"""Gaussian process-based minimization algorithms."""
import numpy as np
from sklearn.utils import check_random_state
from .base import base_minimize
from ..utils import cook_estimator
from ..utils import normalize_dimensions
[docs]def gp_minimize(func, dimensions, base_estimator=None,
n_calls=100, n_random_starts=None,
n_initial_points=10,
initial_point_generator="random",
acq_func="gp_hedge", acq_optimizer="auto", 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, model_queue_size=None):
"""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_initial_points` evaluations.
Finally, `n_calls - len(x0) - n_initial_points` evaluations are
made guided by the surrogate model. If `x0` and `y0` are both
provided then `n_initial_points` evaluations are first made then
`n_calls - n_initial_points` subsequent evaluations are made
guided by the surrogate model.
The first `n_initial_points` are generated by the
`initial_point_generator`.
Parameters
----------
func : callable
Function to minimize. Should take a single list of parameters
and return the objective value.
If you have a search-space where all dimensions have names,
then you can use :func:`skopt.utils.use_named_args` as a decorator
on your objective function, in order to call it directly
with the named arguments. See `use_named_args` for an example.
dimensions : [list, shape (n_dims,)
List of search space dimensions.
Each search dimension can be defined either as
- a `(lower_bound, upper_bound)` tuple (for `Real` or `Integer`
dimensions),
- a `(lower_bound, upper_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: None
Number of evaluations of `func` with random points before
approximating it with `base_estimator`.
.. deprecated:: 0.8
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`. Initial point
generator can be changed by setting `initial_point_generator`.
initial_point_generator : str, InitialPointGenerator instance, \
default: 'random'
Sets a initial points generator. Can be either
- `"random"` for uniform random numbers,
- `"sobol"` for a Sobol sequence,
- `"halton"` for a Halton sequence,
- `"hammersly"` for a Hammersly sequence,
- `"lhs"` for a latin hypercube sequence,
acq_func : string, default: `"gp_hedge"`
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 :math:`\\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 :math:`softmax(\\eta g_i)`
- After fitting the surrogate model with `(X_best, y_best)`,
the gains are updated such that :math:`g_i -= \\mu(X_i)`
- `"EIps"` for negated expected improvement per second to take into
account the function compute time. Then, the objective function is
assumed to return two values, the first being the objective value and
the second being the time taken in seconds.
- `"PIps"` for negated probability of improvement per second. The
return type of the objective function is assumed to be similar to
that of `"EIps"`
acq_optimizer : string, `"sampling"` or `"lbfgs"`, default: `"lbfgs"`
Method to minimize the acquisition 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 `"auto"`, then `acq_optimizer` is configured on the
basis of the space searched over.
If the space is Categorical then this is set to be `"sampling"`.
- 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.
model_queue_size : int or None, default: None
Keeps list of models only as long as the argument given. In the
case of None, the list has no capped length.
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
.. seealso:: functions :class:`skopt.forest_minimize`,
:class:`skopt.dummy_minimize`, :class:`skopt.gbrt_minimize`
"""
# Check params
rng = check_random_state(random_state)
space = normalize_dimensions(dimensions)
if base_estimator is None:
base_estimator = cook_estimator(
"GP", space=space, random_state=rng.randint(0, np.iinfo(np.int32).max),
noise=noise)
return base_minimize(
func, space, 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_initial_points=n_initial_points,
initial_point_generator=initial_point_generator,
n_restarts_optimizer=n_restarts_optimizer,
x0=x0, y0=y0, random_state=rng, verbose=verbose,
callback=callback, n_jobs=n_jobs, model_queue_size=model_queue_size)
