max_qEI: Maximization of Multipoint Expected Improvement Criterion...

In libKriging/dolka: Utility functions for Bayesian optimization

Maximization of Multipoint Expected Improvement Criterion (qEI)

Description

Maximization of the qEI criterion. This is a refactoring of the function from the DiceOptim package. However only the "genoud" method can be used for now. This function calls the function genoud using analytical formulae of qEI and its gradient qEI.grad.

Usage

max_qEI(
  model,
  npoints,
  lower,
  upper,
  crit = c("exact", "CL"),
  minimization = TRUE,
  optimcontrol = NULL,
  trace = 1
)

Arguments

model	An object in km-class as created by using km.
npoints	Integer representing the desired number of iterations.
lower, upper	Numeric vectors of lower and upper bounds.
crit	Character "exact" or "CL" specifying the criterion used. "exact" triggers the maximization of the multipoint expected improvement at each iteration (see max_qEI), "CL" applies the Constant Liar heuristic.
minimization	Logical specifying if the qEI to be maximized is used in minimization or in maximization.
optimcontrol	an optional list of control parameters for optimization. See details.
trace	Integer. Level of verbosity.

Details

The parameters of list optimcontrol include optimcontrol$method, with character value "BFGS" (default) or "genoud", specifying the method used to maximize the criterion. This is irrelevant when crit is "CL" because this method always uses genoud.

• When crit == "CL"

  The elements of optimcontrol can have the following names and values.

  • parinit Optional numeric matrix of initial values. Must have model@d columns, the number of rows is not constrained.

  • L The "liar". Either a character in c("max", "min", "mean"), or a scalar value specifying the liar. For the character values: "min" sets the liar to min(model@y), "max" set the liar to max(model@y), and "mean" set it to the prediction of the model. When L is NULL, it is replaced by "min" if minimization is TRUE, or by "max" otherwise.

  • Formal arguments of genoud. These include pop.size with default : 3 * 2^d when d < 6 and 32 * d otherwise, where d is the number of inputs. One can also use max.generations (default: 12), wait.generations (default: 2) and BFGSburnin (default: 2).

• When optimcontrol$method == "BFGS"

  The elements of optimcontrol can have the following names and values.

  • nStarts (default: 4).

  • fastCompute Logical. If TRUE (default), a fast approximation method based on a semi-analytic formula is used, see Marmin (2014) for details.

  • samplingFun A function which samples a batch of starting point Default : sampleFromEI.

  • parinit Optional 3d-array of initial (or candidate) batches. For each k, the slice parinit[ , , k] is a matrix of size c(npoints, d) representing one batch. The number of initial batches dim(parinit)[3] is not contrained and does not have to be equal to nStarts. If there is too few initial batches for nStarts, missing batches are drawn with samplingFun (default NULL).

• When optimcontrol$method = "genoud"

  • optimcontrol$fastCompute Logical. If TRUE (default), a fast approximation method based on a semi-analytic formula is used, see Marmin(2014) for details.

  • parinit Optional numeric matrix of candidate starting points (one row corresponds to one point).

  • The formal arguments of the genoud function Main parameters are pop.size (default: 50 * d * npoints), max.generations (default: 5), wait.generations (default: 2), BFGSburnin (default: 2).

Value

A list with components:

• par A matrix with its rows containing the npoints input vectors found.

• value A value giving the qEI computed in par.

Caution

this function may well be renamed max_qEI_genoud in a future version.

Author(s)

Sébastien Marmin, Clément Chevalier and David Ginsbourger.

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.

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_with_grad.

Examples

set.seed(0)
## 3-points EI maximization.
## 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)) 
response.branin <- apply(design.fact, 1, branin)
lower <- c(0, 0); upper <- c(1, 1)

## number of point in the batch
batchSize <- 3

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

# maximization of qEI

## With a multistarted BFGS algorithm
## ==================================
## Not run: 
    maxBFGS <- max_qEI(model = fitted.model, npoints = batchSize,
                       lower = lower, upper = upper, 
                       crit = "exact",
                       optimcontrol = list(nStarts = 3, method = "BFGS"))

    print(maxBFGS$value)

## End(Not run)

## With a genetic algorithm using derivatives
## ==========================================
maxGen  <- max_qEI(model = fitted.model, npoints = batchSize,
                   lower = lower, upper = upper, 
                   crit = "exact",
                   optimcontrol = list(nStarts = 3, method = "genoud",
                                       pop.size = 100, max.generations = 15))
print(maxGen$value)

## With the constant liar heuristic
## ================================
## Not run: 
    maxCL   <- max_qEI(model = fitted.model, npoints = batchSize,
                       lower = lower, upper = upper,
                       crit = "CL",
                       optimcontrol = list(pop.size = 20))
    print(maxCL$value)

## End(Not run)

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