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

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 <i>duality function</i>. The required imputation method is also developed using the same estimation technique with an adaptive <i>m</i>-out-of-<i>n</i> bootstrap approach. We illustrate its applications with the many-normal-means problem, n&lt;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 article conducts an in-depth investigation into the theoretical aspects of the topic. It concludes with remarks on some open problems. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.