Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization

TL;DR

This paper extends the Hard Thresholding Pursuit algorithm to general sparsity-constrained convex optimization, providing theoretical guarantees and demonstrating superior performance in sparse logistic regression and precision matrix estimation.

Contribution

It generalizes HTP to a broader class of problems and proves convergence and accuracy guarantees for the new method.

Findings

The proposed method converges at a strong rate.

It achieves higher accuracy in sparse logistic regression.

It outperforms existing greedy methods in experiments.

Abstract

Hard Thresholding Pursuit (HTP) is an iterative greedy selection procedure for finding sparse solutions of underdetermined linear systems. This method has been shown to have strong theoretical guarantee and impressive numerical performance. In this paper, we generalize HTP from compressive sensing to a generic problem setup of sparsity-constrained convex optimization. The proposed algorithm iterates between a standard gradient descent step and a hard thresholding step with or without debiasing. We prove that our method enjoys the strong guarantees analogous to HTP in terms of rate of convergence and parameter estimation accuracy. Numerical evidences show that our method is superior to the state-of-the-art greedy selection methods in sparse logistic regression and sparse precision matrix estimation tasks.