import numpy as np
import warnings
from scipy.stats import norm
[docs]def gaussian_acquisition_1D(X, model, y_opt=None, acq_func="LCB",
acq_func_kwargs=None, return_grad=True):
"""
A wrapper around the acquisition function that is called by fmin_l_bfgs_b.
This is because lbfgs allows only 1-D input.
"""
return _gaussian_acquisition(np.expand_dims(X, axis=0),
model, y_opt, acq_func=acq_func,
acq_func_kwargs=acq_func_kwargs,
return_grad=return_grad)
def _gaussian_acquisition(X, model, y_opt=None, acq_func="LCB",
return_grad=False, acq_func_kwargs=None):
"""
Wrapper so that the output of this function can be
directly passed to a minimizer.
"""
# Check inputs
X = np.asarray(X)
if X.ndim != 2:
raise ValueError("X is {}-dimensional, however,"
" it must be 2-dimensional.".format(X.ndim))
if acq_func_kwargs is None:
acq_func_kwargs = dict()
xi = acq_func_kwargs.get("xi", 0.01)
kappa = acq_func_kwargs.get("kappa", 1.96)
# Evaluate acquisition function
per_second = acq_func.endswith("ps")
if per_second:
model, time_model = model.estimators_
if acq_func == "LCB":
func_and_grad = gaussian_lcb(X, model, kappa, return_grad)
if return_grad:
acq_vals, acq_grad = func_and_grad
else:
acq_vals = func_and_grad
elif acq_func in ["EI", "PI", "EIps", "PIps"]:
if acq_func in ["EI", "EIps"]:
func_and_grad = gaussian_ei(X, model, y_opt, xi, return_grad)
else:
func_and_grad = gaussian_pi(X, model, y_opt, xi, return_grad)
if return_grad:
acq_vals = -func_and_grad[0]
acq_grad = -func_and_grad[1]
else:
acq_vals = -func_and_grad
if acq_func in ["EIps", "PIps"]:
if return_grad:
mu, std, mu_grad, std_grad = time_model.predict(
X, return_std=True, return_mean_grad=True,
return_std_grad=True)
else:
mu, std = time_model.predict(X, return_std=True)
# acq = acq / E(t)
inv_t = np.exp(-mu + 0.5*std**2)
acq_vals *= inv_t
# grad = d(acq_func) * inv_t + (acq_vals *d(inv_t))
# inv_t = exp(g)
# d(inv_t) = inv_t * grad(g)
# d(inv_t) = inv_t * (-mu_grad + std * std_grad)
if return_grad:
acq_grad *= inv_t
acq_grad += acq_vals * (-mu_grad + std*std_grad)
else:
raise ValueError("Acquisition function not implemented.")
if return_grad:
return acq_vals, acq_grad
return acq_vals
[docs]def gaussian_lcb(X, model, kappa=1.96, return_grad=False):
"""
Use the lower confidence bound to estimate the acquisition
values.
The trade-off between exploitation and exploration is left to
be controlled by the user through the parameter ``kappa``.
Parameters
----------
X : array-like, shape (n_samples, n_features)
Values where the acquisition function should be computed.
model : sklearn estimator that implements predict with ``return_std``
The fit estimator that approximates the function through the
method ``predict``.
It should have a ``return_std`` parameter that returns the standard
deviation.
kappa : float, default 1.96 or 'inf'
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.
If set to 'inf', the acquisition function will only use the variance
which is useful in a pure exploration setting.
Useless if ``method`` is not set to "LCB".
return_grad : boolean, optional
Whether or not to return the grad. Implemented only for the case where
``X`` is a single sample.
Returns
-------
values : array-like, shape (X.shape[0],)
Acquisition function values computed at X.
grad : array-like, shape (n_samples, n_features)
Gradient at X.
"""
# Compute posterior.
with warnings.catch_warnings():
warnings.simplefilter("ignore")
if return_grad:
mu, std, mu_grad, std_grad = model.predict(
X, return_std=True, return_mean_grad=True,
return_std_grad=True)
if kappa == "inf":
return -std, -std_grad
return mu - kappa * std, mu_grad - kappa * std_grad
else:
mu, std = model.predict(X, return_std=True)
if kappa == "inf":
return -std
return mu - kappa * std
[docs]def gaussian_pi(X, model, y_opt=0.0, xi=0.01, return_grad=False):
"""
Use the probability of improvement to calculate the acquisition values.
The conditional probability `P(y=f(x) | x)` form a gaussian with a
certain mean and standard deviation approximated by the model.
The PI condition is derived by computing ``E[u(f(x))]``
where ``u(f(x)) = 1``, if ``f(x) < y_opt`` and ``u(f(x)) = 0``,
if``f(x) > y_opt``.
This means that the PI condition does not care about how "better" the
predictions are than the previous values, since it gives an equal reward
to all of them.
Note that the value returned by this function should be maximized to
obtain the ``X`` with maximum improvement.
Parameters
----------
X : array-like, shape=(n_samples, n_features)
Values where the acquisition function should be computed.
model : sklearn estimator that implements predict with ``return_std``
The fit estimator that approximates the function through the
method ``predict``.
It should have a ``return_std`` parameter that returns the standard
deviation.
y_opt : float, default 0
Previous minimum value which we would like to improve upon.
xi : float, default=0.01
Controls how much improvement one wants over the previous best
values. Useful only when ``method`` is set to "EI"
return_grad : boolean, optional
Whether or not to return the grad. Implemented only for the case where
``X`` is a single sample.
Returns
-------
values : [array-like, shape=(X.shape[0],)
Acquisition function values computed at X.
"""
with warnings.catch_warnings():
warnings.simplefilter("ignore")
if return_grad:
mu, std, mu_grad, std_grad = model.predict(
X, return_std=True, return_mean_grad=True,
return_std_grad=True)
else:
mu, std = model.predict(X, return_std=True)
# check dimensionality of mu, std so we can divide them below
if (mu.ndim != 1) or (std.ndim != 1):
raise ValueError("mu and std are {}-dimensional and {}-dimensional, "
"however both must be 1-dimensional. Did you train "
"your model with an (N, 1) vector instead of an "
"(N,) vector?"
.format(mu.ndim, std.ndim))
values = np.zeros_like(mu)
mask = std > 0
improve = y_opt - xi - mu[mask]
scaled = improve / std[mask]
values[mask] = norm.cdf(scaled)
if return_grad:
if not np.all(mask):
return values, np.zeros_like(std_grad)
# Substitute (y_opt - xi - mu) / sigma = t and apply chain rule.
# improve_grad is the gradient of t wrt x.
improve_grad = -mu_grad * std - std_grad * improve
improve_grad /= std**2
return values, improve_grad * norm.pdf(scaled)
return values
[docs]def gaussian_ei(X, model, y_opt=0.0, xi=0.01, return_grad=False):
"""
Use the expected improvement to calculate the acquisition values.
The conditional probability `P(y=f(x) | x)` form a gaussian with a certain
mean and standard deviation approximated by the model.
The EI condition is derived by computing ``E[u(f(x))]``
where ``u(f(x)) = 0``, if ``f(x) > y_opt`` and ``u(f(x)) = y_opt - f(x)``,
if``f(x) < y_opt``.
This solves one of the issues of the PI condition by giving a reward
proportional to the amount of improvement got.
Note that the value returned by this function should be maximized to
obtain the ``X`` with maximum improvement.
Parameters
----------
X : array-like, shape=(n_samples, n_features)
Values where the acquisition function should be computed.
model : sklearn estimator that implements predict with ``return_std``
The fit estimator that approximates the function through the
method ``predict``.
It should have a ``return_std`` parameter that returns the standard
deviation.
y_opt : float, default 0
Previous minimum value which we would like to improve upon.
xi : float, default=0.01
Controls how much improvement one wants over the previous best
values. Useful only when ``method`` is set to "EI"
return_grad : boolean, optional
Whether or not to return the grad. Implemented only for the case where
``X`` is a single sample.
Returns
-------
values : array-like, shape=(X.shape[0],)
Acquisition function values computed at X.
"""
with warnings.catch_warnings():
warnings.simplefilter("ignore")
if return_grad:
mu, std, mu_grad, std_grad = model.predict(
X, return_std=True, return_mean_grad=True,
return_std_grad=True)
else:
mu, std = model.predict(X, return_std=True)
# check dimensionality of mu, std so we can divide them below
if (mu.ndim != 1) or (std.ndim != 1):
raise ValueError("mu and std are {}-dimensional and {}-dimensional, "
"however both must be 1-dimensional. Did you train "
"your model with an (N, 1) vector instead of an "
"(N,) vector?"
.format(mu.ndim, std.ndim))
values = np.zeros_like(mu)
mask = std > 0
improve = y_opt - xi - mu[mask]
scaled = improve / std[mask]
cdf = norm.cdf(scaled)
pdf = norm.pdf(scaled)
exploit = improve * cdf
explore = std[mask] * pdf
values[mask] = exploit + explore
if return_grad:
if not np.all(mask):
return values, np.zeros_like(std_grad)
# Substitute (y_opt - xi - mu) / sigma = t and apply chain rule.
# improve_grad is the gradient of t wrt x.
improve_grad = -mu_grad * std - std_grad * improve
improve_grad /= std ** 2
cdf_grad = improve_grad * pdf
pdf_grad = -improve * cdf_grad
exploit_grad = -mu_grad * cdf - pdf_grad
explore_grad = std_grad * pdf + pdf_grad
grad = exploit_grad + explore_grad
return values, grad
return values
