A general framework for deep learning
This provides a theoretical foundation for deep learning with dependent data, addressing a broad class of mixing processes, but it is incremental as it extends existing frameworks to more general settings.
The paper tackles the problem of developing a general deep learning framework for nonparametric regression and classification under various data dependencies, proposing two estimators (NPDNN and SPDNN) and showing they achieve minimax optimal expected excess risk bounds (up to logarithmic factors) for Hölder smooth functions.
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.