  From a model-building perspective, we propose a paradigm shift for fitting
over-parameterized models. Philosophically, the mindset is to fit models to
future observations rather than to the observed sample. Technically, given an
imputation method to generate future observations, we fit over-parameterized
models to these future observations by optimizing an approximation of the
desired expected loss function based on its sample counterpart and an adaptive
$\textit{duality function}$. The required imputation method is also developed
using the same estimation technique with an adaptive $m$-out-of-$n$ bootstrap
approach. We illustrate its applications with the many-normal-means problem, $n
< p$ linear regression, and neural network-based image classification of MNIST
digits. The numerical results demonstrate its superior performance across these
diverse applications. While primarily expository, the paper conducts an
in-depth investigation into the theoretical aspects of the topic. It concludes
with remarks on some open problems.
