On Iterative Hard Thresholding Methods for High-dimensional M-Estimation
This work addresses a gap in theoretical understanding for scalable optimization methods in high-dimensional statistics, though it is incremental as it extends existing analysis to more realistic settings.
The paper tackles the lack of theoretical analysis for iterative hard thresholding (IHT) methods in high-dimensional statistical models, providing the first tight bounds that match minimax lower bounds for sparse regression and low-rank matrix recovery.
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.