Statistical Properties of Deep Neural Networks with Dependent Data

arXiv:2410.11113v4 Announce Type: replace-cross Abstract: This paper develops theory for deep neural network (DNN) estimators under dependent data. To provide theory applicable to a variety of DNN-based estimators, I first establish nonasymptotic probability bounds on the theoretical and empirical L^{2}-errors of nonparametric sieve estimators for a general class of estimation problems under possibly nonstationary eta-mixing data taking values in unbounded sets. I then apply the theory to fully connected and convolutional DNN estimators without bounds or sparsity restrictions on the DNN weights. For both DNN classes, I derive general results when the function to be estimated is Holder smooth and the data are nonstationary, subgaussian, and eta-mixing with either exponential or polynomial decay. I then specialize these to nonparametric regression, logistic regression, and quantile regression settings. Under exponential eta-mixing, the resulting estimators attain the nonparametric minimax rate of Stone (1982) up to logarithmic factors.

Source: arXiv cs.LG | 2026-09-24