Computation-Risk Tradeoffs for Covariance-Thresholded Regression

We present a family of linear regression estimators that provides a fine-grained tradeoff between statistical accuracy and computational efficiency. The estimators are based on hard thresholding of the sample covariance matrix entries together with `2-regularizion (ridge regression). We analyze the predictive risk of this family of estimators as a function of the threshold and regularization parameter. With appropriate parameter choices, the estimate is the solution to a sparse, diagonally dominant linear system, solvable in near-linear time. Our analysis shows how the risk varies with the sparsity and regularization level, thus establishing a statistical estimation setting for which there is an explicit, smooth tradeoff between risk and computation. Simulations are provided to support the theoretical analyses.