Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality

• URL: http://arxiv.org/abs/2406.08321v1
• Date: Wed, 12 Jun 2024 15:21:51 GMT
• Title: Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality
• Authors: William Kengne, Modou Wade,
• Abstract summary: We consider sparse-penalized regularization for deep neural network predictor. We deal with the squared and a broad class of loss functions.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: The explicit regularization and optimality of deep neural networks estimators from independent data have made considerable progress recently. The study of such properties on dependent data is still a challenge. In this paper, we carry out deep learning from strongly mixing observations, and deal with the squared and a broad class of loss functions. We consider sparse-penalized regularization for deep neural network predictor. For a general framework that includes, regression estimation, classification, time series prediction,$\cdots$, oracle inequality for the expected excess risk is established and a bound on the class of H\"older smooth functions is provided. For nonparametric regression from strong mixing data and sub-exponentially error, we provide an oracle inequality for the $L_2$ error and investigate an upper bound of this error on a class of H\"older composition functions. For the specific case of nonparametric autoregression with Gaussian and Laplace errors, a lower bound of the $L_2$ error on this H\"older composition class is established. Up to logarithmic factor, this bound matches its upper bound; so, the deep neural network estimator attains the minimax optimal rate.

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