Statistical Properties of Deep Neural Networks with Dependent Data

However, the statistical properties of DNN estimators with dependent data are largely unknown, and existing results for general nonparametric estimators are often inapplicable to DNN estimators. As a result, empirical use of DNN estimators often lacks a theoretical foundation. This paper aims to address this deficiency by first providing general results for nonparametric sieve estimators that offer a framework that is flexible enough for studying DNN estimators under dependent data. These results are then applied to both nonparametric regression and classification contexts, yielding theoretical properties for a class of DNN architectures commonly used in applications. Notably, Brown (2024) demonstrates the practical implications of these results in a partially linear regression model with dependent data by obtaining n-asymptotic normality of the estimator for the finite dimensional parameter after first-stage DNN estimation of infinite dimensional parameters. DNN estimators can be viewed as adaptive linear sieve estimators, where inputs are passed through hidden layers that'learn' basis functions from the data by optimizing over compositions of simpler functions.