Benign Overfitting in Time Series Linear Models with Over-Parameterization

The success of large-scale models in recent years has increased the importance of statistical models with numerous parameters. Several studies have analyzed over-parameterized linear models with high-dimensional data, which may not be sparse; however, existing results rely on the assumption of sample independence. In this study, we analyze a linear regression model with dependent time-series data in an over-parameterized setting. We consider an estimator using interpolation and develop a theory for the excess risk of the estimator. Then, we derive non-asymptotic risk bounds for the estimator for cases with dependent data. This analysis reveals that the coherence of the temporal covariance plays a key role; the risk bound is influenced by the product of temporal covariance matrices at different time steps. Moreover, we show the convergence rate of the risk bound and demonstrate that it is also influenced by the coherence of the temporal covariance. Finally, we provide several examples of specific dependent processes applicable to our setting.