  This work presents a new procedure for obtaining predictive distributions in
the context of Gaussian process (GP) modeling, with a relaxation of the
interpolation constraints outside some ranges of interest: the mean of the
predictive distributions no longer necessarily interpolates the observed values
when they are outside ranges of interest, but are simply constrained to remain
outside. This method called relaxed Gaussian process (reGP) interpolation
provides better predictive distributions in ranges of interest, especially in
cases where a stationarity assumption for the GP model is not appropriate. It
can be viewed as a goal-oriented method and becomes particularly interesting in
Bayesian optimization, for example, for the minimization of an objective
function, where good predictive distributions for low function values are
important. When the expected improvement criterion and reGP are used for
sequentially choosing evaluation points, the convergence of the resulting
optimization algorithm is theoretically guaranteed (provided that the function
to be optimized lies in the reproducing kernel Hilbert spaces attached to the
known covariance of the underlying Gaussian process). Experiments indicate that
using reGP instead of stationary GP models in Bayesian optimization is
beneficial.
