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

TL;DR

An adaptive algorithm is proposed to optimally balance the resources between oracle statistics estimation and final multi-fidelity estimator construction, leveraging ideas from multilevel best linear unbiased estimators and a bandit-learning procedure in Xu et al. (2022).

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.

Figures (15)

• Figure 1: Demonstration of the relationship between ${\mathsf{LRMC}_{\mathrm{opt}}}$, ${\mathsf{LRMC}^*_{\mathrm{opt}}}$, ACVs, and MLBLUEs. In particular, the AETC-OPT algorithm adaptively optimizes and outputs an ${\mathsf{LRMC}_{\mathrm{opt}}}$ (the blue diamond) for the high-fidelity mean. Up to a first-order error whose coefficient is sharply characterized by multi-fidelity structures, the ${\mathsf{LRMC}_{\mathrm{opt}}}$ coincides with the MLBLUE under the same exploration and exploitation samples. Moreover, this MLBLUE exhibits a comparable MSE to the MLBLUE for the high-fidelity mean under the globally optimal sample allocation (the red diamond). Specifically, the inclusion of ${\mathsf{LRMC}^*_{\mathrm{opt}}}$ within ACVs is due to \ref{['lemma:acv']}; Arrow (1) is due to \ref{['addback']}, Arrow (2) is due to \ref{['khty']}, and Arrow (3) is due to \ref{['ajan']}. The dashed arrow is empirical but supported by the evidence provided in (1)--(3).
• Figure 1: ( Left) Comparison of the empirical MSE of various estimators of compliance when selecting from 5 elasticity models as the total budget increases from $4\times 10^5$ to $2\times 10^6$. ( Right) The percentage of the total budget given to exploration sampling of AETC, AETC-OPT, and AETC-OPT-E across the budgets.
• Figure 1: The computational costs of the ice-sheet models used in this study.
• Figure 1: Geometry, boundary conditions, and loading for a linear elastic structure with the square domain.
• Figure 2: ( Left) The impact of the number of exploration samples on the MSE of the AETC-OPT algorithm in blue. The number of exploration samples chosen by AETC-OPT-E is denoted by the red-shaded region. ( Right) Evolution of the estimated loss function as the AETC-OPT-E algorithm takes more exploration samples. Lighter curves represent loss functions with fewer exploration samples.

Theorems & Definitions (13)

• Definition 3.1: Exploration-unbiased exploitation
• Lemma 3.2
• Definition 4.1
• Lemma 4.5: Exploration unbiasedness of $\widehat{\mu}_S$
• Lemma 4.6: Existence of $\gamma_S \in (0, \infty)$
• Lemma 4.7: Estimability of $\gamma_S$
• Theorem 4.8
• Corollary 4.9
• Theorem 4.10
• Theorem 4.11