Benign Overfitting in Time Series Linear Model 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 that may not be sparse; however, existing results depend on the independent setting of samples. In this study, we analyze a linear regression model with dependent time series data under over-parameterization settings. We consider an estimator via interpolation and developed a theory for the excess risk of the estimator. Then, we derive bounds of risks by the estimator for the cases where the temporal correlation of each coordinate of dependent data is homogeneous and heterogeneous, respectively. The derived bounds reveal that a temporal covariance of the data plays a key role; its strength affects the bias of the risk, and its nondegeneracy affects the variance of the risk. Moreover, for the heterogeneous correlation case, we show that the convergence rate of risks with short-memory processes is identical to that of cases with independent data, and the risk can converge to zero even with long-memory processes. Our theory can be extended to infinite-dimensional data in a unified manner. We also present several examples of specific dependent processes that can be applied to our setting.