def _tree_minimize(func, dimensions, base_estimator, n_calls,
n_points, n_random_starts, random_state=None):
rng = check_random_state(random_state)
space = Space(dimensions)
# Initialize with random points
if n_random_starts <= 0:
raise ValueError(
"Expected n_random_starts > 0, got %d" % n_random_starts)
if n_calls <= 0:
raise ValueError(
"Expected n_calls > 0, got %d" % n_random_starts)
if n_calls < n_random_starts:
raise ValueError(
"Expected n_calls >= %d, got %d" % (n_random_starts, n_calls))
Xi = space.rvs(n_samples=n_random_starts, random_state=rng)
yi = [func(x) for x in Xi]
if np.ndim(yi) != 1:
raise ValueError(
"The function to be optimized should return a scalar")
# Tree-based optimization loop
models = []
n_model_iter = n_calls - n_random_starts
for i in range(n_model_iter):
rgr = clone(base_estimator)
rgr.fit(space.transform(Xi), yi)
models.append(rgr)
# `rgr` predicts constants for each leaf which means that the EI
# has zero gradient over large distances. As a result we can not
# use gradient based optimizers like BFGS, so using random sampling
# for the moment.
X = space.transform(space.rvs(n_samples=n_points,
random_state=rng))
values = -gaussian_ei(X, rgr, np.min(yi))
next_x = X[np.argmin(values)]
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi = np.vstack((Xi, next_x))
yi.append(next_y)
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
return res
def create_result(Xi,
yi,
space=None,
rng=None,
specs=None,
models=None,
model_mu=None,
model_std=None,
gurobi_mipgap=None):
"""
Initialize an `OptimizeResult` object.
Parameters
----------
Xi : list of lists, shape (n_iters, n_features)
Location of the minimum at every iteration.
yi : array-like, shape (n_iters,)
Minimum value obtained at every iteration.
space : Space instance, optional
Search space.
rng : RandomState instance, optional
State of the random state.
specs : dict, optional
Call specifications.
models : list, optional
List of fit surrogate models.
Returns
-------
res : `OptimizeResult`, scipy object
OptimizeResult instance with the required information.
"""
res = OptimizeResult()
yi = np.asarray(yi)
if np.ndim(yi) == 2:
res.log_time = np.ravel(yi[:, 1])
yi = np.ravel(yi[:, 0])
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = yi
res.x_iters = Xi
res.models = models
res.model_mu = model_mu
res.model_std = model_std
res.gurobi_mipgap = gurobi_mipgap
res.space = space
res.random_state = rng
res.specs = specs
return res
def create_result(xi, yi, space=None, rs=None, specs=None, models=None):
res = OptimizeResult()
yi = np.asarray(yi)
if np.ndim(yi) == 2:
res.log_time = np.ravel(yi[:, 1])
yi = np.ravel(yi[:, 0])
best = np.argmin(yi)
res.x = xi[best]
res.fun = yi[best]
res.func_vals = yi
res.x_iters = xi
res.models = models
res.space = space
res.random_state = rs
res.specs = specs
return res
def create_result(Xi, yi, space=None, rng=None, specs=None, models=None):
"""
Initialize an `OptimizeResult` object.
Parameters
----------
* `Xi` [list of lists, shape=(n_iters, n_features)]:
Location of the minimum at every iteration.
* `yi` [array-like, shape=(n_iters,)]:
Minimum value obtained at every iteration.
* `space` [Space instance, optional]:
Search space.
* `rng` [RandomState instance, optional]:
State of the random state.
* `specs` [dict, optional]:
Call specifications.
* `models` [list, optional]:
List of fit surrogate models.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
OptimizeResult instance with the required information.
"""
res = OptimizeResult()
yi = np.asarray(yi)
if np.ndim(yi) == 2:
res.log_time = np.ravel(yi[:, 1])
yi = np.ravel(yi[:, 0])
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = yi
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
res.specs = specs
return res
def create_result(Xi, yi, space=None, rng=None, specs=None, models=None):
"""
Initialize an `OptimizeResult` object.
Parameters
----------
* `Xi` [list of lists, shape=(n_iters, n_features)]:
Location of the minimum at every iteration.
* `yi` [array-like, shape=(n_iters,)]:
Minimum value obtained at every iteration.
* `space` [Space instance, optional]:
Search space.
* `rng` [RandomState instance, optional]:
State of the random state.
* `specs` [dict, optional]:
Call specifications.
* `models` [list, optional]:
List of fit surrogate models.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
OptimizeResult instance with the required information.
"""
res = OptimizeResult()
yi = np.asarray(yi)
if np.ndim(yi) == 2:
res.log_time = np.ravel(yi[:, 1])
yi = np.ravel(yi[:, 0])
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = yi
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
res.specs = specs
return res
def dummy_minimize(func,
dimensions,
n_calls=100,
x0=None,
y0=None,
random_state=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.
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))
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)
elif len(x0) == 0 and y0 is not None:
raise ValueError("`x0`cannot be `None` when `y0` is provided")
else: # len(x0) == 0 and y0 is None
y0 = []
X = x0
y = y0
# Random search
X = X + space.rvs(n_samples=n_calls, random_state=rng)
first = True
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)
y = np.array(y)
# Pack results
res = OptimizeResult()
best = np.argmin(y)
res.x = X[best]
res.fun = y[best]
res.func_vals = y
res.x_iters = X
res.models = [] # Create attribute even though it is empty
res.space = space
res.random_state = rng
res.specs = specs
return res
def gbrt_minimize(func,
bounds,
base_estimator=None,
maxiter=100,
n_points=20,
n_start=10,
random_state=None):
"""Sequential optimisation 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.
Parameters
----------
* `func` [callable]:
Function to minimize. Should take a array of parameters and
return the function values.
* `bounds` [array-like, shape=(n_parameters, 2)]:
- ``bounds[i][0]`` should give the lower bound of each parameter and
- ``bounds[i][1]`` should give the upper bound of each parameter.
* `base_estimator` [`GradientBoostingQuantileRegressor`]:
The regressor to use as surrogate model
* `maxiter` [int, default=100]:
Number of iterations used to find the minimum. This corresponds
to the total number of evaluations of `func`. If `n_start` > 0
only `maxiter - n_start` iterations are used.
* `n_start` [int, default=10]:
Number of random points to draw before fitting `base_estimator`
for the first time. If `n_start > maxiter` this degrades to
a random search for the minimum.
* `n_points` [int, default=20]:
Number of points to sample when minimizing the acquisition function.
* `random_state` [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:
- `x` [float]: location of the minimum.
- `fun` [float]: function value at the minimum.
- `models`: surrogate models used for each iteration.
- `x_iters` [array]: location of function evaluation for each
iteration.
- `func_vals` [array]: function value for each iteration.
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)
# Bounds
num_params = len(bounds)
lower_bounds, upper_bounds = extract_bounds(bounds)
# Default estimator
if base_estimator is None:
base_estimator = GradientBoostingQuantileRegressor(random_state=rng)
# Record the points and function values evaluated as part of
# the minimization
Xi = np.zeros((maxiter, num_params))
yi = np.zeros(maxiter)
# Initialize with random points
if n_start == 0:
raise ValueError("Need at least one starting point.")
if maxiter == 0:
raise ValueError("Need to perform at least one iteration.")
n_start = min(n_start, maxiter)
Xi[:n_start] = _random_points(lower_bounds,
upper_bounds,
n_points=n_start,
random_state=rng)
best_x = Xi[:n_start].ravel()
yi[:n_start] = [func(xi) for xi in Xi[:n_start]]
best_y = np.min(yi[:n_start])
models = []
for i in range(n_start, maxiter):
rgr = clone(base_estimator)
# only the first i points are meaningful
rgr.fit(Xi[:i, :], yi[:i])
models.append(rgr)
# `rgr` predicts constants for each leaf which means that the EI
# has zero gradient over large distances. As a result we can not
# use gradient based optimisers like BFGS, use random sampling
# for the moment.
x0 = _random_points(lower_bounds,
upper_bounds,
n_points=n_points,
random_state=rng)
aq = _expected_improvement(x0, rgr, best_y)
best = np.argmin(aq)
Xi[i] = x0[best].ravel()
yi[i] = func(x0[best])
if yi[i] < best_y:
best_y = yi[i]
best_x = Xi[i]
res = OptimizeResult()
res.x = best_x
res.fun = best_y
res.func_vals = yi
res.x_iters = Xi
res.models = models
return res
def create_result(Xi,
yi,
n_evaluations=None,
space=None,
rng=None,
specs=None,
models=None,
maximize=False):
"""
Initialize an `OptimizeResult` object.
Parameters
----------
* `Xi` [list of lists, shape=(n_iters, n_features)]:
Location of the minimum at every iteration.
* `yi` [array-like, shape=(n_iters,)]:
Minimum value obtained at every iteration.
* `space` [Space instance, optional]:
Search space.
* `rng` [RandomState instance, optional]:
State of the random state.
* `specs` [dict, optional]:
Call specifications.
* `models` [list, optional]:
List of fit surrogate models.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
OptimizeResult instance with the required information.
"""
res = OptimizeResult()
try:
# Hyperband returns evaluations as lists of lists.
# We want to store the results as a single array.
yi = list(itertools.chain.from_iterable(yi))
Xi = list(itertools.chain.from_iterable(Xi))
except TypeError:
# All algorithms other than Hyperband already return a single list.
pass
yi = np.asarray(yi)
if np.ndim(yi) == 2:
res.log_time = np.ravel(yi[:, 1])
yi = np.ravel(yi[:, 0])
if maximize:
best = np.argmax(yi)
else:
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
if n_evaluations:
unique, sort_indices = np.unique(yi, return_index=True)
if len(unique) < n_evaluations:
func_sort_idx = np.argsort(yi)
func_vals = sorted(yi)
res.func_vals = np.asarray(func_vals[:n_evaluations])
x_iter_sort = []
for idx in func_sort_idx:
x_iter_sort.append(Xi[idx])
res.x_iters = np.asarray(x_iter_sort[:n_evaluations])
res.all_func_vals = np.asarray(yi)
res.all_x_iters = np.asarray(Xi)
else:
func_vals = sorted(unique)
res.func_vals = np.asarray(func_vals[:n_evaluations])
x_iter_sort = []
for idx in sort_indices:
x_iter_sort.append(Xi[idx])
res.x_iters = np.asarray(x_iter_sort[:n_evaluations])
res.all_func_vals = np.asarray(yi)
res.all_x_iters = np.asarray(Xi)
else:
res.func_vals = np.asarray(yi)
res.x_iters = np.asarray(Xi)
res.models = models
res.space = space
res.random_state = rng
res.specs = specs
return res
def gp_minimize(func,
bounds=None,
search="sampling",
random_state=None,
maxiter=1000,
acq="UCB",
num_points=500):
"""
Black-box 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 standared 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 acquistion function
over the Gaussian posterior which is much quicker to evaluate.
Parameters
----------
func: callable
Function to minimize. Should take a array of parameters and
return the function value.
bounds: array-like, shape (n_parameters, 2)
``bounds[i][0]`` should give the lower bound of each parameter and
``bounds[i][1]`` should give the upper bound of each parameter.
search: string, "sampling" or "lbfgs"
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to "sampling", ``num_points`` are sampled randomly
and the Gaussian Process prior is updated with that point that gives
the best acquision value over the Gaussian posterior.
If search is set to "lbfgs", then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquistion function
over the Gaussian posterior.
random_state: int, RandomState instance, or None (default)
Set random state to something other than None for reproducible
results.
maxiter: int, default 1000
Number of iterations to find the minimum. In other words, the
number of function evaluations.
acq: string, default "UCB"
Function to minimize over the gaussian posterior. Can be either
the "UCB" which refers to the UpperConfidenceBound or "EI" which
is the Expected Improvement.
num_points: int, default 500
Number of points to sample to determine the next "best" point.
Useless if search is set to "lbfgs".
Returns
-------
res: OptimizeResult, scipy object
The optimization result returned as a OptimizeResult object.
Important attributes are
``x`` - float, the optimization solution,
``fun`` - float, the value of the function at the optimum,
``models``- gp_models[i]. the prior on the function fit at
iteration[i].
``func_vals`` - the function value at the ith iteration.
``x_iters`` - the value of ``x`` corresponding to the function value
at the ith iteration.
For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
rng = np.random.RandomState(random_state)
num_params = len(bounds)
lower_bounds, upper_bounds = zip(*bounds)
upper_bounds = np.asarray(upper_bounds)
lower_bounds = np.asarray(lower_bounds)
x0 = rng.rand(num_params)
func_val = [func(lower_bounds + (upper_bounds - lower_bounds) * x0)]
length_scale = np.ones(num_params)
gp_params = {
'kernel': Matern(length_scale=length_scale, nu=2.5),
'normalize_y': True,
'random_state': random_state
}
lbfgs_bounds = np.tile((0, 1), (num_params, 1))
gp_models = []
x = np.reshape(x0, (1, -1))
for i in range(maxiter):
gpr = GaussianProcessRegressor(**gp_params)
gpr.fit(x, func_val)
if search == "sampling":
sampling = rng.rand(num_points, num_params)
acquis = acquisition_func(sampling, gpr, np.min(func_val), acq)
best_arg = np.argmin(acquis)
best_x = sampling[best_arg]
elif search == "lbfgs":
init = rng.rand(num_params)
best_x, _, _ = fmin_l_bfgs_b(acquisition_func,
np.asfortranarray(init),
args=(gpr, np.min(func_val), acq),
bounds=lbfgs_bounds,
approx_grad=True,
maxiter=10)
gp_models.append(gpr)
best_f = func(lower_bounds + (upper_bounds - lower_bounds) * best_x)
x_list = x.tolist()
x_list.append(best_x)
x = np.asarray(x_list)
func_val.append(best_f)
x = lower_bounds + (upper_bounds - lower_bounds) * x
func_ind = np.argmin(func_val)
x_val = x[func_ind]
best_func_val = func_val[func_ind]
res = OptimizeResult()
res.models = gp_models
res.x = x_val
res.fun = best_func_val
res.func_vals = func_val
res.x_iters = x
return res
def gp_minimize(func, dimensions, base_estimator=None, alpha=10e-10,
acq="EI", xi=0.01, kappa=1.96, search="auto", n_calls=100,
n_points=500, n_random_starts=10, n_restarts_optimizer=5,
x0=None, y0=None, random_state=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_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` [a Gaussian process estimator]:
The Gaussian process estimator to use for optimization.
* `alpha` [float, default=1e-10]:
Value added to the diagonal of the kernel matrix during fitting.
Larger values correspond to an increased noise level in the
observations and reduce potential numerical issues during fitting.
* `acq` [string, default=`"EI"`]:
Function to minimize over the gaussian prior. Can be either
- `"LCB"` for lower confidence bound,
- `"EI"` for expected improvement,
- `"PI"` for probability of improvement.
* `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"`.
* `search` [string, `"auto"`, `"sampling"` or `"lbfgs"`, default=`"auto"`]:
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to `"auto"`, then it is set to `"lbfgs"`` if
all the search dimensions are Real(continuous). It defaults to
`"sampling"` for all other cases.
If search is set to `"sampling"`, `n_points` are sampled randomly
and the Gaussian Process prior is updated with the point that gives
the best acquisition value over the Gaussian prior.
If search is set to `"lbfgs"`, then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquisition function
over the Gaussian prior.
* `n_calls` [int, default=100]:
Number of calls to `func`.
* `n_points` [int, default=500]:
Number of points to sample to determine the next "best" point.
Useless if search is set to `"lbfgs"`.
* `n_random_starts` [int, default=10]:
Number of evaluations of `func` with random initialization points
before approximating the `func` with `base_estimator`.
* `n_restarts_optimizer` [int, default=10]:
The number of restarts of the optimizer when `search` is `"lbfgs"`.
* `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.
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
"""
# 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)
# Default GP
if base_estimator is None:
base_estimator = GaussianProcessRegressor(
kernel=(ConstantKernel(1.0, (0.01, 1000.0)) *
Matern(length_scale=np.ones(space.transformed_n_dims),
length_scale_bounds=[(0.01, 100)] * space.transformed_n_dims,
nu=2.5)),
normalize_y=True, alpha=alpha, random_state=random_state)
# Initialize with provided points (x0 and y0) and/or random points
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, but got %s" % type(x0))
n_init_func_calls = len(x0) if y0 is not None else 0
n_total_init_calls = n_random_starts + n_init_func_calls
if n_total_init_calls <= 0:
# if x0 is not provided and n_random_starts is 0 then
# it will ask for n_random_starts to be > 0.
raise ValueError(
"Expected `n_random_starts` > 0, got %d" % n_random_starts)
if n_calls < n_total_init_calls:
raise ValueError(
"Expected `n_calls` >= %d, got %d" % (n_total_init_calls, n_calls))
if y0 is None and x0:
y0 = [func(x) for x in x0]
elif x0:
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")
else:
y0 = []
Xi = x0 + space.rvs(n_samples=n_random_starts, random_state=rng)
yi = y0 + [func(x) for x in Xi[len(x0):]]
if np.ndim(yi) != 1:
raise ValueError("`func` should return a scalar")
if search == "auto":
if space.is_real:
search = "lbfgs"
else:
search = "sampling"
elif search not in ["lbfgs", "sampling"]:
raise ValueError(
"Expected search to be 'lbfgs', 'sampling' or 'auto', "
"got %s" % search)
# Bayesian optimization loop
models = []
n_model_iter = n_calls - n_total_init_calls
for i in range(n_model_iter):
gp = clone(base_estimator)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
gp.fit(space.transform(Xi), yi)
models.append(gp)
if search == "sampling":
X = space.transform(space.rvs(n_samples=n_points,
random_state=rng))
values = _gaussian_acquisition(
X=X, model=gp, y_opt=np.min(yi), method=acq,
xi=xi, kappa=kappa)
next_x = X[np.argmin(values)]
elif search == "lbfgs":
best = np.inf
for j in range(n_restarts_optimizer):
x0 = space.transform(space.rvs(n_samples=1,
random_state=rng))[0]
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, a, _ = fmin_l_bfgs_b(
_acquisition, x0,
args=(gp, np.min(yi), acq, xi, kappa),
bounds=space.transformed_bounds,
approx_grad=True, maxiter=20)
if a < best:
next_x, best = x, a
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi.append(next_x)
yi.append(next_y)
# Pack results
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
res.specs = specs
return res
def gp_minimize(func,
bounds,
base_estimator=None,
acq="LCB",
xi=0.01,
kappa=1.96,
search="sampling",
maxiter=1000,
n_points=500,
n_start=10,
n_restarts_optimizer=5,
random_state=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 standared 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.
Parameters
----------
* `func` [callable]:
Function to minimize. Should take a array of parameters and
return the function values.
* `bounds` [array-like, shape=(n_parameters, 2)]:
- ``bounds[i][0]`` should give the lower bound of each parameter and
- ``bounds[i][1]`` should give the upper bound of each parameter.
* `base_estimator` [a Gaussian process estimator]:
The Gaussian process estimator to use for optimization.
* `acq` [string, default=`"LCB"`]:
Function to minimize over the gaussian prior. Can be either
- `"LCB"` for lower confidence bound,
- `"EI"` for expected improvement,
- `"PI"` for probability of improvement.
* `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"`.
* `search` [string, `"sampling"` or `"lbfgs"`]:
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to `"sampling"`, `n_points` are sampled randomly
and the Gaussian Process prior is updated with the point that gives
the best acquisition value over the Gaussian prior.
If search is set to `"lbfgs"`, then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquisition function
over the Gaussian prior.
* `maxiter` [int, default=1000]:
Number of iterations to find the minimum. Note that `n_start`
iterations are effectively discounted, such that total number of
function evaluations is at most `maxiter`.
* `n_points` [int, default=500]:
Number of points to sample to determine the next "best" point.
Useless if search is set to `"lbfgs"`.
* `n_start` [int, default=10]:
Number of random initialization points.
* `n_restarts_optimizer` [int, default=10]:
The number of restarts of the optimizer when `search` is `"lbfgs"`.
* `random_state` [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:
- `x` [float]: location of the minimum.
- `fun` [float]: function value at the minimum.
- `models`: surrogate models used for each iteration.
- `x_iters` [array]: location of function evaluation for each
iteration.
- `func_vals` [array]: function value for each iteration.
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)
# Bounds
n_params = len(bounds)
lb, ub = extract_bounds(bounds)
# Default GP
if base_estimator is None:
base_estimator = GaussianProcessRegressor(
kernel=(ConstantKernel(1.0, (0.01, 1000.0)) *
Matern(length_scale=np.ones(n_params),
length_scale_bounds=[(0.01, 100)] * n_params,
nu=2.5)),
normalize_y=True,
alpha=10e-6,
random_state=random_state)
# First points
Xi = lb + (ub - lb) * rng.rand(n_start, n_params)
yi = [func(x) for x in Xi]
if np.ndim(yi) != 1:
raise ValueError("The function to be optimized should return a scalar")
# Bayesian optimization loop
models = []
for i in range(maxiter - n_start):
gp = clone(base_estimator)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
gp.fit(Xi, yi)
models.append(gp)
if search == "sampling":
X = lb + (ub - lb) * rng.rand(n_points, n_params)
values = acquisition(X=X,
model=gp,
y_opt=np.min(yi),
method=acq,
xi=xi,
kappa=kappa)
next_x = X[np.argmin(values)]
elif search == "lbfgs":
best = np.inf
for j in range(n_restarts_optimizer):
x0 = lb + (ub - lb) * rng.rand(n_params)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, a, _ = fmin_l_bfgs_b(_acquisition,
x0,
args=(gp, np.min(yi), acq, xi,
kappa),
bounds=bounds,
approx_grad=True,
maxiter=10)
if a < best:
next_x, best = x, a
next_y = func(next_x)
Xi = np.vstack((Xi, next_x))
yi.append(next_y)
# Pack results
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
return res
def gp_minimize(func, dimensions, base_estimator=None, acq="LCB", xi=0.01,
kappa=1.96, search="sampling", maxiter=1000, n_points=500,
n_start=10, n_restarts_optimizer=5, random_state=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 standared 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.
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` [a Gaussian process estimator]:
The Gaussian process estimator to use for optimization.
* `acq` [string, default=`"LCB"`]:
Function to minimize over the gaussian prior. Can be either
- `"LCB"` for lower confidence bound,
- `"EI"` for expected improvement,
- `"PI"` for probability of improvement.
* `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"`.
* `search` [string, `"sampling"` or `"lbfgs"`]:
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to `"sampling"`, `n_points` are sampled randomly
and the Gaussian Process prior is updated with the point that gives
the best acquisition value over the Gaussian prior.
If search is set to `"lbfgs"`, then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquisition function
over the Gaussian prior.
* `maxiter` [int, default=1000]:
Number of iterations to find the minimum. Note that `n_start`
iterations are effectively discounted, such that total number of
function evaluations is at most `maxiter`.
* `n_points` [int, default=500]:
Number of points to sample to determine the next "best" point.
Useless if search is set to `"lbfgs"`.
* `n_start` [int, default=10]:
Number of random initialization points.
* `n_restarts_optimizer` [int, default=10]:
The number of restarts of the optimizer when `search` is `"lbfgs"`.
* `random_state` [int, RandomState instance, or None (default)]:
Set random state to something other than None for reproducible
results.
Returns
-------
* `res` [`OptimizeResult`, scipy object]:
The optimization result returned as a OptimizeResult object.
Important attributes are:
- `x` [float]: location of the minimum.
- `fun` [float]: function value at the minimum.
- `models`: surrogate models used for each iteration.
- `x_iters` [array]: location of function evaluation for each
iteration.
- `func_vals` [array]: function value for each iteration.
- `space` [Space]: the optimisation space.
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)
space = Space(dimensions)
# Default GP
if base_estimator is None:
base_estimator = GaussianProcessRegressor(
kernel=(ConstantKernel(1.0, (0.01, 1000.0)) *
Matern(length_scale=np.ones(space.transformed_n_dims),
length_scale_bounds=[(0.01, 100)] * space.transformed_n_dims,
nu=2.5)),
normalize_y=True, alpha=10e-6, random_state=random_state)
# First points
Xi = space.rvs(n_samples=n_start, random_state=rng)
yi = [func(x) for x in Xi]
if np.ndim(yi) != 1:
raise ValueError(
"The function to be optimized should return a scalar")
# Bayesian optimization loop
models = []
for i in range(maxiter - n_start):
gp = clone(base_estimator)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
gp.fit(space.transform(Xi), yi)
models.append(gp)
if search == "sampling":
X = space.transform(space.rvs(n_samples=n_points,
random_state=rng))
values = _gaussian_acquisition(
X=X, model=gp, y_opt=np.min(yi), method=acq,
xi=xi, kappa=kappa)
next_x = X[np.argmin(values)]
elif search == "lbfgs":
best = np.inf
for j in range(n_restarts_optimizer):
x0 = space.transform(space.rvs(n_samples=1,
random_state=rng))[0]
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, a, _ = fmin_l_bfgs_b(
_acquisition, x0,
args=(gp, np.min(yi), acq, xi, kappa),
bounds=space.transformed_bounds,
approx_grad=True, maxiter=10)
if a < best:
next_x, best = x, a
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi = np.vstack((Xi, next_x))
yi.append(next_y)
# Pack results
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
return res
def _tree_minimize(func,
dimensions,
base_estimator,
n_calls,
n_points,
n_random_starts,
x0=None,
y0=None,
random_state=None,
acq="EI",
xi=0.01,
kappa=1.96):
rng = check_random_state(random_state)
space = Space(dimensions)
# Initialize with provided points (x0 and y0) and/or random points
if n_calls <= 0:
raise ValueError("Expected `n_calls` > 0, got %d" % n_random_starts)
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, but got %s" % type(x0))
n_init_func_calls = len(x0) if y0 is not None else 0
n_total_init_calls = n_random_starts + n_init_func_calls
if n_total_init_calls <= 0:
# if x0 is not provided and n_random_starts is 0 then
# it will ask for n_random_starts to be > 0.
raise ValueError("Expected `n_random_starts` > 0, got %d" %
n_random_starts)
if n_calls < n_total_init_calls:
raise ValueError("Expected `n_calls` >= %d, got %d" %
(n_total_init_calls, n_calls))
if y0 is None and x0:
y0 = [func(x) for x in x0]
elif x0:
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")
else:
y0 = []
Xi = x0 + space.rvs(n_samples=n_random_starts, random_state=rng)
yi = y0 + [func(x) for x in Xi[len(x0):]]
if np.ndim(yi) != 1:
raise ValueError("`func` should return a scalar")
# Tree-based optimization loop
models = []
n_model_iter = n_calls - n_total_init_calls
for i in range(n_model_iter):
rgr = clone(base_estimator)
rgr.fit(space.transform(Xi), yi)
models.append(rgr)
# `rgr` predicts constants for each leaf which means that the EI
# has zero gradient over large distances. As a result we can not
# use gradient based optimizers like BFGS, so using random sampling
# for the moment.
X = space.transform(space.rvs(n_samples=n_points, random_state=rng))
values = _gaussian_acquisition(X=X,
model=rgr,
y_opt=np.min(yi),
method=acq,
xi=xi,
kappa=kappa)
next_x = X[np.argmin(values)]
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi.append(next_x)
yi.append(next_y)
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
return res
def dummy_minimize(func, dimensions, n_calls=100,
x0=None, y0=None, random_state=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.
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))
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)
elif len(x0) == 0 and y0 is not None:
raise ValueError("`x0`cannot be `None` when `y0` is provided")
else: # len(x0) == 0 and y0 is None
y0 = []
X = x0
y = y0
# Random search
X = X + space.rvs(n_samples=n_calls, random_state=rng)
first = True
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)
y = np.array(y)
# Pack results
res = OptimizeResult()
best = np.argmin(y)
res.x = X[best]
res.fun = y[best]
res.func_vals = y
res.x_iters = X
res.models = [] # Create attribute even though it is empty
res.space = space
res.random_state = rng
res.specs = specs
return res
def gp_minimize(func,
dimensions,
base_estimator=None,
alpha=10e-10,
acq="EI",
xi=0.01,
kappa=1.96,
search="auto",
n_calls=100,
n_points=500,
n_random_starts=10,
n_restarts_optimizer=5,
x0=None,
y0=None,
random_state=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_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` [a Gaussian process estimator]:
The Gaussian process estimator to use for optimization.
* `alpha` [float, default=1e-10]:
Value added to the diagonal of the kernel matrix during fitting.
Larger values correspond to an increased noise level in the
observations and reduce potential numerical issues during fitting.
* `acq` [string, default=`"EI"`]:
Function to minimize over the gaussian prior. Can be either
- `"LCB"` for lower confidence bound,
- `"EI"` for expected improvement,
- `"PI"` for probability of improvement.
* `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"`.
* `search` [string, `"auto"`, `"sampling"` or `"lbfgs"`, default=`"auto"`]:
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to `"auto"`, then it is set to `"lbfgs"`` if
all the search dimensions are Real(continuous). It defaults to
`"sampling"` for all other cases.
If search is set to `"sampling"`, `n_points` are sampled randomly
and the Gaussian Process prior is updated with the point that gives
the best acquisition value over the Gaussian prior.
If search is set to `"lbfgs"`, then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquisition function
over the Gaussian prior.
* `n_calls` [int, default=100]:
Number of calls to `func`.
* `n_points` [int, default=500]:
Number of points to sample to determine the next "best" point.
Useless if search is set to `"lbfgs"`.
* `n_random_starts` [int, default=10]:
Number of evaluations of `func` with random initialization points
before approximating the `func` with `base_estimator`.
* `n_restarts_optimizer` [int, default=10]:
The number of restarts of the optimizer when `search` is `"lbfgs"`.
* `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.
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
"""
# 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)
# Default GP
if base_estimator is None:
base_estimator = GaussianProcessRegressor(
kernel=(ConstantKernel(1.0, (0.01, 1000.0)) * Matern(
length_scale=np.ones(space.transformed_n_dims),
length_scale_bounds=[(0.01, 100)] * space.transformed_n_dims,
nu=2.5)),
normalize_y=True,
alpha=alpha,
random_state=random_state)
# Initialize with provided points (x0 and y0) and/or random points
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, but got %s" % type(x0))
n_init_func_calls = len(x0) if y0 is not None else 0
n_total_init_calls = n_random_starts + n_init_func_calls
if n_total_init_calls <= 0:
# if x0 is not provided and n_random_starts is 0 then
# it will ask for n_random_starts to be > 0.
raise ValueError("Expected `n_random_starts` > 0, got %d" %
n_random_starts)
if n_calls < n_total_init_calls:
raise ValueError("Expected `n_calls` >= %d, got %d" %
(n_total_init_calls, n_calls))
if y0 is None and x0:
y0 = [func(x) for x in x0]
elif x0:
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")
else:
y0 = []
Xi = x0 + space.rvs(n_samples=n_random_starts, random_state=rng)
yi = y0 + [func(x) for x in Xi[len(x0):]]
if np.ndim(yi) != 1:
raise ValueError("`func` should return a scalar")
if search == "auto":
if space.is_real:
search = "lbfgs"
else:
search = "sampling"
elif search not in ["lbfgs", "sampling"]:
raise ValueError(
"Expected search to be 'lbfgs', 'sampling' or 'auto', "
"got %s" % search)
# Bayesian optimization loop
models = []
n_model_iter = n_calls - n_total_init_calls
for i in range(n_model_iter):
gp = clone(base_estimator)
with warnings.catch_warnings():
warnings.simplefilter("ignore")
gp.fit(space.transform(Xi), yi)
models.append(gp)
if search == "sampling":
X = space.transform(space.rvs(n_samples=n_points,
random_state=rng))
values = _gaussian_acquisition(X=X,
model=gp,
y_opt=np.min(yi),
method=acq,
xi=xi,
kappa=kappa)
next_x = X[np.argmin(values)]
elif search == "lbfgs":
best = np.inf
for j in range(n_restarts_optimizer):
x0 = space.transform(space.rvs(n_samples=1,
random_state=rng))[0]
with warnings.catch_warnings():
warnings.simplefilter("ignore")
x, a, _ = fmin_l_bfgs_b(_acquisition,
x0,
args=(gp, np.min(yi), acq, xi,
kappa),
bounds=space.transformed_bounds,
approx_grad=True,
maxiter=20)
if a < best:
next_x, best = x, a
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi.append(next_x)
yi.append(next_y)
# Pack results
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
res.specs = specs
return res
def gp_minimize(func, bounds=None, search="sampling", random_state=None,
maxiter=1000, acq="UCB", num_points=500):
"""
Black-box 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 standared 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 acquistion function
over the Gaussian posterior which is much quicker to evaluate.
Parameters
----------
func: callable
Function to minimize. Should take a array of parameters and
return the function value.
bounds: array-like, shape (n_parameters, 2)
``bounds[i][0]`` should give the lower bound of each parameter and
``bounds[i][1]`` should give the upper bound of each parameter.
search: string, "sampling" or "lbfgs"
Searching for the next possible candidate to update the Gaussian prior
with.
If search is set to "sampling", ``num_points`` are sampled randomly
and the Gaussian Process prior is updated with that point that gives
the best acquision value over the Gaussian posterior.
If search is set to "lbfgs", then a point is sampled randomly, and
lbfgs is run for 10 iterations optimizing the acquistion function
over the Gaussian posterior.
random_state: int, RandomState instance, or None (default)
Set random state to something other than None for reproducible
results.
maxiter: int, default 1000
Number of iterations to find the minimum. In other words, the
number of function evaluations.
acq: string, default "UCB"
Function to minimize over the gaussian posterior. Can be either
the "UCB" which refers to the UpperConfidenceBound or "EI" which
is the Expected Improvement.
num_points: int, default 500
Number of points to sample to determine the next "best" point.
Useless if search is set to "lbfgs".
Returns
-------
res: OptimizeResult, scipy object
The optimization result returned as a OptimizeResult object.
Important attributes are
``x`` - float, the optimization solution,
``fun`` - float, the value of the function at the optimum,
``models``- gp_models[i]. the prior on the function fit at
iteration[i].
``func_vals`` - the function value at the ith iteration.
``x_iters`` - the value of ``x`` corresponding to the function value
at the ith iteration.
For more details related to the OptimizeResult object, refer
http://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.OptimizeResult.html
"""
rng = np.random.RandomState(random_state)
num_params = len(bounds)
lower_bounds, upper_bounds = zip(*bounds)
upper_bounds = np.asarray(upper_bounds)
lower_bounds = np.asarray(lower_bounds)
x0 = rng.rand(num_params)
func_val = [func(lower_bounds + (upper_bounds - lower_bounds) * x0)]
length_scale = np.ones(num_params)
gp_params = {
'kernel': Matern(length_scale=length_scale, nu=2.5),
'normalize_y': True,
'random_state': random_state
}
lbfgs_bounds = np.tile((0, 1), (num_params, 1))
gp_models = []
x = np.reshape(x0, (1, -1))
for i in range(maxiter):
gpr = GaussianProcessRegressor(**gp_params)
gpr.fit(x, func_val)
if search == "sampling":
sampling = rng.rand(num_points, num_params)
acquis = acquisition_func(sampling, gpr, np.min(func_val), acq)
best_arg = np.argmin(acquis)
best_x = sampling[best_arg]
elif search == "lbfgs":
init = rng.rand(num_params)
best_x, _, _ = fmin_l_bfgs_b(
acquisition_func,
np.asfortranarray(init),
args=(gpr, np.min(func_val), acq),
bounds=lbfgs_bounds, approx_grad=True, maxiter=10)
gp_models.append(gpr)
best_f = func(lower_bounds + (upper_bounds - lower_bounds) * best_x)
x_list = x.tolist()
x_list.append(best_x)
x = np.asarray(x_list)
func_val.append(best_f)
x = lower_bounds + (upper_bounds - lower_bounds) * x
func_ind = np.argmin(func_val)
x_val = x[func_ind]
best_func_val = func_val[func_ind]
res = OptimizeResult()
res.models = gp_models
res.x = x_val
res.fun = best_func_val
res.func_vals = func_val
res.x_iters = x
return res
def _tree_minimize(func, dimensions, base_estimator, n_calls,
n_points, n_random_starts, x0=None, y0=None,
random_state=None, acq="EI", xi=0.01, kappa=1.96):
rng = check_random_state(random_state)
space = Space(dimensions)
# Initialize with provided points (x0 and y0) and/or random points
if n_calls <= 0:
raise ValueError(
"Expected `n_calls` > 0, got %d" % n_random_starts)
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, but got %s" % type(x0))
n_init_func_calls = len(x0) if y0 is not None else 0
n_total_init_calls = n_random_starts + n_init_func_calls
if n_total_init_calls <= 0:
# if x0 is not provided and n_random_starts is 0 then
# it will ask for n_random_starts to be > 0.
raise ValueError(
"Expected `n_random_starts` > 0, got %d" % n_random_starts)
if n_calls < n_total_init_calls:
raise ValueError(
"Expected `n_calls` >= %d, got %d" % (n_total_init_calls, n_calls))
if y0 is None and x0:
y0 = [func(x) for x in x0]
elif x0:
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")
else:
y0 = []
Xi = x0 + space.rvs(n_samples=n_random_starts, random_state=rng)
yi = y0 + [func(x) for x in Xi[len(x0):]]
if np.ndim(yi) != 1:
raise ValueError("`func` should return a scalar")
# Tree-based optimization loop
models = []
n_model_iter = n_calls - n_total_init_calls
for i in range(n_model_iter):
rgr = clone(base_estimator)
rgr.fit(space.transform(Xi), yi)
models.append(rgr)
# `rgr` predicts constants for each leaf which means that the EI
# has zero gradient over large distances. As a result we can not
# use gradient based optimizers like BFGS, so using random sampling
# for the moment.
X = space.transform(space.rvs(n_samples=n_points,
random_state=rng))
values = _gaussian_acquisition(
X=X, model=rgr, y_opt=np.min(yi), method=acq,
xi=xi, kappa=kappa)
next_x = X[np.argmin(values)]
next_x = space.inverse_transform(next_x.reshape((1, -1)))[0]
next_y = func(next_x)
Xi.append(next_x)
yi.append(next_y)
res = OptimizeResult()
best = np.argmin(yi)
res.x = Xi[best]
res.fun = yi[best]
res.func_vals = np.array(yi)
res.x_iters = Xi
res.models = models
res.space = space
res.random_state = rng
return res
