提出通用深度学习框架，适用于多种数据依赖场景并实现最优误差边界。
A general framework for deep learning
- 构建基于广义伯恩斯坦不等式的通用深度学习框架
- 在霍尔德光滑函数上证明两种估计器的期望过失风险上界
- 适用于独立及混合过程数据，对研究者具理论参考价值
本文针对非参数回归与分类场景，提出一种通用深度学习方法框架。该框架适用于满足广义伯恩斯坦型不等式的观测数据，包括独立、ϕ-混合、强混合及C-混合过程。提出两种估计器：非惩罚式深度神经网络（NPDNN）和稀疏惩罚式深度神经网络（SPDNN）。针对霍尔德光滑函数类与复合霍尔德函数类，建立了两类估计器的期望过失风险上界。在独立数据以及ϕ-混合、强混合、C-混合过程等情形下，均推导出NPDNN与SPDNN预测器的期望过失风险上界。结果表明，在许多经典设定中，两者均为极小极大最优（至多对数因子差距）。
原文摘要 · Abstract (English)
This paper develops a general approach for deep learning for a setting that includes nonparametric regression and classification. We perform a framework from data that fulfills a generalized Bernstein-type inequality, including independent, $ϕ$-mixing, strongly mixing and $\mathcal{C}$-mixing observations. Two estimators are proposed: a non-penalized deep neural network estimator (NPDNN) and a sparse-penalized deep neural network estimator (SPDNN). For each of these estimators, bounds of the expected excess risk on the class of Hölder smooth functions and composition Hölder functions are established. Applications to independent data, as well as to $ϕ$-mixing, strongly mixing, $\mathcal{C}$-mixing processes are considered. For each of these examples, the upper bounds of the expected excess risk of the proposed NPDNN and SPDNN predictors are derived. It is shown that both the NPDNN and SPDNN estimators are minimax optimal (up to a logarithmic factor) in many classical settings.
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