Gradient Hard Thresholding Pursuit for Sparsity-Constrained Optimization
This addresses the problem of finding sparse solutions in optimization for researchers and practitioners in machine learning, though it is incremental as it extends an existing method to a broader setup.
The paper generalizes Hard Thresholding Pursuit (HTP) from compressive sensing to sparsity-constrained convex optimization, proposing an algorithm that iterates between gradient descent and hard thresholding, and demonstrates superior performance to state-of-the-art methods in tasks like sparse logistic regression and precision matrix estimation.
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.