Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality
<|reference_start|>Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality: 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.<|reference_end|>
@article{kengne2024deep,
    title={Deep learning from strongly mixing observations: Sparse-penalized
  regularization and minimax optimality},
    author={William Kengne and Modou Wade},
    journal={arXiv preprint arXiv:2406.08321},
    year={2024},
    archivePrefix={arXiv},
    eprint={2406.08321},
    primaryClass={stat.ML cs.LG}
}