EI: Expected Improvement Criterion and its Gradient

In libKriging/dolka: Utility functions for Bayesian optimization

Expected Improvement Criterion and its Gradient

Description

The function EI computes the Expected Improvement at current location x while EI.grad compute the gradient at x. 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(
  x,
  model,
  plugin = NULL,
  type = c("UK", "SK"),
  minimization = TRUE,
  envir = NULL,
  proxy = FALSE
)

EI.grad(
  x,
  model,
  plugin = NULL,
  type = c("UK", "SK"),
  minimization = TRUE,
  envir = NULL,
  proxy = FALSE
)

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.
envir	An optional environment specifying where to assign intermediate values for future gradient calculations. Default is NULL.
proxy	Optional logical. If TRUE, EI is replaced by the kriging mean, to be minimized.

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

Examples

set.seed(123)
## =========================================================================
## 	EI Surface Associated with an Ordinary Kriging Model for the Branin    
##  Function Known at a 9-Points Factorial Design  
## =========================================================================

## a 9-points factorial design, and the corresponding response
## ===========================================================
d <- 2; n <- 9
design.fact <-
    expand.grid(x1 = seq(0, 1, length = 3), x2 = seq(0, 1, length = 3))
y <- apply(design.fact, 1, branin) 

## model fit
## =========
fit1 <- km(~1, design = design.fact, response = y, covtype = "gauss",
           control = list(pop.size = 50, trace = FALSE), parinit = c(0.5, 0.5))

## computing the EI
## ================
x <- c(x1 = 0.2, x2 = 0.4)
EI(x, model = fit1)
EI.grad(x, model = fit1)

## graphics
## ========
contours(object = fit1, which = character(0), grad = TRUE,
         other = "EI", otherGrad = "EI.grad",
         whereGrad = "grid", nGrid = 30) +
    ggtitle("Expected Improvement and its gradient")

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