class ProbabilityOfImprovement(Acquisition):
"""
Probability of Improvement acquisition function for single-objective global optimization.
.. math::
\\alpha(\\mathbf x_{\\star}) = \\int_{-\\infty}^{f_{\\min}} \\, p( f_{\\star}\\,|\\, \\mathbf x, \\mathbf y, \\mathbf x_{\\star} ) \\, d f_{\\star}
"""
def __init__(self, model):
"""
:param model: GPflow model (single output) representing our belief of the objective
"""
super(ProbabilityOfImprovement, self).__init__(model)
self.fmin = DataHolder(np.zeros(1))
self.setup()
def setup(self):
super(ProbabilityOfImprovement, self).setup()
samples_mean, _ = self.models[0].predict_f(self.data[0])
self.fmin.set_data(np.min(samples_mean, axis=0))
def build_acquisition(self, Xcand):
candidate_mean, candidate_var = self.models[0].build_predict(Xcand)
candidate_var = tf.maximum(candidate_var, stability)
normal = tf.contrib.distributions.Normal(candidate_mean,
tf.sqrt(candidate_var))
return normal.cdf(self.fmin, name=self.__class__.__name__)
class ExpectedImprovement(Acquisition):
"""
Expected Improvement acquisition function for single-objective global optimization.
Introduced by (Mockus et al, 1975).
Key reference:
::
@article{Jones:1998,
title={Efficient global optimization of expensive black-box functions},
author={Jones, Donald R and Schonlau, Matthias and Welch, William J},
journal={Journal of Global optimization},
volume={13},
number={4},
pages={455--492},
year={1998},
publisher={Springer}
}
This acquisition function is the expectation of the improvement over the current best observation
w.r.t. the predictive distribution. The definition is closely related to the :class:`.ProbabilityOfImprovement`,
but adds a multiplication with the improvement w.r.t the current best observation to the integral.
.. math::
\\alpha(\\mathbf x_{\\star}) = \\int \\max(f_{\\min} - f_{\\star}, 0) \\, p( f_{\\star}\\,|\\, \\mathbf x, \\mathbf y, \\mathbf x_{\\star} ) \\, d f_{\\star}
"""
def __init__(self, model):
"""
:param model: GPflow model (single output) representing our belief of the objective
"""
super(ExpectedImprovement, self).__init__(model)
assert (isinstance(model, Model))
self.fmin = DataHolder(np.zeros(1))
self.setup()
def setup(self):
super(ExpectedImprovement, self).setup()
# Obtain the lowest posterior mean for the previous - feasible - evaluations
feasible_samples = self.data[0][
self.highest_parent.feasible_data_index(), :]
samples_mean, _ = self.models[0].predict_f(feasible_samples)
self.fmin.set_data(np.min(samples_mean, axis=0))
def build_acquisition(self, Xcand):
# Obtain predictive distributions for candidates
candidate_mean, candidate_var = self.models[0].build_predict(Xcand)
candidate_var = tf.maximum(candidate_var, stability)
# Compute EI
normal = tf.contrib.distributions.Normal(candidate_mean,
tf.sqrt(candidate_var))
t1 = (self.fmin - candidate_mean) * normal.cdf(self.fmin)
t2 = candidate_var * normal.prob(self.fmin)
return tf.add(t1, t2, name=self.__class__.__name__)
class LinearTransform(DataTransform):
"""
A simple linear transform of the form
.. math::
\\mathbf Y = (\\mathbf A \\mathbf X^{T})^{T} + \\mathbf b \\otimes \\mathbf 1_{N}^{T}
"""
def __init__(self, A, b):
"""
:param A: scaling matrix. Either a P-dimensional vector, or a P x P transformation matrix. For the latter,
the inverse and backward methods are not guaranteed to work as A must be invertible.
It is also possible to specify a matrix with size P x Q with Q != P to achieve
a lower dimensional representation of X.
In this case, A is not invertible, hence inverse and backward transforms are not supported.
:param b: A P-dimensional offset vector.
"""
super(LinearTransform, self).__init__()
assert A is not None
assert b is not None
b = np.atleast_1d(b)
A = np.atleast_1d(A)
if len(A.shape) == 1:
A = np.diag(A)
assert (len(b.shape) == 1)
assert (len(A.shape) == 2)
self.A = DataHolder(A)
self.b = DataHolder(b)
def build_forward(self, X):
return tf.matmul(X, tf.transpose(self.A)) + self.b
@AutoFlow((float_type, [None, None]))
def backward(self, Y):
"""
Overwrites the default backward approach, to avoid an explicit matrix inversion.
"""
return self.build_backward(Y)
def build_backward(self, Y):
"""
TensorFlow implementation of the inverse mapping
"""
L = tf.cholesky(tf.transpose(self.A))
XT = tf.cholesky_solve(L, tf.transpose(Y-self.b))
return tf.transpose(XT)
def build_backward_variance(self, Yvar):
"""
Additional method for scaling variance backward (used in :class:`.Normalizer`). Can process both the diagonal
variances returned by predict_f, as well as full covariance matrices.
:param Yvar: size N x N x P or size N x P
:return: Yvar scaled, same rank and size as input
"""
rank = tf.rank(Yvar)
# Because TensorFlow evaluates both fn1 and fn2, the transpose can't be in the same line. If a full cov
# matrix is provided fn1 turns it into a rank 4, then tries to transpose it as a rank 3.
# Splitting it in two steps however works fine.
Yvar = tf.cond(tf.equal(rank, 2), lambda: tf.matrix_diag(tf.transpose(Yvar)), lambda: Yvar)
Yvar = tf.cond(tf.equal(rank, 2), lambda: tf.transpose(Yvar, perm=[1, 2, 0]), lambda: Yvar)
N = tf.shape(Yvar)[0]
D = tf.shape(Yvar)[2]
L = tf.cholesky(tf.square(tf.transpose(self.A)))
Yvar = tf.reshape(Yvar, [N * N, D])
scaled_var = tf.reshape(tf.transpose(tf.cholesky_solve(L, tf.transpose(Yvar))), [N, N, D])
return tf.cond(tf.equal(rank, 2), lambda: tf.reduce_sum(scaled_var, axis=1), lambda: scaled_var)
def assign(self, other):
"""
Assign the parameters of another :class:`LinearTransform`. Can be useful to avoid graph
re-compilation.
:param other: :class:`.LinearTransform` object
"""
assert other is not None
assert isinstance(other, LinearTransform)
self.A.set_data(other.A.value)
self.b.set_data(other.b.value)
def __invert__(self):
A_inv = np.linalg.inv(self.A.value.T)
return LinearTransform(A_inv, -np.dot(self.b.value, A_inv))
def __str__(self):
return 'XA + b'