论文标题
良性过度适应时间序列的线性模型过度参数化
Benign Overfitting in Time Series Linear Models with Over-Parameterization
论文作者
论文摘要
近年来，大型模型的成功提高了具有许多参数的统计模型的重要性。几项研究已经分析了具有高维数据的过度参数的线性模型，这可能并不稀疏。但是，现有结果取决于样本独立性的假设。在这项研究中，我们分析了一个线性回归模型，该模型在过度参数设置中使用依赖的时间序列数据。我们考虑使用插值的估计量，并为估计量的多余风险开发理论。然后，对于具有依赖数据的病例，我们为估计器得出了非反应风险界限。该分析表明，时间协方差的连贯性起着关键作用。风险结合在不同时间步骤的时间协方差矩阵的乘积受到影响。此外，我们显示了风险绑定的收敛速率，并证明它也受到时间协方差的连贯性的影响。最后，我们提供了适用于我们设置的特定依赖过程的几个示例。
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.
