  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.
