hetGP (version 1.1.7)

crit_qEI: Parallel Expected improvement

Description

Fast approximated batch-Expected Improvement criterion (for minimization)

Usage

crit_qEI(x, model, cst = NULL, preds = NULL)

Arguments

x

matrix of new designs representing the batch of q points, one point per row (size q x d)

model

homGP or hetGP model, including inverse matrices.

cst

optional plugin value used in the EI, see details

preds

optional predictions at x to avoid recomputing if already done (must include the predictive covariance, i.e., the cov slot)

Details

cst is classically the observed minimum in the deterministic case. In the noisy case, the min of the predictive mean works fine.

References

M. Binois (2015), Uncertainty quantification on Pareto fronts and high-dimensional strategies in Bayesian optimization, with applications in multi-objective automotive design. Ecole Nationale Superieure des Mines de Saint-Etienne, PhD thesis.

Examples

## Optimization example (noiseless)
set.seed(42)

## Test function defined in [0,1]
ftest <- f1d

n_init <- 5 # number of unique designs
X <- seq(0, 1, length.out = n_init)
X <- matrix(X, ncol = 1)
Z <- ftest(X)

## Predictive grid
ngrid <- 51
xgrid <- seq(0,1, length.out = ngrid)
Xgrid <- matrix(xgrid, ncol = 1)

model <- mleHomGP(X = X, Z = Z, lower = 0.01, upper = 1, known = list(g = 2e-8))

# Regular EI function
cst <- min(model$Z0)
EIgrid <- crit_EI(Xgrid, model, cst = cst)
plot(xgrid, EIgrid, type = "l")
abline(v = X, lty = 2) # observations

# Create batch (based on regular EI peaks)
xbatch <- matrix(c(0.37, 0.17, 0.7), 3, 1)
abline(v = xbatch, col = "red")
fqEI <- crit_qEI(xbatch, model, cst = cst)

# Compare with Monte Carlo qEI
preds <- predict(model, xbatch, xprime = xbatch)
nsim <- 1e4
simus <- matrix(rnorm(3 * nsim), nsim) %*% chol(preds$cov)
simus <- simus + matrix(preds$mean, nrow = nsim, ncol = 3, byrow = TRUE)
MCqEI <- mean(apply(cst - simus, 1, function(x) max(c(x, 0))))