qEI.grad: Gradient of the multipoint expected improvement (qEI) criterion

Description

Computes an exact or approximate gradient of the multipoint expected improvement criterion

Usage

qEI.grad(x, model, plugin = NULL, type = "UK", minimization = TRUE, fastCompute = TRUE, eps = 10^(-6), envir = NULL)

Arguments

a matrix representing the set of input points (one row corresponds to one point) where to evaluate the gradient,

model

an object of class km,

plugin

optional scalar: if provided, it replaces the minimum of the current observations,

type

"SK" or "UK" (by default), depending whether uncertainty related to trend estimation has to be taken into account,

minimization

logical specifying if EI is used in minimiziation or in maximization,

fastCompute

if TRUE, a fast approximation method based on a semi-analytic formula is used (see [Marmin 2014] for details),

eps

the value of epsilon of the fast computation trick. Relevant only if fastComputation is TRUE,

envir

an optional environment specifying where to get intermediate values calculated in qEI.

Value

The gradient of the multipoint expected improvement criterion with respect to x. A 0-matrix is returned if the batch of input points contains twice the same point or a point from the design experiment of the km object (the gradient does not exist in these cases).

References

C. Chevalier and D. Ginsbourger (2013), Fast computation of the multi-points expected improvement with applications in batch selection. In G. Nicosia and P. Pardalos (Eds.) Learning and Intelligent Optimization, Lecture Notes in Computer Science, pp 59-69, Springer.

D. Ginsbourger, R. Le Riche, L. Carraro (2007), A Multipoint Criterion for Deterministic Parallel Global Optimization based on Kriging. The International Conference on Non Convex Programming, 2007.

D. Ginsbourger, R. Le Riche, and L. Carraro. Kriging is well-suited to parallelize optimization (2010), In Lim Meng Hiot, Yew Soon Ong, Yoel Tenne, and Chi-Keong Goh, editors, Computational Intelligence in Expensive Optimization Problems, Adaptation Learning and Optimization, pages 131-162. Springer Berlin Heidelberg.

S. Marmin, C. Chevalier and D. Ginsbourger (2015), Differentiating the multipoint Expected Improvement for optimal batch design. In P. Pardalos et al. (Eds.) Machine Learning, Optimization, and Big Data, Lecture Notes in Computer Science 9432, pp 37-48.

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

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

Examples

Run this code

## Not run: 
# set.seed(15)
# # Example 1 - validation by comparison to finite difference approximations
# 
# # a 10-points optimum LHS design and the corresponding responses
# d <- 2;n <- d*5
# design <- maximinESE_LHS(lhsDesign(n,d)$design,1)$design
# colnames(design)<-c("x1", "x2")
# lower <- c(0,0)
# upper <- c(1,1)
# y <- data.frame(apply(design, 1, branin))
# names(y) <- "y"
# 
# # learning
# model <- km(~1, design=design, response=y)
# 
# # pick up 4 points sampled from the simple expected improvement
# q <- 4
# X <- sampleFromEI(model,n=q)
# 
# # compute the gradient at the 4-point batch
# grad.analytic <- qEI.grad(X,model)
# # numerically compute the gradient
# grad.numeric <- matrix(NaN,q,d)
# eps <- 10^(-6)
# EPS <- matrix(0,q,d)
# for (i in 1:q) {
#   for (j in 1:d) {
#     EPS[i,j] <- eps
#     grad.numeric[i,j] <- 1/eps*(qEI(X+EPS,model,fastCompute=FALSE)-qEI(X,model,fastCompute=FALSE))
#     EPS[i,j] <- 0
#   }
# }
# print(grad.numeric)
# print(grad.analytic)
# 
# # graphics: displays the EI criterion, the design points in black,
# # the batch points in red and the gradient in blue.
# nGrid <- 15
# gridAxe1 <- seq(lower[1],upper[1],length=nGrid)
# gridAxe2 <- seq(lower[2],upper[2],length=nGrid)
# grid <- expand.grid(gridAxe1,gridAxe2)
# aa <- apply(grid,1,EI,model=model)
# myMat <- matrix(aa,nrow=nGrid)
# image(x = gridAxe1, y = gridAxe2, z = myMat,
#       col = colorRampPalette(c("darkgray","white"))(5*10),
#       ylab = names(design)[1], xlab=names(design)[2],
#       main = "qEI-gradient of a batch of 4 points", axes = TRUE,
#       zlim = c(min(myMat), max(myMat)))
# contour(x = gridAxe1, y = gridAxe2, z = myMat,
#         add = TRUE, nlevels = 10)
# points(X[,1],X[,2],pch=19,col='red')
# points(model@X[,1],model@X[,2],pch=19)
# arrows(X[,1],X[,2],X[,1]+0.012*grad.analytic[,1],X[,2]+0.012*grad.analytic[,2],col='blue')
# ## End(Not run)