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This work presents a new procedure for obtaining predictive distributions inthe context of Gaussian process (GP) modeling, with a relaxation of theinterpolation constraints outside some ranges of interest: the mean of thepredictive distributions no longer necessarily interpolates the observed valueswhen they are outside ranges of interest, but are simply constrained to remainoutside. This method called relaxed Gaussian process (reGP) interpolationprovides better predictive distributions in ranges of interest, especially incases where a stationarity assumption for the GP model is not appropriate. Itcan be viewed as a goal-oriented method and becomes particularly interesting inBayesian optimization, for example, for the minimization of an objectivefunction, where good predictive distributions for low function values areimportant. When the expected improvement criterion and reGP are used forsequentially choosing evaluation points, the convergence of the resultingoptimization algorithm is theoretically guaranteed (provided that the functionto be optimized lies in the reproducing kernel Hilbert spaces attached to theknown covariance of the underlying Gaussian process). Experiments indicate thatusing reGP instead of stationary GP models in Bayesian optimization isbeneficial.

Relaxed Gaussian process interpolation: a goal-oriented approach to Bayesian optimization