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

TL;DR

This paper introduces an adaptive algorithm that balances the estimation of covariance statistics and the construction of multi-fidelity estimators, improving efficiency in complex statistical models.

Contribution

It proposes a novel adaptive method that optimally allocates resources between estimating oracle statistics and building multi-fidelity estimators, accounting for estimation errors.

Findings

The proposed estimator achieves mean-squared error comparable to the optimal oracle-based estimator.

Numerical experiments validate the effectiveness of the adaptive algorithm in PDE and ice-sheet modeling.

The method reduces computational costs while maintaining estimator accuracy.

Abstract

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.