Non-convex Global Minimization and False Discovery Rate Control for the TREX

TL;DR

This paper introduces a polynomial-time algorithm for the non-convex TREX regression method, enabling global optimization, improved variable selection, and false discovery rate control in high-dimensional sparse regression.

Contribution

It presents the first polynomial-time algorithm guaranteeing global optimality for TREX, enhancing variable ranking and enabling FDR control via the knockoff filter.

Findings

The new algorithm guarantees finding the global minimum of TREX.

TREX's heuristic performance depends on problem difficulty.

The approach allows FDR control in high-dimensional regression.

Abstract

The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a non-convex optimization problem. This paper shows a remarkable result: despite the non-convexity of the TREX problem, there exists a polynomial-time algorithm that is guaranteed to find the global minimum. This result adds the TREX to a very short list of non-convex optimization problems that can be globally optimized (principal components analysis being a famous example). After deriving and developing this new approach, we demonstrate that (i) the ability of the preexisting TREX heuristic to reach the global minimum is strongly dependent on the difficulty of the underlying statistical problem, (ii) the new polynomial-time algorithm for TREX permits a novel variable ranking and selection scheme, (iii) this scheme can be incorporated into a rule that controls the false discovery rate (FDR) of included features in the model. To achieve this last aim, we provide an extension of the results of Barber & Candes (2015) to establish that the knockoff filter framework can be applied to the TREX. This investigation thus provides both a rare case study of a heuristic for non-convex optimization and a novel way of exploiting non-convexity for statistical inference.