Iterative Hard Thresholding with Adaptive Regularization: Sparser Solutions Without Sacrificing Runtime

Abstract

We propose a simple modification to the iterative hard thresholding (IHT) algorithm, which recovers asymptotically sparser solutions as a function of the condition number. When aiming to minimize a convex function f(x) with condition number $\kappa$ subject to x being an s-sparse vector, the standard IHT guarantee is a solution with relaxed sparsity $O(s\kappa^2)$, while our proposed algorithm, regularized IHT, returns a solution with sparsity $O(s\kappa)$. Our algorithm significantly improves over ARHT [Axiotis & Sviridenko, 2021] which also achieves $O(s\kappa)$, as it does not require re-optimization in each iteration (and so is much faster), is deterministic, and does not require knowledge of the optimal solution value f(x*) or the optimal sparsity level s. Our main technical tool is an adaptive regularization framework, in which the algorithm progressively learns the weights of an l_2 regularization term that will allow convergence to sparser solutions. We also apply this framework to low rank optimization, where we achieve a similar improvement of the best known condition number dependence from $\kappa^2$ to $\kappa$.