Algorithmic Analysis and Statistical Estimation of SLOPE via Approximate Message Passing

TL;DR

This paper provides an asymptotically exact analysis of the SLOPE estimator in high-dimensional linear regression using approximate message passing (AMP), overcoming challenges posed by the non-separable sorted l1 penalty.

Contribution

It develops a novel AMP-based framework with state evolution analysis for the SLOPE problem, enabling precise characterization and fast convergence insights.

Findings

AMP iterates converge to the SLOPE solution asymptotically

The analysis yields explicit asymptotic dynamics of the SLOPE estimator

Numerical results demonstrate rapid convergence of the AMP algorithm

Abstract

SLOPE is a relatively new convex optimization procedure for high-dimensional linear regression via the sorted l1 penalty: the larger the rank of the fitted coefficient, the larger the penalty. This non-separable penalty renders many existing techniques invalid or inconclusive in analyzing the SLOPE solution. In this paper, we develop an asymptotically exact characterization of the SLOPE solution under Gaussian random designs through solving the SLOPE problem using approximate message passing (AMP). This algorithmic approach allows us to approximate the SLOPE solution via the much more amenable AMP iterates. Explicitly, we characterize the asymptotic dynamics of the AMP iterates relying on a recently developed state evolution analysis for non-separable penalties, thereby overcoming the difficulty caused by the sorted l1 penalty. Moreover, we prove that the AMP iterates converge to the SLOPE solution in an asymptotic sense, and numerical simulations show that the convergence is surprisingly fast. Our proof rests on a novel technique that specifically leverages the SLOPE problem. In contrast to prior literature, our work not only yields an asymptotically sharp analysis but also offers an algorithmic, flexible, and constructive approach to understanding the SLOPE problem.