  We consider the problem of finding an input to a stochastic black box
function such that the scalar output of the black box function is as close as
possible to a target value in the sense of the expected squared error. While
the optimization of stochastic black boxes is classic in (robust) Bayesian
optimization, the current approaches based on Gaussian processes predominantly
focus either on i) maximization/minimization rather than target value
optimization or ii) on the expectation, but not the variance of the output,
ignoring output variations due to stochasticity in uncontrollable environmental
variables. In this work, we fill this gap and derive acquisition functions for
common criteria such as the expected improvement, the probability of
improvement, and the lower confidence bound, assuming that aleatoric effects
are Gaussian with known variance. Our experiments illustrate that this setting
is compatible with certain extensions of Gaussian processes, and show that the
thus derived acquisition functions can outperform classical Bayesian
optimization even if the latter assumptions are violated. An industrial use
case in billet forging is presented.
