The Mean Squared Error of the Ridgeless Least Squares Estimator under General Assumptions on Regression Errors

In recent years, there has been a significant growth in research focusing on minimum $\ell_2$ norm (ridgeless) interpolation least squares estimators. However, the majority of these analyses have been limited to a simple regression error structure, assuming independent and identically distributed errors with zero mean and common variance, independent of the feature vectors. Additionally, the main focus of these theoretical analyses has been on the out-of-sample prediction risk. This paper breaks away from the existing literature by examining the mean squared error of the ridgeless interpolation least squares estimator, allowing for more general assumptions about the regression errors. Specifically, we investigate the potential benefits of overparameterization by characterizing the mean squared error in a finite sample. Our findings reveal that including a large number of unimportant parameters relative to the sample size can effectively reduce the mean squared error of the estimator. Notably, we establish that the estimation difficulties associated with the variance term can be summarized through the trace of the variance-covariance matrix of the regression errors.