Deep learning from strongly mixing observations: Sparse-penalized regularization and minimax optimality

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.

Robust deep learning from weakly dependent data

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.

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.