Dimension-free bounds in high-dimensional linear regression via error-in-operator approach
This provides theoretical guarantees for high-dimensional regression practitioners, though it appears incremental as it builds on existing empirical risk minimization frameworks.
The paper tackles high-dimensional linear regression with random design by introducing an error-in-operator approach that avoids direct covariance estimation, deriving non-asymptotic dimension-free bounds on excess prediction risk and showing auxiliary variables do not increase effective dimension with proper tuning.
We consider a problem of high-dimensional linear regression with random design. We suggest a novel approach referred to as error-in-operator which does not estimate the design covariance $Σ$ directly but incorporates it into empirical risk minimization. We provide an expansion of the excess prediction risk and derive non-asymptotic dimension-free bounds on the leading term and the remainder. This helps us to show that auxiliary variables do not increase the effective dimension of the problem, provided that parameters of the procedure are tuned properly. We also discuss computational aspects of our method and illustrate its performance with numerical experiments.