Bayesian optimisation: expected improvement under a GP posterior

The acquisition implemented is standard expected improvement; the usual label ("Wang-Frazier 2017") matches no traceable paper and is not relied on. Expected improvement vanishes exactly where the posterior variance is zero, which is what stops the search re-evaluating a point it already knows.

Usage

Gpvbo(X, y, X_grid, lengthscale = 1, variance = 1, noise = 1e-06, xi = 0)

Arguments

X: Evaluated inputs.
y: Observed objective values (minimisation).
X_grid: Candidate points.
lengthscale, variance, noise: GP hyperparameters.
xi: Exploration offset.

Value

List with estimate, acquisition, mean, sd, next_index, next_point, f_min, n, method.

Details

Formula: EI(x) = (f_min - mu(x)) Phi(z) + sigma(x) phi(z), z = (f_min - mu(x)) / sigma(x); EI = 0 when sigma(x) = 0.

References

Jones, Schonlau and Welch (1998), Efficient global optimization of expensive black-box functions, Journal of Global Optimization 13(4):455-492, eq. (15). doi:10.1023/A:1008306431147

Examples

Gpvbo(X = c(1, 2, 3, 4, 5, 6, 7, 8), y = c(1, 2, 3, 4, 5, 6, 7, 8),
  X_grid = c(1, 2, 3, 4, 5, 6, 7, 8))
#> $estimate
#> [1] 0.0003990359
#> 
#> $acquisition
#> [1] 0.0003990359 0.0000000000 0.0000000000 0.0000000000 0.0000000000
#> [6] 0.0000000000 0.0000000000 0.0000000000
#> 
#> $mean
#> [1] 0.9999998 1.9999989 2.9999993 3.9999976 4.9999994 5.9999954 7.0000006
#> [8] 7.9999923
#> 
#> $sd
#> [1] 0.0009999990 0.0009999981 0.0009999977 0.0009999975 0.0009999975
#> [6] 0.0009999977 0.0009999981 0.0009999990
#> 
#> $next_index
#> [1] 1
#> 
#> $next_point
#> [1] 1
#> 
#> $f_min
#> [1] 1
#> 
#> $n
#> [1] 8
#> 
#> $method
#> [1] "expected improvement, Jones, Schonlau & Welch (1998) eq. (15)"
#> 
#> attr(,"class")
#> [1] "morie_rich_result" "list"