A New Perspective on Debiasing Linear Regressions

Abstract: In this paper, we propose an abstract procedure for debiasing constrained or regularized potentially high-dimensional linear models. It is elementary to show that the proposed procedure can produce $\frac{1}{\sqrt{n}}$-confidence intervals for individual coordinates (or even bounded contrasts) in models with unknown covariance, provided that the covariance has bounded spectrum. While the proof of the statistical guarantees of our procedure is simple, its implementation requires more care due to the complexity of the optimization programs we need to solve. We spend the bulk of this paper giving examples in which the proposed algorithm can be implemented in practice. One fairly general class of instances which are amenable to applications of our procedure include convex constrained least squares. We are able to translate the procedure to an abstract algorithm over this class of models, and we give concrete examples where efficient polynomial time methods for debiasing exist. Those include the constrained version of the group LASSO, regression under monotone constraints, regression with positive monotone constraints and non-negative least squares. We also demonstrate that our method can debias Minkowski gauge selectors such as the ones proposed by Cai et al. (2016) under a certain condition. This solves an open problem posed by Cai et al. (2016) on how to debias such selectors when the covariance is unknown. In addition, we show that our abstract procedure can be applied to efficiently debias group LASSO, SLOPE and square-root SLOPE, among other popular regularized procedures under certain assumptions. We provide thorough simulation results in support of our theoretical findings.