Penalized deep neural networks estimator with general loss functions under weak dependence

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.

Sparse-penalized deep neural networks estimator under weak dependence

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.

Excess risk bound for deep learning under weak dependence

This paper considers deep neural networks for learning weakly dependent processes in a general framework that includes, for instance, regression estimation, time series prediction, time series classification. The $\psi$-weak dependence structure considered is quite large and covers other conditions such as mixing, association,$\ldots$ Firstly, the approximation of smooth functions by deep neural networks with a broad class of activation functions is considered. We derive the required depth, width and sparsity of a deep neural network to approximate any H\"{o}lder smooth function, defined on any compact set $\mx$. Secondly, we establish a bound of the excess risk for the learning of weakly dependent observations by deep neural networks. When the target function is sufficiently smooth, this bound is close to the usual $\mathcal{O}(n^{-1/2})$.

Deep learning for $ψ$-weakly dependent processes

In this paper, we perform deep neural networks for learning $\psi$-weakly dependent processes. Such weak-dependence property includes a class of weak dependence conditions such as mixing, association,$\cdots$ and the setting considered here covers many commonly used situations such as: regression estimation, time series prediction, time series classification,$\cdots$ The consistency of the empirical risk minimization algorithm in the class of deep neural networks predictors is established. We achieve the generalization bound and obtain a learning rate, which is less than $\mathcal{O}(n^{-1/\alpha})$, for all $\alpha > 2 $. Applications to binary time series classification and prediction in affine causal models with exogenous covariates are carried out. Some simulation results are provided, as well as an application to the US recession data.