Deep learning for $\psi$-weakly dependent processes

TL;DR

This paper investigates the use of deep neural networks to learn and predict $$-weakly dependent processes, establishing theoretical guarantees and demonstrating applications in time series classification and economic data analysis.

Contribution

It proves the consistency and generalization bounds of deep neural networks for $$-weakly dependent processes, covering various dependence conditions and applications.

Findings

Established learning rates less than (n^{-1/}) for deep neural networks.

Proved the consistency of empirical risk minimization in this setting.

Demonstrated practical applications in time series classification and US recession data.

Abstract

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.