Dimension-free bounds in high-dimensional linear regression via error-in-operator approach

TL;DR

This paper introduces an error-in-operator approach for high-dimensional linear regression that avoids estimating the design covariance directly, providing dimension-free bounds and demonstrating effective performance through theoretical analysis and experiments.

Contribution

The paper proposes a novel error-in-operator method that yields non-asymptotic, dimension-free bounds in high-dimensional regression without estimating the covariance matrix.

Findings

Dimension-free bounds on excess prediction risk

Auxiliary variables do not increase effective dimension

Method performs well in numerical experiments

Abstract

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 $\Sigma$ 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.