:
The path of the file from which to load the optimization result.

* **kwargs [other keyword arguments]:
All other keyword arguments will be passed to joblib.load.

Returns
-------
* res [OptimizeResult, scipy object]:
Reconstructed OptimizeResult instance.
"""


## Classes

class Optimizer

Run bayesian optimisation loop.

An Optimizer represents the steps of a bayesian optimisation loop. To use it you need to provide your own loop mechanism. The various optimisers provided by skopt use this class under the hood.

Use this class directly if you want to control the iterations of your bayesian optimisation loop.

## Parameters

• dimensions [list, shape=(n_dims,)]: List of search space dimensions. Each search dimension can be defined either as

• a (upper_bound, lower_bound) tuple (for Real or Integer dimensions),
• a (upper_bound, lower_bound, "prior") tuple (for Real dimensions),
• as a list of categories (for Categorical dimensions), or
• an instance of a Dimension object (Real, Integer or Categorical).
• base_estimator [sklearn regressor]: Should inherit from sklearn.base.RegressorMixin. In addition the predict method, should have an optional return_std argument, which returns std(Y | x) along withE[Y | x].

• n_random_starts [int, default=10]: DEPRECATED, use n_initial_points instead.

• n_initial_points [int, default=10]: Number of evaluations of func with initialization points before approximating it with base_estimator. Points provided as x0 count as initialization points. If len(x0) < n_initial_points additional points are sampled at random.

• acq_func [string, default="EI"]: Function to minimize over the posterior distribution. Can be either

• "LCB" for lower confidence bound.
• "EI" for negative expected improvement.
• "PI" for negative probability of improvement.
• "gp_hedge" Probabilistically choose one of the above three acquisition functions at every iteration.
• The gains g_i are initialized to zero.
• At every iteration,
• Each acquisition function is optimised independently to propose an candidate point X_i.
• Out of all these candidate points, the next point X_best is chosen by $softmax(\eta g_i)$
• After fitting the surrogate model with (X_best, y_best), the gains are updated such that $g_i -= \mu(X_i)$
• acq_optimizer [string, "sampling" or "lbfgs", default="lbfgs"]: Method to minimize the acquistion function. The fit model is updated with the optimal value obtained by optimizing acq_func with acq_optimizer.

• If set to "sampling", then acq_func is optimized by computing acq_func at n_points randomly sampled points.
• If set to "lbfgs", then acq_func is optimized by
• Sampling n_restarts_optimizer points randomly.
• "lbfgs" is run for 20 iterations with these points as initial points to find local minima.
• The optimal of these local minima is used to update the prior.
• random_state [int, RandomState instance, or None (default)]: Set random state to something other than None for reproducible results.

• acq_func_kwargs [dict]: Additional arguments to be passed to the acquistion function.

• acq_optimizer_kwargs [dict]: Additional arguments to be passed to the acquistion optimizer.

## Attributes

• Xi [list]: Points at which objective has been evaluated.
• yi [scalar]: Values of objective at corresponding points in Xi.
• models [list]: Regression models used to fit observations and compute acquisition function.
• space An instance of skopt.space.Space. Stores parameter search space used to sample points, bounds, and type of parameters.
class Optimizer(object):
"""Run bayesian optimisation loop.

An Optimizer represents the steps of a bayesian optimisation loop. To
use it you need to provide your own loop mechanism. The various
optimisers provided by skopt use this class under the hood.

Use this class directly if you want to control the iterations of your
bayesian optimisation loop.

Parameters
----------
* dimensions [list, shape=(n_dims,)]:
List of search space dimensions.
Each search dimension can be defined either as

- a (upper_bound, lower_bound) tuple (for Real or Integer
dimensions),
- a (upper_bound, lower_bound, "prior") tuple (for Real
dimensions),
- as a list of categories (for Categorical dimensions), or
- an instance of a Dimension object (Real, Integer or
Categorical).

* base_estimator [sklearn regressor]:
Should inherit from sklearn.base.RegressorMixin.
In addition the predict method, should have an optional return_std
argument, which returns std(Y | x) along with E[Y | x].

* n_random_starts [int, default=10]:
DEPRECATED, use n_initial_points instead.

* n_initial_points [int, default=10]:
Number of evaluations of func with initialization points
before approximating it with base_estimator. Points provided as
x0 count as initialization points. If len(x0) < n_initial_points
additional points are sampled at random.

* acq_func [string, default="EI"]:
Function to minimize over the posterior distribution. Can be either

- "LCB" for lower confidence bound.
- "EI" for negative expected improvement.
- "PI" for negative probability of improvement.
- "gp_hedge" Probabilistically choose one of the above three
acquisition functions at every iteration.
- The gains g_i are initialized to zero.
- At every iteration,
- Each acquisition function is optimised independently to
propose an candidate point X_i.
- Out of all these candidate points, the next point X_best is
chosen by $softmax(\eta g_i)$
- After fitting the surrogate model with (X_best, y_best),
the gains are updated such that $g_i -= \mu(X_i)$

* acq_optimizer [string, "sampling" or "lbfgs", default="lbfgs"]:
Method to minimize the acquistion function. The fit model
is updated with the optimal value obtained by optimizing acq_func
with acq_optimizer.

- If set to "sampling", then acq_func is optimized by computing
acq_func at n_points randomly sampled points.
- If set to "lbfgs", then acq_func is optimized by
- Sampling n_restarts_optimizer points randomly.
- "lbfgs" is run for 20 iterations with these points as initial
points to find local minima.
- The optimal of these local minima is used to update the prior.

* random_state [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.

* acq_func_kwargs [dict]:
Additional arguments to be passed to the acquistion function.

* acq_optimizer_kwargs [dict]:
Additional arguments to be passed to the acquistion optimizer.

Attributes
----------
* Xi [list]:
Points at which objective has been evaluated.
* yi [scalar]:
Values of objective at corresponding points in Xi.
* models [list]:
Regression models used to fit observations and compute acquisition
function.
* space
An instance of skopt.space.Space. Stores parameter search space used
to sample points, bounds, and type of parameters.

"""
def __init__(self, dimensions, base_estimator,
n_random_starts=None, n_initial_points=10,
acq_func="gp_hedge",
acq_optimizer="lbfgs",
random_state=None, acq_func_kwargs=None,
acq_optimizer_kwargs=None):
# Arguments that are just stored not checked
self.acq_func = acq_func
self.rng = check_random_state(random_state)
self.acq_func_kwargs = acq_func_kwargs

if self.acq_func == "gp_hedge":
self.cand_acq_funcs_ = ["EI", "LCB", "PI"]
self.gains_ = np.zeros(3)
else:
self.cand_acq_funcs_ = [self.acq_func]

if acq_func_kwargs is None:
acq_func_kwargs = dict()
self.eta = acq_func_kwargs.get("eta", 1.0)

if acq_optimizer_kwargs is None:
acq_optimizer_kwargs = dict()

self.n_points = acq_optimizer_kwargs.get("n_points", 10000)
self.n_restarts_optimizer = acq_optimizer_kwargs.get(
"n_restarts_optimizer", 5)
n_jobs = acq_optimizer_kwargs.get("n_jobs", 1)
self.acq_optimizer_kwargs = acq_optimizer_kwargs

self.space = Space(dimensions)
self.models = []
self.Xi = []
self.yi = []

self._cat_inds = []
self._non_cat_inds = []
for ind, dim in enumerate(self.space.dimensions):
if isinstance(dim, Categorical):
self._cat_inds.append(ind)
else:
self._non_cat_inds.append(ind)

if n_random_starts is not None:
warnings.warn(("n_random_starts will be removed in favour of "
"n_initial_points."),
DeprecationWarning)
n_initial_points = n_random_starts

self._check_arguments(base_estimator, n_initial_points, acq_optimizer)

self.n_jobs = n_jobs

# The cache of responses of ask method for n_points not None.
# This ensures that multiple calls to ask with n_points set
# return same sets of points.
# The cache is reset to {} at every call to tell.
self.cache_ = {}

def _check_arguments(self, base_estimator, n_initial_points,
acq_optimizer):
"""Check arguments for sanity."""
if not is_regressor(base_estimator):
raise ValueError(
"%s has to be a regressor." % base_estimator)
self.base_estimator = base_estimator

if n_initial_points < 0:
raise ValueError(
"Expected n_initial_points >= 0, got %d" % n_initial_points)
self._n_initial_points = n_initial_points
self.n_initial_points_ = n_initial_points

if (isinstance(base_estimator, (ExtraTreesRegressor,
RandomForestRegressor,
not acq_optimizer == "sampling"):
raise ValueError(
"The tree-based regressor {0} should run with "
"acq_optimizer='sampling'".format(type(base_estimator)))

if acq_optimizer not in ["lbfgs", "sampling"]:
raise ValueError("Expected acq_optimizer to be 'lbfgs' or "
"'sampling', got {0}".format(acq_optimizer))
self.acq_optimizer = acq_optimizer

def copy(self, random_state=None):
"""Create a shallow copy of an instance of the optimizer.

Parameters
----------
* random_state [int, RandomState instance, or None (default)]:
Set the random state of the copy.
"""

optimizer = Optimizer(
dimensions=self.space.dimensions,
base_estimator=self.base_estimator,
n_initial_points=self.n_initial_points_,
acq_func=self.acq_func,
acq_optimizer=self.acq_optimizer,
acq_func_kwargs=self.acq_func_kwargs,
acq_optimizer_kwargs=self.acq_optimizer_kwargs,
random_state=random_state,
)

if hasattr(self, "gains_"):
optimizer.gains_ = np.copy(self.gains_)

if self.Xi:
optimizer.tell(self.Xi, self.yi)

return optimizer

"""Query point or multiple points at which objective should be evaluated.

* n_points [int or None, default=None]:
Number of points returned by the ask method.
If the value is None, a single point to evaluate is returned.
Otherwise a list of points to evaluate is returned of size
n_points. This is useful if you can evaluate your objective in
parallel, and thus obtain more objective function evaluations per
unit of time.

* strategy [string, default="cl_min"]:
Method to use to sample multiple points (see also n_points
description). This parameter is ignored if n_points = None.
Supported options are "cl_min", "cl_mean" or "cl_max".

- If set to "cl_min", then constant liar strtategy is used
with lie objective value being minimum of observed objective
values. "cl_mean" and "cl_max" means mean and max of values
respectively. For details on this strategy see:

https://hal.archives-ouvertes.fr/hal-00732512/document

With this strategy a copy of optimizer is created, which is
then asked for a point, and the point is told to the copy of
optimizer with some fake objective (lie), the next point is
asked from copy, it is also told to the copy with fake
objective and so on. The type of lie defines different
flavours of cl_x strategies.

"""
if n_points is None:

supported_strategies = ["cl_min", "cl_mean", "cl_max"]

if not (isinstance(n_points, int) and n_points > 0):
raise ValueError(
"n_points should be int > 0, got " + str(n_points)
)

if strategy not in supported_strategies:
raise ValueError(
"Expected parallel_strategy to be one of " +
str(supported_strategies) + ", " + "got %s" % strategy
)

# Caching the result with n_points not None. If some new parameters
# are provided to the ask, the cache_ is not used.
if (n_points, strategy) in self.cache_:
return self.cache_[(n_points, strategy)]

# Copy of the optimizer is made in order to manage the
# deletion of points with "lie" objective (the copy of
opt = self.copy()

X = []
for i in range(n_points):
X.append(x)
if strategy == "cl_min":
y_lie = np.min(opt.yi) if opt.yi else 0.0  # CL-min lie
elif strategy == "cl_mean":
y_lie = np.mean(opt.yi) if opt.yi else 0.0  # CL-max lie
else:
y_lie = np.max(opt.yi) if opt.yi else 0.0  # CL-max lie
opt.tell(x, y_lie)  # lie to the optimizer

self.cache_ = {(n_points, strategy): X}  # cache_ the result

return X

"""Suggest next point at which to evaluate the objective.

Return a random point while not at least n_initial_points
observations have been telled, after that base_estimator is used
to determine the next point.
"""
if self._n_initial_points > 0:
# this will not make a copy of self.rng and hence keep advancing
# our random state.
return self.space.rvs(random_state=self.rng)[0]

else:
if not self.models:
raise RuntimeError("Random evaluations exhausted and no "
"model has been fit.")

next_x = self._next_x
min_delta_x = min([self.space.distance(next_x, xi)
for xi in self.Xi])
if abs(min_delta_x) <= 1e-8:
warnings.warn("The objective has been evaluated "
"at this point before.")

# return point computed from last call to tell()
return next_x

def tell(self, x, y, fit=True):
"""Record an observation (or several) of the objective function.

Provide values of the objective function at points suggested by ask()
or other points. By default a new model will be fit to all
observations. The new model is used to suggest the next point at
which to evaluate the objective. This point can be retrieved by calling
ask().

To add observations without fitting a new model set fit to False.

To add multiple observations in a batch pass a list-of-lists for x
and a list of scalars for y.

Parameters
----------
* x [list or list-of-lists]:
Point at which objective was evaluated.

* y [scalar or list]:
Value of objective at x.

* fit [bool, default=True]
Fit a model to observed evaluations of the objective. A model will
only be fitted after n_initial_points points have been told to the
optimizer irrespective of the value of fit.
"""
# if y isn't a scalar it means we have been handed a batch of points
if (isinstance(y, Iterable) and all(isinstance(point, Iterable)
for point in x)):
if not np.all([p in self.space for p in x]):
raise ValueError("Not all points are within the bounds of"
" the space.")
self.Xi.extend(x)
self.yi.extend(y)
self._n_initial_points -= len(y)

elif isinstance(x, Iterable) and isinstance(y, Number):
if x not in self.space:
raise ValueError("Point (%s) is not within the bounds of"
" the space (%s)."
% (x, self.space.bounds))
self.Xi.append(x)
self.yi.append(y)
self._n_initial_points -= 1

else:
raise ValueError("Type of arguments x (%s) and y (%s) "
"not compatible." % (type(x), type(y)))

# optimizer learned somethnig new - discard cache
self.cache_ = {}

# after being "told" n_initial_points we switch from sampling
# random points to using a surrogate model
if fit and self._n_initial_points <= 0:
transformed_bounds = np.array(self.space.transformed_bounds)
est = clone(self.base_estimator)

with warnings.catch_warnings():
warnings.simplefilter("ignore")
est.fit(self.space.transform(self.Xi), self.yi)

if hasattr(self, "next_xs_") and self.acq_func == "gp_hedge":
self.gains_ -= est.predict(np.vstack(self.next_xs_))
self.models.append(est)

X = self.space.transform(self.space.rvs(
n_samples=self.n_points, random_state=self.rng))
self.next_xs_ = []
for cand_acq_func in self.cand_acq_funcs_:
values = _gaussian_acquisition(
X=X, model=est, y_opt=np.min(self.yi),
acq_func=cand_acq_func,
acq_func_kwargs=self.acq_func_kwargs)
# Find the minimum of the acquisition function by randomly
# sampling points from the space
if self.acq_optimizer == "sampling":
next_x = X[np.argmin(values)]

# Use BFGS to find the mimimum of the acquisition function, the
# minimization starts from n_restarts_optimizer different
# points and the best minimum is used
elif self.acq_optimizer == "lbfgs":
x0 = X[np.argsort(values)[:self.n_restarts_optimizer]]

with warnings.catch_warnings():
warnings.simplefilter("ignore")
results = Parallel(n_jobs=self.n_jobs)(
delayed(fmin_l_bfgs_b)(
gaussian_acquisition_1D, x,
args=(est, np.min(self.yi), cand_acq_func,
self.acq_func_kwargs),
bounds=self.space.transformed_bounds,
maxiter=20)
for x in x0)

cand_xs = np.array([r[0] for r in results])
cand_acqs = np.array([r[1] for r in results])
next_x = cand_xs[np.argmin(cand_acqs)]

# lbfgs should handle this but just in case there are
# precision errors.
if not self.space.is_categorical:
next_x = np.clip(
next_x, transformed_bounds[:, 0],
transformed_bounds[:, 1])
self.next_xs_.append(next_x)

if self.acq_func == "gp_hedge":
logits = np.array(self.gains_)
logits -= np.max(logits)
exp_logits = np.exp(self.eta * logits)
probs = exp_logits / np.sum(exp_logits)
next_x = self.next_xs_[np.argmax(self.rng.multinomial(1,
probs))]
else:
next_x = self.next_xs_[0]

# note the need for [0] at the end
self._next_x = self.space.inverse_transform(
next_x.reshape((1, -1)))[0]

# Pack results
return create_result(self.Xi, self.yi, self.space, self.rng,
models=self.models)

def run(self, func, n_iter=1):
"""Execute ask() + tell() n_iter times"""
for _ in range(n_iter):
self.tell(x, func(x))

return create_result(self.Xi, self.yi, self.space, self.rng,
models=self.models)


### Static methods

def __init__(

self, dimensions, base_estimator, n_random_starts=None, n_initial_points=10, acq_func='gp_hedge', acq_optimizer='lbfgs', random_state=None, acq_func_kwargs=None, acq_optimizer_kwargs=None)

Initialize self. See help(type(self)) for accurate signature.

def __init__(self, dimensions, base_estimator,
n_random_starts=None, n_initial_points=10,
acq_func="gp_hedge",
acq_optimizer="lbfgs",
random_state=None, acq_func_kwargs=None,
acq_optimizer_kwargs=None):
# Arguments that are just stored not checked
self.acq_func = acq_func
self.rng = check_random_state(random_state)
self.acq_func_kwargs = acq_func_kwargs
if self.acq_func == "gp_hedge":
self.cand_acq_funcs_ = ["EI", "LCB", "PI"]
self.gains_ = np.zeros(3)
else:
self.cand_acq_funcs_ = [self.acq_func]
if acq_func_kwargs is None:
acq_func_kwargs = dict()
self.eta = acq_func_kwargs.get("eta", 1.0)
if acq_optimizer_kwargs is None:
acq_optimizer_kwargs = dict()
self.n_points = acq_optimizer_kwargs.get("n_points", 10000)
self.n_restarts_optimizer = acq_optimizer_kwargs.get(
"n_restarts_optimizer", 5)
n_jobs = acq_optimizer_kwargs.get("n_jobs", 1)
self.acq_optimizer_kwargs = acq_optimizer_kwargs
self.space = Space(dimensions)
self.models = []
self.Xi = []
self.yi = []
self._cat_inds = []
self._non_cat_inds = []
for ind, dim in enumerate(self.space.dimensions):
if isinstance(dim, Categorical):
self._cat_inds.append(ind)
else:
self._non_cat_inds.append(ind)
if n_random_starts is not None:
warnings.warn(("n_random_starts will be removed in favour of "
"n_initial_points."),
DeprecationWarning)
n_initial_points = n_random_starts
self._check_arguments(base_estimator, n_initial_points, acq_optimizer)
self.n_jobs = n_jobs
# The cache of responses of ask method for n_points not None.
# This ensures that multiple calls to ask with n_points set
# return same sets of points.
# The cache is reset to {} at every call to tell.
self.cache_ = {}


self, n_points=None, strategy='cl_min')

Query point or multiple points at which objective should be evaluated.

• n_points [int or None, default=None]: Number of points returned by the ask method. If the value is None, a single point to evaluate is returned. Otherwise a list of points to evaluate is returned of size n_points. This is useful if you can evaluate your objective in parallel, and thus obtain more objective function evaluations per unit of time.

• strategy [string, default="cl_min"]: Method to use to sample multiple points (see also n_points description). This parameter is ignored if n_points = None. Supported options are "cl_min", "cl_mean" or "cl_max".

• If set to "cl_min", then constant liar strtategy is used with lie objective value being minimum of observed objective values. "cl_mean" and "cl_max" means mean and max of values respectively. For details on this strategy see:

https://hal.archives-ouvertes.fr/hal-00732512/document

With this strategy a copy of optimizer is created, which is then asked for a point, and the point is told to the copy of optimizer with some fake objective (lie), the next point is asked from copy, it is also told to the copy with fake objective and so on. The type of lie defines different flavours of cl_x strategies.

def ask(self, n_points=None, strategy="cl_min"):
"""Query point or multiple points at which objective should be evaluated.
* n_points [int or None, default=None]:
Number of points returned by the ask method.
If the value is None, a single point to evaluate is returned.
Otherwise a list of points to evaluate is returned of size
n_points. This is useful if you can evaluate your objective in
parallel, and thus obtain more objective function evaluations per
unit of time.
* strategy [string, default="cl_min"]:
Method to use to sample multiple points (see also n_points
description). This parameter is ignored if n_points = None.
Supported options are "cl_min", "cl_mean" or "cl_max".
- If set to "cl_min", then constant liar strtategy is used
with lie objective value being minimum of observed objective
values. "cl_mean" and "cl_max" means mean and max of values
respectively. For details on this strategy see:
https://hal.archives-ouvertes.fr/hal-00732512/document
With this strategy a copy of optimizer is created, which is
then asked for a point, and the point is told to the copy of
optimizer with some fake objective (lie), the next point is
asked from copy, it is also told to the copy with fake
objective and so on. The type of lie defines different
flavours of cl_x strategies.
"""
if n_points is None:
supported_strategies = ["cl_min", "cl_mean", "cl_max"]
if not (isinstance(n_points, int) and n_points > 0):
raise ValueError(
"n_points should be int > 0, got " + str(n_points)
)
if strategy not in supported_strategies:
raise ValueError(
"Expected parallel_strategy to be one of " +
str(supported_strategies) + ", " + "got %s" % strategy
)
# Caching the result with n_points not None. If some new parameters
# are provided to the ask, the cache_ is not used.
if (n_points, strategy) in self.cache_:
return self.cache_[(n_points, strategy)]
# Copy of the optimizer is made in order to manage the
# deletion of points with "lie" objective (the copy of
opt = self.copy()
X = []
for i in range(n_points):
X.append(x)
if strategy == "cl_min":
y_lie = np.min(opt.yi) if opt.yi else 0.0  # CL-min lie
elif strategy == "cl_mean":
y_lie = np.mean(opt.yi) if opt.yi else 0.0  # CL-max lie
else:
y_lie = np.max(opt.yi) if opt.yi else 0.0  # CL-max lie
opt.tell(x, y_lie)  # lie to the optimizer
self.cache_ = {(n_points, strategy): X}  # cache_ the result
return X


def copy(

self, random_state=None)

Create a shallow copy of an instance of the optimizer.

## Parameters

• random_state [int, RandomState instance, or None (default)]: Set the random state of the copy.
def copy(self, random_state=None):
"""Create a shallow copy of an instance of the optimizer.
Parameters
----------
* random_state [int, RandomState instance, or None (default)]:
Set the random state of the copy.
"""
optimizer = Optimizer(
dimensions=self.space.dimensions,
base_estimator=self.base_estimator,
n_initial_points=self.n_initial_points_,
acq_func=self.acq_func,
acq_optimizer=self.acq_optimizer,
acq_func_kwargs=self.acq_func_kwargs,
acq_optimizer_kwargs=self.acq_optimizer_kwargs,
random_state=random_state,
)
if hasattr(self, "gains_"):
optimizer.gains_ = np.copy(self.gains_)
if self.Xi:
optimizer.tell(self.Xi, self.yi)
return optimizer


def run(

self, func, n_iter=1)

Execute ask() + tell() n_iter times

def run(self, func, n_iter=1):
"""Execute ask() + tell() n_iter times"""
for _ in range(n_iter):
self.tell(x, func(x))
return create_result(self.Xi, self.yi, self.space, self.rng,
models=self.models)


def tell(

self, x, y, fit=True)

Record an observation (or several) of the objective function.

Provide values of the objective function at points suggested by ask() or other points. By default a new model will be fit to all observations. The new model is used to suggest the next point at which to evaluate the objective. This point can be retrieved by calling ask().

To add observations without fitting a new model set fit to False.

To add multiple observations in a batch pass a list-of-lists for x and a list of scalars for y.

## Parameters

• x [list or list-of-lists]: Point at which objective was evaluated.

• y [scalar or list]: Value of objective at x.

• fit [bool, default=True] Fit a model to observed evaluations of the objective. A model will only be fitted after n_initial_points points have been told to the optimizer irrespective of the value of fit.

def tell(self, x, y, fit=True):
"""Record an observation (or several) of the objective function.
Provide values of the objective function at points suggested by ask()
or other points. By default a new model will be fit to all
observations. The new model is used to suggest the next point at
which to evaluate the objective. This point can be retrieved by calling
ask().
To add observations without fitting a new model set fit to False.
To add multiple observations in a batch pass a list-of-lists for x
and a list of scalars for y.
Parameters
----------
* x [list or list-of-lists]:
Point at which objective was evaluated.
* y [scalar or list]:
Value of objective at x.
* fit [bool, default=True]
Fit a model to observed evaluations of the objective. A model will
only be fitted after n_initial_points points have been told to the
optimizer irrespective of the value of fit.
"""
# if y isn't a scalar it means we have been handed a batch of points
if (isinstance(y, Iterable) and all(isinstance(point, Iterable)
for point in x)):
if not np.all([p in self.space for p in x]):
raise ValueError("Not all points are within the bounds of"
" the space.")
self.Xi.extend(x)
self.yi.extend(y)
self._n_initial_points -= len(y)
elif isinstance(x, Iterable) and isinstance(y, Number):
if x not in self.space:
raise ValueError("Point (%s) is not within the bounds of"
" the space (%s)."
% (x, self.space.bounds))
self.Xi.append(x)
self.yi.append(y)
self._n_initial_points -= 1
else:
raise ValueError("Type of arguments x (%s) and y (%s) "
"not compatible." % (type(x), type(y)))
# optimizer learned somethnig new - discard cache
self.cache_ = {}
# after being "told" n_initial_points we switch from sampling
# random points to using a surrogate model
if fit and self._n_initial_points <= 0:
transformed_bounds = np.array(self.space.transformed_bounds)
est = clone(self.base_estimator)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
est.fit(self.space.transform(self.Xi), self.yi)
if hasattr(self, "next_xs_") and self.acq_func == "gp_hedge":
self.gains_ -= est.predict(np.vstack(self.next_xs_))
self.models.append(est)
X = self.space.transform(self.space.rvs(
n_samples=self.n_points, random_state=self.rng))
self.next_xs_ = []
for cand_acq_func in self.cand_acq_funcs_:
values = _gaussian_acquisition(
X=X, model=est, y_opt=np.min(self.yi),
acq_func=cand_acq_func,
acq_func_kwargs=self.acq_func_kwargs)
# Find the minimum of the acquisition function by randomly
# sampling points from the space
if self.acq_optimizer == "sampling":
next_x = X[np.argmin(values)]
# Use BFGS to find the mimimum of the acquisition function, the
# minimization starts from n_restarts_optimizer different
# points and the best minimum is used
elif self.acq_optimizer == "lbfgs":
x0 = X[np.argsort(values)[:self.n_restarts_optimizer]]
with warnings.catch_warnings():
warnings.simplefilter("ignore")
results = Parallel(n_jobs=self.n_jobs)(
delayed(fmin_l_bfgs_b)(
gaussian_acquisition_1D, x,
args=(est, np.min(self.yi), cand_acq_func,
self.acq_func_kwargs),
bounds=self.space.transformed_bounds,
maxiter=20)
for x in x0)
cand_xs = np.array([r[0] for r in results])
cand_acqs = np.array([r[1] for r in results])
next_x = cand_xs[np.argmin(cand_acqs)]
# lbfgs should handle this but just in case there are
# precision errors.
if not self.space.is_categorical:
next_x = np.clip(
next_x, transformed_bounds[:, 0],
transformed_bounds[:, 1])
self.next_xs_.append(next_x)
if self.acq_func == "gp_hedge":
logits = np.array(self.gains_)
logits -= np.max(logits)
exp_logits = np.exp(self.eta * logits)
probs = exp_logits / np.sum(exp_logits)
next_x = self.next_xs_[np.argmax(self.rng.multinomial(1,
probs))]
else:
next_x = self.next_xs_[0]
# note the need for [0] at the end
self._next_x = self.space.inverse_transform(
next_x.reshape((1, -1)))[0]
# Pack results
return create_result(self.Xi, self.yi, self.space, self.rng,
models=self.models)


### Instance variables

var Xi

var acq_func

var acq_func_kwargs

var acq_optimizer_kwargs

var cache_

var eta

var models

var n_jobs

var n_points

var n_restarts_optimizer

var rng

var space

var yi