Penalized deep neural networks estimator with general loss functions
  under weak dependence
William Kengne Modou Wade
This paper carries out sparse-penalized deep neural networks predictors for
learning weakly dependent processes, with a broad class of loss functions. We
deal with a general framework that includes, regression estimation,
classification, times series prediction, $\cdots$ The $\psi$-weak dependence
structure is considered, and for the specific case of bounded observations,
$\theta_\infty$-coefficients are also used. In this case of
$\theta_\infty$-weakly dependent, a non asymptotic generalization bound within
the class of deep neural networks predictors is provided. For learning both
$\psi$ and $\theta_\infty$-weakly dependent processes, oracle inequalities for
the excess risk of the sparse-penalized deep neural networks estimators are
established. When the target function is sufficiently smooth, the convergence
rate of these excess risk is close to $\mathcal{O}(n^{-1/3})$. Some simulation
results are provided, and application to the forecast of the particulate matter
in the Vit\'{o}ria metropolitan area is also considered.