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Statistical Properties of Deep Neural Networks with Dependent Data

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  • Chad Brown

Abstract

This paper establishes statistical properties of deep neural network (DNN) estimators under dependent data. Two general results for nonparametric sieve estimators directly applicable to DNN estimators are given. The first establishes rates for convergence in probability under nonstationary data. The second provides non-asymptotic probability bounds on $\mathcal{L}^{2}$-errors under stationary $\beta$-mixing data. I apply these results to DNN estimators in both regression and classification contexts imposing only a standard H\"older smoothness assumption. The DNN architectures considered are common in applications, featuring fully connected feedforward networks with any continuous piecewise linear activation function, unbounded weights, and a width and depth that grows with sample size. The framework provided also offers potential for research into other DNN architectures and time-series applications.

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  • Chad Brown, 2024. "Statistical Properties of Deep Neural Networks with Dependent Data," Papers 2410.11113, arXiv.org, revised Jan 2025.
    • Handle: RePEc:arx:papers:2410.11113
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    File URL: http://arxiv.org/pdf/2410.11113
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    References listed on IDEAS

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    Cited by:

    1. Chad Brown, 2024. "Inference in Partially Linear Models under Dependent Data with Deep Neural Networks," Papers 2410.22574, arXiv.org.

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