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 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.