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

2025-05-14

AI Summary

In multifidelity Monte Carlo (MFMC) estimation, the computational cost of oracle statistics—such as inter-model covariances—and their induced estimation bias are often overlooked. Method: We propose an adaptive resource allocation algorithm that jointly optimizes both the estimation of oracle statistics and the construction of the final MFMC estimator—the first such approach. It integrates the multilevel best linear unbiased estimation (ML-BLUE) framework with a bandit-style sequential learning strategy, requiring no prior knowledge of covariance structure and automatically achieving the theoretical minimum mean-squared error (MSE). The algorithm accommodates nonstationary and heterogeneous model ensembles. Results: On elliptic PDE solving and ice-sheet mass change modeling tasks, our estimator attains the BLU lower bound on MSE convergence rate achievable under ideal oracle allocation, while substantially reducing overall computational cost.

Technology Category

• Machine Learning: Ensemble Methods
• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Sampling/Simulation-based Search

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• User Modeling, Personalization and Recommendation: Federated recommendation systems and personalization
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems

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.

Problem

• Balancing exploration and exploitation in multi-fidelity estimation
• Reducing computational costs by optimizing model sample allocation
• Adaptively estimating oracle statistics for accurate error control

Innovation

• Adaptive algorithm balances oracle estimation and estimator construction
• Leverages multilevel best linear unbiased estimators
• Uses bandit-learning for optimal sample allocation