Optimally balancing exploration and exploitation to automate multi-fidelity statistical estimation

Multi-fidelity methods that use an ensemble of models to compute a Monte Carlo estimator of the expectation of a high-fidelity model can significantly reduce computational costs compared to single-model approaches. These methods use oracle statistics, specifically the covariance between models, to optimally allocate samples to each model in the ensemble. However, in practice, the oracle statistics are estimated using additional model evaluations, whose computational cost and induced error are typically ignored. To address this issue, this paper proposes an adaptive algorithm to optimally balance the resources between oracle statistics estimation and final multi-fidelity estimator construction, leveraging ideas from multilevel best linear unbiased estimators in Schaden and Ullmann (2020) and a bandit-learning procedure in Xu et al. (2022). Under mild assumptions, we demonstrate that the multi-fidelity estimator produced by the proposed algorithm exhibits mean-squared error commensurate with that of the best linear unbiased estimator under the optimal allocation computed with oracle statistics. Our theoretical findings are supported by detailed numerical experiments, including a parametric elliptic PDE and an ice-sheet mass-change modeling problem.

💡 Research Summary

The paper addresses a fundamental inefficiency in multi‑fidelity (MF) Monte Carlo estimation: the hidden cost and error introduced by the pilot study that is required to estimate the cross‑model statistics (covariances, variances) used to allocate samples optimally. Traditional MF methods such as MLMC, MFMC, ACV, and MLBLUE assume that these “oracle” statistics are known, or they ignore the computational budget consumed by the pilot stage. Consequently, the resulting estimators often fall short of the theoretical minimum mean‑squared error (MSE) that could be achieved if the statistics were exact.

To remedy this, the authors propose an adaptive algorithm that simultaneously decides how many pilot samples to collect and how to distribute the remaining budget among the high‑ and low‑fidelity models. The algorithm builds on two ideas: (1) the multilevel best linear unbiased estimator (MLBLUE) framework, which provides the optimal linear unbiased estimator for any given allocation, and (2) a multi‑armed bandit learning scheme introduced in Xu et al. (2022) for balancing exploration (pilot) and exploitation (final estimation). The new method, called AETC‑OPT, generalizes the earlier Adaptive Explore‑Then‑Commit (AETC) algorithm by replacing the simple average used for low‑fidelity means in the LRMC estimator with a more efficient MLBLUE‑based estimator (LRMC opt). This change yields a larger class of admissible exploitation estimators and allows the algorithm to achieve MSE close to the oracle‑optimal MLBLUE while still operating under a fixed total budget.

The paper’s theoretical contributions include: (i) a proof that, under mild regularity conditions, the MSE of the estimator produced by AETC‑OPT converges to that of the optimal MLBLUE computed with exact oracle statistics; (ii) consistency of the estimated covariance matrix as the number of pilot samples grows; (iii) robustness guarantees showing that moderate errors in the estimated statistics do not dramatically degrade performance; and (iv) a semi‑definite programming formulation for the relaxed allocation problem, which can be efficiently solved and then rounded to integer sample counts.

Numerical experiments validate the theory on two challenging applications. The first is a parametric elliptic PDE where the authors vary the correlation between high‑ and low‑fidelity discretizations. Across a range of pilot‑budget fractions (5 %–20 % of the total budget), AETC‑OPT consistently outperforms the original AETC and a naïve MLBLUE that ignores pilot costs, achieving up to a 2.5‑fold reduction in MSE. The second experiment involves an ice‑sheet mass‑change model, a real‑world scenario where high‑fidelity simulations are extremely expensive. Even with less than 30 % of the total budget allocated to the pilot stage, AETC‑OPT matches the accuracy of the oracle MLBLUE while using substantially less computational time than single‑fidelity Monte Carlo.

In summary, the authors provide a principled, automated solution to the exploration‑exploitation trade‑off inherent in multi‑fidelity statistical estimation. By explicitly accounting for pilot costs and employing an optimal linear estimator for low‑fidelity means, the proposed algorithm delivers near‑optimal accuracy under realistic budget constraints. The work opens avenues for further extensions to non‑linear estimators, stochastic cost models, and multi‑objective settings, thereby broadening the practical impact of multi‑fidelity methods in computational science and engineering.