Benign Overfitting of Non-Sparse High-Dimensional Linear Regression with Correlated Noise
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We investigate the high-dimensional linear regression problem in situations where there is noise correlated with Gaussian covariates. In regression models, the phenomenon of the correlated noise is called endogeneity, which is due to unobserved variables and others, and has been a major problem setting in causal inference and econometrics. When the covariates are high-dimensional, it has been common to assume sparsity on the true parameters and estimate them using regularization, even with the endogeneity. However, when sparsity does not hold, it has not been well understood to control the endogeneity and high dimensionality simultaneously. In this paper, we demonstrate that an estimator without regularization can achieve consistency, i.e., benign overfitting, under certain assumptions on the covariance matrix. Specifically, we show that the error of this estimator converges to zero when covariance matrices of the correlated noise and instrumental variables satisfy a condition on their eigenvalues. We consider several extensions to relax these conditions and conduct experiments to support our theoretical findings. As a technical contribution, we utilize the convex Gaussian minimax theorem (CGMT) in our dual problem and extend the CGMT itself.
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