A bandit-learning approach to multifidelity approximation

TL;DR

This paper introduces a bandit-learning method for multifidelity approximation that adaptively leverages data of different fidelities to improve parameter estimates without prior statistical knowledge.

Contribution

It proposes the AETC algorithm, a novel adaptive approach that optimally balances exploration and exploitation in multifidelity settings under a linear model assumption.

Findings

The AETC algorithm is proven to be trajectory-wise optimal.

The method extends efficiently to vector-valued responses.

Numerical experiments confirm the theoretical advantages.

Abstract

Multifidelity approximation is an important technique in scientific computation and simulation. In this paper, we introduce a bandit-learning approach for leveraging data of varying fidelities to achieve precise estimates of the parameters of interest. Under a linear model assumption, we formulate a multifidelity approximation as a modified stochastic bandit, and analyze the loss for a class of policies that uniformly explore each model before exploiting. Utilizing the estimated conditional mean-squared error, we propose a consistent algorithm, adaptive Explore-Then-Commit (AETC), and establish a corresponding trajectory-wise optimality result. These results are then extended to the case of vector-valued responses, where we demonstrate that the algorithm is efficient without the need to worry about estimating high-dimensional parameters. The main advantage of our approach is that we require neither hierarchical model structure nor \textit{a priori} knowledge of statistical information (e.g., correlations) about or between models. Instead, the AETC algorithm requires only knowledge of which model is a trusted high-fidelity model, along with (relative) computational cost estimates of querying each model. Numerical experiments are provided at the end to support our theoretical findings.