Tuned Regularized Estimators for Linear Regression via Covariance Fitting
This work addresses parameter tuning in linear models for practitioners, offering a unified approach but is incremental as it builds on existing estimators.
The paper tackles the problem of selecting tuned regularized parameters for linear regression by proposing a covariance fitting method to derive data-adaptive weights, unifying estimators like ridge regression and LASSO under a common framework and demonstrating practical relevance through numerical examples.
We consider the problem of finding tuned regularized parameter estimators for linear models. We start by showing that three known optimal linear estimators belong to a wider class of estimators that can be formulated as a solution to a weighted and constrained minimization problem. The optimal weights, however, are typically unknown in many applications. This begs the question, how should we choose the weights using only the data? We propose using the covariance fitting SPICE-methodology to obtain data-adaptive weights and show that the resulting class of estimators yields tuned versions of known regularized estimators - such as ridge regression, LASSO, and regularized least absolute deviation. These theoretical results unify several important estimators under a common umbrella. The resulting tuned estimators are also shown to be practically relevant by means of a number of numerical examples.