Fast Saddle-Point Algorithm for Generalized Dantzig Selector and FDR Control with the Ordered l1-Norm

TL;DR

This paper introduces a primal-dual saddle-point algorithm for the generalized Dantzig selector that achieves optimal convergence rates, simplifies parameter tuning, and demonstrates effective false discovery rate control in variable selection.

Contribution

It develops a new saddle-point reformulation for GDS, enabling optimal convergence and local acceleration without requiring strong convexity or smoothness.

Findings

Achieves $O(1/k)$ convergence rate with no sensitive parameter tuning.

Shows potential for $O(1/k^2)$ local acceleration in special cases.

Demonstrates effective false discovery rate control in variable selection.

Abstract

In this paper we propose a primal-dual proximal extragradient algorithm to solve the generalized Dantzig selector (GDS) estimation problem, based on a new convex-concave saddle-point (SP) reformulation. Our new formulation makes it possible to adopt recent developments in saddle-point optimization, to achieve the optimal $O(1/k)$ rate of convergence. Compared to the optimal non-SP algorithms, ours do not require specification of sensitive parameters that affect algorithm performance or solution quality. We also provide a new analysis showing a possibility of local acceleration to achieve the rate of $O(1/k^2)$ in special cases even without strong convexity or strong smoothness. As an application, we propose a GDS equipped with the ordered $\ell_1$-norm, showing its false discovery rate control properties in variable selection. Algorithm performance is compared between ours and other alternatives, including the linearized ADMM, Nesterov's smoothing, Nemirovski's mirror-prox, and the accelerated hybrid proximal extragradient techniques.