Mean-Variance Pareto Learning with Replicate-Aware Bayesian Optimization

Stochastic simulators often exhibit a trade-off between average performance and run-to-run variability. When the preferred balance is not known in advance, the goal is not one weighted compromise but the set of efficient designs spanning this trade-off, namely the mean–variance Pareto frontier. Learning this frontier under a limited simulation budget creates an additional difficulty: at each design, the mean and variance are estimated from the same finite collection of simulator runs. Their estimation errors therefore depend on the number of replications and can be correlated, so a sequential algorithm must decide both where to sample and how much simulation effort to allocate there. We develop a replicate-aware Bayesian optimization framework for this joint learning and allocation problem. We first derive the exact covariance of the sample mean and sample variance for any finite replication count and establish their joint asymptotic distribution. We then embed this sampling structure in a hierarchical multi-output Gaussian process that separates finite-sample estimation dependence from dependence between the underlying mean and variance surfaces. Based on this posterior, hierarchical expected quantile hypervolume improvement (HEQHVI) jointly selects a design and replication count by maximizing the expected hypervolume gain of a posterior-quantile Pareto front per simulator run. This creates an explicit trade-off between exploring new designs and reducing uncertainty at previously evaluated ones. For parallel evaluation, we extend HEQHVI using a minimum-energy design (MED) criterion that retains the best serial candidate while spreading the remaining batch members across promising but distinct regions. Synthetic benchmarks and an F119 aero-engine study illustrate how the framework learns mean–variance trade-offs while adapting replication effort.