From a model-building perspective, in this paper we propose a paradigm shift for fitting over-parameterized models. Philosophically, the mindset is to fit models to future observations rather than the observed sample. Technically, choosing an imputation model for generating future observations, we fit over-parameterized models to future observations via optimizing an approximation to the desired expected loss function based on its sample counterpart and what we call an adaptive \it {dual function}. This technique is discussed in detail for both creating bootstrap imputation and final estimation with bootstrap imputation. The method is illustrated with the many-normal-means problem, $n < p$ linear regression, and deep convolutional neural networks for image classification of MNIST digits. The numerical results demonstrate superior performance across these three different types of applications. For example, For the many-normal-means problem, our method uniformly dominates James-Stein and Efron's $g-$modeling, and for the MNIST image classification, it shows state-of-the-art performance in common model structures. While this paper is largely expository because of the ambitious task of taking a look at over-parameterized models from a new perspective, fundamental theoretical properties are also investigated. We conclude the paper with a few remarks.
翻译:从建模的角度看,本文中我们建议对设计过度参数模型进行范式转变。从哲学角度讲,这种思维模式是使模型适合未来的观测,而不是观察到的样本。从技术上讲,我们选择了用于未来观测的估算模型,我们根据抽样对应方和我们所称的适应性[双功能],优化了预期损失功能的近似值,从而将过度参数模型与未来观测相匹配。这一方法经过详细讨论,既用于创建陷阱估算和最终估算,又用于用靴子套套套套套套套套式套接合。该方法用多种正常手段的问题,即$ < p$ 线性回归,以及用于MNMIST数字图像分类的深刻革命性神经网络来加以说明。数字结果显示这三种不同应用类型的超强性性能。例如,对于许多正常手段问题,我们的方法一致地控制着James-Stein和Efron的美元模型模型,对于MNIST的图像分类,它展示了共同模型结构中的状态和艺术性表现。我们从一个基本的理论角度来审视了几部模型。