On Iterative Hard Thresholding Methods for High-dimensional M-Estimation

TL;DR

This paper provides the first rigorous analysis of iterative hard thresholding methods for high-dimensional M-estimation, demonstrating tight bounds that match minimax lower bounds and applying to sparse regression and low-rank matrix recovery.

Contribution

It introduces a general analysis framework for IHT-style algorithms in high-dimensional statistical models, extending understanding beyond restrictive previous settings.

Findings

Tight bounds for IHT methods in high-dimensional settings

Analysis applies to sparse regression and low-rank matrix recovery

Includes analysis of fully corrective and two-stage hard-thresholding algorithms

Abstract

The use of M-estimators in generalized linear regression models in high dimensional settings requires risk minimization with hard $L_0$ constraints. Of the known methods, the class of projected gradient descent (also known as iterative hard thresholding (IHT)) methods is known to offer the fastest and most scalable solutions. However, the current state-of-the-art is only able to analyze these methods in extremely restrictive settings which do not hold in high dimensional statistical models. In this work we bridge this gap by providing the first analysis for IHT-style methods in the high dimensional statistical setting. Our bounds are tight and match known minimax lower bounds. Our results rely on a general analysis framework that enables us to analyze several popular hard thresholding style algorithms (such as HTP, CoSaMP, SP) in the high dimensional regression setting. We also extend our analysis to a large family of "fully corrective methods" that includes two-stage and partial hard-thresholding algorithms. We show that our results hold for the problem of sparse regression, as well as low-rank matrix recovery.