Black‐Box Optimization With Simultaneous Statistical Inference for Optimal Performance

Black‐box optimization is often encountered for decision‐making in complex systems management, where knowledge of the system is limited. Under these circumstances, it is essential to balance the utilization of new information with computational efficiency. In practice, decision‐makers often face the dual tasks of optimization and statistical inference for the optimal performance, in order to achieve it with high reliability. Our goal is to address the dual tasks in an online fashion. Wu et al. (2025) point out that the sample average of performance estimates generated by the optimization algorithm does not necessarily admit a central limit theorem (CLT). We propose an algorithm that not only tackles this issue, but also provides a consistent online estimator for the variance of the performance. Furthermore, we characterize the convergence rate of the coverage probabilities of the asymptotic confidence intervals.