Deep learning for $ψ$-weakly dependent processes
This provides theoretical guarantees for deep learning on a broad class of weakly dependent time series data, which is incremental but practically relevant for time series analysis.
The paper establishes consistency and generalization bounds for deep neural networks applied to ψ-weakly dependent processes, achieving a learning rate better than O(n^{-1/α}) for all α > 2, with applications to binary time series classification and prediction in affine causal models.
In this paper, we perform deep neural networks for learning $ψ$-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 $\mathcal{O}(n^{-1/α})$, for all $α> 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.