Recent developments on deep learning established some theoretical properties of deep neural networks estimators. However, most of the existing works on this topic are restricted to bounded loss functions or (sub)-Gaussian or bounded input. This paper considers robust deep learning from weakly dependent observations, with unbounded loss function and unbounded input/output. It is only assumed that the output variable has a finite $r$ order moment, with $r >1$. Non asymptotic bounds for the expected excess risk of the deep neural network estimator are established under strong mixing, and $\psi$-weak dependence assumptions on the observations. We derive a relationship between these bounds and $r$, and when the data have moments of any order (that is $r=\infty$), the convergence rate is close to some well-known results. When the target predictor belongs to the class of H\"older smooth functions with sufficiently large smoothness index, the rate of the expected excess risk for exponentially strongly mixing data is close to or as same as those for obtained with i.i.d. samples. Application to robust nonparametric regression and robust nonparametric autoregression are considered. The simulation study for models with heavy-tailed errors shows that, robust estimators with absolute loss and Huber loss function outperform the least squares method.

近期的深度学习研究在有界的损失函数或(亚)高斯或有界输入的情况下建立了深度神经网络估计器的一些理论性质。本文考虑了从弱相关观测中进行鲁棒深度学习，涉及无界的损失函数和无界的输入/输出。仅假设输出变量具有有限的r阶矩，其中r>1。在强混合和ψ-弱相关假设的情况下，建立了深度神经网络估计器的期望超额风险的非渐近界限。我们推导出了这些界限与r之间的关系，并且当数据具有任意阶的矩(即r =∞)时，收敛速度接近于一些著名结果。当目标预测函数属于具有足够大平滑指数的H"older平滑函数类时，期望超额风险的速率对于指数强混合数据接近于或与使用独立同分布样本获得的速率相同。我们考虑了鲁棒非参数回归和鲁棒非参数自回归的应用。对于具有重尾误差的模型的模拟研究表明，具有绝对损失和Huber损失函数的鲁棒估计器优于最小二乘法。

弱相关数据的强健深度学习