EI_with_grad: Expected Improvement Criterion With Gradient

In libKriging/dolka: Utility functions for Bayesian optimization

Expected Improvement Criterion With Gradient

Description

The function EI_with_grad computes the Expected Improvement at current location x and its gradient if wanted. The current minimum of the observations in model can be replaced by an arbitrary value (plugin), which is useful in particular in noisy frameworks.

Usage

EI_with_grad(
  x,
  model,
  plugin = NULL,
  type = c("UK", "SK"),
  minimization = TRUE,
  proxy = FALSE,
  deriv = TRUE,
  out_list = deriv
)

Arguments

x	A numeric vector representing the input for which one wishes to calculate EI. The length d of this vector must be equal to d, the dimension of the input space used for the kriging results in model.
model	An object of class km.
plugin	Optional scalar: if provided, it replaces the minimum of the current observations.
type	"UK" (default) or "SK", depending whether uncertainty related to trend estimation has to be taken into account.
minimization	Logical specifying if EI is used in minimization or in maximization.
proxy	Optional logical. If TRUE, EI is replaced by the kriging mean, to be minimized.
deriv	Logical. If TRUE the result is a list with the two elements objective and gradient. Else the result is the value of the objective.
out_list	Logical When out_list is TRUE the result is a list with one element objective. If deriv is TRUE the list has a second element gradient. When out_list is FALSE the result is the numerical value of the objective, possibly having an attribute named "gradient".

Details

The Expected Improvement (EI) is defined as

E[{ min Y(X) - Y(x) }_+ | Y(X) = y(X)]

where X is the current design of experiments and Y is the random process assumed to have generated the objective function y and z_+ = max(z, 0) denotes the positive part of a real number z. The value of EI is non-negative but can be numerically zero close to the inputs used in model. The EI and its gradient are computed using their closed forms.

Value

The expected improvement as defined in Details (for EI) or its gradient (for EI.grad). If plugin is specified, its provided value will replace min Y(X) in the formula. The EI and its gradient are numeric vectors with length 1 and d.

Author(s)

David Ginsbourger, Olivier Roustant and Victor Picheny.

References

D. Ginsbourger (2009), Multiples métamodèles pour l'approximation et l'optimisation de fonctions numériques multivariables, Ph.D. thesis, Ècole Nationale Supérieure des Mines de Saint-Ètienne.

D.R. Jones, M. Schonlau, and W.J. Welch (1998), Efficient global optimization of expensive black-box functions, Journal of Global Optimization, 13, 455-492.

J. Mockus (1988), Bayesian Approach to Global Optimization. Kluwer academic publishers.

T.J. Santner, B.J. Williams, and W.J. Notz (2003), The design and analysis of computer experiments, Springer.

M. Schonlau (1997), Computer experiments and global optimization, Ph.D. thesis, University of Waterloo.

See Also

max_EI, EGO.nsteps, qEI, EI and EI.grad.

libKriging/dolka documentation built on April 14, 2022, 7:17 a.m.