Estimation of Over-parameterized Models from an Auto-Modeling Perspective

Abstract

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.