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.

翻译：独立数据下深度神经网络估计量的显式正则化与最优性研究近期取得了显著进展，但此类性质在相依数据上的研究仍具挑战性。本文针对强混合观测开展深度学习研究，处理平方损失及一类广泛损失函数。我们考虑深度神经网络预测器的稀疏惩罚正则化。对于包含回归估计、分类、时间序列预测等在内的通用框架，建立了期望超额风险的神谕不等式，并给出了Hölder光滑函数类的边界。针对强混合数据与次指数误差的非参数回归问题，我们建立了$L_2$误差的神谕不等式，并在Hölder复合函数类上探究该误差的上界。对于具有高斯和拉普拉斯误差的非参数自回归特例，在该Hölder复合函数类上建立了$L_2$误差的下界。该下界与上界在对数因子意义下匹配，因此深度神经网络估计量达到了极小极大最优速率。