qEI: Analytical expression of the multipoint expected improvement (qEI) criterion

Description

Computes the multipoint expected improvement criterion.

Usage

qEI(
  x,
  model,
  plugin = NULL,
  type = "UK",
  minimization = TRUE,
  fastCompute = TRUE,
  eps = 10^(-5),
  envir = NULL
)

Arguments

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

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 multipoint Expected Improvement, defined as $$qEI(X_{new}) := E[ ( min(Y(X)) - min(Y( X_{new} )) )_{+} | Y(X)=y(X)],$$

where $X$ is the current design of experiments, $ X_{new} $ is a new candidate design, and $Y$ is a random process assumed to have generated the objective function $y$.

References

C. Chevalier and D. Ginsbourger (2014) Learning and Intelligent Optimization - 7th International Conference, Lion 7, Catania, Italy, January 7-11, 2013, Revised Selected Papers, chapter Fast computation of the multipoint Expected Improvement with applications in batch selection, pages 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.

S. Marmin. Developpements pour l'evaluation et la maximisation du critere d'amelioration esperee multipoint en optimisation globale (2014). Master's thesis, Mines Saint-Etienne (France) and University of Bern (Switzerland).

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.

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 {
# }
# NOT RUN {
set.seed(007)

# Monte-Carlo validation

# a 4-d, 81-points grid design, and the corresponding response
d <- 4; n <- 3^d
design <- do.call(expand.grid,rep(list(seq(0,1,length=3)),d))
names(design) <- paste("x",1:d,sep="")
y <- data.frame(apply(design, 1, hartman4))
names(y) <- "y"

# learning
model <- km(~1, design=design, response=y, control=list(trace=FALSE))

# pick up 10 points sampled from the 1-point expected improvement
q <- 10
X <- sampleFromEI(model,n=q)
# simulation of the minimum of the kriging random vector at X
t1 <- proc.time()
newdata <- as.data.frame(X)
colnames(newdata) <- colnames(model@X)

krig  <- predict(object=model, newdata=newdata,type="UK",se.compute=TRUE, cov.compute=TRUE)
mk <- krig$mean
Sigma.q <- krig$cov
mychol <- chol(Sigma.q)
nsim <- 300000
white.noise <- rnorm(n=nsim*q)
minYsim <- apply(crossprod(mychol,matrix(white.noise,nrow=q)) + mk,2,min)
# simulation of the improvement (minimization)
qImprovement <- (min(model@y)-minYsim)*((min(model@y)-minYsim) > 0)

# empirical expectation of the improvement and confident interval (95%)
eiMC <- mean(qImprovement)
confInterv <- c(eiMC - 1.96*sd(qImprovement)*1/sqrt(nsim),eiMC + 1.96*sd(qImprovement)*1/sqrt(nsim))

# MC estimation of the qEI
print(eiMC) 
t2 <- proc.time()
# qEI with analytical formula
qEI(X,model,fastCompute= FALSE)
t3 <- proc.time()
# qEI with fast computation trick
qEI(X,model)
t4 <- proc.time()
t2-t1  # Time of MC computation
t3-t2  # Time of normal computation
t4-t3  # Time of fast computation
# }
# NOT RUN {
# }