We consider the nonparametric regression and the classification problems for $\psi$-weakly dependent processes. This weak dependence structure is more general than conditions such as, mixing, association, $\ldots$. A penalized estimation method for sparse deep neural networks is performed. In both nonparametric regression and binary classification problems, we establish oracle inequalities for the excess risk of the sparse-penalized deep neural networks estimators. Convergence rates of the excess risk of these estimators are also derived. The simulation results displayed show that, the proposed estimators overall work well than the non penalized estimators.

翻译：我们考虑$\psi$-弱依赖过程的非参数回归和分类问题。这种弱依赖结构比混合性、关联性等条件更具一般性。本文提出了一种针对稀疏深度神经网络的惩罚估计方法。在非参数回归和二分类问题中，我们建立了稀疏惩罚深度神经网络估计器超额风险的Oracle不等式，并推导了这些估计器超额风险的收敛速度。仿真结果表明，所提出的估计器整体上优于非惩罚估计器。