Formal convergence guarantees for the QSI-EEM strategy

Establish formal convergence guarantees for the Quantile Set Inversion Expected Estimator Modification (QSI-EEM) sequential Bayesian active learning strategy based on Gaussian process modeling, by proving that the estimator of the quantile set Γ(f) converges (e.g., in measure or almost surely) to the true set as the number of simulator evaluations increases under specified modeling and acquisition conditions.

Background

The paper introduces the Quantile Set Inversion (QSI) problem, where the goal is to estimate the set of deterministic inputs x such that the probability, with respect to uncertain inputs S, that f(x,S) falls in a specified region C is below a threshold α. To tackle expensive black-box simulators and small quantile sets, the authors propose a Bayesian active learning strategy that combines Gaussian process modeling with a new sampling criterion under the Expected Estimator Modification (EEM) framework, and an SMC-based implementation for small sets.

While empirical results show that QSI-EEM can efficiently estimate quantile sets, the authors highlight that the method currently lacks theoretical guarantees of convergence. This motivates a precise open problem: to provide a rigorous convergence analysis of QSI-EEM (e.g., consistency or convergence rates) under assumptions on the Gaussian process prior, the estimator, and the acquisition mechanism, thereby strengthening the method’s theoretical foundation.