# Gradient of the multipoint expected improvement (qEI) criterion Computes an exact or approximate gradient of the multipoint expected improvement criterion ```r qEI.grad( x, model, plugin = NULL, type = "UK", minimization = TRUE, fastCompute = TRUE, eps = 10^(-6), envir = NULL ) ``` ## Arguments - `x`: 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`. ## Returns 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). ## Examples ```r set.seed(15) # Example 1 - validation by comparison to finite difference approximations # a 9-points factorial design, and the corresponding response d <- 2 n <- 9 design <- expand.grid(seq(0,1,length=3), seq(0,1,length=3)) names(design)<-c("x1", "x2") design <- data.frame(design) names(design)<-c("x1", "x2") y <- apply(design, 1, branin) y <- data.frame(y) names(y) <- "y" # learning model <- km(~1, design=design, response=y) # pick up 2 points sampled from the simple expected improvement q <- 2 # increase to 4 for a more meaningful test 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) ## Not run: # 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) ``` ## 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. 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. 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). 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. ## See Also `qEI` ## Author(s) Sebastien Marmin Clement Chevalier David Ginsbourger

qEI.grad function