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We propose a simple modification to the iterative hard thresholding (IHT)algorithm, which recovers asymptotically sparser solutions as a function of thecondition number. When aiming to minimize a convex function $f(x)$ withcondition number $\kappa$ subject to $x$ being an $s$-sparse vector, thestandard 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 which also finds asolution of sparsity $O(s\kappa)$, as it does not require re-optimization ineach iteration (and so is much faster), is deterministic, and does not requireknowledge of the optimal solution value $f(x^*)$ or the optimal sparsity level$s$. Our main technical tool is an adaptive regularization framework, in whichthe algorithm progressively learns the weights of an $\ell_2$ regularizationterm that will allow convergence to sparser solutions. We also apply thisframework to low rank optimization, where we achieve a similar improvement ofthe best known condition number dependence from $\kappa^2$ to $\kappa$.

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