Robust Bayesian target value optimization

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.

Automated summary

This article discusses how to find an input such that the output of a stochastic black box function is as close as possible to a target value. It fills the gap in current approaches by deriving acquisition functions for common criteria and demonstrating their compatibility with certain extensions of Gaussian processes. The experiments show that these derived acquisition functions can outperform classical Bayesian optimization.